3.18.34 \(\int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx\) [1734]

Optimal. Leaf size=173 \[ \frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2}}{9 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2}}{11 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2}}{13 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5} \]

[Out]

2/9*(-a*e+b*d)^3*(-A*e+B*d)*(e*x+d)^(9/2)/e^5-2/11*(-a*e+b*d)^2*(-3*A*b*e-B*a*e+4*B*b*d)*(e*x+d)^(11/2)/e^5+6/
13*b*(-a*e+b*d)*(-A*b*e-B*a*e+2*B*b*d)*(e*x+d)^(13/2)/e^5-2/15*b^2*(-A*b*e-3*B*a*e+4*B*b*d)*(e*x+d)^(15/2)/e^5
+2/17*b^3*B*(e*x+d)^(17/2)/e^5

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Rubi [A]
time = 0.07, antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {78} \begin {gather*} -\frac {2 b^2 (d+e x)^{15/2} (-3 a B e-A b e+4 b B d)}{15 e^5}+\frac {6 b (d+e x)^{13/2} (b d-a e) (-a B e-A b e+2 b B d)}{13 e^5}-\frac {2 (d+e x)^{11/2} (b d-a e)^2 (-a B e-3 A b e+4 b B d)}{11 e^5}+\frac {2 (d+e x)^{9/2} (b d-a e)^3 (B d-A e)}{9 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(b*d - a*e)^3*(B*d - A*e)*(d + e*x)^(9/2))/(9*e^5) - (2*(b*d - a*e)^2*(4*b*B*d - 3*A*b*e - a*B*e)*(d + e*x)
^(11/2))/(11*e^5) + (6*b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e)*(d + e*x)^(13/2))/(13*e^5) - (2*b^2*(4*b*B*d -
A*b*e - 3*a*B*e)*(d + e*x)^(15/2))/(15*e^5) + (2*b^3*B*(d + e*x)^(17/2))/(17*e^5)

Rule 78

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]))))

Rubi steps

\begin {align*} \int (a+b x)^3 (A+B x) (d+e x)^{7/2} \, dx &=\int \left (\frac {(-b d+a e)^3 (-B d+A e) (d+e x)^{7/2}}{e^4}+\frac {(-b d+a e)^2 (-4 b B d+3 A b e+a B e) (d+e x)^{9/2}}{e^4}-\frac {3 b (b d-a e) (-2 b B d+A b e+a B e) (d+e x)^{11/2}}{e^4}+\frac {b^2 (-4 b B d+A b e+3 a B e) (d+e x)^{13/2}}{e^4}+\frac {b^3 B (d+e x)^{15/2}}{e^4}\right ) \, dx\\ &=\frac {2 (b d-a e)^3 (B d-A e) (d+e x)^{9/2}}{9 e^5}-\frac {2 (b d-a e)^2 (4 b B d-3 A b e-a B e) (d+e x)^{11/2}}{11 e^5}+\frac {6 b (b d-a e) (2 b B d-A b e-a B e) (d+e x)^{13/2}}{13 e^5}-\frac {2 b^2 (4 b B d-A b e-3 a B e) (d+e x)^{15/2}}{15 e^5}+\frac {2 b^3 B (d+e x)^{17/2}}{17 e^5}\\ \end {align*}

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Mathematica [A]
time = 0.20, size = 227, normalized size = 1.31 \begin {gather*} \frac {2 (d+e x)^{9/2} \left (1105 a^3 e^3 (-2 B d+11 A e+9 B e x)+255 a^2 b e^2 \left (13 A e (-2 d+9 e x)+B \left (8 d^2-36 d e x+99 e^2 x^2\right )\right )-51 a b^2 e \left (-5 A e \left (8 d^2-36 d e x+99 e^2 x^2\right )+B \left (16 d^3-72 d^2 e x+198 d e^2 x^2-429 e^3 x^3\right )\right )+b^3 \left (17 A e \left (-16 d^3+72 d^2 e x-198 d e^2 x^2+429 e^3 x^3\right )+B \left (128 d^4-576 d^3 e x+1584 d^2 e^2 x^2-3432 d e^3 x^3+6435 e^4 x^4\right )\right )\right )}{109395 e^5} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^3*(A + B*x)*(d + e*x)^(7/2),x]

