3.55 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{23}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{4 x^4} \]

[Out]

-(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(16
*x^16) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (
b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(4*x^4)

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Rubi [A]  time = 0.208572, antiderivative size = 117, 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 \[ -\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^23,x]

[Out]

-(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(16
*x^16) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (
b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(4*x^4)

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Rubi in Sympy [A]  time = 23.0413, size = 114, normalized size = 0.97 \[ - \frac{A a^{5}}{22 x^{22}} - \frac{B b^{5}}{4 x^{4}} - \frac{a^{4} \left (5 A b + B a\right )}{19 x^{19}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{16 x^{16}} - \frac{10 a^{2} b^{2} \left (A b + B a\right )}{13 x^{13}} - \frac{a b^{3} \left (A b + 2 B a\right )}{2 x^{10}} - \frac{b^{4} \left (A b + 5 B a\right )}{7 x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**23,x)

[Out]

-A*a**5/(22*x**22) - B*b**5/(4*x**4) - a**4*(5*A*b + B*a)/(19*x**19) - 5*a**3*b*
(2*A*b + B*a)/(16*x**16) - 10*a**2*b**2*(A*b + B*a)/(13*x**13) - a*b**3*(A*b + 2
*B*a)/(2*x**10) - b**4*(A*b + 5*B*a)/(7*x**7)

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Mathematica [A]  time = 0.0955924, size = 117, normalized size = 1. \[ -\frac{a^5 A}{22 x^{22}}-\frac{a^4 (a B+5 A b)}{19 x^{19}}-\frac{5 a^3 b (a B+2 A b)}{16 x^{16}}-\frac{10 a^2 b^2 (a B+A b)}{13 x^{13}}-\frac{b^4 (5 a B+A b)}{7 x^7}-\frac{a b^3 (2 a B+A b)}{2 x^{10}}-\frac{b^5 B}{4 x^4} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^23,x]

[Out]

-(a^5*A)/(22*x^22) - (a^4*(5*A*b + a*B))/(19*x^19) - (5*a^3*b*(2*A*b + a*B))/(16
*x^16) - (10*a^2*b^2*(A*b + a*B))/(13*x^13) - (a*b^3*(A*b + 2*a*B))/(2*x^10) - (
b^4*(A*b + 5*a*B))/(7*x^7) - (b^5*B)/(4*x^4)

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Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{22\,{x}^{22}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{19\,{x}^{19}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{2\,{x}^{10}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{7\,{x}^{7}}}-{\frac{B{b}^{5}}{4\,{x}^{4}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^23,x)

[Out]

-1/22*a^5*A/x^22-1/19*a^4*(5*A*b+B*a)/x^19-5/16*a^3*b*(2*A*b+B*a)/x^16-10/13*a^2
*b^2*(A*b+B*a)/x^13-1/2*a*b^3*(A*b+2*B*a)/x^10-1/7*b^4*(A*b+5*B*a)/x^7-1/4*b^5*B
/x^4

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Maxima [A]  time = 1.37735, size = 163, normalized size = 1.39 \[ -\frac{76076 \, B b^{5} x^{18} + 43472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 152152 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 234080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 95095 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 13832 \, A a^{5} + 16016 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{304304 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^23,x, algorithm="maxima")

[Out]

-1/304304*(76076*B*b^5*x^18 + 43472*(5*B*a*b^4 + A*b^5)*x^15 + 152152*(2*B*a^2*b
^3 + A*a*b^4)*x^12 + 234080*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 95095*(B*a^4*b + 2*A*a
^3*b^2)*x^6 + 13832*A*a^5 + 16016*(B*a^5 + 5*A*a^4*b)*x^3)/x^22

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Fricas [A]  time = 0.221386, size = 163, normalized size = 1.39 \[ -\frac{76076 \, B b^{5} x^{18} + 43472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 152152 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 234080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 95095 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 13832 \, A a^{5} + 16016 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{304304 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^23,x, algorithm="fricas")

[Out]

-1/304304*(76076*B*b^5*x^18 + 43472*(5*B*a*b^4 + A*b^5)*x^15 + 152152*(2*B*a^2*b
^3 + A*a*b^4)*x^12 + 234080*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 95095*(B*a^4*b + 2*A*a
^3*b^2)*x^6 + 13832*A*a^5 + 16016*(B*a^5 + 5*A*a^4*b)*x^3)/x^22

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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**3+a)**5*(B*x**3+A)/x**23,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.21254, size = 171, normalized size = 1.46 \[ -\frac{76076 \, B b^{5} x^{18} + 217360 \, B a b^{4} x^{15} + 43472 \, A b^{5} x^{15} + 304304 \, B a^{2} b^{3} x^{12} + 152152 \, A a b^{4} x^{12} + 234080 \, B a^{3} b^{2} x^{9} + 234080 \, A a^{2} b^{3} x^{9} + 95095 \, B a^{4} b x^{6} + 190190 \, A a^{3} b^{2} x^{6} + 16016 \, B a^{5} x^{3} + 80080 \, A a^{4} b x^{3} + 13832 \, A a^{5}}{304304 \, x^{22}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^23,x, algorithm="giac")

[Out]

-1/304304*(76076*B*b^5*x^18 + 217360*B*a*b^4*x^15 + 43472*A*b^5*x^15 + 304304*B*
a^2*b^3*x^12 + 152152*A*a*b^4*x^12 + 234080*B*a^3*b^2*x^9 + 234080*A*a^2*b^3*x^9
+ 95095*B*a^4*b*x^6 + 190190*A*a^3*b^2*x^6 + 16016*B*a^5*x^3 + 80080*A*a^4*b*x^
3 + 13832*A*a^5)/x^22