3.47 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{15}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{a^5 A}{14 x^{14}}-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+b^4 x (5 a B+A b)+\frac{1}{4} b^5 B x^4 \]

[Out]

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

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Rubi [A]  time = 0.0677994, antiderivative size = 110, 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} \[ -\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{a^5 A}{14 x^{14}}-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+b^4 x (5 a B+A b)+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

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

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^{15}} \, dx &=\int \left (b^4 (A b+5 a B)+\frac{a^5 A}{x^{15}}+\frac{a^4 (5 A b+a B)}{x^{12}}+\frac{5 a^3 b (2 A b+a B)}{x^9}+\frac{10 a^2 b^2 (A b+a B)}{x^6}+\frac{5 a b^3 (A b+2 a B)}{x^3}+b^5 B x^3\right ) \, dx\\ &=-\frac{a^5 A}{14 x^{14}}-\frac{a^4 (5 A b+a B)}{11 x^{11}}-\frac{5 a^3 b (2 A b+a B)}{8 x^8}-\frac{2 a^2 b^2 (A b+a B)}{x^5}-\frac{5 a b^3 (A b+2 a B)}{2 x^2}+b^4 (A b+5 a B) x+\frac{1}{4} b^5 B x^4\\ \end{align*}

Mathematica [A]  time = 0.0412247, size = 110, normalized size = 1. \[ -\frac{2 a^2 b^2 (a B+A b)}{x^5}-\frac{a^4 (a B+5 A b)}{11 x^{11}}-\frac{5 a^3 b (a B+2 A b)}{8 x^8}-\frac{a^5 A}{14 x^{14}}-\frac{5 a b^3 (2 a B+A b)}{2 x^2}+b^4 x (5 a B+A b)+\frac{1}{4} b^5 B x^4 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^15,x]

[Out]

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

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Maple [A]  time = 0.009, size = 102, normalized size = 0.9 \begin{align*}{\frac{{b}^{5}B{x}^{4}}{4}}+{b}^{5}Ax+5\,a{b}^{4}Bx-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{2\,{x}^{2}}}-{\frac{A{a}^{5}}{14\,{x}^{14}}}-2\,{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{5}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{8\,{x}^{8}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^15,x)

[Out]

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

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Maxima [A]  time = 1.39243, size = 161, normalized size = 1.46 \begin{align*} \frac{1}{4} \, B b^{5} x^{4} +{\left (5 \, B a b^{4} + A b^{5}\right )} x - \frac{1540 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 1232 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 385 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 44 \, A a^{5} + 56 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="maxima")

[Out]

1/4*B*b^5*x^4 + (5*B*a*b^4 + A*b^5)*x - 1/616*(1540*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 1232*(B*a^3*b^2 + A*a^2*b^3
)*x^9 + 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 44*A*a^5 + 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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Fricas [A]  time = 1.4132, size = 277, normalized size = 2.52 \begin{align*} \frac{154 \, B b^{5} x^{18} + 616 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 1540 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 1232 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 385 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 44 \, A a^{5} - 56 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="fricas")

[Out]

1/616*(154*B*b^5*x^18 + 616*(5*B*a*b^4 + A*b^5)*x^15 - 1540*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 1232*(B*a^3*b^2 + A
*a^2*b^3)*x^9 - 385*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 44*A*a^5 - 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^14

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Sympy [A]  time = 27.6844, size = 122, normalized size = 1.11 \begin{align*} \frac{B b^{5} x^{4}}{4} + x \left (A b^{5} + 5 B a b^{4}\right ) - \frac{44 A a^{5} + x^{12} \left (1540 A a b^{4} + 3080 B a^{2} b^{3}\right ) + x^{9} \left (1232 A a^{2} b^{3} + 1232 B a^{3} b^{2}\right ) + x^{6} \left (770 A a^{3} b^{2} + 385 B a^{4} b\right ) + x^{3} \left (280 A a^{4} b + 56 B a^{5}\right )}{616 x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**15,x)

[Out]

B*b**5*x**4/4 + x*(A*b**5 + 5*B*a*b**4) - (44*A*a**5 + x**12*(1540*A*a*b**4 + 3080*B*a**2*b**3) + x**9*(1232*A
*a**2*b**3 + 1232*B*a**3*b**2) + x**6*(770*A*a**3*b**2 + 385*B*a**4*b) + x**3*(280*A*a**4*b + 56*B*a**5))/(616
*x**14)

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Giac [A]  time = 1.22931, size = 166, normalized size = 1.51 \begin{align*} \frac{1}{4} \, B b^{5} x^{4} + 5 \, B a b^{4} x + A b^{5} x - \frac{3080 \, B a^{2} b^{3} x^{12} + 1540 \, A a b^{4} x^{12} + 1232 \, B a^{3} b^{2} x^{9} + 1232 \, A a^{2} b^{3} x^{9} + 385 \, B a^{4} b x^{6} + 770 \, A a^{3} b^{2} x^{6} + 56 \, B a^{5} x^{3} + 280 \, A a^{4} b x^{3} + 44 \, A a^{5}}{616 \, x^{14}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^15,x, algorithm="giac")

[Out]

1/4*B*b^5*x^4 + 5*B*a*b^4*x + A*b^5*x - 1/616*(3080*B*a^2*b^3*x^12 + 1540*A*a*b^4*x^12 + 1232*B*a^3*b^2*x^9 +
1232*A*a^2*b^3*x^9 + 385*B*a^4*b*x^6 + 770*A*a^3*b^2*x^6 + 56*B*a^5*x^3 + 280*A*a^4*b*x^3 + 44*A*a^5)/x^14