3.17.25 \(\int (A+B x) (d+e x)^{3/2} (a^2+2 a b x+b^2 x^2)^{3/2} \, dx\)

Optimal. Leaf size=308 \[ -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \]

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Rubi [A]  time = 0.14, antiderivative size = 308, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {770, 77} \begin {gather*} -\frac {2 b^2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{11/2} (-3 a B e-A b e+4 b B d)}{11 e^5 (a+b x)}+\frac {2 b \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{9/2} (b d-a e) (-a B e-A b e+2 b B d)}{3 e^5 (a+b x)}-\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{7/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (a+b x)}+\frac {2 \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{5/2} (b d-a e)^3 (B d-A e)}{5 e^5 (a+b x)}+\frac {2 b^3 B \sqrt {a^2+2 a b x+b^2 x^2} (d+e x)^{13/2}}{13 e^5 (a+b x)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(5/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(5*e^5*(a + b*x)) - (2*(b*d - a*e)
^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)^(7/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(7*e^5*(a + b*x)) + (2*b*(b*d -
a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(9/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(3*e^5*(a + b*x)) - (2*b^2*(4*b*
B*d - A*b*e - 3*a*B*e)*(d + e*x)^(11/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(11*e^5*(a + b*x)) + (2*b^3*B*(d + e*x)
^(13/2)*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(13*e^5*(a + b*x))

Rule 77

Int[((a_.) + (b_.)*(x_))*((c_) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Int[ExpandIntegran
d[(a + b*x)*(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && NeQ[b*c - a*d, 0] && ((ILtQ[
n, 0] && ILtQ[p, 0]) || EqQ[p, 1] || (IGtQ[p, 0] && ( !IntegerQ[n] || LeQ[9*p + 5*(n + 2), 0] || GeQ[n + p + 1
, 0] || (GeQ[n + p + 2, 0] && RationalQ[a, b, c, d, e, f]))))

Rule 770

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Dis
t[(a + b*x + c*x^2)^FracPart[p]/(c^IntPart[p]*(b/2 + c*x)^(2*FracPart[p])), Int[(d + e*x)^m*(f + g*x)*(b/2 + c
*x)^(2*p), x], x] /; FreeQ[{a, b, c, d, e, f, g, m}, x] && EqQ[b^2 - 4*a*c, 0]

Rubi steps

\begin {align*} \int (A+B x) (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a b+b^2 x\right )^3 (A+B x) (d+e x)^{3/2} \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (-\frac {b^3 (b d-a e)^3 (-B d+A e) (d+e x)^{3/2}}{e^4}+\frac {b^3 (b d-a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{5/2}}{e^4}-\frac {3 b^4 (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{7/2}}{e^4}+\frac {b^5 (-4 b B d+A b e+3 a B e) (d+e x)^{9/2}}{e^4}+\frac {b^6 B (d+e x)^{11/2}}{e^4}\right ) \, dx}{b^2 \left (a b+b^2 x\right )}\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 e^5 (a+b x)}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 e^5 (a+b x)}+\frac {2 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 e^5 (a+b x)}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 e^5 (a+b x)}+\frac {2 b^3 B (d+e x)^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 e^5 (a+b x)}\\ \end {align*}

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Mathematica [A]  time = 0.18, size = 163, normalized size = 0.53 \begin {gather*} \frac {2 \left ((a+b x)^2\right )^{3/2} (d+e x)^{5/2} \left (-1365 b^2 (d+e x)^3 (-3 a B e-A b e+4 b B d)+5005 b (d+e x)^2 (b d-a e) (-a B e-A b e+2 b B d)-2145 (d+e x) (b d-a e)^2 (-a B e-3 A b e+4 b B d)+3003 (b d-a e)^3 (B d-A e)+1155 b^3 B (d+e x)^4\right )}{15015 e^5 (a+b x)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*((a + b*x)^2)^(3/2)*(d + e*x)^(5/2)*(3003*(b*d - a*e)^3*(B*d - A*e) - 2145*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e
- a*B*e)*(d + e*x) + 5005*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^2 - 1365*b^2*(4*b*B*d - A*b*e - 3
*a*B*e)*(d + e*x)^3 + 1155*b^3*B*(d + e*x)^4))/(15015*e^5*(a + b*x)^3)