[Out]

(2*(d + e*x)^(9/2)*(1105*a^3*e^3*(-2*B*d + 11*A*e + 9*B*e*x) + 255*a^2*b*e^2*(13*A*e*(-2*d + 9*e*x) + B*(8*d^2
- 36*d*e*x + 99*e^2*x^2)) - 51*a*b^2*e*(-5*A*e*(8*d^2 - 36*d*e*x + 99*e^2*x^2) + B*(16*d^3 - 72*d^2*e*x + 198
*d*e^2*x^2 - 429*e^3*x^3)) + b^3*(17*A*e*(-16*d^3 + 72*d^2*e*x - 198*d*e^2*x^2 + 429*e^3*x^3) + B*(128*d^4 - 5
76*d^3*e*x + 1584*d^2*e^2*x^2 - 3432*d*e^3*x^3 + 6435*e^4*x^4))))/(109395*e^5)

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Maple [A]
time = 0.09, size = 171, normalized size = 0.99 Too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^3*(B*x+A)*(e*x+d)^(7/2),x,method=_RETURNVERBOSE)

[Out]

2/e^5*(1/17*b^3*B*(e*x+d)^(17/2)+1/15*(3*(a*e-b*d)*b^2*B+b^3*(A*e-B*d))*(e*x+d)^(15/2)+1/13*(3*(a*e-b*d)^2*b*B
+3*(a*e-b*d)*b^2*(A*e-B*d))*(e*x+d)^(13/2)+1/11*((a*e-b*d)^3*B+3*(a*e-b*d)^2*b*(A*e-B*d))*(e*x+d)^(11/2)+1/9*(
a*e-b*d)^3*(A*e-B*d)*(e*x+d)^(9/2))

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Maxima [A]
time = 0.30, size = 279, normalized size = 1.61 \begin {gather*} \frac {2}{109395} \, {\left (6435 \, {\left (x e + d\right )}^{\frac {17}{2}} B b^{3} - 7293 \, {\left (4 \, B b^{3} d - 3 \, B a b^{2} e - A b^{3} e\right )} {\left (x e + d\right )}^{\frac {15}{2}} + 25245 \, {\left (2 \, B b^{3} d^{2} + B a^{2} b e^{2} + A a b^{2} e^{2} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d\right )} {\left (x e + d\right )}^{\frac {13}{2}} - 9945 \, {\left (4 \, B b^{3} d^{3} - B a^{3} e^{3} - 3 \, A a^{2} b e^{3} - 3 \, {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{2} + 6 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d\right )} {\left (x e + d\right )}^{\frac {11}{2}} + 12155 \, {\left (B b^{3} d^{4} + A a^{3} e^{4} - {\left (3 \, B a b^{2} e + A b^{3} e\right )} d^{3} + 3 \, {\left (B a^{2} b e^{2} + A a b^{2} e^{2}\right )} d^{2} - {\left (B a^{3} e^{3} + 3 \, A a^{2} b e^{3}\right )} d\right )} {\left (x e + d\right )}^{\frac {9}{2}}\right )} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(7/2),x, algorithm="maxima")

[Out]

