3.52 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{20}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^5 A}{19 x^{19}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^2 b^2 (a B+A b)}{x^{10}}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{x} \]

[Out]

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

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Rubi [A]  time = 0.209927, antiderivative size = 113, 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 \[ -\frac{a^5 A}{19 x^{19}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^2 b^2 (a B+A b)}{x^{10}}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^5 B}{x} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^20,x]

[Out]

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

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Rubi in Sympy [A]  time = 22.7828, size = 109, normalized size = 0.96 \[ - \frac{A a^{5}}{19 x^{19}} - \frac{B b^{5}}{x} - \frac{a^{4} \left (5 A b + B a\right )}{16 x^{16}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{13 x^{13}} - \frac{a^{2} b^{2} \left (A b + B a\right )}{x^{10}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{7 x^{7}} - \frac{b^{4} \left (A b + 5 B a\right )}{4 x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**20,x)

[Out]

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

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Mathematica [A]  time = 0.0660739, size = 119, normalized size = 1.05 \[ -\frac{91 a^5 \left (16 A+19 B x^3\right )+665 a^4 b x^3 \left (13 A+16 B x^3\right )+2128 a^3 b^2 x^6 \left (10 A+13 B x^3\right )+3952 a^2 b^3 x^9 \left (7 A+10 B x^3\right )+4940 a b^4 x^{12} \left (4 A+7 B x^3\right )+6916 b^5 x^{15} \left (A+4 B x^3\right )}{27664 x^{19}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^20,x]

[Out]

-(6916*b^5*x^15*(A + 4*B*x^3) + 4940*a*b^4*x^12*(4*A + 7*B*x^3) + 3952*a^2*b^3*x
^9*(7*A + 10*B*x^3) + 2128*a^3*b^2*x^6*(10*A + 13*B*x^3) + 665*a^4*b*x^3*(13*A +
16*B*x^3) + 91*a^5*(16*A + 19*B*x^3))/(27664*x^19)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^20,x)

[Out]

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

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Maxima [A]  time = 1.37421, size = 163, normalized size = 1.44 \[ -\frac{27664 \, B b^{5} x^{18} + 6916 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 19760 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 27664 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 10640 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 1456 \, A a^{5} + 1729 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{27664 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^20,x, algorithm="maxima")

[Out]

-1/27664*(27664*B*b^5*x^18 + 6916*(5*B*a*b^4 + A*b^5)*x^15 + 19760*(2*B*a^2*b^3
+ A*a*b^4)*x^12 + 27664*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 10640*(B*a^4*b + 2*A*a^3*b
^2)*x^6 + 1456*A*a^5 + 1729*(B*a^5 + 5*A*a^4*b)*x^3)/x^19

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Fricas [A]  time = 0.215082, size = 163, normalized size = 1.44 \[ -\frac{27664 \, B b^{5} x^{18} + 6916 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 19760 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 27664 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 10640 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 1456 \, A a^{5} + 1729 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{27664 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^20,x, algorithm="fricas")

[Out]

-1/27664*(27664*B*b^5*x^18 + 6916*(5*B*a*b^4 + A*b^5)*x^15 + 19760*(2*B*a^2*b^3
+ A*a*b^4)*x^12 + 27664*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 10640*(B*a^4*b + 2*A*a^3*b
^2)*x^6 + 1456*A*a^5 + 1729*(B*a^5 + 5*A*a^4*b)*x^3)/x^19

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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**20,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.213534, size = 171, normalized size = 1.51 \[ -\frac{27664 \, B b^{5} x^{18} + 34580 \, B a b^{4} x^{15} + 6916 \, A b^{5} x^{15} + 39520 \, B a^{2} b^{3} x^{12} + 19760 \, A a b^{4} x^{12} + 27664 \, B a^{3} b^{2} x^{9} + 27664 \, A a^{2} b^{3} x^{9} + 10640 \, B a^{4} b x^{6} + 21280 \, A a^{3} b^{2} x^{6} + 1729 \, B a^{5} x^{3} + 8645 \, A a^{4} b x^{3} + 1456 \, A a^{5}}{27664 \, x^{19}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^20,x, algorithm="giac")

[Out]

-1/27664*(27664*B*b^5*x^18 + 34580*B*a*b^4*x^15 + 6916*A*b^5*x^15 + 39520*B*a^2*
b^3*x^12 + 19760*A*a*b^4*x^12 + 27664*B*a^3*b^2*x^9 + 27664*A*a^2*b^3*x^9 + 1064
0*B*a^4*b*x^6 + 21280*A*a^3*b^2*x^6 + 1729*B*a^5*x^3 + 8645*A*a^4*b*x^3 + 1456*A
*a^5)/x^19