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IntegrateAlgebraic [A]  time = 53.02, size = 374, normalized size = 1.21 \begin {gather*} \frac {2 (d+e x)^{5/2} \sqrt {\frac {(a e+b e x)^2}{e^2}} \left (3003 a^3 A e^4+2145 a^3 B e^3 (d+e x)-3003 a^3 B d e^3+6435 a^2 A b e^3 (d+e x)-9009 a^2 A b d e^3+9009 a^2 b B d^2 e^2-12870 a^2 b B d e^2 (d+e x)+5005 a^2 b B e^2 (d+e x)^2+9009 a A b^2 d^2 e^2-12870 a A b^2 d e^2 (d+e x)+5005 a A b^2 e^2 (d+e x)^2-9009 a b^2 B d^3 e+19305 a b^2 B d^2 e (d+e x)-15015 a b^2 B d e (d+e x)^2+4095 a b^2 B e (d+e x)^3-3003 A b^3 d^3 e+6435 A b^3 d^2 e (d+e x)-5005 A b^3 d e (d+e x)^2+1365 A b^3 e (d+e x)^3+3003 b^3 B d^4-8580 b^3 B d^3 (d+e x)+10010 b^3 B d^2 (d+e x)^2-5460 b^3 B d (d+e x)^3+1155 b^3 B (d+e x)^4\right )}{15015 e^4 (a e+b e x)} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(A + B*x)*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^(3/2),x]

[Out]

(2*(d + e*x)^(5/2)*Sqrt[(a*e + b*e*x)^2/e^2]*(3003*b^3*B*d^4 - 3003*A*b^3*d^3*e - 9009*a*b^2*B*d^3*e + 9009*a*
A*b^2*d^2*e^2 + 9009*a^2*b*B*d^2*e^2 - 9009*a^2*A*b*d*e^3 - 3003*a^3*B*d*e^3 + 3003*a^3*A*e^4 - 8580*b^3*B*d^3
*(d + e*x) + 6435*A*b^3*d^2*e*(d + e*x) + 19305*a*b^2*B*d^2*e*(d + e*x) - 12870*a*A*b^2*d*e^2*(d + e*x) - 1287
0*a^2*b*B*d*e^2*(d + e*x) + 6435*a^2*A*b*e^3*(d + e*x) + 2145*a^3*B*e^3*(d + e*x) + 10010*b^3*B*d^2*(d + e*x)^
2 - 5005*A*b^3*d*e*(d + e*x)^2 - 15015*a*b^2*B*d*e*(d + e*x)^2 + 5005*a*A*b^2*e^2*(d + e*x)^2 + 5005*a^2*b*B*e
^2*(d + e*x)^2 - 5460*b^3*B*d*(d + e*x)^3 + 1365*A*b^3*e*(d + e*x)^3 + 4095*a*b^2*B*e*(d + e*x)^3 + 1155*b^3*B
*(d + e*x)^4))/(15015*e^4*(a*e + b*e*x))

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fricas [A]  time = 0.43, size = 446, normalized size = 1.45 \begin {gather*} \frac {2 \, {\left (1155 \, B b^{3} e^{6} x^{6} + 128 \, B b^{3} d^{6} + 3003 \, A a^{3} d^{2} e^{4} - 208 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} e + 1144 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} e^{2} - 858 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} e^{3} + 105 \, {\left (14 \, B b^{3} d e^{5} + 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} e^{6}\right )} x^{5} + 35 \, {\left (B b^{3} d^{2} e^{4} + 52 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d e^{5} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} e^{6}\right )} x^{4} - 5 \, {\left (8 \, B b^{3} d^{3} e^{3} - 13 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{4} - 1430 \, {\left (B a^{2} b + A a b^{2}\right )} d e^{5} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} e^{6}\right )} x^{3} + 3 \, {\left (16 \, B b^{3} d^{4} e^{2} + 1001 \, A a^{3} e^{6} - 26 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e^{3} + 143 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{4} + 1144 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{5}\right )} x^{2} - {\left (64 \, B b^{3} d^{5} e - 6006 \, A a^{3} d e^{5} - 104 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} e^{2} + 572 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} e^{3} - 429 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d}}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="fricas")

[Out]