2/109395*(6435*(x*e + d)^(17/2)*B*b^3 - 7293*(4*B*b^3*d - 3*B*a*b^2*e - A*b^3*e)*(x*e + d)^(15/2) + 25245*(2*B
*b^3*d^2 + B*a^2*b*e^2 + A*a*b^2*e^2 - (3*B*a*b^2*e + A*b^3*e)*d)*(x*e + d)^(13/2) - 9945*(4*B*b^3*d^3 - B*a^3
*e^3 - 3*A*a^2*b*e^3 - 3*(3*B*a*b^2*e + A*b^3*e)*d^2 + 6*(B*a^2*b*e^2 + A*a*b^2*e^2)*d)*(x*e + d)^(11/2) + 121
55*(B*b^3*d^4 + A*a^3*e^4 - (3*B*a*b^2*e + A*b^3*e)*d^3 + 3*(B*a^2*b*e^2 + A*a*b^2*e^2)*d^2 - (B*a^3*e^3 + 3*A
*a^2*b*e^3)*d)*(x*e + d)^(9/2))*e^(-5)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 611 vs. \(2 (163) = 326\).
time = 0.74, size = 611, normalized size = 3.53 \begin {gather*} \frac {2}{109395} \, {\left (128 \, B b^{3} d^{8} + {\left (6435 \, B b^{3} x^{8} + 12155 \, A a^{3} x^{4} + 7293 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{7} + 25245 \, {\left (B a^{2} b + A a b^{2}\right )} x^{6} + 9945 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{5}\right )} e^{8} + 2 \, {\left (11154 \, B b^{3} d x^{7} + 24310 \, A a^{3} d x^{3} + 12903 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d x^{6} + 45900 \, {\left (B a^{2} b + A a b^{2}\right )} d x^{5} + 18785 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d x^{4}\right )} e^{7} + 2 \, {\left (13233 \, B b^{3} d^{2} x^{6} + 36465 \, A a^{3} d^{2} x^{2} + 15759 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} x^{5} + 58395 \, {\left (B a^{2} b + A a b^{2}\right )} d^{2} x^{4} + 25415 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{2} x^{3}\right )} e^{6} + 4 \, {\left (2727 \, B b^{3} d^{3} x^{5} + 12155 \, A a^{3} d^{3} x + 3400 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} x^{4} + 13515 \, {\left (B a^{2} b + A a b^{2}\right )} d^{3} x^{3} + 6630 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{3} x^{2}\right )} e^{5} + 5 \, {\left (7 \, B b^{3} d^{4} x^{4} + 2431 \, A a^{3} d^{4} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{4} x^{3} + 