3.1032 \(\int \frac{(a+b x)^3 (A+B x)}{(d+e x)^9} \, dx\)

Optimal. Leaf size=163 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{5 e^5 (d+e x)^5}-\frac{b (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (d+e x)^6}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (d+e x)^7}-\frac{(b d-a e)^3 (B d-A e)}{8 e^5 (d+e x)^8}-\frac{b^3 B}{4 e^5 (d+e x)^4} \]

[Out]

-((b*d - a*e)^3*(B*d - A*e))/(8*e^5*(d + e*x)^8) + ((b*d - a*e)^2*(4*b*B*d - 3*A
*b*e - a*B*e))/(7*e^5*(d + e*x)^7) - (b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(
2*e^5*(d + e*x)^6) + (b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(5*e^5*(d + e*x)^5) - (b^
3*B)/(4*e^5*(d + e*x)^4)

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Rubi [A]  time = 0.37256, antiderivative size = 163, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05 \[ \frac{b^2 (-3 a B e-A b e+4 b B d)}{5 e^5 (d+e x)^5}-\frac{b (b d-a e) (-a B e-A b e+2 b B d)}{2 e^5 (d+e x)^6}+\frac{(b d-a e)^2 (-a B e-3 A b e+4 b B d)}{7 e^5 (d+e x)^7}-\frac{(b d-a e)^3 (B d-A e)}{8 e^5 (d+e x)^8}-\frac{b^3 B}{4 e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x)^3*(A + B*x))/(d + e*x)^9,x]

[Out]

-((b*d - a*e)^3*(B*d - A*e))/(8*e^5*(d + e*x)^8) + ((b*d - a*e)^2*(4*b*B*d - 3*A
*b*e - a*B*e))/(7*e^5*(d + e*x)^7) - (b*(b*d - a*e)*(2*b*B*d - A*b*e - a*B*e))/(
2*e^5*(d + e*x)^6) + (b^2*(4*b*B*d - A*b*e - 3*a*B*e))/(5*e^5*(d + e*x)^5) - (b^
3*B)/(4*e^5*(d + e*x)^4)

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Rubi in Sympy [A]  time = 60.0028, size = 155, normalized size = 0.95 \[ - \frac{B b^{3}}{4 e^{5} \left (d + e x\right )^{4}} - \frac{b^{2} \left (A b e + 3 B a e - 4 B b d\right )}{5 e^{5} \left (d + e x\right )^{5}} - \frac{b \left (a e - b d\right ) \left (A b e + B a e - 2 B b d\right )}{2 e^{5} \left (d + e x\right )^{6}} - \frac{\left (a e - b d\right )^{2} \left (3 A b e + B a e - 4 B b d\right )}{7 e^{5} \left (d + e x\right )^{7}} - \frac{\left (A e - B d\right ) \left (a e - b d\right )^{3}}{8 e^{5} \left (d + e x\right )^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x+a)**3*(B*x+A)/(e*x+d)**9,x)

[Out]

-B*b**3/(4*e**5*(d + e*x)**4) - b**2*(A*b*e + 3*B*a*e - 4*B*b*d)/(5*e**5*(d + e*
x)**5) - b*(a*e - b*d)*(A*b*e + B*a*e - 2*B*b*d)/(2*e**5*(d + e*x)**6) - (a*e -
b*d)**2*(3*A*b*e + B*a*e - 4*B*b*d)/(7*e**5*(d + e*x)**7) - (A*e - B*d)*(a*e - b
*d)**3/(8*e**5*(d + e*x)**8)

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Mathematica [A]  time = 0.180137, size = 211, normalized size = 1.29 \[ -\frac{5 a^3 e^3 (7 A e+B (d+8 e x))+5 a^2 b e^2 \left (3 A e (d+8 e x)+B \left (d^2+8 d e x+28 e^2 x^2\right )\right )+a b^2 e \left (5 A e \left (d^2+8 d e x+28 e^2 x^2\right )+3 B \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )\right )+b^3 \left (A e \left (d^3+8 d^2 e x+28 d e^2 x^2+56 e^3 x^3\right )+B \left (d^4+8 d^3 e x+28 d^2 e^2 x^2+56 d e^3 x^3+70 e^4 x^4\right )\right )}{280 e^5 (d+e x)^8} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x)^3*(A + B*x))/(d + e*x)^9,x]

[Out]

