3.46 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^{14}} \, dx\)

Optimal. Leaf size=115 \[ -\frac{a^5 A}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{10 x^{10}}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+\frac{1}{2} b^4 x^2 (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{5} b^5 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.215489, antiderivative size = 115, 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 \[ -\frac{a^5 A}{13 x^{13}}-\frac{a^4 (a B+5 A b)}{10 x^{10}}-\frac{5 a^3 b (a B+2 A b)}{7 x^7}-\frac{5 a^2 b^2 (a B+A b)}{2 x^4}+\frac{1}{2} b^4 x^2 (5 a B+A b)-\frac{5 a b^3 (2 a B+A b)}{x}+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^14,x]

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{13 x^{13}} + \frac{B b^{5} x^{5}}{5} - \frac{a^{4} \left (5 A b + B a\right )}{10 x^{10}} - \frac{5 a^{3} b \left (2 A b + B a\right )}{7 x^{7}} - \frac{5 a^{2} b^{2} \left (A b + B a\right )}{2 x^{4}} - \frac{5 a b^{3} \left (A b + 2 B a\right )}{x} + b^{4} \left (A b + 5 B a\right ) \int x\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**14,x)

[Out]

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

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Mathematica [A]  time = 0.0746194, size = 117, normalized size = 1.02 \[ -\frac{a^5 \left (70 A+91 B x^3\right )+65 a^4 b x^3 \left (7 A+10 B x^3\right )+325 a^3 b^2 x^6 \left (4 A+7 B x^3\right )+2275 a^2 b^3 x^9 \left (A+4 B x^3\right )-2275 a b^4 x^{12} \left (B x^3-2 A\right )-91 b^5 x^{15} \left (5 A+2 B x^3\right )}{910 x^{13}} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^14,x]

[Out]

-(-2275*a*b^4*x^12*(-2*A + B*x^3) - 91*b^5*x^15*(5*A + 2*B*x^3) + 2275*a^2*b^3*x
^9*(A + 4*B*x^3) + 325*a^3*b^2*x^6*(4*A + 7*B*x^3) + 65*a^4*b*x^3*(7*A + 10*B*x^
3) + a^5*(70*A + 91*B*x^3))/(910*x^13)

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Maple [A]  time = 0.009, size = 107, normalized size = 0.9 \[{\frac{{b}^{5}B{x}^{5}}{5}}+{\frac{A{x}^{2}{b}^{5}}{2}}+{\frac{5\,B{x}^{2}a{b}^{4}}{2}}-{\frac{A{a}^{5}}{13\,{x}^{13}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{2\,{x}^{4}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{10\,{x}^{10}}}-5\,{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{x}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{7\,{x}^{7}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^14,x)

[Out]

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

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Maxima [A]  time = 1.45299, size = 165, normalized size = 1.43 \[ \frac{1}{5} \, B b^{5} x^{5} + \frac{1}{2} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{2} - \frac{4550 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2275 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 650 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 70 \, A a^{5} + 91 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^14,x, algorithm="maxima")

[Out]

1/5*B*b^5*x^5 + 1/2*(5*B*a*b^4 + A*b^5)*x^2 - 1/910*(4550*(2*B*a^2*b^3 + A*a*b^4
)*x^12 + 2275*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 650*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 70
*A*a^5 + 91*(B*a^5 + 5*A*a^4*b)*x^3)/x^13

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Fricas [A]  time = 0.228634, size = 163, normalized size = 1.42 \[ \frac{182 \, B b^{5} x^{18} + 455 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 4550 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 2275 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 650 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 70 \, A a^{5} - 91 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^14,x, algorithm="fricas")

[Out]

1/910*(182*B*b^5*x^18 + 455*(5*B*a*b^4 + A*b^5)*x^15 - 4550*(2*B*a^2*b^3 + A*a*b
^4)*x^12 - 2275*(B*a^3*b^2 + A*a^2*b^3)*x^9 - 650*(B*a^4*b + 2*A*a^3*b^2)*x^6 -
70*A*a^5 - 91*(B*a^5 + 5*A*a^4*b)*x^3)/x^13

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Sympy [A]  time = 34.1169, size = 128, normalized size = 1.11 \[ \frac{B b^{5} x^{5}}{5} + x^{2} \left (\frac{A b^{5}}{2} + \frac{5 B a b^{4}}{2}\right ) - \frac{70 A a^{5} + x^{12} \left (4550 A a b^{4} + 9100 B a^{2} b^{3}\right ) + x^{9} \left (2275 A a^{2} b^{3} + 2275 B a^{3} b^{2}\right ) + x^{6} \left (1300 A a^{3} b^{2} + 650 B a^{4} b\right ) + x^{3} \left (455 A a^{4} b + 91 B a^{5}\right )}{910 x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**14,x)

[Out]

B*b**5*x**5/5 + x**2*(A*b**5/2 + 5*B*a*b**4/2) - (70*A*a**5 + x**12*(4550*A*a*b*
*4 + 9100*B*a**2*b**3) + x**9*(2275*A*a**2*b**3 + 2275*B*a**3*b**2) + x**6*(1300
*A*a**3*b**2 + 650*B*a**4*b) + x**3*(455*A*a**4*b + 91*B*a**5))/(910*x**13)

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GIAC/XCAS [A]  time = 0.217205, size = 173, normalized size = 1.5 \[ \frac{1}{5} \, B b^{5} x^{5} + \frac{5}{2} \, B a b^{4} x^{2} + \frac{1}{2} \, A b^{5} x^{2} - \frac{9100 \, B a^{2} b^{3} x^{12} + 4550 \, A a b^{4} x^{12} + 2275 \, B a^{3} b^{2} x^{9} + 2275 \, A a^{2} b^{3} x^{9} + 650 \, B a^{4} b x^{6} + 1300 \, A a^{3} b^{2} x^{6} + 91 \, B a^{5} x^{3} + 455 \, A a^{4} b x^{3} + 70 \, A a^{5}}{910 \, x^{13}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^14,x, algorithm="giac")

[Out]

1/5*B*b^5*x^5 + 5/2*B*a*b^4*x^2 + 1/2*A*b^5*x^2 - 1/910*(9100*B*a^2*b^3*x^12 + 4
550*A*a*b^4*x^12 + 2275*B*a^3*b^2*x^9 + 2275*A*a^2*b^3*x^9 + 650*B*a^4*b*x^6 + 1
300*A*a^3*b^2*x^6 + 91*B*a^5*x^3 + 455*A*a^4*b*x^3 + 70*A*a^5)/x^13