3.36 \(\int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^4} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^5 A}{3 x^3}+a^4 \log (x) (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

[Out]

-(a^5*A)/(3*x^3) + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + (5*a^2*b^2*(A*b + a*B)*x^6)/3
+ (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^12)/12 + (b^5*B*x^15)/15
+ a^4*(5*A*b + a*B)*Log[x]

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Rubi [A]  time = 0.335624, antiderivative size = 113, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^5 A}{3 x^3}+a^4 \log (x) (a B+5 A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

[In]  Int[((a + b*x^3)^5*(A + B*x^3))/x^4,x]

[Out]

-(a^5*A)/(3*x^3) + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + (5*a^2*b^2*(A*b + a*B)*x^6)/3
+ (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^12)/12 + (b^5*B*x^15)/15
+ a^4*(5*A*b + a*B)*Log[x]

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{3 x^{3}} + \frac{B b^{5} x^{15}}{15} + \frac{a^{4} \left (5 A b + B a\right ) \log{\left (x^{3} \right )}}{3} + \frac{5 a^{3} b x^{3} \left (2 A b + B a\right )}{3} + \frac{10 a^{2} b^{2} \left (A b + B a\right ) \int ^{x^{3}} x\, dx}{3} + \frac{5 a b^{3} x^{9} \left (A b + 2 B a\right )}{9} + \frac{b^{4} x^{12} \left (A b + 5 B a\right )}{12} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**3+a)**5*(B*x**3+A)/x**4,x)

[Out]

-A*a**5/(3*x**3) + B*b**5*x**15/15 + a**4*(5*A*b + B*a)*log(x**3)/3 + 5*a**3*b*x
**3*(2*A*b + B*a)/3 + 10*a**2*b**2*(A*b + B*a)*Integral(x, (x, x**3))/3 + 5*a*b*
*3*x**9*(A*b + 2*B*a)/9 + b**4*x**12*(A*b + 5*B*a)/12

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Mathematica [A]  time = 0.0803826, size = 115, normalized size = 1.02 \[ -\frac{a^5 A}{3 x^3}+\frac{5}{3} a^3 b x^3 (a B+2 A b)+\frac{5}{3} a^2 b^2 x^6 (a B+A b)+\log (x) \left (a^5 B+5 a^4 A b\right )+\frac{1}{12} b^4 x^{12} (5 a B+A b)+\frac{5}{9} a b^3 x^9 (2 a B+A b)+\frac{1}{15} b^5 B x^{15} \]

Antiderivative was successfully verified.

[In]  Integrate[((a + b*x^3)^5*(A + B*x^3))/x^4,x]

[Out]

-(a^5*A)/(3*x^3) + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + (5*a^2*b^2*(A*b + a*B)*x^6)/3
+ (5*a*b^3*(A*b + 2*a*B)*x^9)/9 + (b^4*(A*b + 5*a*B)*x^12)/12 + (b^5*B*x^15)/15
+ (5*a^4*A*b + a^5*B)*Log[x]

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Maple [A]  time = 0.011, size = 123, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{15}}{15}}+{\frac{A{x}^{12}{b}^{5}}{12}}+{\frac{5\,B{x}^{12}a{b}^{4}}{12}}+{\frac{5\,A{x}^{9}a{b}^{4}}{9}}+{\frac{10\,B{x}^{9}{a}^{2}{b}^{3}}{9}}+{\frac{5\,A{x}^{6}{a}^{2}{b}^{3}}{3}}+{\frac{5\,B{x}^{6}{a}^{3}{b}^{2}}{3}}+{\frac{10\,A{x}^{3}{a}^{3}{b}^{2}}{3}}+{\frac{5\,B{x}^{3}{a}^{4}b}{3}}+5\,A\ln \left ( x \right ){a}^{4}b+B\ln \left ( x \right ){a}^{5}-{\frac{A{a}^{5}}{3\,{x}^{3}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^3+a)^5*(B*x^3+A)/x^4,x)

[Out]

1/15*b^5*B*x^15+1/12*A*x^12*b^5+5/12*B*x^12*a*b^4+5/9*A*x^9*a*b^4+10/9*B*x^9*a^2
*b^3+5/3*A*x^6*a^2*b^3+5/3*B*x^6*a^3*b^2+10/3*A*x^3*a^3*b^2+5/3*B*x^3*a^4*b+5*A*
ln(x)*a^4*b+B*ln(x)*a^5-1/3*a^5*A/x^3

