3.44 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{12}} \, dx\)

Optimal. Leaf size=109 \[ -\frac{5 a^2 b^2 (a B+A b)}{x^2}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^5 A}{11 x^{11}}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5
*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^7)/7

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Rubi [A]  time = 0.064277, antiderivative size = 109, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{x^2}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^5 A}{11 x^{11}}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5
*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^7)/7

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^{12}} \, dx &=\int \left (5 a b^3 (A b+2 a B)+\frac{a^5 A}{x^{12}}+\frac{a^4 (5 A b+a B)}{x^9}+\frac{5 a^3 b (2 A b+a B)}{x^6}+\frac{10 a^2 b^2 (A b+a B)}{x^3}+b^4 (A b+5 a B) x^3+b^5 B x^6\right ) \, dx\\ &=-\frac{a^5 A}{11 x^{11}}-\frac{a^4 (5 A b+a B)}{8 x^8}-\frac{a^3 b (2 A b+a B)}{x^5}-\frac{5 a^2 b^2 (A b+a B)}{x^2}+5 a b^3 (A b+2 a B) x+\frac{1}{4} b^4 (A b+5 a B) x^4+\frac{1}{7} b^5 B x^7\\ \end{align*}

Mathematica [A]  time = 0.0360838, size = 109, normalized size = 1. \[ -\frac{5 a^2 b^2 (a B+A b)}{x^2}-\frac{a^3 b (a B+2 A b)}{x^5}-\frac{a^4 (a B+5 A b)}{8 x^8}-\frac{a^5 A}{11 x^{11}}+\frac{1}{4} b^4 x^4 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^12,x]

[Out]

-(a^5*A)/(11*x^11) - (a^4*(5*A*b + a*B))/(8*x^8) - (a^3*b*(2*A*b + a*B))/x^5 - (5*a^2*b^2*(A*b + a*B))/x^2 + 5
*a*b^3*(A*b + 2*a*B)*x + (b^4*(A*b + 5*a*B)*x^4)/4 + (b^5*B*x^7)/7

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Maple [A]  time = 0.007, size = 108, normalized size = 1. \begin{align*}{\frac{{b}^{5}B{x}^{7}}{7}}+{\frac{A{x}^{4}{b}^{5}}{4}}+{\frac{5\,B{x}^{4}a{b}^{4}}{4}}+5\,a{b}^{4}Ax+10\,{a}^{2}{b}^{3}Bx-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{8\,{x}^{8}}}-{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{{x}^{5}}}-5\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{2}}}-{\frac{A{a}^{5}}{11\,{x}^{11}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^12,x)

[Out]

1/7*b^5*B*x^7+1/4*A*x^4*b^5+5/4*B*x^4*a*b^4+5*a*b^4*A*x+10*a^2*b^3*B*x-1/8*a^4*(5*A*b+B*a)/x^8-a^3*b*(2*A*b+B*
a)/x^5-5*a^2*b^2*(A*b+B*a)/x^2-1/11*a^5*A/x^11

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Maxima [A]  time = 1.18392, size = 162, normalized size = 1.49 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + \frac{1}{4} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{4} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac{440 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 88 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 8 \, A a^{5} + 11 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{88 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="maxima")

[Out]

1/7*B*b^5*x^7 + 1/4*(5*B*a*b^4 + A*b^5)*x^4 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/88*(440*(B*a^3*b^2 + A*a^2*b^3)*
x^9 + 88*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 8*A*a^5 + 11*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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Fricas [A]  time = 1.41697, size = 275, normalized size = 2.52 \begin{align*} \frac{88 \, B b^{5} x^{18} + 154 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 3080 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 3080 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 616 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 56 \, A a^{5} - 77 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="fricas")

[Out]

1/616*(88*B*b^5*x^18 + 154*(5*B*a*b^4 + A*b^5)*x^15 + 3080*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 3080*(B*a^3*b^2 + A*
a^2*b^3)*x^9 - 616*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 56*A*a^5 - 77*(B*a^5 + 5*A*a^4*b)*x^3)/x^11

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Sympy [A]  time = 3.83957, size = 126, normalized size = 1.16 \begin{align*} \frac{B b^{5} x^{7}}{7} + x^{4} \left (\frac{A b^{5}}{4} + \frac{5 B a b^{4}}{4}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{8 A a^{5} + x^{9} \left (440 A a^{2} b^{3} + 440 B a^{3} b^{2}\right ) + x^{6} \left (176 A a^{3} b^{2} + 88 B a^{4} b\right ) + x^{3} \left (55 A a^{4} b + 11 B a^{5}\right )}{88 x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**12,x)

[Out]

B*b**5*x**7/7 + x**4*(A*b**5/4 + 5*B*a*b**4/4) + x*(5*A*a*b**4 + 10*B*a**2*b**3) - (8*A*a**5 + x**9*(440*A*a**
2*b**3 + 440*B*a**3*b**2) + x**6*(176*A*a**3*b**2 + 88*B*a**4*b) + x**3*(55*A*a**4*b + 11*B*a**5))/(88*x**11)

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Giac [A]  time = 1.1897, size = 167, normalized size = 1.53 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + \frac{5}{4} \, B a b^{4} x^{4} + \frac{1}{4} \, A b^{5} x^{4} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac{440 \, B a^{3} b^{2} x^{9} + 440 \, A a^{2} b^{3} x^{9} + 88 \, B a^{4} b x^{6} + 176 \, A a^{3} b^{2} x^{6} + 11 \, B a^{5} x^{3} + 55 \, A a^{4} b x^{3} + 8 \, A a^{5}}{88 \, x^{11}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^12,x, algorithm="giac")

[Out]

1/7*B*b^5*x^7 + 5/4*B*a*b^4*x^4 + 1/4*A*b^5*x^4 + 10*B*a^2*b^3*x + 5*A*a*b^4*x - 1/88*(440*B*a^3*b^2*x^9 + 440
*A*a^2*b^3*x^9 + 88*B*a^4*b*x^6 + 176*A*a^3*b^2*x^6 + 11*B*a^5*x^3 + 55*A*a^4*b*x^3 + 8*A*a^5)/x^11