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

Optimal. Leaf size=1916 \[ \frac {-3 a^2 c^2 x^2 b^6+6 c^3 x b^6+6 a^2 c d x b^6-2 a^3 c^2 x^3 b^5-3 a c^3 x^2 b^5+3 a^3 c d x^2 b^5-6 a^3 d^2 x b^5-6 a c^2 d x b^5+2 a^2 c^3 x^3 b^4+9 a^3 c^3 x^2 b^4+3 a^2 c^2 d x^2 b^4-18 a^3 c^2 d x b^4+4 a^4 c^3 x^3 b^3+3 a^2 c^4 x^2 b^3-9 a^4 c^2 d x^2 b^3-6 a^3 c^4 x b^3+18 a^4 c d^2 x b^3-6 a^2 c^3 d x b^3-4 a^3 c^4 x^3 b^2-6 a^4 c^4 x^2 b^2-3 a^3 c^3 d x^2 b^2+6 a^3 c^2 d^2 x b^2+24 a^4 c^3 d x b^2-2 a^5 c^4 x^3 b+6 a^5 c^3 d x^2 b-18 a^5 c^2 d^2 x b+2 a^4 c^5 x^3}{6 a^2 c^2 \left (a c-b^2\right )^3}+\frac {\sqrt {\frac {b+a x}{d+c x}} \left (15 c^2 d b^7+15 c^3 x b^7-4 a c d^2 b^6-10 a c^3 x^2 b^6+3 a^2 c^2 d b^6+3 a^2 c^3 x b^6-14 a c^2 d x b^6-3 a^2 d^3 b^5+8 a^2 c^3 x^3 b^5+4 a^3 c d^2 b^5-2 a^3 c^3 x^2 b^5+10 a^2 c^2 d x^2 b^5+12 a c^3 d b^5+12 a c^4 x b^5-a^2 c d^2 x b^5+2 a^3 c^2 d x b^5-15 a^4 d^3 b^4-8 a^4 c^3 x^3 b^4-40 a^2 c^2 d^2 b^4+8 a^2 c^4 x^2 b^4+2 a^4 c^2 d x^2 b^4-24 a^3 c^3 d b^4-24 a^3 c^4 x b^4-5 a^4 c d^2 x b^4-32 a^2 c^3 d x b^4+12 a^3 c d^3 b^3-16 a^3 c^4 x^3 b^3-8 a^4 c^2 d^2 b^3+16 a^4 c^4 x^2 b^3-8 a^3 c^3 d x^2 b^3-3 a^2 c^4 d b^3-3 a^2 c^5 x b^3+20 a^3 c^2 d^2 x b^3+8 a^4 c^3 d x b^3+48 a^5 c d^3 b^2+16 a^5 c^4 x^3 b^2-4 a^3 c^3 d^2 b^2+2 a^3 c^5 x^2 b^2-16 a^5 c^3 d x^2 b^2-3 a^4 c^4 d b^2-3 a^4 c^5 x b^2+16 a^5 c^2 d^2 x b^2-2 a^3 c^4 d x b^2+15 a^4 c^2 d^3 b+8 a^4 c^5 x^3 b+52 a^5 c^3 d^2 b-14 a^5 c^5 x^2 b-2 a^4 c^4 d x^2 b+5 a^4 c^3 d^2 x b+38 a^5 c^4 d x b-57 a^6 c^2 d^3-8 a^6 c^5 x^3+14 a^6 c^4 d x^2-35 a^6 c^3 d^2 x\right )}{24 a^3 c^2 \left (a c-b^2\right )^3}+\frac {\left (5 c^3 b^{10}+a^2 c^3 b^9-3 a c^2 d b^9+15 a c^4 b^8-a^2 c d^2 b^8+a^3 c^2 d b^8-9 a^3 c^4 b^7-a^3 d^3 b^7+3 a^4 c d^2 b^7-33 a^2 c^3 d b^7-5 a^2 c^5 b^6-5 a^5 d^3 b^6+13 a^3 c^2 d^2 b^6-a^4 c^3 d b^6-9 a^4 c^5 b^5+5 a^4 c d^3 b^5-11 a^5 c^2 d^2 b^5-13 a^3 c^4 d b^5+a^3 c^6 b^4+21 a^6 c d^3 b^4+33 a^4 c^3 d^2 b^4+39 a^5 c^4 d b^4+a^5 c^6 b^3-15 a^5 c^2 d^3 b^3+5 a^6 c^3 d^2 b^3+a^4 c^5 d b^3-35 a^7 c^2 d^3 b^2+3 a^5 c^4 d^2 b^2+9 a^6 c^5 d b^2-5 a^6 c^3 d^3 b-45 a^7 c^4 d^2 b+35 a^8 c^3 d^3\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{8 a^{7/2} c^{5/2} \left (a c-b^2\right )^4}+\frac {\left (c^4 b^9+a^3 c d^2 b^8-a c^3 d b^8-a^4 d^3 b^7-a^3 c^5 b^6-4 a^4 c^2 d^2 b^6-2 a^2 c^4 d b^6+4 a^5 c d^3 b^5+2 a^3 c^3 d^2 b^5+a^4 c^4 d b^5+6 a^5 c^3 d^2 b^4-6 a^6 c^2 d^3 b^3+a^4 c^4 d^2 b^3+2 a^5 c^5 d b^3-a^5 c^3 d^3 b^2-6 a^6 c^4 d^2 b^2+4 a^7 c^3 d^3 b\right ) \log (d+c x)}{a^3 c^3 \left (a c-b^2\right )^4}+\frac {2 \left (c b^{10}-a d b^9-a^3 c^2 b^7-2 a^2 c d b^7+2 a^3 d^2 b^6+a^4 c d b^6+a^4 c d^2 b^4+2 a^5 c^2 d b^4-a^5 d^3 b^3-2 a^6 c d^2 b^3-a^7 c^2 d^2 b+a^8 c d^3\right ) \log \left (a-b \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 b \left (a c-b^2\right )^4} \]

