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

Optimal. Leaf size=112 \[ -\frac{a^5 A}{2 x^2}+a^4 x (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

[Out]

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

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Rubi [A]  time = 0.188466, 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 \[ -\frac{a^5 A}{2 x^2}+a^4 x (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ - \frac{A a^{5}}{2 x^{2}} + \frac{B b^{5} x^{16}}{16} + \frac{5 a^{3} b x^{4} \left (2 A b + B a\right )}{4} + \frac{10 a^{2} b^{2} x^{7} \left (A b + B a\right )}{7} + \frac{a b^{3} x^{10} \left (A b + 2 B a\right )}{2} + \frac{b^{4} x^{13} \left (A b + 5 B a\right )}{13} + \frac{a^{4} \left (5 A b + B a\right ) \int B\, dx}{B} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.062037, size = 112, normalized size = 1. \[ -\frac{a^5 A}{2 x^2}+a^4 x (a B+5 A b)+\frac{5}{4} a^3 b x^4 (a B+2 A b)+\frac{10}{7} a^2 b^2 x^7 (a B+A b)+\frac{1}{13} b^4 x^{13} (5 a B+A b)+\frac{1}{2} a b^3 x^{10} (2 a B+A b)+\frac{1}{16} b^5 B x^{16} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.005, size = 120, normalized size = 1.1 \[{\frac{{b}^{5}B{x}^{16}}{16}}+{\frac{A{x}^{13}{b}^{5}}{13}}+{\frac{5\,B{x}^{13}a{b}^{4}}{13}}+{\frac{A{x}^{10}a{b}^{4}}{2}}+B{x}^{10}{a}^{2}{b}^{3}+{\frac{10\,A{x}^{7}{a}^{2}{b}^{3}}{7}}+{\frac{10\,B{x}^{7}{a}^{3}{b}^{2}}{7}}+{\frac{5\,A{x}^{4}{a}^{3}{b}^{2}}{2}}+{\frac{5\,B{x}^{4}{a}^{4}b}{4}}+5\,Ax{a}^{4}b+Bx{a}^{5}-{\frac{A{a}^{5}}{2\,{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.39172, size = 157, normalized size = 1.4 \[ \frac{1}{16} \, B b^{5} x^{16} + \frac{1}{13} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{13} + \frac{1}{2} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{10} + \frac{10}{7} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{7} + \frac{5}{4} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - \frac{A a^{5}}{2 \, x^{2}} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.212429, size = 163, normalized size = 1.46 \[ \frac{91 \, B b^{5} x^{18} + 112 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 728 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 1820 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 728 \, A a^{5} + 1456 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{1456 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1456*(91*B*b^5*x^18 + 112*(5*B*a*b^4 + A*b^5)*x^15 + 728*(2*B*a^2*b^3 + A*a*b^
4)*x^12 + 2080*(B*a^3*b^2 + A*a^2*b^3)*x^9 + 1820*(B*a^4*b + 2*A*a^3*b^2)*x^6 -
728*A*a^5 + 1456*(B*a^5 + 5*A*a^4*b)*x^3)/x^2

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Sympy [A]  time = 1.54518, size = 128, normalized size = 1.14 \[ - \frac{A a^{5}}{2 x^{2}} + \frac{B b^{5} x^{16}}{16} + x^{13} \left (\frac{A b^{5}}{13} + \frac{5 B a b^{4}}{13}\right ) + x^{10} \left (\frac{A a b^{4}}{2} + B a^{2} b^{3}\right ) + x^{7} \left (\frac{10 A a^{2} b^{3}}{7} + \frac{10 B a^{3} b^{2}}{7}\right ) + x^{4} \left (\frac{5 A a^{3} b^{2}}{2} + \frac{5 B a^{4} b}{4}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.217075, size = 161, normalized size = 1.44 \[ \frac{1}{16} \, B b^{5} x^{16} + \frac{5}{13} \, B a b^{4} x^{13} + \frac{1}{13} \, A b^{5} x^{13} + B a^{2} b^{3} x^{10} + \frac{1}{2} \, A a b^{4} x^{10} + \frac{10}{7} \, B a^{3} b^{2} x^{7} + \frac{10}{7} \, A a^{2} b^{3} x^{7} + \frac{5}{4} \, B a^{4} b x^{4} + \frac{5}{2} \, A a^{3} b^{2} x^{4} + B a^{5} x + 5 \, A a^{4} b x - \frac{A a^{5}}{2 \, x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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