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Maxima [A]  time = 1.37017, size = 162, normalized size = 1.43 \[ \frac{1}{15} \, B b^{5} x^{15} + \frac{1}{12} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{12} + \frac{5}{9} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{9} + \frac{5}{3} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} - \frac{A a^{5}}{3 \, x^{3}} + \frac{1}{3} \,{\left (B a^{5} + 5 \, A a^{4} b\right )} \log \left (x^{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^4,x, algorithm="maxima")

[Out]

1/15*B*b^5*x^15 + 1/12*(5*B*a*b^4 + A*b^5)*x^12 + 5/9*(2*B*a^2*b^3 + A*a*b^4)*x^
9 + 5/3*(B*a^3*b^2 + A*a^2*b^3)*x^6 + 5/3*(B*a^4*b + 2*A*a^3*b^2)*x^3 - 1/3*A*a^
5/x^3 + 1/3*(B*a^5 + 5*A*a^4*b)*log(x^3)

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Fricas [A]  time = 0.235678, size = 166, normalized size = 1.47 \[ \frac{12 \, B b^{5} x^{18} + 15 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 100 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 300 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 300 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 60 \, A a^{5} + 180 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3} \log \left (x\right )}{180 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^4,x, algorithm="fricas")

[Out]

1/180*(12*B*b^5*x^18 + 15*(5*B*a*b^4 + A*b^5)*x^15 + 100*(2*B*a^2*b^3 + A*a*b^4)
*x^12 + 300*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 300*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 60*A
*a^5 + 180*(B*a^5 + 5*A*a^4*b)*x^3*log(x))/x^3

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Sympy [A]  time = 2.00845, size = 133, normalized size = 1.18 \[ - \frac{A a^{5}}{3 x^{3}} + \frac{B b^{5} x^{15}}{15} + a^{4} \left (5 A b + B a\right ) \log{\left (x \right )} + x^{12} \left (\frac{A b^{5}}{12} + \frac{5 B a b^{4}}{12}\right ) + x^{9} \left (\frac{5 A a b^{4}}{9} + \frac{10 B a^{2} b^{3}}{9}\right ) + x^{6} \left (\frac{5 A a^{2} b^{3}}{3} + \frac{5 B a^{3} b^{2}}{3}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x**3+a)**5*(B*x**3+A)/x**4,x)

[Out]

-A*a**5/(3*x**3) + B*b**5*x**15/15 + a**4*(5*A*b + B*a)*log(x) + x**12*(A*b**5/1
2 + 5*B*a*b**4/12) + x**9*(5*A*a*b**4/9 + 10*B*a**2*b**3/9) + x**6*(5*A*a**2*b**
3/3 + 5*B*a**3*b**2/3) + x**3*(10*A*a**3*b**2/3 + 5*B*a**4*b/3)

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GIAC/XCAS [A]  time = 0.219947, size = 193, normalized size = 1.71 \[ \frac{1}{15} \, B b^{5} x^{15} + \frac{5}{12} \, B a b^{4} x^{12} + \frac{1}{12} \, A b^{5} x^{12} + \frac{10}{9} \, B a^{2} b^{3} x^{9} + \frac{5}{9} \, A a b^{4} x^{9} + \frac{5}{3} \, B a^{3} b^{2} x^{6} + \frac{5}{3} \, A a^{2} b^{3} x^{6} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} +{\left (B a^{5} + 5 \, A a^{4} b\right )}{\rm ln}\left ({\left | x \right |}\right ) - \frac{B a^{5} x^{3} + 5 \, A a^{4} b x^{3} + A a^{5}}{3 \, x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x^3 + A)*(b*x^3 + a)^5/x^4,x, algorithm="giac")

[Out]

1/15*B*b^5*x^15 + 5/12*B*a*b^4*x^12 + 1/12*A*b^5*x^12 + 10/9*B*a^2*b^3*x^9 + 5/9
*A*a*b^4*x^9 + 5/3*B*a^3*b^2*x^6 + 5/3*A*a^2*b^3*x^6 + 5/3*B*a^4*b*x^3 + 10/3*A*
a^3*b^2*x^3 + (B*a^5 + 5*A*a^4*b)*ln(abs(x)) - 1/3*(B*a^5*x^3 + 5*A*a^4*b*x^3 +
A*a^5)/x^3