3.50 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{18}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{2 x^2}-\frac{a b^3 (2 a B+A b)}{x^5}+b^5 B x \]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11
*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b
+ 5*a*B))/(2*x^2) + b^5*B*x

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Rubi [A]  time = 0.199415, 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 \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{2 x^2}-\frac{a b^3 (2 a B+A b)}{x^5}+b^5 B x \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11
*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b
+ 5*a*B))/(2*x^2) + b^5*B*x

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{17 x^{17}} - \frac{a^{4} \left (5 A b + B a\right )}{14 x^{14}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{11 x^{11}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{4 x^{8}} - \frac{a b^{3} \left (A b + 2 B a\right )}{x^{5}} + b^{5} \int B\, dx - \frac{b^{4} \left (A b + 5 B a\right )}{2 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**18,x)

[Out]

-A*a**5/(17*x**17) - a**4*(5*A*b + B*a)/(14*x**14) - 5*a**3*b*(2*A*b + B*a)/(11*
x**11) - 5*a**2*b**2*(A*b + B*a)/(4*x**8) - a*b**3*(A*b + 2*B*a)/x**5 + b**5*Int
egral(B, x) - b**4*(A*b + 5*B*a)/(2*x**2)

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Mathematica [A]  time = 0.100783, size = 110, normalized size = 1. \[ -\frac{a^5 A}{17 x^{17}}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{b^4 (5 a B+A b)}{2 x^2}-\frac{a b^3 (2 a B+A b)}{x^5}+b^5 B x \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11
*x^11) - (5*a^2*b^2*(A*b + a*B))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b
+ 5*a*B))/(2*x^2) + b^5*B*x

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Maple [A]  time = 0.009, size = 101, normalized size = 0.9 \[ -{\frac{A{a}^{5}}{17\,{x}^{17}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{4\,{x}^{8}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{{x}^{5}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{2\,{x}^{2}}}+{b}^{5}Bx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^18,x)

[Out]

-1/17*a^5*A/x^17-1/14*a^4*(5*A*b+B*a)/x^14-5/11*a^3*b*(2*A*b+B*a)/x^11-5/4*a^2*b
^2*(A*b+B*a)/x^8-a*b^3*(A*b+2*B*a)/x^5-1/2*b^4*(A*b+5*B*a)/x^2+b^5*B*x

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Maxima [A]  time = 1.39531, size = 161, normalized size = 1.46 \[ B b^{5} x - \frac{2618 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 5236 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 6545 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 2380 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 308 \, A a^{5} + 374 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^18,x, algorithm="maxima")

[Out]

B*b^5*x - 1/5236*(2618*(5*B*a*b^4 + A*b^5)*x^15 + 5236*(2*B*a^2*b^3 + A*a*b^4)*x
^12 + 6545*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 2380*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 308*
A*a^5 + 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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Fricas [A]  time = 0.212468, size = 163, normalized size = 1.48 \[ \frac{5236 \, B b^{5} x^{18} - 2618 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 5236 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 6545 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 2380 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 308 \, A a^{5} - 374 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^18,x, algorithm="fricas")

[Out]

1/5236*(5236*B*b^5*x^18 - 2618*(5*B*a*b^4 + A*b^5)*x^15 - 5236*(2*B*a^2*b^3 + A*
a*b^4)*x^12 - 6545*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 2380*(B*a^4*b + 2*A*a^3*b^2)*x^
6 - 308*A*a^5 - 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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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**18,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.212127, size = 169, normalized size = 1.54 \[ B b^{5} x - \frac{13090 \, B a b^{4} x^{15} + 2618 \, A b^{5} x^{15} + 10472 \, B a^{2} b^{3} x^{12} + 5236 \, A a b^{4} x^{12} + 6545 \, B a^{3} b^{2} x^{9} + 6545 \, A a^{2} b^{3} x^{9} + 2380 \, B a^{4} b x^{6} + 4760 \, A a^{3} b^{2} x^{6} + 374 \, B a^{5} x^{3} + 1870 \, A a^{4} b x^{3} + 308 \, A a^{5}}{5236 \, x^{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^18,x, algorithm="giac")

[Out]

B*b^5*x - 1/5236*(13090*B*a*b^4*x^15 + 2618*A*b^5*x^15 + 10472*B*a^2*b^3*x^12 +
5236*A*a*b^4*x^12 + 6545*B*a^3*b^2*x^9 + 6545*A*a^2*b^3*x^9 + 2380*B*a^4*b*x^6 +
4760*A*a^3*b^2*x^6 + 374*B*a^5*x^3 + 1870*A*a^4*b*x^3 + 308*A*a^5)/x^17