\(\int \frac {x^2-c x^2 (\frac {b+a x}{d+c x})^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx\) [3153]

Optimal result

Integrand size = 52, antiderivative size = 1916 \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {6 b^6 c^3 x-6 a^3 b^3 c^4 x+6 a^2 b^6 c d x-18 a^3 b^4 c^2 d x-6 a b^5 c^2 d x+24 a^4 b^2 c^3 d x-6 a^2 b^3 c^3 d x-6 a^3 b^5 d^2 x+18 a^4 b^3 c d^2 x-18 a^5 b c^2 d^2 x+6 a^3 b^2 c^2 d^2 x-3 a^2 b^6 c^2 x^2+9 a^3 b^4 c^3 x^2-3 a b^5 c^3 x^2-6 a^4 b^2 c^4 x^2+3 a^2 b^3 c^4 x^2+3 a^3 b^5 c d x^2-9 a^4 b^3 c^2 d x^2+3 a^2 b^4 c^2 d x^2+6 a^5 b c^3 d x^2-3 a^3 b^2 c^3 d x^2-2 a^3 b^5 c^2 x^3+4 a^4 b^3 c^3 x^3+2 a^2 b^4 c^3 x^3-2 a^5 b c^4 x^3-4 a^3 b^2 c^4 x^3+2 a^4 c^5 x^3}{6 a^2 c^2 \left (-b^2+a c\right )^3}+\frac {\sqrt {\frac {b+a x}{d+c x}} \left (3 a^2 b^6 c^2 d+15 b^7 c^2 d-24 a^3 b^4 c^3 d+12 a b^5 c^3 d-3 a^4 b^2 c^4 d-3 a^2 b^3 c^4 d+4 a^3 b^5 c d^2-4 a b^6 c d^2-8 a^4 b^3 c^2 d^2-40 a^2 b^4 c^2 d^2+52 a^5 b c^3 d^2-4 a^3 b^2 c^3 d^2-15 a^4 b^4 d^3-3 a^2 b^5 d^3+48 a^5 b^2 c d^3+12 a^3 b^3 c d^3-57 a^6 c^2 d^3+15 a^4 b c^2 d^3+3 a^2 b^6 c^3 x+15 b^7 c^3 x-24 a^3 b^4 c^4 x+12 a b^5 c^4 x-3 a^4 b^2 c^5 x-3 a^2 b^3 c^5 x+2 a^3 b^5 c^2 d x-14 a b^6 c^2 d x+8 a^4 b^3 c^3 d x-32 a^2 b^4 c^3 d x+38 a^5 b c^4 d x-2 a^3 b^2 c^4 d x-5 a^4 b^4 c d^2 x-a^2 b^5 c d^2 x+16 a^5 b^2 c^2 d^2 x+20 a^3 b^3 c^2 d^2 x-35 a^6 c^3 d^2 x+5 a^4 b c^3 d^2 x-2 a^3 b^5 c^3 x^2-10 a b^6 c^3 x^2+16 a^4 b^3 c^4 x^2+8 a^2 b^4 c^4 x^2-14 a^5 b c^5 x^2+2 a^3 b^2 c^5 x^2+2 a^4 b^4 c^2 d x^2+10 a^2 b^5 c^2 d x^2-16 a^5 b^2 c^3 d x^2-8 a^3 b^3 c^3 d x^2+14 a^6 c^4 d x^2-2 a^4 b c^4 d x^2-8 a^4 b^4 c^3 x^3+8 a^2 b^5 c^3 x^3+16 a^5 b^2 c^4 x^3-16 a^3 b^3 c^4 x^3-8 a^6 c^5 x^3+8 a^4 b c^5 x^3\right )}{24 a^3 c^2 \left (-b^2+a c\right )^3}+\frac {\left (a^2 b^9 c^3+5 b^{10} c^3-9 a^3 b^7 c^4+15 a b^8 c^4-9 a^4 b^5 c^5-5 a^2 b^6 c^5+a^5 b^3 c^6+a^3 b^4 c^6+a^3 b^8 c^2 d-3 a b^9 c^2 d-a^4 b^6 c^3 d-33 a^2 b^7 c^3 d+39 a^5 b^4 c^4 d-13 a^3 b^5 c^4 d+9 a^6 b^2 c^5 d+a^4 b^3 c^5 d+3 a^4 b^7 c d^2-a^2 b^8 c d^2-11 a^5 b^5 c^2 d^2+13 a^3 b^6 c^2 d^2+5 a^6 b^3 c^3 d^2+33 a^4 b^4 c^3 d^2-45 a^7 b c^4 d^2+3 a^5 b^2 c^4 d^2-5 a^5 b^6 d^3-a^3 b^7 d^3+21 a^6 b^4 c d^3+5 a^4 b^5 c d^3-35 a^7 b^2 c^2 d^3-15 a^5 b^3 c^2 d^3+35 a^8 c^3 d^3-5 a^6 b c^3 d^3\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2} \left (-b^2+a c\right )^4}+\frac {\left (b^9 c^4-a^3 b^6 c^5-a b^8 c^3 d+a^4 b^5 c^4 d-2 a^2 b^6 c^4 d+2 a^5 b^3 c^5 d+a^3 b^8 c d^2-4 a^4 b^6 c^2 d^2+6 a^5 b^4 c^3 d^2+2 a^3 b^5 c^3 d^2-6 a^6 b^2 c^4 d^2+a^4 b^3 c^4 d^2-a^4 b^7 d^3+4 a^5 b^5 c d^3-6 a^6 b^3 c^2 d^3+4 a^7 b c^3 d^3-a^5 b^2 c^3 d^3\right ) \log (d+c x)}{a^3 c^3 \left (-b^2+a c\right )^4}+\frac {2 \left (b^{10} c-a^3 b^7 c^2-a b^9 d+a^4 b^6 c d-2 a^2 b^7 c d+2 a^5 b^4 c^2 d+2 a^3 b^6 d^2-2 a^6 b^3 c d^2+a^4 b^4 c d^2-a^7 b c^2 d^2-a^5 b^3 d^3+a^8 c d^3\right ) \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (-b^2+a c\right )^4} \]

[Out]

