3.47 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{15}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{a^5 A}{14 x^{14}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{4} b^5 B x^4 \]

[Out]

-(a^5*A)/(14*x^14) - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*
x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b
+ 5*a*B)*x + (b^5*B*x^4)/4

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Rubi [A]  time = 0.217637, antiderivative size = 110, 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 \[ -\frac{a^5 A}{14 x^{14}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

-(a^5*A)/(14*x^14) - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*
x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b
+ 5*a*B)*x + (b^5*B*x^4)/4

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{14 x^{14}} + \frac{B b^{5} x^{4}}{4} - \frac{a^{4} \left (5 A b + B a\right )}{11 x^{11}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{8 x^{8}} - \frac{2 a^{2} b^{2} \left (A b + B a\right )}{x^{5}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{2 x^{2}} + \frac{b^{4} \left (A b + 5 B a\right ) \int A\, dx}{A} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**15,x)

[Out]

-A*a**5/(14*x**14) + B*b**5*x**4/4 - a**4*(5*A*b + B*a)/(11*x**11) - 5*a**3*b*(2
*A*b + B*a)/(8*x**8) - 2*a**2*b**2*(A*b + B*a)/x**5 - 5*a*b**3*(A*b + 2*B*a)/(2*
x**2) + b**4*(A*b + 5*B*a)*Integral(A, x)/A

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Mathematica [A]  time = 0.0838845, size = 110, normalized size = 1. \[ -\frac{a^5 A}{14 x^{14}}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{2 a^2 b^2 (a B+A b)}{x^5}+b^4 x (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

-(a^5*A)/(14*x^14) - (a^4*(5*A*b + a*B))/(11*x^11) - (5*a^3*b*(2*A*b + a*B))/(8*
x^8) - (2*a^2*b^2*(A*b + a*B))/x^5 - (5*a*b^3*(A*b + 2*a*B))/(2*x^2) + b^4*(A*b
+ 5*a*B)*x + (b^5*B*x^4)/4

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Maple [A]  time = 0.01, size = 102, normalized size = 0.9 \[{\frac{{b}^{5}B{x}^{4}}{4}}+Ax{b}^{5}+5\,Bxa{b}^{4}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{A{a}^{5}}{14\,{x}^{14}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{2\,{x}^{2}}}-2\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{5}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^15,x)

[Out]

1/4*b^5*B*x^4+A*x*b^5+5*B*x*a*b^4-5/8*a^3*b*(2*A*b+B*a)/x^8-1/11*a^4*(5*A*b+B*a)
/x^11-1/14*a^5*A/x^14-5/2*a*b^3*(A*b+2*B*a)/x^2-2*a^2*b^2*(A*b+B*a)/x^5

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Maxima [A]  time = 1.36994, size = 161, normalized size = 1.46 \[ \frac{1}{4} \, B b^{5} x^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac{1540 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 385 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 44 \, A a^{5} + 56 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^15,x, algorithm="maxima")

[Out]

1/4*B*b^5*x^4 + (5*B*a*b^4 + A*b^5)*x - 1/616*(1540*(2*B*a^2*b^3 + A*a*b^4)*x^12
+ 1232*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 44*A*a^5
+ 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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Fricas [A]  time = 0.217464, size = 163, normalized size = 1.48 \[ \frac{154 \, B b^{5} x^{18} + 616 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1540 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 1232 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 385 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 44 \, A a^{5} - 56 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^15,x, algorithm="fricas")

[Out]

1/616*(154*B*b^5*x^18 + 616*(5*B*a*b^4 + A*b^5)*x^15 - 1540*(2*B*a^2*b^3 + A*a*b
^4)*x^12 - 1232*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 -
44*A*a^5 - 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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Sympy [A]  time = 54.383, size = 122, normalized size = 1.11 \[ \frac{B b^{5} x^{4}}{4} + x \left (A b^{5} + 5 B a b^{4}\right ) - \frac{44 A a^{5} + x^{12} \left (1540 A a b^{4} + 3080 B a^{2} b^{3}\right ) + x^{9} \left (1232 A a^{2} b^{3} + 1232 B a^{3} b^{2}\right ) + x^{6} \left (770 A a^{3} b^{2} + 385 B a^{4} b\right ) + x^{3} \left (280 A a^{4} b + 56 B a^{5}\right )}{616 x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**15,x)

[Out]

B*b**5*x**4/4 + x*(A*b**5 + 5*B*a*b**4) - (44*A*a**5 + x**12*(1540*A*a*b**4 + 30
80*B*a**2*b**3) + x**9*(1232*A*a**2*b**3 + 1232*B*a**3*b**2) + x**6*(770*A*a**3*
b**2 + 385*B*a**4*b) + x**3*(280*A*a**4*b + 56*B*a**5))/(616*x**14)

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GIAC/XCAS [A]  time = 0.216209, size = 166, normalized size = 1.51 \[ \frac{1}{4} \, B b^{5} x^{4} + 5 \, B a b^{4} x + A b^{5} x - \frac{3080 \, B a^{2} b^{3} x^{12} + 1540 \, A a b^{4} x^{12} + 1232 \, B a^{3} b^{2} x^{9} + 1232 \, A a^{2} b^{3} x^{9} + 385 \, B a^{4} b x^{6} + 770 \, A a^{3} b^{2} x^{6} + 56 \, B a^{5} x^{3} + 280 \, A a^{4} b x^{3} + 44 \, A a^{5}}{616 \, x^{14}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^15,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5*B*a*b^4*x + A*b^5*x - 1/616*(3080*B*a^2*b^3*x^12 + 1540*A*a*b^
4*x^12 + 1232*B*a^3*b^2*x^9 + 1232*A*a^2*b^3*x^9 + 385*B*a^4*b*x^6 + 770*A*a^3*b
^2*x^6 + 56*B*a^5*x^3 + 280*A*a^4*b*x^3 + 44*A*a^5)/x^14