153 \, {\left (B a^{2} b + A a b^{2}\right )} d^{4} x^{2} + 221 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{4} x\right )} e^{4} - 2 \, {\left (20 \, B b^{3} d^{5} x^{3} + 51 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{5} x^{2} + 510 \, {\left (B a^{2} b + A a b^{2}\right )} d^{5} x + 1105 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} d^{5}\right )} e^{3} + 8 \, {\left (6 \, B b^{3} d^{6} x^{2} + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{6} x + 255 \, {\left (B a^{2} b + A a b^{2}\right )} d^{6}\right )} e^{2} - 16 \, {\left (4 \, B b^{3} d^{7} x + 17 \, {\left (3 \, B a b^{2} + A b^{3}\right )} d^{7}\right )} e\right )} \sqrt {x e + d} e^{\left (-5\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(7/2),x, algorithm="fricas")

[Out]

2/109395*(128*B*b^3*d^8 + (6435*B*b^3*x^8 + 12155*A*a^3*x^4 + 7293*(3*B*a*b^2 + A*b^3)*x^7 + 25245*(B*a^2*b +
A*a*b^2)*x^6 + 9945*(B*a^3 + 3*A*a^2*b)*x^5)*e^8 + 2*(11154*B*b^3*d*x^7 + 24310*A*a^3*d*x^3 + 12903*(3*B*a*b^2
+ A*b^3)*d*x^6 + 45900*(B*a^2*b + A*a*b^2)*d*x^5 + 18785*(B*a^3 + 3*A*a^2*b)*d*x^4)*e^7 + 2*(13233*B*b^3*d^2*
x^6 + 36465*A*a^3*d^2*x^2 + 15759*(3*B*a*b^2 + A*b^3)*d^2*x^5 + 58395*(B*a^2*b + A*a*b^2)*d^2*x^4 + 25415*(B*a
^3 + 3*A*a^2*b)*d^2*x^3)*e^6 + 4*(2727*B*b^3*d^3*x^5 + 12155*A*a^3*d^3*x + 3400*(3*B*a*b^2 + A*b^3)*d^3*x^4 +
13515*(B*a^2*b + A*a*b^2)*d^3*x^3 + 6630*(B*a^3 + 3*A*a^2*b)*d^3*x^2)*e^5 + 5*(7*B*b^3*d^4*x^4 + 2431*A*a^3*d^
4 + 17*(3*B*a*b^2 + A*b^3)*d^4*x^3 + 153*(B*a^2*b + A*a*b^2)*d^4*x^2 + 221*(B*a^3 + 3*A*a^2*b)*d^4*x)*e^4 - 2*
(20*B*b^3*d^5*x^3 + 51*(3*B*a*b^2 + A*b^3)*d^5*x^2 + 510*(B*a^2*b + A*a*b^2)*d^5*x + 1105*(B*a^3 + 3*A*a^2*b)*
d^5)*e^3 + 8*(6*B*b^3*d^6*x^2 + 17*(3*B*a*b^2 + A*b^3)*d^6*x + 255*(B*a^2*b + A*a*b^2)*d^6)*e^2 - 16*(4*B*b^3*
d^7*x + 17*(3*B*a*b^2 + A*b^3)*d^7)*e)*sqrt(x*e + d)*e^(-5)