2/15015*(1155*B*b^3*e^6*x^6 + 128*B*b^3*d^6 + 3003*A*a^3*d^2*e^4 - 208*(3*B*a*b^2 + A*b^3)*d^5*e + 1144*(B*a^2
*b + A*a*b^2)*d^4*e^2 - 858*(B*a^3 + 3*A*a^2*b)*d^3*e^3 + 105*(14*B*b^3*d*e^5 + 13*(3*B*a*b^2 + A*b^3)*e^6)*x^
5 + 35*(B*b^3*d^2*e^4 + 52*(3*B*a*b^2 + A*b^3)*d*e^5 + 143*(B*a^2*b + A*a*b^2)*e^6)*x^4 - 5*(8*B*b^3*d^3*e^3 -
13*(3*B*a*b^2 + A*b^3)*d^2*e^4 - 1430*(B*a^2*b + A*a*b^2)*d*e^5 - 429*(B*a^3 + 3*A*a^2*b)*e^6)*x^3 + 3*(16*B*
b^3*d^4*e^2 + 1001*A*a^3*e^6 - 26*(3*B*a*b^2 + A*b^3)*d^3*e^3 + 143*(B*a^2*b + A*a*b^2)*d^2*e^4 + 1144*(B*a^3
+ 3*A*a^2*b)*d*e^5)*x^2 - (64*B*b^3*d^5*e - 6006*A*a^3*d*e^5 - 104*(3*B*a*b^2 + A*b^3)*d^4*e^2 + 572*(B*a^2*b
+ A*a*b^2)*d^3*e^3 - 429*(B*a^3 + 3*A*a^2*b)*d^2*e^4)*x)*sqrt(e*x + d)/e^5

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giac [B]  time = 0.34, size = 1524, normalized size = 4.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="giac")

[Out]

2/45045*(15015*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d^2*e^(-1)*sgn(b*x + a) + 45045*((x*e + d)^(3/2) -
3*sqrt(x*e + d)*d)*A*a^2*b*d^2*e^(-1)*sgn(b*x + a) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(
x*e + d)*d^2)*B*a^2*b*d^2*e^(-2)*sgn(b*x + a) + 9009*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e +
d)*d^2)*A*a*b^2*d^2*e^(-2)*sgn(b*x + a) + 3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)
*d^2 - 35*sqrt(x*e + d)*d^3)*B*a*b^2*d^2*e^(-3)*sgn(b*x + a) + 1287*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d
+ 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*b^3*d^2*e^(-3)*sgn(b*x + a) + 143*(35*(x*e + d)^(9/2) - 180
*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*b^3*d^2*e^(-
4)*sgn(b*x + a) + 6006*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*B*a^3*d*e^(-1)*sgn(b*
x + a) + 18018*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^2*b*d*e^(-1)*sgn(b*x + a)
+ 7722*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^2*b*d*e
^(-2)*sgn(b*x + a) + 7722*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d
)*d^3)*A*a*b^2*d*e^(-2)*sgn(b*x + a) + 858*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d
^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a*b^2*d*e^(-3)*sgn(b*x + a) + 286*(35*(x*e + d)^(9/2)
- 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*b^3*d*e
^(-3)*sgn(b*x + a) + 130*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e +
d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*b^3*d*e^(-4)*sgn(b*x + a) + 45045*sqrt(x*e
+ d)*A*a^3*d^2*sgn(b*x + a) + 30030*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*A*a^3*d*sgn(b*x + a) + 1287*(5*(x*e
+ d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*B*a^3*e^(-1)*sgn(b*x + a) +
3861*(5*(x*e + d)^(7/2) - 21*(x*e + d)^(5/2)*d + 35*(x*e + d)^(3/2)*d^2 - 35*sqrt(x*e + d)*d^3)*A*a^2*b*e^(-1
)*sgn(b*x + a) + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/
2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a^2*b*e^(-2)*sgn(b*x + a) + 429*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d
+ 378*(x*e + d)^(5/2)*d^2 - 420*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*A*a*b^2*e^(-2)*sgn(b*x + a) + 195
*(63*(x*e + d)^(11/2) - 385*(x*e + d)^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e
+ d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d^5)*B*a*b^2*e^(-3)*sgn(b*x + a) + 65*(63*(x*e + d)^(11/2) - 385*(x*e + d)
^(9/2)*d + 990*(x*e + d)^(7/2)*d^2 - 1386*(x*e + d)^(5/2)*d^3 + 1155*(x*e + d)^(3/2)*d^4 - 693*sqrt(x*e + d)*d
^5)*A*b^3*e^(-3)*sgn(b*x + a) + 15*(231*(x*e + d)^(13/2) - 1638*(x*e + d)^(11/2)*d + 5005*(x*e + d)^(9/2)*d^2
- 8580*(x*e + d)^(7/2)*d^3 + 9009*(x*e + d)^(5/2)*d^4 - 6006*(x*e + d)^(3/2)*d^5 + 3003*sqrt(x*e + d)*d^6)*B*b
^3*e^(-4)*sgn(b*x + a) + 3003*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*sgn(b*x
+ a))*e^(-1)