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Rubi [A]  time = 4.68, antiderivative size = 1063, normalized size of antiderivative = 0.55, number of steps used = 9, number of rules used = 5, integrand size = 52, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.096, Rules used = {1647, 1629, 635, 208, 260} \begin {gather*} -\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 ) \tanh ^{-1}\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 \left (\frac {1}{d+c x}\right ) (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} \end {gather*}

Antiderivative was successfully verified.

[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)^(-1)])/(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 208

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

Rule 260

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 635

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 1629

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 1647

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*} \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 &=(2 (b c-a d)) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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) \operatorname {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 ) \operatorname {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 ) \operatorname {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 ) \tanh ^{-1}\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 \left (\frac {1}{d+c x}\right )}{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*}

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Mathematica [A]  time = 5.33, size = 1379, normalized size = 0.72 \begin {gather*} \frac {1}{24} (a d-b c) \left (-\frac {8 \left (a^2-b\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)^3}{a c^2 \left (a c-b^2\right ) (a d-b c)}+\frac {8 (a b-c) (d+c x)^3}{c^3 \left (a c-b^2\right ) (b c-a d)}+\frac {12 \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 ) (d+c x)^2}{a c^3 \left (b^2-a c\right )^2 (a d-b c)}+\frac {12 \left (4 c d a^3-b \left (2 c^2+3 b d\right ) a^2+b c \left (b^2-3 d\right ) a+b^2 \left (c^2+2 b d\right )\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)^2}{a c^2 \left (b^2-a c\right )^2 (a d-b c)}+\frac {24 \left (\left (c^3+2 a^2 d c\right ) b^6-3 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 ) (d+c x)}{a^2 c^3 \left (a c-b^2\right )^3 (a d-b c)}+\frac {24 \left (-6 c^2 d^2 a^5-3 b^4 d^2 a^3+3 b c^2 d (2 a c+d) a^3-3 b^3 c d (2 a c+d) a^2+b^2 c \left (-a c^3-2 d c^2+8 a^2 d^2\right ) a^2+b^5 \left (c^3+2 a^2 d c+a d^2\right )\right ) \sqrt {\frac {b+a x}{d+c x}} (d+c x)}{a^2 c^2 \left (a c-b^2\right )^3 (a d-b c)}+\frac {24 \left (6 c^2 d^2 a^5+3 b^4 d^2 a^3-3 b c^2 d (2 a c+d) a^3+3 b^3 c d (2 a c+d) a^2+b^2 c \left (a c^3+2 d c^2-8 a^2 d^2\right ) a^2-b^5 \left (c^3+2 a^2 d c+a d^2\right )\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )}{a^{5/2} c^{5/2} \left (b^2-a c\right )^3}-\frac {5 \left (a^2-b\right ) \left (3 \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right ) (b c-a d)^2+\sqrt {a} \sqrt {c} \sqrt {\frac {b+a x}{d+c x}} (d+c x) (-3 b c+2 a x c+5 a d)\right )}{a^{7/2} c^{5/2} \left (a c-b^2\right )}+\frac {18 \left (4 c d a^3-b \left (2 c^2+3 b d\right ) a^2+b c \left (b^2-3 d\right ) a+b^2 \left (c^2+2 b d\right )\right ) \left (\sqrt {a} \sqrt {c} \sqrt {\frac {b+a x}{d+c x}} (d+c x)+(a d-b c) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}}{\sqrt {a}}\right )\right )}{a^{5/2} c^{5/2} \left (b^2-a c\right )^2}-\frac {48 \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 )}{a^3 b \left (b^2-a c\right )^4}+\frac {24 \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 (\sqrt {a}-\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}\right )}{a^3 \left (b-\sqrt {a} \sqrt {c}\right )^4 \left (b+\sqrt {a} \sqrt {c}\right )^3 c^3}-\frac {24 \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 (\sqrt {a}+\sqrt {c} \sqrt {\frac {b+a x}{d+c x}}\right )}{\left (b+\sqrt {a} \sqrt {c}\right )^4 \left (a^{3/2} c^{3/2}-a b c\right )^3}\right ) \end {gather*}