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 1523 vs. \(2 (170) = 340\).
time = 1.70, size = 1523, normalized size = 8.80 \begin {gather*} \begin {cases} \frac {2 A a^{3} d^{4} \sqrt {d + e x}}{9 e} + \frac {8 A a^{3} d^{3} x \sqrt {d + e x}}{9} + \frac {4 A a^{3} d^{2} e x^{2} \sqrt {d + e x}}{3} + \frac {8 A a^{3} d e^{2} x^{3} \sqrt {d + e x}}{9} + \frac {2 A a^{3} e^{3} x^{4} \sqrt {d + e x}}{9} - \frac {4 A a^{2} b d^{5} \sqrt {d + e x}}{33 e^{2}} + \frac {2 A a^{2} b d^{4} x \sqrt {d + e x}}{33 e} + \frac {16 A a^{2} b d^{3} x^{2} \sqrt {d + e x}}{11} + \frac {92 A a^{2} b d^{2} e x^{3} \sqrt {d + e x}}{33} + \frac {68 A a^{2} b d e^{2} x^{4} \sqrt {d + e x}}{33} + \frac {6 A a^{2} b e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 A a b^{2} d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {8 A a b^{2} d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {2 A a b^{2} d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {424 A a b^{2} d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {916 A a b^{2} d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {240 A a b^{2} d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {6 A a b^{2} e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 A b^{3} d^{7} \sqrt {d + e x}}{6435 e^{4}} + \frac {16 A b^{3} d^{6} x \sqrt {d + e x}}{6435 e^{3}} - \frac {4 A b^{3} d^{5} x^{2} \sqrt {d + e x}}{2145 e^{2}} + \frac {2 A b^{3} d^{4} x^{3} \sqrt {d + e x}}{1287 e} + \frac {320 A b^{3} d^{3} x^{4} \sqrt {d + e x}}{1287} + \frac {412 A b^{3} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 A b^{3} d e^{2} x^{6} \sqrt {d + e x}}{195} + \frac {2 A b^{3} e^{3} x^{7} \sqrt {d + e x}}{15} - \frac {4 B a^{3} d^{5} \sqrt {d + e x}}{99 e^{2}} + \frac {2 B a^{3} d^{4} x \sqrt {d + e x}}{99 e} + \frac {16 B a^{3} d^{3} x^{2} \sqrt {d + e x}}{33} + \frac {92 B a^{3} d^{2} e x^{3} \sqrt {d + e x}}{99} + \frac {68 B a^{3} d e^{2} x^{4} \sqrt {d + e x}}{99} + \frac {2 B a^{3} e^{3} x^{5} \sqrt {d + e x}}{11} + \frac {16 B a^{2} b d^{6} \sqrt {d + e x}}{429 e^{3}} - \frac {8 B a^{2} b d^{5} x \sqrt {d + e x}}{429 e^{2}} + \frac {2 B a^{2} b d^{4} x^{2} \sqrt {d + e x}}{143 e} + \frac {424 B a^{2} b d^{3} x^{3} \sqrt {d + e x}}{429} + \frac {916 B a^{2} b d^{2} e x^{4} \sqrt {d + e x}}{429} + \frac {240 B a^{2} b d e^{2} x^{5} \sqrt {d + e x}}{143} + \frac {6 B a^{2} b e^{3} x^{6} \sqrt {d + e x}}{13} - \frac {32 B a b^{2} d^{7} \sqrt {d + e x}}{2145 e^{4}} + \frac {16 B a b^{2} d^{6} x \sqrt {d + e x}}{2145 e^{3}} - \frac {4 B a b^{2} d^{5} x^{2} \sqrt {d + e x}}{715 e^{2}} + \frac {2 B a b^{2} d^{4} x^{3} \sqrt {d + e x}}{429 e} + \frac {320 B a b^{2} d^{3} x^{4} \sqrt {d + e x}}{429} + \frac {1236 B a b^{2} d^{2} e x^{5} \sqrt {d + e x}}{715} + \frac {92 B a b^{2} d e^{2} x^{6} \sqrt {d + e x}}{65} + \frac {2 B a b^{2} e^{3} x^{7} \sqrt {d + e x}}{5} + \frac {256 B b^{3} d^{8} \sqrt {d + e x}}{109395 e^{5}} - \frac {128 B b^{3} d^{7} x \sqrt {d + e x}}{109395 e^{4}} + \frac {32 B b^{3} d^{6} x^{2} \sqrt {d + e x}}{36465 e^{3}} - \frac {16 B b^{3} d^{5} x^{3} \sqrt {d + e x}}{21879 e^{2}} + \frac {14 B b^{3} d^{4} x^{4} \sqrt {d + e x}}{21879 e} + \frac {2424 B b^{3} d^{3} x^{5} \sqrt {d + e x}}{12155} + \frac {1604 B b^{3} d^{2} e x^{6} \sqrt {d + e x}}{3315} + \frac {104 B b^{3} d e^{2} x^{7} \sqrt {d + e x}}{255} + \frac {2 B b^{3} e^{3} x^{8} \sqrt {d + e x}}{17} & \text {for}\: e \neq 0 \\d^{\frac {7}{2}} \left (A a^{3} x + \frac {3 A a^{2} b x^{2}}{2} + A a b^{2} x^{3} + \frac {A b^{3} x^{4}}{4} + \frac {B a^{3} x^{2}}{2} + B a^{2} b x^{3} + \frac {3 B a b^{2} x^{4}}{4} + \frac {B b^{3} x^{5}}{5}\right ) & \text {otherwise} \end {cases} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**3*(B*x+A)*(e*x+d)**(7/2),x)