1/6*(-2*a^5*b*c^4*x^3+4*a^4*b^3*c^3*x^3-2*a^3*b^5*c^2*x^3+6*a^5*b*c^3*d*x^2-9*a^4*b^3*c^2*d*x^2-6*a^4*b^2*c^4*
x^2+2*a^4*c^5*x^3+3*a^3*b^5*c*d*x^2+9*a^3*b^4*c^3*x^2-4*a^3*b^2*c^4*x^3-3*a^2*b^6*c^2*x^2+2*a^2*b^4*c^3*x^3-18
*a^5*b*c^2*d^2*x+18*a^4*b^3*c*d^2*x+24*a^4*b^2*c^3*d*x-6*a^3*b^5*d^2*x-18*a^3*b^4*c^2*d*x-6*a^3*b^3*c^4*x-3*a^
3*b^2*c^3*d*x^2+6*a^2*b^6*c*d*x+3*a^2*b^4*c^2*d*x^2+3*a^2*b^3*c^4*x^2-3*a*b^5*c^3*x^2+6*a^3*b^2*c^2*d^2*x-6*a^
2*b^3*c^3*d*x-6*a*b^5*c^2*d*x+6*b^6*c^3*x)/a^2/c^2/(a*c-b^2)^3+1/24*((a*x+b)/(c*x+d))^(1/2)*(-8*a^6*c^5*x^3+16
*a^5*b^2*c^4*x^3-8*a^4*b^4*c^3*x^3+14*a^6*c^4*d*x^2-16*a^5*b^2*c^3*d*x^2-14*a^5*b*c^5*x^2+2*a^4*b^4*c^2*d*x^2+
16*a^4*b^3*c^4*x^2+8*a^4*b*c^5*x^3-2*a^3*b^5*c^3*x^2-16*a^3*b^3*c^4*x^3+8*a^2*b^5*c^3*x^3-35*a^6*c^3*d^2*x+16*
a^5*b^2*c^2*d^2*x+38*a^5*b*c^4*d*x-5*a^4*b^4*c*d^2*x+8*a^4*b^3*c^3*d*x-3*a^4*b^2*c^5*x-2*a^4*b*c^4*d*x^2+2*a^3
*b^5*c^2*d*x-24*a^3*b^4*c^4*x-8*a^3*b^3*c^3*d*x^2+2*a^3*b^2*c^5*x^2+3*a^2*b^6*c^3*x+10*a^2*b^5*c^2*d*x^2+8*a^2
*b^4*c^4*x^2-10*a*b^6*c^3*x^2-57*a^6*c^2*d^3+48*a^5*b^2*c*d^3+52*a^5*b*c^3*d^2-15*a^4*b^4*d^3-8*a^4*b^3*c^2*d^
2-3*a^4*b^2*c^4*d+5*a^4*b*c^3*d^2*x+4*a^3*b^5*c*d^2-24*a^3*b^4*c^3*d+20*a^3*b^3*c^2*d^2*x-2*a^3*b^2*c^4*d*x+3*
a^2*b^6*c^2*d-a^2*b^5*c*d^2*x-32*a^2*b^4*c^3*d*x-3*a^2*b^3*c^5*x-14*a*b^6*c^2*d*x+12*a*b^5*c^4*x+15*b^7*c^3*x+
15*a^4*b*c^2*d^3+12*a^3*b^3*c*d^3-4*a^3*b^2*c^3*d^2-3*a^2*b^5*d^3-40*a^2*b^4*c^2*d^2-3*a^2*b^3*c^4*d-4*a*b^6*c
*d^2+12*a*b^5*c^3*d+15*b^7*c^2*d)/a^3/c^2/(a*c-b^2)^3+1/8*(35*a^8*c^3*d^3-35*a^7*b^2*c^2*d^3-45*a^7*b*c^4*d^2+
21*a^6*b^4*c*d^3+5*a^6*b^3*c^3*d^2+9*a^6*b^2*c^5*d-5*a^5*b^6*d^3-11*a^5*b^5*c^2*d^2+39*a^5*b^4*c^4*d+a^5*b^3*c
^6+3*a^4*b^7*c*d^2-a^4*b^6*c^3*d-9*a^4*b^5*c^5+a^3*b^8*c^2*d-9*a^3*b^7*c^4+a^2*b^9*c^3-5*a^6*b*c^3*d^3-15*a^5*
b^3*c^2*d^3+3*a^5*b^2*c^4*d^2+5*a^4*b^5*c*d^3+33*a^4*b^4*c^3*d^2+a^4*b^3*c^5*d-a^3*b^7*d^3+13*a^3*b^6*c^2*d^2-
13*a^3*b^5*c^4*d+a^3*b^4*c^6-a^2*b^8*c*d^2-33*a^2*b^7*c^3*d-5*a^2*b^6*c^5-3*a*b^9*c^2*d+15*a*b^8*c^4+5*b^10*c^
3)*arctanh(c^(1/2)*((a*x+b)/(c*x+d))^(1/2)/a^(1/2))/a^(7/2)/c^(5/2)/(a*c-b^2)^4+(4*a^7*b*c^3*d^3-6*a^6*b^3*c^2
*d^3-6*a^6*b^2*c^4*d^2+4*a^5*b^5*c*d^3+6*a^5*b^4*c^3*d^2+2*a^5*b^3*c^5*d-a^4*b^7*d^3-4*a^4*b^6*c^2*d^2+a^4*b^5
*c^4*d+a^3*b^8*c*d^2-a^3*b^6*c^5-a^5*b^2*c^3*d^3+a^4*b^3*c^4*d^2+2*a^3*b^5*c^3*d^2-2*a^2*b^6*c^4*d-a*b^8*c^3*d
+b^9*c^4)*ln(c*x+d)/a^3/c^3/(a*c-b^2)^4+2*(a^8*c*d^3-a^7*b*c^2*d^2-2*a^6*b^3*c*d^2+2*a^5*b^4*c^2*d+a^4*b^6*c*d
-a^3*b^7*c^2-a^5*b^3*d^3+a^4*b^4*c*d^2+2*a^3*b^6*d^2-2*a^2*b^7*c*d-a*b^9*d+b^10*c)*ln(a-b*((a*x+b)/(c*x+d))^(1
/2))/a^3/b/(a*c-b^2)^4

Rubi [A] (verified)

Time = 2.83 (sec) , antiderivative size = 1060, normalized size of antiderivative = 0.55, number of steps used = 9, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {1661, 1643, 649, 214, 266} \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=-\frac {\left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right ) (b c-a d)^3}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {\left (5 c^2 b^9+a c \left (15 c^2+2 b d\right ) b^7+a^2 \left (-5 c^4+b^3 c^2-18 b d c^2+b^2 d^2\right ) b^5-a^3 c \left (-2 d b^4+9 c^2 b^3+5 d^2 b^2+18 c^2 d b-c^4\right ) b^3-a^4 \left (-5 d^2 b^4+10 c^2 d b^3+9 c^4 b^2-15 c^2 d^2 b-2 c^4 d\right ) b^2+5 a^6 c^2 d \left (2 c^2+7 b d\right ) b+a^5 c \left (b c^4+5 d^2 c^2+30 b^2 d c^2-21 b^3 d^2\right ) b-35 a^7 c^3 d^2\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right ) (b c-a d)}{8 a^{7/2} c^{5/2} \left (b^2-a c\right )^4}+\frac {b \left (c^3 b^7+a^3 d^2 b^6-a^2 c \left (a c^3+2 d c^2+4 a^2 d^2\right ) b^4+6 a^5 c^2 d^2 b^2+a^4 c^3 d (2 a c+d) b-4 a^6 c^3 d^2\right ) \log (d+c x) (b c-a d)}{a^3 c^3 \left (b^2-a c\right )^4}+\frac {2 \left (b^3-a^3 c\right ) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) (b c-a d)}{a^3 b \left (b^2-a c\right )^4}+\frac {(d+c x) \left (8 a \left (-\left (\left (c^3+2 a^2 d c\right ) b^6\right )+3 a^3 d^2 b^5+6 a^3 c^2 d b^4+a^2 c \left (a c^3+2 d c^2-8 a^2 d^2\right ) b^3-6 a^4 c^3 d b^2+6 a^5 c^2 d^2 b-a^4 c^3 d^2\right )-c \left (5 c^2 b^7+2 a c \left (2 c^2+b d\right ) b^5+a^2 \left (-c^4+b^3 c^2-16 b d c^2+b^2 d^2\right ) b^3-2 a^3 c \left (-d b^3+4 c^2 b^2+2 d^2 b+c^2 d\right ) b^2+2 a^5 c d \left (11 c^2+16 b d\right ) b-a^4 \left (b c^4-11 d^2 c^2+8 b^2 d c^2+11 b^3 d^2\right ) b-29 a^6 c^2 d^2\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}+\frac {6 a \left (4 b c d a^3-\left (3 d b^3+2 c^2 b^2+2 c^2 d\right ) a^2+b^2 c \left (b^2+d\right ) a+b^3 c^2\right ) (b c-a d)^2+c \left (19 c d a^4-b \left (7 c^2+13 b d\right ) a^3+b c \left (b^2-13 d\right ) a^2+b^2 \left (c^2+7 b d\right ) a+5 b^4 c\right ) \sqrt {\frac {b+a x}{d+c x}} (b c-a d)^2}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2} \]

