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

3.32.53.1 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} \]

output

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+4 
8*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-...
 

3.32.53.2 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} \]

input

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

output

(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)])/Sq 
rt[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)*S 
qrt[(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*Sqr...
 

3.32.53.3 Rubi [A] (verified)

Time = 4.95 (sec) , antiderivative size = 1116, normalized size of antiderivative = 0.58, number of steps used = 10, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.173, Rules used = {7268, 25, 2178, 25, 2178, 27, 2178, 2160, 2009}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {x^2-c x^2 \left (\frac {a x+b}{c x+d}\right )^{3/2}}{a-b \sqrt {\frac {a x+b}{c x+d}}} \, dx\)

\(\Big \downarrow \) 7268

\(\displaystyle 2 (b c-a d) \int -\frac {\sqrt {\frac {b+a x}{d+c x}} \left (1-c \left (\frac {b+a x}{d+c x}\right )^{3/2}\right ) \left (b-\frac {d (b+a x)}{d+c x}\right )^2}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^4}d\sqrt {\frac {b+a x}{d+c x}}\)

\(\Big \downarrow \) 25

\(\displaystyle -2 (b c-a d) \int \frac {\sqrt {\frac {b+a x}{d+c x}} \left (1-c \left (\frac {b+a x}{d+c x}\right )^{3/2}\right ) \left (b-\frac {d (b+a x)}{d+c x}\right )^2}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^4}d\sqrt {\frac {b+a x}{d+c x}}\)

\(\Big \downarrow \) 2178

\(\displaystyle 2 (b c-a d) \left (-\frac {\int -\frac {-\frac {6 a c d^2 (b+a x)^3}{(d+c x)^3}+\frac {6 a d (2 b c-a d) (b+a x)^2}{(d+c x)^2}-\frac {6 a (b c-a d)^2 (b+a x)}{c (d+c x)}+\frac {a \left (a^2-b\right ) (b c-a d)^2}{c \left (b^2-a c\right )}+6 a d^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}-\frac {\left (-5 b d^2 a^4+2 c d \left (5 b^2+3 d\right ) a^3-b \left (5 b^2 c^2+12 d c^2+b d^2\right ) a^2+2 b^3 c d a+5 b^4 c^2\right ) \sqrt {\frac {b+a x}{d+c x}}}{c \left (b^2-a c\right )}}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}d\sqrt {\frac {b+a x}{d+c x}}}{6 a c}-\frac {(b c-a d)^2 \left (c \left (a^2-b\right ) \sqrt {\frac {a x+b}{c x+d}}+a (a b-c)\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (a x+b)}{c x+d}\right )^3}\right )\)

\(\Big \downarrow \) 25

\(\displaystyle 2 (b c-a d) \left (\frac {\int \frac {-\frac {6 a c d^2 (b+a x)^3}{(d+c x)^3}+\frac {6 a d (2 b c-a d) (b+a x)^2}{(d+c x)^2}-\frac {6 a (b c-a d)^2 (b+a x)}{c (d+c x)}+\frac {a \left (a^2-b\right ) (b c-a d)^2}{c \left (b^2-a c\right )}+6 a d^2 \left (\frac {b+a x}{d+c x}\right )^{3/2}-\frac {\left (-5 b d^2 a^4+2 c d \left (5 b^2+3 d\right ) a^3-b \left (5 b^2 c^2+12 d c^2+b d^2\right ) a^2+2 b^3 c d a+5 b^4 c^2\right ) \sqrt {\frac {b+a x}{d+c x}}}{c \left (b^2-a c\right )}}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}d\sqrt {\frac {b+a x}{d+c x}}}{6 a c}-\frac {(b c-a d)^2 \left (c \left (a^2-b\right ) \sqrt {\frac {a x+b}{c x+d}}+a (a b-c)\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (a x+b)}{c x+d}\right )^3}\right )\)