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maple [A]  time = 0.05, size = 317, normalized size = 1.03 \begin {gather*} \frac {2 \left (e x +d \right )^{\frac {5}{2}} \left (1155 b^{3} B \,x^{4} e^{4}+1365 A \,b^{3} e^{4} x^{3}+4095 B a \,b^{2} e^{4} x^{3}-840 B \,b^{3} d \,e^{3} x^{3}+5005 A a \,b^{2} e^{4} x^{2}-910 A \,b^{3} d \,e^{3} x^{2}+5005 B \,a^{2} b \,e^{4} x^{2}-2730 B a \,b^{2} d \,e^{3} x^{2}+560 B \,b^{3} d^{2} e^{2} x^{2}+6435 A \,a^{2} b \,e^{4} x -2860 A a \,b^{2} d \,e^{3} x +520 A \,b^{3} d^{2} e^{2} x +2145 B \,a^{3} e^{4} x -2860 B \,a^{2} b d \,e^{3} x +1560 B a \,b^{2} d^{2} e^{2} x -320 B \,b^{3} d^{3} e x +3003 A \,a^{3} e^{4}-2574 A \,a^{2} b d \,e^{3}+1144 A a \,b^{2} d^{2} e^{2}-208 A \,b^{3} d^{3} e -858 B \,a^{3} d \,e^{3}+1144 B \,a^{2} b \,d^{2} e^{2}-624 B a \,b^{2} d^{3} e +128 B \,b^{3} d^{4}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15015 \left (b x +a \right )^{3} e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x)

[Out]

2/15015*(e*x+d)^(5/2)*(1155*B*b^3*e^4*x^4+1365*A*b^3*e^4*x^3+4095*B*a*b^2*e^4*x^3-840*B*b^3*d*e^3*x^3+5005*A*a
*b^2*e^4*x^2-910*A*b^3*d*e^3*x^2+5005*B*a^2*b*e^4*x^2-2730*B*a*b^2*d*e^3*x^2+560*B*b^3*d^2*e^2*x^2+6435*A*a^2*
b*e^4*x-2860*A*a*b^2*d*e^3*x+520*A*b^3*d^2*e^2*x+2145*B*a^3*e^4*x-2860*B*a^2*b*d*e^3*x+1560*B*a*b^2*d^2*e^2*x-
320*B*b^3*d^3*e*x+3003*A*a^3*e^4-2574*A*a^2*b*d*e^3+1144*A*a*b^2*d^2*e^2-208*A*b^3*d^3*e-858*B*a^3*d*e^3+1144*
B*a^2*b*d^2*e^2-624*B*a*b^2*d^3*e+128*B*b^3*d^4)*((b*x+a)^2)^(3/2)/e^5/(b*x+a)^3