[In]

Int[(x^2 - c*x^2*((b + a*x)/(d + c*x))^(3/2))/(a - b*Sqrt[(b + a*x)/(d + c*x)]),x]

[Out]

((d + c*x)*(8*a*(6*a^3*b^4*c^2*d - 6*a^4*b^2*c^3*d + 3*a^3*b^5*d^2 + 6*a^5*b*c^2*d^2 - a^4*c^3*d^2 - b^6*(c^3
+ 2*a^2*c*d) + a^2*b^3*c*(a*c^3 + 2*c^2*d - 8*a^2*d^2)) - c*(5*b^7*c^2 - 29*a^6*c^2*d^2 + 2*a*b^5*c*(2*c^2 + b
*d) + 2*a^5*b*c*d*(11*c^2 + 16*b*d) - 2*a^3*b^2*c*(4*b^2*c^2 - b^3*d + c^2*d + 2*b*d^2) + a^2*b^3*(b^3*c^2 - c
^4 - 16*b*c^2*d + b^2*d^2) - a^4*b*(b*c^4 + 8*b^2*c^2*d + 11*b^3*d^2 - 11*c^2*d^2))*Sqrt[(b + a*x)/(d + c*x)])
)/(8*a^3*c^3*(b^2 - a*c)^3) - ((b*c - a*d)^3*(a*(a*b - c) + (a^2 - b)*c*Sqrt[(b + a*x)/(d + c*x)]))/(3*a*c^3*(
b^2 - a*c)*(a - (c*(b + a*x))/(d + c*x))^3) + (6*a*(b*c - a*d)^2*(b^3*c^2 + 4*a^3*b*c*d + a*b^2*c*(b^2 + d) -
a^2*(2*b^2*c^2 + 3*b^3*d + 2*c^2*d)) + c*(b*c - a*d)^2*(5*b^4*c + a^2*b*c*(b^2 - 13*d) + 19*a^4*c*d + a*b^2*(c
^2 + 7*b*d) - a^3*b*(7*c^2 + 13*b*d))*Sqrt[(b + a*x)/(d + c*x)])/(12*a^2*c^3*(b^2 - a*c)^2*(a - (c*(b + a*x))/
(d + c*x))^2) + ((b*c - a*d)*(5*b^9*c^2 - 35*a^7*c^3*d^2 + a*b^7*c*(15*c^2 + 2*b*d) + 5*a^6*b*c^2*d*(2*c^2 + 7
*b*d) + a^2*b^5*(b^3*c^2 - 5*c^4 - 18*b*c^2*d + b^2*d^2) - a^3*b^3*c*(9*b^3*c^2 - c^4 - 2*b^4*d + 18*b*c^2*d +
5*b^2*d^2) + a^5*b*c*(b*c^4 + 30*b^2*c^2*d - 21*b^3*d^2 + 5*c^2*d^2) - a^4*b^2*(9*b^2*c^4 + 10*b^3*c^2*d - 2*
c^4*d - 5*b^4*d^2 - 15*b*c^2*d^2))*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]])/(8*a^(7/2)*c^(5/2)*(b
^2 - a*c)^4) + (b*(b*c - a*d)*(b^7*c^3 + a^3*b^6*d^2 + 6*a^5*b^2*c^2*d^2 - 4*a^6*c^3*d^2 + a^4*b*c^3*d*(2*a*c
+ d) - a^2*b^4*c*(a*c^3 + 2*c^2*d + 4*a^2*d^2))*Log[d + c*x])/(a^3*c^3*(b^2 - a*c)^4) + (2*(b^3 - a^3*c)*(b*c
- a*d)*(b^3 - a^2*d)^2*Log[a - b*Sqrt[(b + a*x)/(d + c*x)]])/(a^3*b*(b^2 - a*c)^4)

Rule 214

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]

Rule 266

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rule 649

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[(-a)*c]

Rule 1643

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*
Pq*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 1661

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[(d +
e*x)^m*Pq, a + c*x^2, x], f = Coeff[PolynomialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 0], g = Coeff[Polyn
omialRemainder[(d + e*x)^m*Pq, a + c*x^2, x], x, 1]}, Simp[(a*g - c*f*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)))
, x] + Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*ExpandToSum[(2*a*c*(p + 1)*Q)/(d + e*x)^m +
(c*f*(2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && ILtQ[m, 0]

