3.38 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^6} \, dx\)

Optimal. Leaf size=113 \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)-\frac{a^5 A}{5 x^5}+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

[Out]

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

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Rubi [A]  time = 0.062649, antiderivative size = 113, 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, Rules used = {448} \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)-\frac{a^5 A}{5 x^5}+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^6,x]

[Out]

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

Rule 448

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Int[ExpandI
ntegrand[(e*x)^m*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, m, n}, x] && NeQ[b*c - a*d, 0] &
& IGtQ[p, 0] && IGtQ[q, 0]

Rubi steps

\begin{align*} \int \frac{\left (a+b x^3\right )^5 \left (A+B x^3\right )}{x^6} \, dx &=\int \left (5 a^3 b (2 A b+a B)+\frac{a^5 A}{x^6}+\frac{a^4 (5 A b+a B)}{x^3}+10 a^2 b^2 (A b+a B) x^3+5 a b^3 (A b+2 a B) x^6+b^4 (A b+5 a B) x^9+b^5 B x^{12}\right ) \, dx\\ &=-\frac{a^5 A}{5 x^5}-\frac{a^4 (5 A b+a B)}{2 x^2}+5 a^3 b (2 A b+a B) x+\frac{5}{2} a^2 b^2 (A b+a B) x^4+\frac{5}{7} a b^3 (A b+2 a B) x^7+\frac{1}{10} b^4 (A b+5 a B) x^{10}+\frac{1}{13} b^5 B x^{13}\\ \end{align*}

Mathematica [A]  time = 0.0355137, size = 113, normalized size = 1. \[ \frac{5}{2} a^2 b^2 x^4 (a B+A b)-\frac{a^4 (a B+5 A b)}{2 x^2}+5 a^3 b x (a B+2 A b)-\frac{a^5 A}{5 x^5}+\frac{1}{10} b^4 x^{10} (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{13} b^5 B x^{13} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^6,x]

[Out]

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

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Maple [A]  time = 0.006, size = 119, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{13}}{13}}+{\frac{A{x}^{10}{b}^{5}}{10}}+{\frac{B{x}^{10}a{b}^{4}}{2}}+{\frac{5\,A{x}^{7}a{b}^{4}}{7}}+{\frac{10\,B{x}^{7}{a}^{2}{b}^{3}}{7}}+{\frac{5\,A{x}^{4}{a}^{2}{b}^{3}}{2}}+{\frac{5\,B{x}^{4}{a}^{3}{b}^{2}}{2}}+10\,{a}^{3}{b}^{2}Ax+5\,{a}^{4}bBx-{\frac{A{a}^{5}}{5\,{x}^{5}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{2\,{x}^{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^6,x)

[Out]

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

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Maxima [A]  time = 1.00114, size = 162, normalized size = 1.43 \begin{align*} \frac{1}{13} \, B b^{5} x^{13} + \frac{1}{10} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + \frac{5}{7} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + \frac{5}{2} \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + 5 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x - \frac{2 \, A a^{5} + 5 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.36795, size = 275, normalized size = 2.43 \begin{align*} \frac{70 \, B b^{5} x^{18} + 91 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 650 \,{\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} + 4550 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 182 \, A a^{5} - 455 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{910 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.596036, size = 131, normalized size = 1.16 \begin{align*} \frac{B b^{5} x^{13}}{13} + x^{10} \left (\frac{A b^{5}}{10} + \frac{B a b^{4}}{2}\right ) + x^{7} \left (\frac{5 A a b^{4}}{7} + \frac{10 B a^{2} b^{3}}{7}\right ) + x^{4} \left (\frac{5 A a^{2} b^{3}}{2} + \frac{5 B a^{3} b^{2}}{2}\right ) + x \left (10 A a^{3} b^{2} + 5 B a^{4} b\right ) - \frac{2 A a^{5} + x^{3} \left (25 A a^{4} b + 5 B a^{5}\right )}{10 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**6,x)

[Out]

B*b**5*x**13/13 + x**10*(A*b**5/10 + B*a*b**4/2) + x**7*(5*A*a*b**4/7 + 10*B*a**2*b**3/7) + x**4*(5*A*a**2*b**
3/2 + 5*B*a**3*b**2/2) + x*(10*A*a**3*b**2 + 5*B*a**4*b) - (2*A*a**5 + x**3*(25*A*a**4*b + 5*B*a**5))/(10*x**5
)

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Giac [A]  time = 1.20654, size = 167, normalized size = 1.48 \begin{align*} \frac{1}{13} \, B b^{5} x^{13} + \frac{1}{2} \, B a b^{4} x^{10} + \frac{1}{10} \, A b^{5} x^{10} + \frac{10}{7} \, B a^{2} b^{3} x^{7} + \frac{5}{7} \, A a b^{4} x^{7} + \frac{5}{2} \, B a^{3} b^{2} x^{4} + \frac{5}{2} \, A a^{2} b^{3} x^{4} + 5 \, B a^{4} b x + 10 \, A a^{3} b^{2} x - \frac{5 \, B a^{5} x^{3} + 25 \, A a^{4} b x^{3} + 2 \, A a^{5}}{10 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^6,x, algorithm="giac")

[Out]

1/13*B*b^5*x^13 + 1/2*B*a*b^4*x^10 + 1/10*A*b^5*x^10 + 10/7*B*a^2*b^3*x^7 + 5/7*A*a*b^4*x^7 + 5/2*B*a^3*b^2*x^
4 + 5/2*A*a^2*b^3*x^4 + 5*B*a^4*b*x + 10*A*a^3*b^2*x - 1/10*(5*B*a^5*x^3 + 25*A*a^4*b*x^3 + 2*A*a^5)/x^5