Antiderivative was successfully verified.

[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]

((-(b*c) + a*d)*((24*(-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))*(d + c*x))/(a^2*c^3*(-b^2 + a*c)^3*(-(b*c) +
a*d)) + (24*(-3*a^3*b^4*d^2 - 6*a^5*c^2*d^2 - 3*a^2*b^3*c*d*(2*a*c + d) + 3*a^3*b*c^2*d*(2*a*c + d) + b^5*(c^3
+ 2*a^2*c*d + a*d^2) + a^2*b^2*c*(-(a*c^3) - 2*c^2*d + 8*a^2*d^2))*Sqrt[(b + a*x)/(d + c*x)]*(d + c*x))/(a^2*
c^2*(-b^2 + a*c)^3*(-(b*c) + a*d)) + (12*(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))*(d + c*x)^2)/(a*c^3*(b^2 - a*c)^2*(-(b*c) + a*d)) + (12*(a*b*c*(b^2 - 3*d) + 4*a^3*c*d + b^2*(c^2
+ 2*b*d) - a^2*b*(2*c^2 + 3*b*d))*Sqrt[(b + a*x)/(d + c*x)]*(d + c*x)^2)/(a*c^2*(b^2 - a*c)^2*(-(b*c) + a*d))
+ (8*(a*b - c)*(d + c*x)^3)/(c^3*(-b^2 + a*c)*(b*c - a*d)) - (8*(a^2 - b)*Sqrt[(b + a*x)/(d + c*x)]*(d + c*x)
^3)/(a*c^2*(-b^2 + a*c)*(-(b*c) + a*d)) + (24*(3*a^3*b^4*d^2 + 6*a^5*c^2*d^2 + 3*a^2*b^3*c*d*(2*a*c + d) - 3*a
^3*b*c^2*d*(2*a*c + d) - b^5*(c^3 + 2*a^2*c*d + a*d^2) + a^2*b^2*c*(a*c^3 + 2*c^2*d - 8*a^2*d^2))*ArcTanh[(Sqr
t[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]])/(a^(5/2)*c^(5/2)*(b^2 - a*c)^3) - (5*(a^2 - b)*(Sqrt[a]*Sqrt[c]*Sqrt
[(b + a*x)/(d + c*x)]*(d + c*x)*(-3*b*c + 5*a*d + 2*a*c*x) + 3*(b*c - a*d)^2*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(
d + c*x)])/Sqrt[a]]))/(a^(7/2)*c^(5/2)*(-b^2 + a*c)) + (18*(a*b*c*(b^2 - 3*d) + 4*a^3*c*d + b^2*(c^2 + 2*b*d)
- a^2*b*(2*c^2 + 3*b*d))*(Sqrt[a]*Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)]*(d + c*x) + (-(b*c) + a*d)*ArcTanh[(Sqrt[c
]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]]))/(a^(5/2)*c^(5/2)*(b^2 - a*c)^2) - (48*(b^3 - a^3*c)*(b^3 - a^2*d)^2*Lo
g[a - b*Sqrt[(b + a*x)/(d + c*x)]])/(a^3*b*(b^2 - a*c)^4) + (24*(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[Sqrt[a] - Sqrt[c]*Sqrt[(
b + a*x)/(d + c*x)]])/(a^3*(b - Sqrt[a]*Sqrt[c])^4*(b + Sqrt[a]*Sqrt[c])^3*c^3) - (24*(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[Sq
rt[a] + Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)]])/((b + Sqrt[a]*Sqrt[c])^4*(-(a*b*c) + a^(3/2)*c^(3/2))^3)))/24