Rubi steps \begin{align*} \text {integral}& = (2 (b c-a d)) \text {Subst}\left (\int \frac {x \left (b-d x^2\right )^2 \left (-1+c x^3\right )}{(a-b x) \left (a-c x^2\right )^4} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right ) \\ & = -\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\frac {a \left (a^2-b\right ) (b c-a d)^2}{c \left (b^2-a c\right )}-\frac {\left (5 b^4 c^2+2 a b^3 c d-5 a^4 b d^2+2 a^3 c d \left (5 b^2+3 d\right )-a^2 b \left (5 b^2 c^2+12 c^2 d+b d^2\right )\right ) x}{c \left (b^2-a c\right )}-\frac {6 a (b c-a d)^2 x^2}{c}+6 a d^2 x^3+6 a d (2 b c-a d) x^4-6 a c d^2 x^6}{(a-b x) \left (a-c x^2\right )^3} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c} \\ & = -\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \text {Subst}\left (\int \frac {\frac {3 a (b c-a d) \left (a^2 b^3 c-3 b^4 c+a^3 b c^2+a b^2 c^2+3 a^3 b^2 d-a b^3 d-5 a^4 c d+3 a^2 b c d\right )}{\left (b^2-a c\right )^2}-\frac {3 \left (5 b^6 c^2-19 a^5 b c d^2+a b^4 c \left (c^2+2 b d\right )+a^4 d \left (26 b^2 c^2+13 b^3 d+8 c^2 d\right )-a^3 b^2 c \left (7 b c^2+14 b^2 d+3 d^2\right )+a^2 b^3 \left (b^2 c^2-14 c^2 d+b d^2\right )\right ) x}{\left (b^2-a c\right )^2}-48 a^2 d (b c-a d) x^2+24 a^2 c d^2 x^4}{(a-b x) \left (a-c x^2\right )^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^2} \\ & = \frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \text {Subst}\left (\int \frac {-\frac {3 a c \left (11 b^7 c^2-19 a^6 c^2 d^2-2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (5 c^2+8 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2+b^3 d-c^2 d-2 b d^2\right )-a^2 b^3 \left (b^3 c^2-c^4+16 b c^2 d+b^2 d^2\right )+a^4 b \left (b c^4+8 b^2 c^2 d-5 b^3 d^2+5 c^2 d^2\right )\right )}{\left (b^2-a c\right )^3}-\frac {3 b c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) x}{\left (b^2-a c\right )^3}-48 a^3 c d^2 x^2}{(a-b x) \left (a-c x^2\right )} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{24 a^3 c^3} \\ & = \frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \text {Subst}\left (\int \left (-\frac {48 c^3 \left (-b^3+a^3 c\right ) \left (b^3-a^2 d\right )^2}{\left (b^2-a c\right )^4 (-a+b x)}+\frac {3 c \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )+16 b \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) x\right )}{\left (b^2-a c\right )^4 \left (a-c x^2\right )}\right ) \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{24 a^3 c^3} \\ & = \frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}+\frac {(b c-a d) \text {Subst}\left (\int \frac {5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )+16 b \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) x}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^2 \left (b^2-a c\right )^4} \\ & = \frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4}+\frac {\left (2 b (b c-a d) \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {x}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 c^2 \left (b^2-a c\right )^4}+\frac {\left ((b c-a d) \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )\right )\right ) \text {Subst}\left (\int \frac {1}{a-c x^2} \, dx,x,\sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^2 \left (b^2-a c\right )^4} \\ & = \frac {(d+c x) \left (8 a \left (6 a^3 b^4 c^2 d-6 a^4 b^2 c^3 d+3 a^3 b^5 d^2+6 a^5 b c^2 d^2-a^4 c^3 d^2-b^6 \left (c^3+2 a^2 c d\right )+a^2 b^3 c \left (a c^3+2 c^2 d-8 a^2 d^2\right )\right )-c \left (5 b^7 c^2-29 a^6 c^2 d^2+2 a b^5 c \left (2 c^2+b d\right )+2 a^5 b c d \left (11 c^2+16 b d\right )-2 a^3 b^2 c \left (4 b^2 c^2-b^3 d+c^2 d+2 b d^2\right )+a^2 b^3 \left (b^3 c^2-c^4-16 b c^2 d+b^2 d^2\right )-a^4 b \left (b c^4+8 b^2 c^2 d+11 b^3 d^2-11 c^2 d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{8 a^3 c^3 \left (b^2-a c\right )^3}-\frac {(b c-a d)^3 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{3 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}+\frac {(b c-a d)^2 \left (6 a \left (b^3 c^2+4 a^3 b c d+a b^2 c \left (b^2+d\right )-a^2 \left (2 b^2 c^2+3 b^3 d+2 c^2 d\right )\right )+c \left (5 b^4 c+a^2 b c \left (b^2-13 d\right )+19 a^4 c d+a b^2 \left (c^2+7 b d\right )-a^3 b \left (7 c^2+13 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}\right )}{12 a^2 c^3 \left (b^2-a c\right )^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}+\frac {(b c-a d) \left (5 b^9 c^2-35 a^7 c^3 d^2+a b^7 c \left (15 c^2+2 b d\right )+5 a^6 b c^2 d \left (2 c^2+7 b d\right )+a^2 b^5 \left (b^3 c^2-5 c^4-18 b c^2 d+b^2 d^2\right )-a^3 b^3 c \left (9 b^3 c^2-c^4-2 b^4 d+18 b c^2 d+5 b^2 d^2\right )+a^5 b c \left (b c^4+30 b^2 c^2 d-21 b^3 d^2+5 c^2 d^2\right )-a^4 b^2 \left (9 b^2 c^4+10 b^3 c^2 d-2 c^4 d-5 b^4 d^2-15 b c^2 d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2} \left (b^2-a c\right )^4}+\frac {b (b c-a d) \left (b^7 c^3+a^3 b^6 d^2+6 a^5 b^2 c^2 d^2-4 a^6 c^3 d^2+a^4 b c^3 d (2 a c+d)-a^2 b^4 c \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) \log (d+c x)}{a^3 c^3 \left (b^2-a c\right )^4}+\frac {2 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (b^2-a c\right )^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 8.25 (sec) , antiderivative size = 1590, normalized size of antiderivative = 0.83 \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\frac {3 b \sqrt {c} \left (5 b^{10} c^3+35 a^8 c^3 d^3-3 a b^8 c^2 \left (-5 c^2+b d\right )-5 a^7 b c^2 d^2 \left (9 c^2+7 b d\right )+a^2 b^6 c \left (b^3 c^2-5 c^4-33 b c^2 d-b^2 d^2\right )+a^6 b c d \left (9 b c^4+5 b^2 c^2 d+21 b^3 d^2-5 c^2 d^2\right )+a^3 b^4 \left (c^6+b^4 c^2 d-13 b c^4 d+13 b^2 c^2 d^2-b^3 \left (9 c^4+d^3\right )\right )+a^4 b^3 c \left (-b^3 c^2 d+c^4 d+3 b^4 d^2+33 b c^2 d^2+b^2 \left (-9 c^4+5 d^3\right )\right )+a^5 b^2 \left (39 b^2 c^4 d-11 b^3 c^2 d^2+3 c^4 d^2-5 b^4 d^3+b \left (c^6-15 c^2 d^3\right )\right )\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )+\sqrt {a} \left (-b \left (b^2-a c\right ) (d+c x) \left (15 b^7 c^3 \sqrt {\frac {b+a x}{d+c x}}-a^6 c^2 \left (8 b \left (13 d^2-4 c d x+c^2 x^2\right )+c \sqrt {\frac {b+a x}{d+c x}} \left (57 d^2-22 c d x+8 c^2 x^2\right )\right )-2 a b^5 c^2 \left (-6 c^2 \sqrt {\frac {b+a x}{d+c x}}+2 b d \sqrt {\frac {b+a x}{d+c x}}+b c \left (-12+5 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )+2 a^3 b^2 c \left (6 b^4 (3 d-c x)+b^3 c (2 d-c x) \sqrt {\frac {b+a x}{d+c x}}+c^3 (-2 d+c x) \sqrt {\frac {b+a x}{d+c x}}-2 b^2 \left (d^2-c d x-2 c^2 \left (x^2-3 c \sqrt {\frac {b+a x}{d+c x}}\right )\right )+2 b c \left (3 d^2 \sqrt {\frac {b+a x}{d+c x}}+c^2 x \left (3-4 x \sqrt {\frac {b+a x}{d+c x}}\right )+c d \left (-9+2 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )\right )+2 a^5 c \left (b c^3 (26 d-7 c x) \sqrt {\frac {b+a x}{d+c x}}+4 c^2 \left (d^2-c d x+c^2 x^2\right )+b^3 \left (62 d^2-26 c d x+8 c^2 x^2\right )+4 b^2 c \left (6 d^2 \sqrt {\frac {b+a x}{d+c x}}+c d \left (15-4 x \sqrt {\frac {b+a x}{d+c x}}\right )+c^2 x \left (-3+2 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )\right )+a^2 b^3 c \left (3 b^3 c^2 \sqrt {\frac {b+a x}{d+c x}}-3 c^4 \sqrt {\frac {b+a x}{d+c x}}+8 b c^2 (-5 d+c x) \sqrt {\frac {b+a x}{d+c x}}+b^2 \left (-3 d^2 \sqrt {\frac {b+a x}{d+c x}}+2 c d \left (-6+x \sqrt {\frac {b+a x}{d+c x}}\right )+4 c^2 x \left (-3+2 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )\right )-a^4 b \left (c^3 \sqrt {\frac {b+a x}{d+c x}} \left (-15 d^2+10 c d x-8 c^2 x^2\right )+4 b^4 \left (11 d^2-5 c d x+2 c^2 x^2\right )+b c^2 \left (-20 d^2-4 c d x+c^2 \left (16 x^2+3 c \sqrt {\frac {b+a x}{d+c x}}\right )\right )+8 b^2 c^3 \left (d \sqrt {\frac {b+a x}{d+c x}}+c \left (3-2 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )+b^3 c \left (15 d^2 \sqrt {\frac {b+a x}{d+c x}}+4 c^2 x \left (-9+2 x \sqrt {\frac {b+a x}{d+c x}}\right )-2 c d \left (-54+5 x \sqrt {\frac {b+a x}{d+c x}}\right )\right )\right )\right )-24 b^2 \left (b^8 c^4+6 a^5 b^3 c^3 d^2-a^4 b^6 d^3+4 a^7 c^3 d^3-a^5 b c^3 d^2 (6 a c+d)+a b^7 c d \left (-c^2+a^2 d\right )+a^4 b^2 c^2 d \left (2 a c^3+c^2 d-6 a^2 d^2\right )-a^2 b^5 c^2 \left (a c^3+2 c^2 d+4 a^2 d^2\right )+a^3 b^4 c d \left (a c^3+2 c^2 d+4 a^2 d^2\right )\right ) \log \left (\frac {-b c+a d}{d+c x}\right )+48 c^3 \left (b^3-a^3 c\right ) (b c-a d) \left (b^3-a^2 d\right )^2 \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )\right )}{24 a^{7/2} b c^3 \left (b^2-a c\right )^4} \]