\(\Big \downarrow \) 2178

\(\displaystyle 2 (b c-a d) \left (\frac {\frac {\int -\frac {3 \left (\frac {16 d (b c-a d) (b+a x) a^2}{d+c x}-\frac {8 c d^2 (b+a x)^2 a^2}{(d+c x)^2}+\frac {(b c-a d) \left (5 c d a^4-b \left (c^2+3 b d\right ) a^3-b c \left (b^2+3 d\right ) a^2-b^2 \left (c^2-b d\right ) a+3 b^4 c\right ) a}{\left (b^2-a c\right )^2}+\frac {\left (5 c^2 b^6+a c \left (c^2+2 b d\right ) b^4+a^2 \left (b^2 c^2-14 d c^2+b d^2\right ) b^3-a^3 c \left (14 d b^2+7 c^2 b+3 d^2\right ) b^2-19 a^5 c d^2 b+a^4 d \left (13 d b^3+26 c^2 b^2+8 c^2 d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}}{\left (b^2-a c\right )^2}\right )}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^2}d\sqrt {\frac {b+a x}{d+c x}}}{4 a c}+\frac {\frac {6 a (b c-a d) \left (4 a^3 b c d-3 a^2 b^3 d-2 a^2 b^2 c^2-2 a^2 c^2 d+a b^4 c+a b^2 c d+b^3 c^2\right )}{\left (b^2-a c\right )^2}+\frac {c (b c-a d) \left (19 a^4 c d-13 a^3 b^2 d-7 a^3 b c^2+a^2 b^3 c-13 a^2 b c d+7 a b^3 d+a b^2 c^2+5 b^4 c\right ) \sqrt {\frac {a x+b}{c x+d}}}{\left (b^2-a c\right )^2}}{4 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}}{6 a c}-\frac {(b c-a d)^2 \left (c \left (a^2-b\right ) \sqrt {\frac {a x+b}{c x+d}}+a (a b-c)\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (a x+b)}{c x+d}\right )^3}\right )\)

\(\Big \downarrow \) 27

\(\displaystyle 2 (b c-a d) \left (\frac {\frac {\frac {6 a (b c-a d) \left (4 a^3 b c d-3 a^2 b^3 d-2 a^2 b^2 c^2-2 a^2 c^2 d+a b^4 c+a b^2 c d+b^3 c^2\right )}{\left (b^2-a c\right )^2}+\frac {c (b c-a d) \left (19 a^4 c d-13 a^3 b^2 d-7 a^3 b c^2+a^2 b^3 c-13 a^2 b c d+7 a b^3 d+a b^2 c^2+5 b^4 c\right ) \sqrt {\frac {a x+b}{c x+d}}}{\left (b^2-a c\right )^2}}{4 a c^2 \left (a-\frac {c (a x+b)}{c x+d}\right )^2}-\frac {3 \int \frac {\frac {16 d (b c-a d) (b+a x) a^2}{d+c x}-\frac {8 c d^2 (b+a x)^2 a^2}{(d+c x)^2}+\frac {(b c-a d) \left (5 c d a^4-b \left (c^2+3 b d\right ) a^3-b c \left (b^2+3 d\right ) a^2-b^2 \left (c^2-b d\right ) a+3 b^4 c\right ) a}{\left (b^2-a c\right )^2}+\frac {\left (5 c^2 b^6+a c \left (c^2+2 b d\right ) b^4+a^2 \left (b^2 c^2-14 d c^2+b d^2\right ) b^3-a^3 c \left (14 d b^2+7 c^2 b+3 d^2\right ) b^2-19 a^5 c d^2 b+a^4 d \left (13 d b^3+26 c^2 b^2+8 c^2 d\right )\right ) \sqrt {\frac {b+a x}{d+c x}}}{\left (b^2-a c\right )^2}}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^2}d\sqrt {\frac {b+a x}{d+c x}}}{4 a c}}{6 a c}-\frac {(b c-a d)^2 \left (c \left (a^2-b\right ) \sqrt {\frac {a x+b}{c x+d}}+a (a b-c)\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (a x+b)}{c x+d}\right )^3}\right )\)

