3.34 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^2} \, dx\)

Optimal. Leaf size=112 \[ -\frac{a^5 A}{x}+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{5}{4} a^2 b^2 x^8 (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

[Out]

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

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Rubi [A]  time = 0.198324, antiderivative size = 112, 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 \[ -\frac{a^5 A}{x}+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{5}{4} a^2 b^2 x^8 (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^2,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.0669558, size = 112, normalized size = 1. \[ -\frac{a^5 A}{x}+\frac{1}{2} a^4 x^2 (a B+5 A b)+a^3 b x^5 (a B+2 A b)+\frac{5}{4} a^2 b^2 x^8 (a B+A b)+\frac{1}{14} b^4 x^{14} (5 a B+A b)+\frac{5}{11} a b^3 x^{11} (2 a B+A b)+\frac{1}{17} b^5 B x^{17} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^2,x]

[Out]

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

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Maple [A]  time = 0.006, size = 125, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{17}}{17}}+{\frac{A{x}^{14}{b}^{5}}{14}}+{\frac{5\,B{x}^{14}a{b}^{4}}{14}}+{\frac{5\,A{x}^{11}a{b}^{4}}{11}}+{\frac{10\,B{x}^{11}{a}^{2}{b}^{3}}{11}}+{\frac{5\,A{x}^{8}{a}^{2}{b}^{3}}{4}}+{\frac{5\,B{x}^{8}{a}^{3}{b}^{2}}{4}}+2\,A{x}^{5}{a}^{3}{b}^{2}+B{x}^{5}{a}^{4}b+{\frac{5\,A{x}^{2}{a}^{4}b}{2}}+{\frac{B{x}^{2}{a}^{5}}{2}}-{\frac{A{a}^{5}}{x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^2,x)

[Out]

1/17*b^5*B*x^17+1/14*A*x^14*b^5+5/14*B*x^14*a*b^4+5/11*A*x^11*a*b^4+10/11*B*x^11
*a^2*b^3+5/4*A*x^8*a^2*b^3+5/4*B*x^8*a^3*b^2+2*A*x^5*a^3*b^2+B*x^5*a^4*b+5/2*A*x
^2*a^4*b+1/2*B*x^2*a^5-a^5*A/x

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Maxima [A]  time = 1.37657, size = 159, normalized size = 1.42 \[ \frac{1}{17} \, B b^{5} x^{17} + \frac{1}{14} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{14} + \frac{5}{11} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{11} + \frac{5}{4} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{8} +{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{5} - \frac{A a^{5}}{x} + \frac{1}{2} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.219968, size = 163, normalized size = 1.46 \[ \frac{308 \, B b^{5} x^{18} + 374 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 2380 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 6545 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 5236 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 5236 \, A a^{5} + 2618 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 1.5383, size = 129, normalized size = 1.15 \[ - \frac{A a^{5}}{x} + \frac{B b^{5} x^{17}}{17} + x^{14} \left (\frac{A b^{5}}{14} + \frac{5 B a b^{4}}{14}\right ) + x^{11} \left (\frac{5 A a b^{4}}{11} + \frac{10 B a^{2} b^{3}}{11}\right ) + x^{8} \left (\frac{5 A a^{2} b^{3}}{4} + \frac{5 B a^{3} b^{2}}{4}\right ) + x^{5} \left (2 A a^{3} b^{2} + B a^{4} b\right ) + x^{2} \left (\frac{5 A a^{4} b}{2} + \frac{B a^{5}}{2}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**2,x)

[Out]

-A*a**5/x + B*b**5*x**17/17 + x**14*(A*b**5/14 + 5*B*a*b**4/14) + x**11*(5*A*a*b
**4/11 + 10*B*a**2*b**3/11) + x**8*(5*A*a**2*b**3/4 + 5*B*a**3*b**2/4) + x**5*(2
*A*a**3*b**2 + B*a**4*b) + x**2*(5*A*a**4*b/2 + B*a**5/2)

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GIAC/XCAS [A]  time = 0.218487, size = 167, normalized size = 1.49 \[ \frac{1}{17} \, B b^{5} x^{17} + \frac{5}{14} \, B a b^{4} x^{14} + \frac{1}{14} \, A b^{5} x^{14} + \frac{10}{11} \, B a^{2} b^{3} x^{11} + \frac{5}{11} \, A a b^{4} x^{11} + \frac{5}{4} \, B a^{3} b^{2} x^{8} + \frac{5}{4} \, A a^{2} b^{3} x^{8} + B a^{4} b x^{5} + 2 \, A a^{3} b^{2} x^{5} + \frac{1}{2} \, B a^{5} x^{2} + \frac{5}{2} \, A a^{4} b x^{2} - \frac{A a^{5}}{x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^2,x, algorithm="giac")

[Out]

1/17*B*b^5*x^17 + 5/14*B*a*b^4*x^14 + 1/14*A*b^5*x^14 + 10/11*B*a^2*b^3*x^11 + 5
/11*A*a*b^4*x^11 + 5/4*B*a^3*b^2*x^8 + 5/4*A*a^2*b^3*x^8 + B*a^4*b*x^5 + 2*A*a^3
*b^2*x^5 + 1/2*B*a^5*x^2 + 5/2*A*a^4*b*x^2 - A*a^5/x