[In]

Integrate[(x^2 - c*x^2*((b + a*x)/(d + c*x))^(3/2))/(a - b*Sqrt[(b + a*x)/(d + c*x)]),x]

[Out]

(3*b*Sqrt[c]*(5*b^10*c^3 + 35*a^8*c^3*d^3 - 3*a*b^8*c^2*(-5*c^2 + b*d) - 5*a^7*b*c^2*d^2*(9*c^2 + 7*b*d) + a^2
*b^6*c*(b^3*c^2 - 5*c^4 - 33*b*c^2*d - b^2*d^2) + a^6*b*c*d*(9*b*c^4 + 5*b^2*c^2*d + 21*b^3*d^2 - 5*c^2*d^2) +
a^3*b^4*(c^6 + b^4*c^2*d - 13*b*c^4*d + 13*b^2*c^2*d^2 - b^3*(9*c^4 + d^3)) + a^4*b^3*c*(-(b^3*c^2*d) + c^4*d
+ 3*b^4*d^2 + 33*b*c^2*d^2 + b^2*(-9*c^4 + 5*d^3)) + a^5*b^2*(39*b^2*c^4*d - 11*b^3*c^2*d^2 + 3*c^4*d^2 - 5*b
^4*d^3 + b*(c^6 - 15*c^2*d^3)))*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]] + Sqrt[a]*(-(b*(b^2 - a*c
)*(d + c*x)*(15*b^7*c^3*Sqrt[(b + a*x)/(d + c*x)] - a^6*c^2*(8*b*(13*d^2 - 4*c*d*x + c^2*x^2) + c*Sqrt[(b + a*
x)/(d + c*x)]*(57*d^2 - 22*c*d*x + 8*c^2*x^2)) - 2*a*b^5*c^2*(-6*c^2*Sqrt[(b + a*x)/(d + c*x)] + 2*b*d*Sqrt[(b
+ a*x)/(d + c*x)] + b*c*(-12 + 5*x*Sqrt[(b + a*x)/(d + c*x)])) + 2*a^3*b^2*c*(6*b^4*(3*d - c*x) + b^3*c*(2*d
- c*x)*Sqrt[(b + a*x)/(d + c*x)] + c^3*(-2*d + c*x)*Sqrt[(b + a*x)/(d + c*x)] - 2*b^2*(d^2 - c*d*x - 2*c^2*(x^
2 - 3*c*Sqrt[(b + a*x)/(d + c*x)])) + 2*b*c*(3*d^2*Sqrt[(b + a*x)/(d + c*x)] + c^2*x*(3 - 4*x*Sqrt[(b + a*x)/(
d + c*x)]) + c*d*(-9 + 2*x*Sqrt[(b + a*x)/(d + c*x)]))) + 2*a^5*c*(b*c^3*(26*d - 7*c*x)*Sqrt[(b + a*x)/(d + c*
x)] + 4*c^2*(d^2 - c*d*x + c^2*x^2) + b^3*(62*d^2 - 26*c*d*x + 8*c^2*x^2) + 4*b^2*c*(6*d^2*Sqrt[(b + a*x)/(d +
c*x)] + c*d*(15 - 4*x*Sqrt[(b + a*x)/(d + c*x)]) + c^2*x*(-3 + 2*x*Sqrt[(b + a*x)/(d + c*x)]))) + a^2*b^3*c*(
3*b^3*c^2*Sqrt[(b + a*x)/(d + c*x)] - 3*c^4*Sqrt[(b + a*x)/(d + c*x)] + 8*b*c^2*(-5*d + c*x)*Sqrt[(b + a*x)/(d
+ c*x)] + b^2*(-3*d^2*Sqrt[(b + a*x)/(d + c*x)] + 2*c*d*(-6 + x*Sqrt[(b + a*x)/(d + c*x)]) + 4*c^2*x*(-3 + 2*
x*Sqrt[(b + a*x)/(d + c*x)]))) - a^4*b*(c^3*Sqrt[(b + a*x)/(d + c*x)]*(-15*d^2 + 10*c*d*x - 8*c^2*x^2) + 4*b^4
*(11*d^2 - 5*c*d*x + 2*c^2*x^2) + b*c^2*(-20*d^2 - 4*c*d*x + c^2*(16*x^2 + 3*c*Sqrt[(b + a*x)/(d + c*x)])) + 8
*b^2*c^3*(d*Sqrt[(b + a*x)/(d + c*x)] + c*(3 - 2*x*Sqrt[(b + a*x)/(d + c*x)])) + b^3*c*(15*d^2*Sqrt[(b + a*x)/
(d + c*x)] + 4*c^2*x*(-9 + 2*x*Sqrt[(b + a*x)/(d + c*x)]) - 2*c*d*(-54 + 5*x*Sqrt[(b + a*x)/(d + c*x)]))))) -
24*b^2*(b^8*c^4 + 6*a^5*b^3*c^3*d^2 - a^4*b^6*d^3 + 4*a^7*c^3*d^3 - a^5*b*c^3*d^2*(6*a*c + d) + a*b^7*c*d*(-c^
2 + a^2*d) + a^4*b^2*c^2*d*(2*a*c^3 + c^2*d - 6*a^2*d^2) - a^2*b^5*c^2*(a*c^3 + 2*c^2*d + 4*a^2*d^2) + a^3*b^4
*c*d*(a*c^3 + 2*c^2*d + 4*a^2*d^2))*Log[(-(b*c) + a*d)/(d + c*x)] + 48*c^3*(b^3 - a^3*c)*(b*c - a*d)*(b^3 - a^
2*d)^2*Log[a - b*Sqrt[(b + a*x)/(d + c*x)]]))/(24*a^(7/2)*b*c^3*(b^2 - a*c)^4)

Maple [B] (warning: unable to verify)

Leaf count of result is larger than twice the leaf count of optimal. \(168214\) vs. \(2(1896)=3792\).

Time = 0.35 (sec) , antiderivative size = 168215, normalized size of antiderivative = 87.79

method	result	size
default	\(\text {Expression too large to display}\)	\(168215\)

[In]

int((x^2-c*x^2*((a*x+b)/(c*x+d))^(3/2))/(a-b*((a*x+b)/(c*x+d))^(1/2)),x,method=_RETURNVERBOSE)

[Out]

result too large to display

Fricas [A] (verification not implemented)

none

Time = 131.38 (sec) , antiderivative size = 4048, normalized size of antiderivative = 2.11 \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]

[In]

integrate((x^2-c*x^2*((a*x+b)/(c*x+d))^(3/2))/(a-b*((a*x+b)/(c*x+d))^(1/2)),x, algorithm="fricas")

[Out]