[Out]

Piecewise((2*A*a**3*d**4*sqrt(d + e*x)/(9*e) + 8*A*a**3*d**3*x*sqrt(d + e*x)/9 + 4*A*a**3*d**2*e*x**2*sqrt(d +
e*x)/3 + 8*A*a**3*d*e**2*x**3*sqrt(d + e*x)/9 + 2*A*a**3*e**3*x**4*sqrt(d + e*x)/9 - 4*A*a**2*b*d**5*sqrt(d +
e*x)/(33*e**2) + 2*A*a**2*b*d**4*x*sqrt(d + e*x)/(33*e) + 16*A*a**2*b*d**3*x**2*sqrt(d + e*x)/11 + 92*A*a**2*
b*d**2*e*x**3*sqrt(d + e*x)/33 + 68*A*a**2*b*d*e**2*x**4*sqrt(d + e*x)/33 + 6*A*a**2*b*e**3*x**5*sqrt(d + e*x)
/11 + 16*A*a*b**2*d**6*sqrt(d + e*x)/(429*e**3) - 8*A*a*b**2*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*A*a*b**2*d**4
*x**2*sqrt(d + e*x)/(143*e) + 424*A*a*b**2*d**3*x**3*sqrt(d + e*x)/429 + 916*A*a*b**2*d**2*e*x**4*sqrt(d + e*x
)/429 + 240*A*a*b**2*d*e**2*x**5*sqrt(d + e*x)/143 + 6*A*a*b**2*e**3*x**6*sqrt(d + e*x)/13 - 32*A*b**3*d**7*sq
rt(d + e*x)/(6435*e**4) + 16*A*b**3*d**6*x*sqrt(d + e*x)/(6435*e**3) - 4*A*b**3*d**5*x**2*sqrt(d + e*x)/(2145*
e**2) + 2*A*b**3*d**4*x**3*sqrt(d + e*x)/(1287*e) + 320*A*b**3*d**3*x**4*sqrt(d + e*x)/1287 + 412*A*b**3*d**2*
e*x**5*sqrt(d + e*x)/715 + 92*A*b**3*d*e**2*x**6*sqrt(d + e*x)/195 + 2*A*b**3*e**3*x**7*sqrt(d + e*x)/15 - 4*B
*a**3*d**5*sqrt(d + e*x)/(99*e**2) + 2*B*a**3*d**4*x*sqrt(d + e*x)/(99*e) + 16*B*a**3*d**3*x**2*sqrt(d + e*x)/
33 + 92*B*a**3*d**2*e*x**3*sqrt(d + e*x)/99 + 68*B*a**3*d*e**2*x**4*sqrt(d + e*x)/99 + 2*B*a**3*e**3*x**5*sqrt
(d + e*x)/11 + 16*B*a**2*b*d**6*sqrt(d + e*x)/(429*e**3) - 8*B*a**2*b*d**5*x*sqrt(d + e*x)/(429*e**2) + 2*B*a*
*2*b*d**4*x**2*sqrt(d + e*x)/(143*e) + 424*B*a**2*b*d**3*x**3*sqrt(d + e*x)/429 + 916*B*a**2*b*d**2*e*x**4*sqr
t(d + e*x)/429 + 240*B*a**2*b*d*e**2*x**5*sqrt(d + e*x)/143 + 6*B*a**2*b*e**3*x**6*sqrt(d + e*x)/13 - 32*B*a*b
**2*d**7*sqrt(d + e*x)/(2145*e**4) + 16*B*a*b**2*d**6*x*sqrt(d + e*x)/(2145*e**3) - 4*B*a*b**2*d**5*x**2*sqrt(
d + e*x)/(715*e**2) + 2*B*a*b**2*d**4*x**3*sqrt(d + e*x)/(429*e) + 320*B*a*b**2*d**3*x**4*sqrt(d + e*x)/429 +
1236*B*a*b**2*d**2*e*x**5*sqrt(d + e*x)/715 + 92*B*a*b**2*d*e**2*x**6*sqrt(d + e*x)/65 + 2*B*a*b**2*e**3*x**7*
sqrt(d + e*x)/5 + 256*B*b**3*d**8*sqrt(d + e*x)/(109395*e**5) - 128*B*b**3*d**7*x*sqrt(d + e*x)/(109395*e**4)
+ 32*B*b**3*d**6*x**2*sqrt(d + e*x)/(36465*e**3) - 16*B*b**3*d**5*x**3*sqrt(d + e*x)/(21879*e**2) + 14*B*b**3*
d**4*x**4*sqrt(d + e*x)/(21879*e) + 2424*B*b**3*d**3*x**5*sqrt(d + e*x)/12155 + 1604*B*b**3*d**2*e*x**6*sqrt(d
+ e*x)/3315 + 104*B*b**3*d*e**2*x**7*sqrt(d + e*x)/255 + 2*B*b**3*e**3*x**8*sqrt(d + e*x)/17, Ne(e, 0)), (d**
(7/2)*(A*a**3*x + 3*A*a**2*b*x**2/2 + A*a*b**2*x**3 + A*b**3*x**4/4 + B*a**3*x**2/2 + B*a**2*b*x**3 + 3*B*a*b*
*2*x**4/4 + B*b**3*x**5/5), True))

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 2848 vs. \(2 (163) = 326\).
time = 0.65, size = 2848, normalized size = 16.46 \begin {gather*} \text {Too large to display} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^3*(B*x+A)*(e*x+d)^(7/2),x, algorithm="giac")

[Out]