\(\Big \downarrow \) 2178

\(\displaystyle 2 (b c-a d) \left (\frac {\frac {\frac {c (b c-a d) \sqrt {\frac {b+a x}{d+c x}} \left (19 c d a^4-7 b c^2 a^3-13 b^2 d a^3+b^3 c a^2-13 b c d a^2+b^2 c^2 a+7 b^3 d a+5 b^4 c\right )}{\left (b^2-a c\right )^2}+\frac {6 a (b c-a d) \left (a c b^4+c^2 b^3-3 a^2 d b^3-2 a^2 c^2 b^2+a c d b^2+4 a^3 c d b-2 a^2 c^2 d\right )}{\left (b^2-a c\right )^2}}{4 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 \left (\frac {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}}}{2 a c \left (b^2-a c\right )^3 \left (a-\frac {c (b+a x)}{d+c x}\right )}+\frac {\int \frac {\frac {16 c d^2 (b+a x) a^3}{d+c x}+\frac {c \left (11 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 (5 c^2+8 b d\right ) b+a^4 \left (b c^4+5 d^2 c^2+8 b^2 d c^2-5 b^3 d^2\right ) b-19 a^6 c^2 d^2\right ) a}{\left (b^2-a c\right )^3}+\frac {b 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}}}{\left (b^2-a c\right )^3}}{\left (a-b \sqrt {\frac {b+a x}{d+c x}}\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )}d\sqrt {\frac {b+a x}{d+c x}}}{2 a c}\right )}{4 a c}}{6 a c}-\frac {(b c-a d)^2 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}\right )\)

\(\Big \downarrow \) 2160

\(\displaystyle 2 (b c-a d) \left (\frac {\frac {\frac {c (b c-a d) \sqrt {\frac {b+a x}{d+c x}} \left (19 c d a^4-7 b c^2 a^3-13 b^2 d a^3+b^3 c a^2-13 b c d a^2+b^2 c^2 a+7 b^3 d a+5 b^4 c\right )}{\left (b^2-a c\right )^2}+\frac {6 a (b c-a d) \left (a c b^4+c^2 b^3-3 a^2 d b^3-2 a^2 c^2 b^2+a c d b^2+4 a^3 c d b-2 a^2 c^2 d\right )}{\left (b^2-a c\right )^2}}{4 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 \left (\frac {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}}}{2 a c \left (b^2-a c\right )^3 \left (a-\frac {c (b+a x)}{d+c x}\right )}+\frac {\int \left (\frac {16 \left (a^3 c-b^3\right ) \left (b^3-a^2 d\right )^2 c^3}{\left (b^2-a c\right )^4 \left (b \sqrt {\frac {b+a x}{d+c x}}-a\right )}+\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-16 \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 ) \sqrt {\frac {b+a x}{d+c x}} b+35 a^7 c^3 d^2\right ) c}{\left (b^2-a c\right )^4 \left (a-\frac {c (b+a x)}{d+c x}\right )}\right )d\sqrt {\frac {b+a x}{d+c x}}}{2 a c}\right )}{4 a c}}{6 a c}-\frac {(b c-a d)^2 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}\right )\)

\(\Big \downarrow \) 2009

\(\displaystyle 2 (b c-a d) \left (\frac {\frac {\frac {c (b c-a d) \sqrt {\frac {b+a x}{d+c x}} \left (19 c d a^4-7 b c^2 a^3-13 b^2 d a^3+b^3 c a^2-13 b c d a^2+b^2 c^2 a+7 b^3 d a+5 b^4 c\right )}{\left (b^2-a c\right )^2}+\frac {6 a (b c-a d) \left (a c b^4+c^2 b^3-3 a^2 d b^3-2 a^2 c^2 b^2+a c d b^2+4 a^3 c d b-2 a^2 c^2 d\right )}{\left (b^2-a c\right )^2}}{4 a c^2 \left (a-\frac {c (b+a x)}{d+c x}\right )^2}-\frac {3 \left (\frac {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}}}{2 a c \left (b^2-a c\right )^3 \left (a-\frac {c (b+a x)}{d+c x}\right )}+\frac {-\frac {16 \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 ) c^3}{b \left (b^2-a c\right )^4}-\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 ) \sqrt {c}}{\sqrt {a} \left (b^2-a c\right )^4}+\frac {8 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 \left (a-\frac {c (b+a x)}{d+c x}\right )}{\left (b^2-a c\right )^4}}{2 a c}\right )}{4 a c}}{6 a c}-\frac {(b c-a d)^2 \left (a (a b-c)+\left (a^2-b\right ) c \sqrt {\frac {b+a x}{d+c x}}\right )}{6 a c^3 \left (b^2-a c\right ) \left (a-\frac {c (b+a x)}{d+c x}\right )^3}\right )\)