[1/48*(16*(a^5*b^8*c^3 + a^7*b*c^7 - (a^8*b^2 + 3*a^6*b^3)*c^6 + 3*(a^7*b^4 + a^5*b^5)*c^5 - (3*a^6*b^6 + a^4*
b^7)*c^4)*x^3 + 24*(a^4*b^9*c^3 - (2*a^7*b^3 - a^5*b^4)*c^6 + (5*a^6*b^5 - 2*a^4*b^6)*c^5 - (4*a^5*b^7 - a^3*b
^8)*c^4 - (a^5*b^8*c^2 - (2*a^8*b^2 - a^6*b^3)*c^5 + (5*a^7*b^4 - 2*a^5*b^5)*c^4 - (4*a^6*b^6 - a^4*b^7)*c^3)*
d)*x^2 - 3*((a^5*b^4 + a^3*b^5)*c^6 - (9*a^4*b^6 + 5*a^2*b^7)*c^5 - 3*(3*a^3*b^8 - 5*a*b^9)*c^4 + (a^2*b^10 +
5*b^11)*c^3 - (5*a^5*b^7 + a^3*b^8 - 5*(7*a^8*b - a^6*b^2)*c^3 + 5*(7*a^7*b^3 + 3*a^5*b^4)*c^2 - (21*a^6*b^5 +
5*a^4*b^6)*c)*d^3 - (3*(15*a^7*b^2 - a^5*b^3)*c^4 - (5*a^6*b^4 + 33*a^4*b^5)*c^3 + (11*a^5*b^6 - 13*a^3*b^7)*
c^2 - (3*a^4*b^8 - a^2*b^9)*c)*d^2 + ((9*a^6*b^3 + a^4*b^4)*c^5 + 13*(3*a^5*b^5 - a^3*b^6)*c^4 - (a^4*b^7 + 33
*a^2*b^8)*c^3 + (a^3*b^9 - 3*a*b^10)*c^2)*d)*sqrt(a*c)*log(-2*a*c*x - b*c - a*d + 2*sqrt(a*c)*(c*x + d)*sqrt((
a*x + b)/(c*x + d))) - 48*(a^6*b^4*c^6 + a^2*b^9*c^4 - (a^5*b^6 + a^3*b^7)*c^5 - (a^5*b^8*c - 4*a^6*b^6*c^2 -
(3*a^8*b^2 - a^6*b^3)*c^4 + (6*a^7*b^4 - a^5*b^5)*c^3)*d^2 + (a^4*b^9*c^2 + 7*a^6*b^5*c^4 - (4*a^7*b^3 - a^5*b
^4)*c^5 - (4*a^5*b^7 + a^3*b^8)*c^3)*d)*x - 96*(a^4*b^7*c^5 - a*b^10*c^4 - (a^9*c^4 - a^6*b^3*c^3)*d^3 + (a^8*
b*c^5 - 2*a^4*b^6*c^3 + (2*a^7*b^3 - a^5*b^4)*c^4)*d^2 - (2*a^6*b^4*c^5 - a^2*b^9*c^3 + (a^5*b^6 - 2*a^3*b^7)*
c^4)*d)*log(b*sqrt((a*x + b)/(c*x + d)) - a) + 48*(a^4*b^7*c^5 - a*b^10*c^4 + (a^5*b^8 - 4*a^6*b^6*c + 6*a^7*b
^4*c^2 - (4*a^8*b^2 - a^6*b^3)*c^3)*d^3 - (a^4*b^9*c - 4*a^5*b^7*c^2 - (6*a^7*b^3 - a^5*b^4)*c^4 + 2*(3*a^6*b^
5 + a^4*b^6)*c^3)*d^2 - (2*a^6*b^4*c^5 - a^2*b^9*c^3 + (a^5*b^6 - 2*a^3*b^7)*c^4)*d)*log(1/(c*x + d)) - 2*(3*(
(19*a^8*b - 5*a^6*b^2)*c^4 - (35*a^7*b^3 - a^5*b^4)*c^3 + (21*a^6*b^5 + 5*a^4*b^6)*c^2 - (5*a^5*b^7 + a^3*b^8)
*c)*d^3 + 8*((a^8*b - a^6*b^2)*c^7 - 3*(a^7*b^3 - a^5*b^4)*c^6 + 3*(a^6*b^5 - a^4*b^6)*c^5 - (a^5*b^7 - a^3*b^
8)*c^4)*x^3 - 4*((13*a^7*b^2 - a^5*b^3)*c^5 - 3*(5*a^6*b^4 + 3*a^4*b^5)*c^4 + 3*(a^5*b^6 + 3*a^3*b^7)*c^3 - (a
^4*b^8 - a^2*b^9)*c^2)*d^2 + 2*((7*a^7*b^2 - a^5*b^3)*c^7 - 3*(5*a^6*b^4 + a^4*b^5)*c^6 + 9*(a^5*b^6 + a^3*b^7
)*c^5 - (a^4*b^8 + 5*a^2*b^9)*c^4 - ((7*a^8*b - a^6*b^2)*c^6 - 3*(5*a^7*b^3 + a^5*b^4)*c^5 + 9*(a^6*b^5 + a^4*
b^6)*c^4 - (a^5*b^7 + 5*a^3*b^8)*c^3)*d)*x^2 + 3*((a^6*b^3 + a^4*b^4)*c^6 + (7*a^5*b^5 - 5*a^3*b^6)*c^5 - (9*a
^4*b^7 + a^2*b^8)*c^4 + (a^3*b^9 + 5*a*b^10)*c^3)*d + (3*(a^6*b^3 + a^4*b^4)*c^7 + 3*(7*a^5*b^5 - 5*a^3*b^6)*c
^6 - 3*(9*a^4*b^7 + a^2*b^8)*c^5 + 3*(a^3*b^9 + 5*a*b^10)*c^4 + (5*(7*a^8*b - a^6*b^2)*c^5 - 3*(17*a^7*b^3 + 5
*a^5*b^4)*c^4 + 21*(a^6*b^5 + a^4*b^6)*c^3 - (5*a^5*b^7 + a^3*b^8)*c^2)*d^2 - 2*((19*a^7*b^2 - a^5*b^3)*c^6 -
15*(a^6*b^4 + a^4*b^5)*c^5 - 3*(a^5*b^6 - 3*a^3*b^7)*c^4 - (a^4*b^8 - 7*a^2*b^9)*c^3)*d)*x)*sqrt((a*x + b)/(c*
x + d)))/(a^4*b^9*c^3 - 4*a^5*b^7*c^4 + 6*a^6*b^5*c^5 - 4*a^7*b^3*c^6 + a^8*b*c^7), 1/24*(8*(a^5*b^8*c^3 + a^7