-(5*a^3*e^3*(7*A*e + B*(d + 8*e*x)) + 5*a^2*b*e^2*(3*A*e*(d + 8*e*x) + B*(d^2 +
8*d*e*x + 28*e^2*x^2)) + a*b^2*e*(5*A*e*(d^2 + 8*d*e*x + 28*e^2*x^2) + 3*B*(d^3
+ 8*d^2*e*x + 28*d*e^2*x^2 + 56*e^3*x^3)) + b^3*(A*e*(d^3 + 8*d^2*e*x + 28*d*e^2
*x^2 + 56*e^3*x^3) + B*(d^4 + 8*d^3*e*x + 28*d^2*e^2*x^2 + 56*d*e^3*x^3 + 70*e^4
*x^4)))/(280*e^5*(d + e*x)^8)

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Maple [A]  time = 0.01, size = 281, normalized size = 1.7 \[ -{\frac{b \left ( Aab{e}^{2}-Ad{b}^{2}e+B{a}^{2}{e}^{2}-3\,Bdabe+2\,{b}^{2}B{d}^{2} \right ) }{2\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{a}^{3}A{e}^{4}-3\,A{a}^{2}bd{e}^{3}+3\,A{d}^{2}a{b}^{2}{e}^{2}-A{d}^{3}{b}^{3}e-B{a}^{3}d{e}^{3}+3\,B{d}^{2}{a}^{2}b{e}^{2}-3\,B{d}^{3}a{b}^{2}e+{b}^{3}B{d}^{4}}{8\,{e}^{5} \left ( ex+d \right ) ^{8}}}-{\frac{3\,A{a}^{2}b{e}^{3}-6\,Ada{b}^{2}{e}^{2}+3\,A{d}^{2}{b}^{3}e+B{a}^{3}{e}^{3}-6\,Bd{a}^{2}b{e}^{2}+9\,B{d}^{2}a{b}^{2}e-4\,{b}^{3}B{d}^{3}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{B{b}^{3}}{4\,{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{{b}^{2} \left ( Abe+3\,Bae-4\,Bbd \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x+a)^3*(B*x+A)/(e*x+d)^9,x)

[Out]

-1/2*b*(A*a*b*e^2-A*b^2*d*e+B*a^2*e^2-3*B*a*b*d*e+2*B*b^2*d^2)/e^5/(e*x+d)^6-1/8
*(A*a^3*e^4-3*A*a^2*b*d*e^3+3*A*a*b^2*d^2*e^2-A*b^3*d^3*e-B*a^3*d*e^3+3*B*a^2*b*
d^2*e^2-3*B*a*b^2*d^3*e+B*b^3*d^4)/e^5/(e*x+d)^8-1/7*(3*A*a^2*b*e^3-6*A*a*b^2*d*
e^2+3*A*b^3*d^2*e+B*a^3*e^3-6*B*a^2*b*d*e^2+9*B*a*b^2*d^2*e-4*B*b^3*d^3)/e^5/(e*
x+d)^7-1/4*b^3*B/e^5/(e*x+d)^4-1/5*b^2*(A*b*e+3*B*a*e-4*B*b*d)/e^5/(e*x+d)^5

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Maxima [A]  time = 1.39244, size = 452, normalized size = 2.77 \[ -\frac{70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \,{\left (B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \,{\left (B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \,{\left (B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^9,x, algorithm="maxima")

[Out]

-1/280*(70*B*b^3*e^4*x^4 + B*b^3*d^4 + 35*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e
+ 5*(B*a^2*b + A*a*b^2)*d^2*e^2 + 5*(B*a^3 + 3*A*a^2*b)*d*e^3 + 56*(B*b^3*d*e^3
+ (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 28*(B*b^3*d^2*e^2 + (3*B*a*b^2 + A*b^3)*d*e^3 +
5*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 8*(B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 +
5*(B*a^2*b + A*a*b^2)*d*e^3 + 5*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^13*x^8 + 8*d*e^1
2*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^4*e^9*x^4 + 56*d^5*e^8*x^3 + 28
*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

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Fricas [A]  time = 0.206879, size = 452, normalized size = 2.77 \[ -\frac{70 \, B b^{3} e^{4} x^{4} + B b^{3} d^{4} + 35 \, A a^{3} e^{4} +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{3} e + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d^{2} e^{2} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} d e^{3} + 56 \,{\left (B b^{3} d e^{3} +{\left (3 \, B a b^{2} + A b^{3}\right )} e^{4}\right )} x^{3} + 28 \,{\left (B b^{3} d^{2} e^{2} +{\left (3 \, B a b^{2} + A b^{3}\right )} d e^{3} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} e^{4}\right )} x^{2} + 8 \,{\left (B b^{3} d^{3} e +{\left (3 \, B a b^{2} + A b^{3}\right )} d^{2} e^{2} + 5 \,{\left (B a^{2} b + A a b^{2}\right )} d e^{3} + 5 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} e^{4}\right )} x}{280 \,{\left (e^{13} x^{8} + 8 \, d e^{12} x^{7} + 28 \, d^{2} e^{11} x^{6} + 56 \, d^{3} e^{10} x^{5} + 70 \, d^{4} e^{9} x^{4} + 56 \, d^{5} e^{8} x^{3} + 28 \, d^{6} e^{7} x^{2} + 8 \, d^{7} e^{6} x + d^{8} e^{5}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^9,x, algorithm="fricas")