input

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

output

2*(b*c - a*d)*(-1/6*((b*c - a*d)^2*(a*(a*b - c) + (a^2 - b)*c*Sqrt[(b + a* 
x)/(d + c*x)]))/(a*c^3*(b^2 - a*c)*(a - (c*(b + a*x))/(d + c*x))^3) + (((6 
*a*(b*c - a*d)*(a*b^4*c - 2*a^2*b^2*c^2 + b^3*c^2 - 3*a^2*b^3*d + 4*a^3*b* 
c*d + a*b^2*c*d - 2*a^2*c^2*d))/(b^2 - a*c)^2 + (c*(b*c - a*d)*(a^2*b^3*c 
+ 5*b^4*c - 7*a^3*b*c^2 + a*b^2*c^2 - 13*a^3*b^2*d + 7*a*b^3*d + 19*a^4*c* 
d - 13*a^2*b*c*d)*Sqrt[(b + a*x)/(d + c*x)])/(b^2 - a*c)^2)/(4*a*c^2*(a - 
(c*(b + a*x))/(d + c*x))^2) - (3*((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)])/(2*a*c*(b^2 - a*c)^3*(a - (c*(b + a*x))/(d + c*x))) + 
(-((Sqrt[c]*(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]])/(Sqrt[a]*(b^2 - a*c)^4)) - (16*c^3*(b^3 - 
a^3*c)*(b^3 - a^2*d)^2*Log[a - b*Sqrt[(b + a*x)/(d + c*x)]])/(b*(b^2 - ...
 

3.32.53.3.1 Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2160

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

rule 2178

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{Qx = PolynomialQuotient[(d + e*x)^m*Pq, a + b*x^2, x], R = Coeff[Po 
lynomialRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 0], S = Coeff[Polynomia 
lRemainder[(d + e*x)^m*Pq, a + b*x^2, x], x, 1]}, Simp[(a*S - b*R*x)*((a + 
b*x^2)^(p + 1)/(2*a*b*(p + 1))), x] + Simp[1/(2*a*b*(p + 1))   Int[(d + e*x 
)^m*(a + b*x^2)^(p + 1)*ExpandToSum[(2*a*b*(p + 1)*Qx)/(d + e*x)^m + (b*R*( 
2*p + 3))/(d + e*x)^m, x], x], x]] /; FreeQ[{a, b, d, e}, x] && PolyQ[Pq, x 
] && NeQ[b*d^2 + a*e^2, 0] && LtQ[p, -1] && ILtQ[m, 0]
 

rule 7268

Int[u_, x_Symbol] :> With[{lst = SubstForFractionalPowerOfQuotientOfLinears 
[u, x]}, Simp[lst[[2]]*lst[[4]]   Subst[Int[lst[[1]], x], x, lst[[3]]^(1/ls 
t[[2]])], x] /;  !FalseQ[lst]]
 

3.32.53.4 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	\(\text {Expression too large to display}\)	\(168215\)

input

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,me 
thod=_RETURNVERBOSE)
 

output

result too large to display
 

3.32.53.5 Fricas [A] (verification not implemented)

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

input

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")
 

output

[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*...
 

3.32.53.6 Sympy [F(-1)]

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

input

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)
 

output

Timed out
 

3.32.53.7 Maxima [A] (verification not implemented)

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

input

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")
 

output

-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*sqrt((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*...
 

3.32.53.8 Giac [F(-1)]

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

input

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")
 

output

Timed out
 

3.32.53.9 Mupad [F(-1)]

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

input

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)
 

output

\text{Hanged}