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IntegrateAlgebraic [B]  time = 3.46, size = 4104, normalized size = 2.14 \begin {gather*} \text {Result too large to show} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*a^4*b^8*c^3 + 20*a^5*b^6*c^4 - 44*a^3*b^7*c^4 + 8*a^6*b^4*c^5 + 28*a^4*b^5*c^5 - 8*a^5*b^3*c^6 - 12*a^5*b^
7*c^2*d + 36*a^6*b^5*c^3*d + 60*a^4*b^6*c^3*d - 96*a^7*b^3*c^4*d + 12*a^5*b^4*c^4*d + 60*a^6*b^6*c*d^2 - 180*a
^7*b^4*c^2*d^2 - 12*a^5*b^5*c^2*d^2 + 192*a^8*b^2*c^3*d^2 - 60*a^6*b^3*c^3*d^2 - 44*a^7*b^5*d^3 + 124*a^8*b^3*
c*d^3 - 4*a^6*b^4*c*d^3 - 104*a^9*b*c^2*d^3 + 20*a^7*b^2*c^2*d^3 + 8*a^8*c^3*d^3 + 3*a^4*b^7*c^4*Sqrt[(b + a*x
)/(d + c*x)] - 33*a^2*b^8*c^4*Sqrt[(b + a*x)/(d + c*x)] + 24*a^5*b^5*c^5*Sqrt[(b + a*x)/(d + c*x)] + 12*a^3*b^
6*c^5*Sqrt[(b + a*x)/(d + c*x)] - 3*a^6*b^3*c^6*Sqrt[(b + a*x)/(d + c*x)] - 3*a^4*b^4*c^6*Sqrt[(b + a*x)/(d +
c*x)] + 3*a^5*b^6*c^3*d*Sqrt[(b + a*x)/(d + c*x)] + 39*a^3*b^7*c^3*d*Sqrt[(b + a*x)/(d + c*x)] - 48*a^6*b^4*c^
4*d*Sqrt[(b + a*x)/(d + c*x)] + 36*a^4*b^5*c^4*d*Sqrt[(b + a*x)/(d + c*x)] - 27*a^7*b^2*c^5*d*Sqrt[(b + a*x)/(
d + c*x)] - 3*a^5*b^3*c^5*d*Sqrt[(b + a*x)/(d + c*x)] + 9*a^6*b^5*c^2*d^2*Sqrt[(b + a*x)/(d + c*x)] - 3*a^4*b^
6*c^2*d^2*Sqrt[(b + a*x)/(d + c*x)] - 24*a^7*b^3*c^3*d^2*Sqrt[(b + a*x)/(d + c*x)] - 60*a^5*b^4*c^3*d^2*Sqrt[(
b + a*x)/(d + c*x)] + 87*a^8*b*c^4*d^2*Sqrt[(b + a*x)/(d + c*x)] - 9*a^6*b^2*c^4*d^2*Sqrt[(b + a*x)/(d + c*x)]
- 15*a^7*b^4*c*d^3*Sqrt[(b + a*x)/(d + c*x)] - 3*a^5*b^5*c*d^3*Sqrt[(b + a*x)/(d + c*x)] + 48*a^8*b^2*c^2*d^3
*Sqrt[(b + a*x)/(d + c*x)] + 12*a^6*b^3*c^2*d^3*Sqrt[(b + a*x)/(d + c*x)] - 57*a^9*c^3*d^3*Sqrt[(b + a*x)/(d +
c*x)] + 15*a^7*b*c^3*d^3*Sqrt[(b + a*x)/(d + c*x)] + 8*a^3*b^7*c^5*((b + a*x)/(d + c*x))^(3/2) + 40*a*b^8*c^5
*((b + a*x)/(d + c*x))^(3/2) - 64*a^4*b^5*c^6*((b + a*x)/(d + c*x))^(3/2) + 16*a^2*b^6*c^6*((b + a*x)/(d + c*x
))^(3/2) + 8*a^5*b^3*c^7*((b + a*x)/(d + c*x))^(3/2) - 8*a^3*b^4*c^7*((b + a*x)/(d + c*x))^(3/2) - 24*a^4*b^6*
c^4*d*((b + a*x)/(d + c*x))^(3/2) - 24*a^2*b^7*c^4*d*((b + a*x)/(d + c*x))^(3/2) + 96*a^5*b^4*c^5*d*((b + a*x)
/(d + c*x))^(3/2) - 144*a^3*b^5*c^5*d*((b + a*x)/(d + c*x))^(3/2) + 72*a^6*b^2*c^6*d*((b + a*x)/(d + c*x))^(3/
2) + 24*a^4*b^3*c^6*d*((b + a*x)/(d + c*x))^(3/2) - 24*a^5*b^5*c^3*d^2*((b + a*x)/(d + c*x))^(3/2) - 24*a^3*b^
6*c^3*d^2*((b + a*x)/(d + c*x))^(3/2) + 96*a^6*b^3*c^4*d^2*((b + a*x)/(d + c*x))^(3/2) + 144*a^4*b^4*c^4*d^2*(