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maxima [B]  time = 0.63, size = 488, normalized size = 1.58 \begin {gather*} \frac {2 \, {\left (105 \, b^{3} e^{5} x^{5} - 16 \, b^{3} d^{5} + 88 \, a b^{2} d^{4} e - 198 \, a^{2} b d^{3} e^{2} + 231 \, a^{3} d^{2} e^{3} + 35 \, {\left (4 \, b^{3} d e^{4} + 11 \, a b^{2} e^{5}\right )} x^{4} + 5 \, {\left (b^{3} d^{2} e^{3} + 110 \, a b^{2} d e^{4} + 99 \, a^{2} b e^{5}\right )} x^{3} - 3 \, {\left (2 \, b^{3} d^{3} e^{2} - 11 \, a b^{2} d^{2} e^{3} - 264 \, a^{2} b d e^{4} - 77 \, a^{3} e^{5}\right )} x^{2} + {\left (8 \, b^{3} d^{4} e - 44 \, a b^{2} d^{3} e^{2} + 99 \, a^{2} b d^{2} e^{3} + 462 \, a^{3} d e^{4}\right )} x\right )} \sqrt {e x + d} A}{1155 \, e^{4}} + \frac {2 \, {\left (1155 \, b^{3} e^{6} x^{6} + 128 \, b^{3} d^{6} - 624 \, a b^{2} d^{5} e + 1144 \, a^{2} b d^{4} e^{2} - 858 \, a^{3} d^{3} e^{3} + 105 \, {\left (14 \, b^{3} d e^{5} + 39 \, a b^{2} e^{6}\right )} x^{5} + 35 \, {\left (b^{3} d^{2} e^{4} + 156 \, a b^{2} d e^{5} + 143 \, a^{2} b e^{6}\right )} x^{4} - 5 \, {\left (8 \, b^{3} d^{3} e^{3} - 39 \, a b^{2} d^{2} e^{4} - 1430 \, a^{2} b d e^{5} - 429 \, a^{3} e^{6}\right )} x^{3} + 3 \, {\left (16 \, b^{3} d^{4} e^{2} - 78 \, a b^{2} d^{3} e^{3} + 143 \, a^{2} b d^{2} e^{4} + 1144 \, a^{3} d e^{5}\right )} x^{2} - {\left (64 \, b^{3} d^{5} e - 312 \, a b^{2} d^{4} e^{2} + 572 \, a^{2} b d^{3} e^{3} - 429 \, a^{3} d^{2} e^{4}\right )} x\right )} \sqrt {e x + d} B}{15015 \, e^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^(3/2),x, algorithm="maxima")

[Out]

2/1155*(105*b^3*e^5*x^5 - 16*b^3*d^5 + 88*a*b^2*d^4*e - 198*a^2*b*d^3*e^2 + 231*a^3*d^2*e^3 + 35*(4*b^3*d*e^4
+ 11*a*b^2*e^5)*x^4 + 5*(b^3*d^2*e^3 + 110*a*b^2*d*e^4 + 99*a^2*b*e^5)*x^3 - 3*(2*b^3*d^3*e^2 - 11*a*b^2*d^2*e
^3 - 264*a^2*b*d*e^4 - 77*a^3*e^5)*x^2 + (8*b^3*d^4*e - 44*a*b^2*d^3*e^2 + 99*a^2*b*d^2*e^3 + 462*a^3*d*e^4)*x
)*sqrt(e*x + d)*A/e^4 + 2/15015*(1155*b^3*e^6*x^6 + 128*b^3*d^6 - 624*a*b^2*d^5*e + 1144*a^2*b*d^4*e^2 - 858*a
^3*d^3*e^3 + 105*(14*b^3*d*e^5 + 39*a*b^2*e^6)*x^5 + 35*(b^3*d^2*e^4 + 156*a*b^2*d*e^5 + 143*a^2*b*e^6)*x^4 -
5*(8*b^3*d^3*e^3 - 39*a*b^2*d^2*e^4 - 1430*a^2*b*d*e^5 - 429*a^3*e^6)*x^3 + 3*(16*b^3*d^4*e^2 - 78*a*b^2*d^3*e
^3 + 143*a^2*b*d^2*e^4 + 1144*a^3*d*e^5)*x^2 - (64*b^3*d^5*e - 312*a*b^2*d^4*e^2 + 572*a^2*b*d^3*e^3 - 429*a^3
*d^2*e^4)*x)*sqrt(e*x + d)*B/e^5

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \left (A+B\,x\right )\,{\left (d+e\,x\right )}^{3/2}\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2),x)

[Out]

int((A + B*x)*(d + e*x)^(3/2)*(a^2 + b^2*x^2 + 2*a*b*x)^(3/2), x)

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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((B*x+A)*(e*x+d)**(3/2)*(b**2*x**2+2*a*b*x+a**2)**(3/2),x)

[Out]

Timed out

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