2/765765*(255255*((x*e + d)^(3/2) - 3*sqrt(x*e + d)*d)*B*a^3*d^4*e^(-1) + 765765*((x*e + d)^(3/2) - 3*sqrt(x*e
+ d)*d)*A*a^2*b*d^4*e^(-1) + 153153*(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^4*e^(-2) + 153153*(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^4*e^(-2) + 65
637*(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^4*e^(
-3) + 21879*(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
^4*e^(-3) + 2431*(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^4*e^(-4) + 204204*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e
+ d)*d^2)*B*a^3*d^3*e^(-1) + 612612*(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^3*e^(-1) + 262548*(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^3*e^(-2) + 262548*(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^3*e^(-2) + 29172*(35*(x*e + d)^(9/2) - 180*(x*e + d)^(7/2)*d + 378*(x*e + d)^(5/2)*d^2 - 4
20*(x*e + d)^(3/2)*d^3 + 315*sqrt(x*e + d)*d^4)*B*a*b^2*d^3*e^(-3) + 9724*(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^3*e^(-3) + 4420*(
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^3*e^(-4) + 765765*sqrt(x*e + d)*A*a^3*d^4 + 1021020*((x*e + d)^
(3/2) - 3*sqrt(x*e + d)*d)*A*a^3*d^3 + 131274*(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*d^2*e^(-1) + 393822*(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*d^2*e^(-1) + 43758*(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*d^2*e^(-2) + 43758*(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*d^2*e^(-2) + 19890*(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*d^2*e^(-3) + 6630*(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*d^2*e^(-3) + 1530*(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*d^2*e^(-4) + 306306*(3*(x*e + d)^(5/2) - 10*(x*e + d)^(3/2)*d + 15*sqrt(x*e + d)*d^2)*A*a^3*d^2
+ 9724*(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^3*d*e^(-1) + 29172*(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^2*b*d*e^(-1) + 13260*(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^2*b*d*e^(-2) + 13260*(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*a*b^2*d*e^(-2) + 3060*(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*a*b^2*d*e^(-3) + 1020*(231*(x*e + d)^(13/2) - 163
8*(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)*A*b^3*d*e^(-3) + 476*(429*(x*e + d)^(15/2) - 3465*(x*e + d)^(13/2
)*d + 12285*(x*e + d)^(11/2)*d^2 - 25025*(x*e + d)^(9/2)*d^3 + 32175*(x*e + d)^(7/2)*d^4 - 27027*(x*e + d)^(5/
2)*d^5 + 15015*(x*e + d)^(3/2)*d^6 - 6435*sqrt(x*e + d)*d^7)*B*b^3*d*e^(-4) + 87516*(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^3*d + 1105*(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^3*e^(-1) + 3315*(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*a^2*b*e^(-1) + 765*(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*a^2*b*e^(-2) + 765*(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)*A*a*b^2*e^(...

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Mupad [B]
time = 1.26, size = 154, normalized size = 0.89 \begin {gather*} \frac {{\left (d+e\,x\right )}^{15/2}\,\left (2\,A\,b^3\,e-8\,B\,b^3\,d+6\,B\,a\,b^2\,e\right )}{15\,e^5}+\frac {2\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}\,\left (3\,A\,b\,e+B\,a\,e-4\,B\,b\,d\right )}{11\,e^5}+\frac {2\,B\,b^3\,{\left (d+e\,x\right )}^{17/2}}{17\,e^5}+\frac {2\,\left (A\,e-B\,d\right )\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^5}+\frac {6\,b\,\left (a\,e-b\,d\right )\,{\left (d+e\,x\right )}^{13/2}\,\left (A\,b\,e+B\,a\,e-2\,B\,b\,d\right )}{13\,e^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((A + B*x)*(a + b*x)^3*(d + e*x)^(7/2),x)

[Out]

((d + e*x)^(15/2)*(2*A*b^3*e - 8*B*b^3*d + 6*B*a*b^2*e))/(15*e^5) + (2*(a*e - b*d)^2*(d + e*x)^(11/2)*(3*A*b*e
+ B*a*e - 4*B*b*d))/(11*e^5) + (2*B*b^3*(d + e*x)^(17/2))/(17*e^5) + (2*(A*e - B*d)*(a*e - b*d)^3*(d + e*x)^(
9/2))/(9*e^5) + (6*b*(a*e - b*d)*(d + e*x)^(13/2)*(A*b*e + B*a*e - 2*B*b*d))/(13*e^5)

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