(b + a*x)/(d + c*x))^(3/2) - 216*a^7*b*c^5*d^2*((b + a*x)/(d + c*x))^(3/2) + 24*a^5*b^2*c^5*d^2*((b + a*x)/(d
+ c*x))^(3/2) + 40*a^6*b^4*c^2*d^3*((b + a*x)/(d + c*x))^(3/2) + 8*a^4*b^5*c^2*d^3*((b + a*x)/(d + c*x))^(3/2)
- 128*a^7*b^2*c^3*d^3*((b + a*x)/(d + c*x))^(3/2) - 16*a^5*b^3*c^3*d^3*((b + a*x)/(d + c*x))^(3/2) + 136*a^8*
c^4*d^3*((b + a*x)/(d + c*x))^(3/2) - 40*a^6*b*c^4*d^3*((b + a*x)/(d + c*x))^(3/2) - 3*a^2*b^7*c^6*((b + a*x)/
(d + c*x))^(5/2) - 15*b^8*c^6*((b + a*x)/(d + c*x))^(5/2) + 24*a^3*b^5*c^7*((b + a*x)/(d + c*x))^(5/2) - 12*a*
b^6*c^7*((b + a*x)/(d + c*x))^(5/2) + 3*a^4*b^3*c^8*((b + a*x)/(d + c*x))^(5/2) + 3*a^2*b^4*c^8*((b + a*x)/(d
+ c*x))^(5/2) - 3*a^3*b^6*c^5*d*((b + a*x)/(d + c*x))^(5/2) + 9*a*b^7*c^5*d*((b + a*x)/(d + c*x))^(5/2) + 60*a
^2*b^5*c^6*d*((b + a*x)/(d + c*x))^(5/2) - 69*a^5*b^2*c^7*d*((b + a*x)/(d + c*x))^(5/2) + 3*a^3*b^3*c^7*d*((b
+ a*x)/(d + c*x))^(5/2) + 39*a^4*b^5*c^4*d^2*((b + a*x)/(d + c*x))^(5/2) + 3*a^2*b^6*c^4*d^2*((b + a*x)/(d + c
*x))^(5/2) - 120*a^5*b^3*c^5*d^2*((b + a*x)/(d + c*x))^(5/2) - 36*a^3*b^4*c^5*d^2*((b + a*x)/(d + c*x))^(5/2)
+ 153*a^6*b*c^6*d^2*((b + a*x)/(d + c*x))^(5/2) - 39*a^4*b^2*c^6*d^2*((b + a*x)/(d + c*x))^(5/2) - 33*a^5*b^4*
c^3*d^3*((b + a*x)/(d + c*x))^(5/2) + 3*a^3*b^5*c^3*d^3*((b + a*x)/(d + c*x))^(5/2) + 96*a^6*b^2*c^4*d^3*((b +
a*x)/(d + c*x))^(5/2) - 12*a^4*b^3*c^4*d^3*((b + a*x)/(d + c*x))^(5/2) - 87*a^7*c^5*d^3*((b + a*x)/(d + c*x))
^(5/2) + 33*a^5*b*c^5*d^3*((b + a*x)/(d + c*x))^(5/2) - (24*a*b^7*c^6*(b + a*x)^2)/(d + c*x)^2 + (24*a^4*b^4*c
^7*(b + a*x)^2)/(d + c*x)^2 - (48*a^3*b^7*c^4*d*(b + a*x)^2)/(d + c*x)^2 + (144*a^4*b^5*c^5*d*(b + a*x)^2)/(d
+ c*x)^2 + (24*a^2*b^6*c^5*d*(b + a*x)^2)/(d + c*x)^2 - (168*a^5*b^3*c^6*d*(b + a*x)^2)/(d + c*x)^2 + (48*a^3*
b^4*c^6*d*(b + a*x)^2)/(d + c*x)^2 + (120*a^4*b^6*c^3*d^2*(b + a*x)^2)/(d + c*x)^2 - (336*a^5*b^4*c^4*d^2*(b +
a*x)^2)/(d + c*x)^2 + (288*a^6*b^2*c^5*d^2*(b + a*x)^2)/(d + c*x)^2 - (48*a^4*b^3*c^5*d^2*(b + a*x)^2)/(d + c
*x)^2 - (24*a^5*b*c^6*d^2*(b + a*x)^2)/(d + c*x)^2 - (72*a^5*b^5*c^2*d^3*(b + a*x)^2)/(d + c*x)^2 + (192*a^6*b
^3*c^3*d^3*(b + a*x)^2)/(d + c*x)^2 - (144*a^7*b*c^4*d^3*(b + a*x)^2)/(d + c*x)^2 + (24*a^6*c^5*d^3*(b + a*x)^
2)/(d + c*x)^2 + (12*a^3*b^8*c^4*(b + a*x))/(d + c*x) - (36*a^4*b^6*c^5*(b + a*x))/(d + c*x) + (60*a^2*b^7*c^5
*(b + a*x))/(d + c*x) - (24*a^5*b^4*c^6*(b + a*x))/(d + c*x) - (12*a^3*b^5*c^6*(b + a*x))/(d + c*x) + (36*a^4*