*b*c^7 - (a^8*b^2 + 3*a^6*b^3)*c^6 + 3*(a^7*b^4 + a^5*b^5)*c^5 - (3*a^6*b^6 + a^4*b^7)*c^4)*x^3 + 12*(a^4*b^9*
c^3 - (2*a^7*b^3 - a^5*b^4)*c^6 + (5*a^6*b^5 - 2*a^4*b^6)*c^5 - (4*a^5*b^7 - a^3*b^8)*c^4 - (a^5*b^8*c^2 - (2*
a^8*b^2 - a^6*b^3)*c^5 + (5*a^7*b^4 - 2*a^5*b^5)*c^4 - (4*a^6*b^6 - a^4*b^7)*c^3)*d)*x^2 - 3*((a^5*b^4 + a^3*b
^5)*c^6 - (9*a^4*b^6 + 5*a^2*b^7)*c^5 - 3*(3*a^3*b^8 - 5*a*b^9)*c^4 + (a^2*b^10 + 5*b^11)*c^3 - (5*a^5*b^7 + a
^3*b^8 - 5*(7*a^8*b - a^6*b^2)*c^3 + 5*(7*a^7*b^3 + 3*a^5*b^4)*c^2 - (21*a^6*b^5 + 5*a^4*b^6)*c)*d^3 - (3*(15*
a^7*b^2 - a^5*b^3)*c^4 - (5*a^6*b^4 + 33*a^4*b^5)*c^3 + (11*a^5*b^6 - 13*a^3*b^7)*c^2 - (3*a^4*b^8 - a^2*b^9)*
c)*d^2 + ((9*a^6*b^3 + a^4*b^4)*c^5 + 13*(3*a^5*b^5 - a^3*b^6)*c^4 - (a^4*b^7 + 33*a^2*b^8)*c^3 + (a^3*b^9 - 3
*a*b^10)*c^2)*d)*sqrt(-a*c)*arctan(sqrt(-a*c)*(c*x + d)*sqrt((a*x + b)/(c*x + d))/(a*c*x + b*c)) - 24*(a^6*b^4
*c^6 + a^2*b^9*c^4 - (a^5*b^6 + a^3*b^7)*c^5 - (a^5*b^8*c - 4*a^6*b^6*c^2 - (3*a^8*b^2 - a^6*b^3)*c^4 + (6*a^7
*b^4 - a^5*b^5)*c^3)*d^2 + (a^4*b^9*c^2 + 7*a^6*b^5*c^4 - (4*a^7*b^3 - a^5*b^4)*c^5 - (4*a^5*b^7 + a^3*b^8)*c^
3)*d)*x - 48*(a^4*b^7*c^5 - a*b^10*c^4 - (a^9*c^4 - a^6*b^3*c^3)*d^3 + (a^8*b*c^5 - 2*a^4*b^6*c^3 + (2*a^7*b^3
- a^5*b^4)*c^4)*d^2 - (2*a^6*b^4*c^5 - a^2*b^9*c^3 + (a^5*b^6 - 2*a^3*b^7)*c^4)*d)*log(b*sqrt((a*x + b)/(c*x
+ d)) - a) + 24*(a^4*b^7*c^5 - a*b^10*c^4 + (a^5*b^8 - 4*a^6*b^6*c + 6*a^7*b^4*c^2 - (4*a^8*b^2 - a^6*b^3)*c^3
)*d^3 - (a^4*b^9*c - 4*a^5*b^7*c^2 - (6*a^7*b^3 - a^5*b^4)*c^4 + 2*(3*a^6*b^5 + a^4*b^6)*c^3)*d^2 - (2*a^6*b^4
*c^5 - a^2*b^9*c^3 + (a^5*b^6 - 2*a^3*b^7)*c^4)*d)*log(1/(c*x + d)) - (3*((19*a^8*b - 5*a^6*b^2)*c^4 - (35*a^7
*b^3 - a^5*b^4)*c^3 + (21*a^6*b^5 + 5*a^4*b^6)*c^2 - (5*a^5*b^7 + a^3*b^8)*c)*d^3 + 8*((a^8*b - a^6*b^2)*c^7 -
3*(a^7*b^3 - a^5*b^4)*c^6 + 3*(a^6*b^5 - a^4*b^6)*c^5 - (a^5*b^7 - a^3*b^8)*c^4)*x^3 - 4*((13*a^7*b^2 - a^5*b
^3)*c^5 - 3*(5*a^6*b^4 + 3*a^4*b^5)*c^4 + 3*(a^5*b^6 + 3*a^3*b^7)*c^3 - (a^4*b^8 - a^2*b^9)*c^2)*d^2 + 2*((7*a
^7*b^2 - a^5*b^3)*c^7 - 3*(5*a^6*b^4 + a^4*b^5)*c^6 + 9*(a^5*b^6 + a^3*b^7)*c^5 - (a^4*b^8 + 5*a^2*b^9)*c^4 -
((7*a^8*b - a^6*b^2)*c^6 - 3*(5*a^7*b^3 + a^5*b^4)*c^5 + 9*(a^6*b^5 + a^4*b^6)*c^4 - (a^5*b^7 + 5*a^3*b^8)*c^3
)*d)*x^2 + 3*((a^6*b^3 + a^4*b^4)*c^6 + (7*a^5*b^5 - 5*a^3*b^6)*c^5 - (9*a^4*b^7 + a^2*b^8)*c^4 + (a^3*b^9 + 5
*a*b^10)*c^3)*d + (3*(a^6*b^3 + a^4*b^4)*c^7 + 3*(7*a^5*b^5 - 5*a^3*b^6)*c^6 - 3*(9*a^4*b^7 + a^2*b^8)*c^5 + 3
*(a^3*b^9 + 5*a*b^10)*c^4 + (5*(7*a^8*b - a^6*b^2)*c^5 - 3*(17*a^7*b^3 + 5*a^5*b^4)*c^4 + 21*(a^6*b^5 + a^4*b^
6)*c^3 - (5*a^5*b^7 + a^3*b^8)*c^2)*d^2 - 2*((19*a^7*b^2 - a^5*b^3)*c^6 - 15*(a^6*b^4 + a^4*b^5)*c^5 - 3*(a^5*
b^6 - 3*a^3*b^7)*c^4 - (a^4*b^8 - 7*a^2*b^9)*c^3)*d)*x)*sqrt((a*x + b)/(c*x + d)))/(a^4*b^9*c^3 - 4*a^5*b^7*c^
4 + 6*a^6*b^5*c^5 - 4*a^7*b^3*c^6 + a^8*b*c^7)]