[Out]

-1/280*(70*B*b^3*e^4*x^4 + B*b^3*d^4 + 35*A*a^3*e^4 + (3*B*a*b^2 + A*b^3)*d^3*e
+ 5*(B*a^2*b + A*a*b^2)*d^2*e^2 + 5*(B*a^3 + 3*A*a^2*b)*d*e^3 + 56*(B*b^3*d*e^3
+ (3*B*a*b^2 + A*b^3)*e^4)*x^3 + 28*(B*b^3*d^2*e^2 + (3*B*a*b^2 + A*b^3)*d*e^3 +
5*(B*a^2*b + A*a*b^2)*e^4)*x^2 + 8*(B*b^3*d^3*e + (3*B*a*b^2 + A*b^3)*d^2*e^2 +
5*(B*a^2*b + A*a*b^2)*d*e^3 + 5*(B*a^3 + 3*A*a^2*b)*e^4)*x)/(e^13*x^8 + 8*d*e^1
2*x^7 + 28*d^2*e^11*x^6 + 56*d^3*e^10*x^5 + 70*d^4*e^9*x^4 + 56*d^5*e^8*x^3 + 28
*d^6*e^7*x^2 + 8*d^7*e^6*x + d^8*e^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x+a)**3*(B*x+A)/(e*x+d)**9,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212734, size = 379, normalized size = 2.33 \[ -\frac{{\left (70 \, B b^{3} x^{4} e^{4} + 56 \, B b^{3} d x^{3} e^{3} + 28 \, B b^{3} d^{2} x^{2} e^{2} + 8 \, B b^{3} d^{3} x e + B b^{3} d^{4} + 168 \, B a b^{2} x^{3} e^{4} + 56 \, A b^{3} x^{3} e^{4} + 84 \, B a b^{2} d x^{2} e^{3} + 28 \, A b^{3} d x^{2} e^{3} + 24 \, B a b^{2} d^{2} x e^{2} + 8 \, A b^{3} d^{2} x e^{2} + 3 \, B a b^{2} d^{3} e + A b^{3} d^{3} e + 140 \, B a^{2} b x^{2} e^{4} + 140 \, A a b^{2} x^{2} e^{4} + 40 \, B a^{2} b d x e^{3} + 40 \, A a b^{2} d x e^{3} + 5 \, B a^{2} b d^{2} e^{2} + 5 \, A a b^{2} d^{2} e^{2} + 40 \, B a^{3} x e^{4} + 120 \, A a^{2} b x e^{4} + 5 \, B a^{3} d e^{3} + 15 \, A a^{2} b d e^{3} + 35 \, A a^{3} e^{4}\right )} e^{\left (-5\right )}}{280 \,{\left (x e + d\right )}^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)*(b*x + a)^3/(e*x + d)^9,x, algorithm="giac")

[Out]

-1/280*(70*B*b^3*x^4*e^4 + 56*B*b^3*d*x^3*e^3 + 28*B*b^3*d^2*x^2*e^2 + 8*B*b^3*d
^3*x*e + B*b^3*d^4 + 168*B*a*b^2*x^3*e^4 + 56*A*b^3*x^3*e^4 + 84*B*a*b^2*d*x^2*e
^3 + 28*A*b^3*d*x^2*e^3 + 24*B*a*b^2*d^2*x*e^2 + 8*A*b^3*d^2*x*e^2 + 3*B*a*b^2*d
^3*e + A*b^3*d^3*e + 140*B*a^2*b*x^2*e^4 + 140*A*a*b^2*x^2*e^4 + 40*B*a^2*b*d*x*
e^3 + 40*A*a*b^2*d*x*e^3 + 5*B*a^2*b*d^2*e^2 + 5*A*a*b^2*d^2*e^2 + 40*B*a^3*x*e^
4 + 120*A*a^2*b*x*e^4 + 5*B*a^3*d*e^3 + 15*A*a^2*b*d*e^3 + 35*A*a^3*e^4)*e^(-5)/
(x*e + d)^8