b^7*c^3*d*(b + a*x))/(d + c*x) - (132*a^5*b^5*c^4*d*(b + a*x))/(d + c*x) - (60*a^3*b^6*c^4*d*(b + a*x))/(d + c
*x) + (240*a^6*b^3*c^5*d*(b + a*x))/(d + c*x) - (108*a^4*b^4*c^5*d*(b + a*x))/(d + c*x) + (24*a^5*b^2*c^6*d*(b
+ a*x))/(d + c*x) - (156*a^5*b^6*c^2*d^2*(b + a*x))/(d + c*x) + (468*a^6*b^4*c^3*d^2*(b + a*x))/(d + c*x) - (
12*a^4*b^5*c^3*d^2*(b + a*x))/(d + c*x) - (456*a^7*b^2*c^4*d^2*(b + a*x))/(d + c*x) + (156*a^5*b^3*c^4*d^2*(b
+ a*x))/(d + c*x) + (108*a^6*b^5*c*d^3*(b + a*x))/(d + c*x) - (300*a^7*b^3*c^2*d^3*(b + a*x))/(d + c*x) + (12*
a^5*b^4*c^2*d^3*(b + a*x))/(d + c*x) + (240*a^8*b*c^3*d^3*(b + a*x))/(d + c*x) - (36*a^6*b^2*c^3*d^3*(b + a*x)
)/(d + c*x) - (24*a^7*c^4*d^3*(b + a*x))/(d + c*x))/(24*a^3*c^3*(-b^2 + a*c)^3*(a - (c*(b + a*x))/(d + c*x))^3
) + ((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)*ArcTanh[(Sqrt[c]*Sqrt[(b + a*x)/(d + c*x)])/Sqrt[a]])/(8*a^(7/2)*c^(5/2)*(-b^2 + a*c)^4) + (2*(-b^
3 + a^2*d)^2*(b^4*c - a^3*b*c^2 - a*b^3*d + a^4*c*d)*Log[a - b*Sqrt[(b + a*x)/(d + c*x)]])/(a^3*b*(-b^2 + a*c)
^4) + ((-(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)*Log[a - (c*(b + a*x))/(d + c
*x)])/(a^3*c^3*(-b^2 + a*c)^4)

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Timed out

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:Warn
ing, integration of abs or sign assumes constant sign by intervals (correct if the argument is real):Check [ab
s(x*c+d)]Evaluation time: 0.42Error: Bad Argument Type

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maple [B]  time = 0.55, size = 384279, normalized size = 200.56

method	result	size
default	\(\text {Expression too large to display}\)	\(384279\)

Verification of antiderivative is not currently implemented for this CAS.

[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

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maxima [A]  time = 0.97, size = 2606, normalized size = 1.36

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[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))

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mupad [F(-1)]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \text {Hanged} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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}

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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