Sympy [F(-1)]

Timed out. \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\text {Timed out} \]

[In]

integrate((x**2-c*x**2*((a*x+b)/(c*x+d))**(3/2))/(a-b*((a*x+b)/(c*x+d))**(1/2)),x)

[Out]

Timed out

Maxima [A] (verification not implemented)

none

Time = 0.36 (sec) , antiderivative size = 2606, normalized size of antiderivative = 1.36 \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\text {Too large to display} \]

[In]

integrate((x^2-c*x^2*((a*x+b)/(c*x+d))^(3/2))/(a-b*((a*x+b)/(c*x+d))^(1/2)),x, algorithm="maxima")

[Out]

-2*(a^3*b^7*c^2 - b^10*c - (a^8*c - a^5*b^3)*d^3 + (a^7*b*c^2 - 2*a^3*b^6 + (2*a^6*b^3 - a^4*b^4)*c)*d^2 - (2*
a^5*b^4*c^2 - a*b^9 + (a^4*b^6 - 2*a^2*b^7)*c)*d)*log(b*sqrt((a*x + b)/(c*x + d)) - a)/(a^3*b^9 - 4*a^4*b^7*c
+ 6*a^5*b^5*c^2 - 4*a^6*b^3*c^3 + a^7*b*c^4) + (a^3*b^6*c^5 - b^9*c^4 + (a^4*b^7 - 4*a^5*b^5*c + 6*a^6*b^3*c^2
- (4*a^7*b - a^5*b^2)*c^3)*d^3 - (a^3*b^8*c - 4*a^4*b^6*c^2 - (6*a^6*b^2 - a^4*b^3)*c^4 + 2*(3*a^5*b^4 + a^3*
b^5)*c^3)*d^2 - (2*a^5*b^3*c^5 - a*b^8*c^3 + (a^4*b^5 - 2*a^2*b^6)*c^4)*d)*log(-a + (a*x + b)*c/(c*x + d))/(a^
3*b^8*c^3 - 4*a^4*b^6*c^4 + 6*a^5*b^4*c^5 - 4*a^6*b^2*c^6 + a^7*c^7) - 1/16*((a^5*b^3 + a^3*b^4)*c^6 - (9*a^4*
b^5 + 5*a^2*b^6)*c^5 - 3*(3*a^3*b^7 - 5*a*b^8)*c^4 + (a^2*b^9 + 5*b^10)*c^3 - (5*a^5*b^6 + a^3*b^7 - 5*(7*a^8
- a^6*b)*c^3 + 5*(7*a^7*b^2 + 3*a^5*b^3)*c^2 - (21*a^6*b^4 + 5*a^4*b^5)*c)*d^3 - (3*(15*a^7*b - a^5*b^2)*c^4 -
(5*a^6*b^3 + 33*a^4*b^4)*c^3 + (11*a^5*b^5 - 13*a^3*b^6)*c^2 - (3*a^4*b^7 - a^2*b^8)*c)*d^2 + ((9*a^6*b^2 + a
^4*b^3)*c^5 + 13*(3*a^5*b^4 - a^3*b^5)*c^4 - (a^4*b^6 + 33*a^2*b^7)*c^3 + (a^3*b^8 - 3*a*b^9)*c^2)*d)*log((c*s
qrt((a*x + b)/(c*x + d)) - sqrt(a*c))/(c*sqrt((a*x + b)/(c*x + d)) + sqrt(a*c)))/((a^3*b^8*c^2 - 4*a^4*b^6*c^3
+ 6*a^5*b^4*c^4 - 4*a^6*b^2*c^5 + a^7*c^6)*sqrt(a*c)) + 1/24*(4*a^4*b^8*c^3 + 8*a^5*b^3*c^6 - 4*(2*a^6*b^4 +
7*a^4*b^5)*c^5 - 4*(5*a^5*b^6 - 11*a^3*b^7)*c^4 + 4*(11*a^7*b^5 - 2*a^8*c^3 + (26*a^9*b - 5*a^7*b^2)*c^2 - (31
*a^8*b^3 - a^6*b^4)*c)*d^3 - 3*((a^4*b^3 + a^2*b^4)*c^8 + 4*(2*a^3*b^5 - a*b^6)*c^7 - (a^2*b^7 + 5*b^8)*c^6 -
((29*a^7 - 11*a^5*b)*c^5 - 4*(8*a^6*b^2 - a^4*b^3)*c^4 + (11*a^5*b^4 - a^3*b^5)*c^3)*d^3 + ((51*a^6*b - 13*a^4
*b^2)*c^6 - 4*(10*a^5*b^3 + 3*a^3*b^4)*c^5 + (13*a^4*b^5 + a^2*b^6)*c^4)*d^2 + (20*a^2*b^5*c^6 - (23*a^5*b^2 -
a^3*b^3)*c^7 - (a^3*b^6 - 3*a*b^7)*c^5)*d)*((a*x + b)/(c*x + d))^(5/2) - 12*(5*a^6*b^6*c + (16*a^8*b^2 - 5*a^
6*b^3)*c^3 - (15*a^7*b^4 + a^5*b^5)*c^2)*d^2 - 8*((a^5*b^3 - a^3*b^4)*c^7 - 2*(4*a^4*b^5 - a^2*b^6)*c^6 + (a^3
*b^7 + 5*a*b^8)*c^5 + ((17*a^8 - 5*a^6*b)*c^4 - 2*(8*a^7*b^2 + a^5*b^3)*c^3 + (5*a^6*b^4 + a^4*b^5)*c^2)*d^3 -
3*((9*a^7*b - a^5*b^2)*c^5 - 2*(2*a^6*b^3 + 3*a^4*b^4)*c^4 + (a^5*b^5 + a^3*b^6)*c^3)*d^2 + 3*((3*a^6*b^2 + a
^4*b^3)*c^6 + 2*(2*a^5*b^4 - 3*a^3*b^5)*c^5 - (a^4*b^6 + a^2*b^7)*c^4)*d)*((a*x + b)/(c*x + d))^(3/2) + 12*(a^
5*b^7*c^2 + (8*a^7*b^3 - a^5*b^4)*c^4 - (3*a^6*b^5 + 5*a^4*b^6)*c^3)*d + 3*((a^6*b^3 + a^4*b^4)*c^6 - 4*(2*a^5
*b^5 + a^3*b^6)*c^5 - (a^4*b^7 - 11*a^2*b^8)*c^4 + ((19*a^9 - 5*a^7*b)*c^3 - 4*(4*a^8*b^2 + a^6*b^3)*c^2 + (5*
a^7*b^4 + a^5*b^5)*c)*d^3 - ((29*a^8*b - 3*a^6*b^2)*c^4 - 4*(2*a^7*b^3 + 5*a^5*b^4)*c^3 + (3*a^6*b^5 - a^4*b^6
)*c^2)*d^2 + ((9*a^7*b^2 + a^5*b^3)*c^5 + 4*(4*a^6*b^4 - 3*a^4*b^5)*c^4 - (a^5*b^6 + 13*a^3*b^7)*c^3)*d)*sqrt(
(a*x + b)/(c*x + d)) - 24*(a^4*b^4*c^7 - a*b^7*c^6 - (3*a^5*b^5*c^2 - 8*a^6*b^3*c^3 + 6*a^7*b*c^4 - a^6*c^5)*d
^3 + (5*a^4*b^6*c^3 - 14*a^5*b^4*c^4 - a^5*b*c^6 + 2*(6*a^6*b^2 - a^4*b^3)*c^5)*d^2 - (2*a^3*b^7*c^4 + (7*a^5*
b^3 - 2*a^3*b^4)*c^6 - (6*a^4*b^5 + a^2*b^6)*c^5)*d)*(a*x + b)^2/(c*x + d)^2 - 12*(a^3*b^8*c^4 - (2*a^5*b^4 +
a^3*b^5)*c^6 - (3*a^4*b^6 - 5*a^2*b^7)*c^5 + (9*a^6*b^5*c - 2*a^7*c^4 + (20*a^8*b - 3*a^6*b^2)*c^3 - (25*a^7*b
^3 - a^5*b^4)*c^2)*d^3 - (13*a^5*b^6*c^2 + (38*a^7*b^2 - 13*a^5*b^3)*c^4 - (39*a^6*b^4 - a^4*b^5)*c^3)*d^2 + (
3*a^4*b^7*c^3 + 2*a^5*b^2*c^6 + (20*a^6*b^3 - 9*a^4*b^4)*c^5 - (11*a^5*b^5 + 5*a^3*b^6)*c^4)*d)*(a*x + b)/(c*x
+ d))/(a^6*b^6*c^3 - 3*a^7*b^4*c^4 + 3*a^8*b^2*c^5 - a^9*c^6 - (a^3*b^6*c^6 - 3*a^4*b^4*c^7 + 3*a^5*b^2*c^8 -
a^6*c^9)*(a*x + b)^3/(c*x + d)^3 + 3*(a^4*b^6*c^5 - 3*a^5*b^4*c^6 + 3*a^6*b^2*c^7 - a^7*c^8)*(a*x + b)^2/(c*x
+ d)^2 - 3*(a^5*b^6*c^4 - 3*a^6*b^4*c^5 + 3*a^7*b^2*c^6 - a^8*c^7)*(a*x + b)/(c*x + d))

Giac [F(-1)]

Timed out. \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\text {Timed out} \]

[In]

integrate((x^2-c*x^2*((a*x+b)/(c*x+d))^(3/2))/(a-b*((a*x+b)/(c*x+d))^(1/2)),x, algorithm="giac")

[Out]

Timed out

Mupad [F(-1)]

Timed out. \[ \int \frac {x^2-c x^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}}{a-b \sqrt {\frac {b+a x}{d+c x}}} \, dx=\text {Hanged} \]

[In]

int((x^2 - c*x^2*((b + a*x)/(d + c*x))^(3/2))/(a - b*((b + a*x)/(d + c*x))^(1/2)),x)

[Out]

\text{Hanged}