3.50 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{18}} \, dx\)

Optimal. Leaf size=110 \[ -\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{a^5 A}{17 x^{17}}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{2 x^2}+b^5 B x \]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (5*a^2*b^2*(A*b + a*B
))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x

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Rubi [A]  time = 0.0613283, 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} \[ -\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{a^5 A}{17 x^{17}}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{2 x^2}+b^5 B x \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (5*a^2*b^2*(A*b + a*B
))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x

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^{18}} \, dx &=\int \left (b^5 B+\frac{a^5 A}{x^{18}}+\frac{a^4 (5 A b+a B)}{x^{15}}+\frac{5 a^3 b (2 A b+a B)}{x^{12}}+\frac{10 a^2 b^2 (A b+a B)}{x^9}+\frac{5 a b^3 (A b+2 a B)}{x^6}+\frac{b^4 (A b+5 a B)}{x^3}\right ) \, dx\\ &=-\frac{a^5 A}{17 x^{17}}-\frac{a^4 (5 A b+a B)}{14 x^{14}}-\frac{5 a^3 b (2 A b+a B)}{11 x^{11}}-\frac{5 a^2 b^2 (A b+a B)}{4 x^8}-\frac{a b^3 (A b+2 a B)}{x^5}-\frac{b^4 (A b+5 a B)}{2 x^2}+b^5 B x\\ \end{align*}

Mathematica [A]  time = 0.0432395, size = 110, normalized size = 1. \[ -\frac{5 a^2 b^2 (a B+A b)}{4 x^8}-\frac{a^4 (a B+5 A b)}{14 x^{14}}-\frac{5 a^3 b (a B+2 A b)}{11 x^{11}}-\frac{a^5 A}{17 x^{17}}-\frac{a b^3 (2 a B+A b)}{x^5}-\frac{b^4 (5 a B+A b)}{2 x^2}+b^5 B x \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^18,x]

[Out]

-(a^5*A)/(17*x^17) - (a^4*(5*A*b + a*B))/(14*x^14) - (5*a^3*b*(2*A*b + a*B))/(11*x^11) - (5*a^2*b^2*(A*b + a*B
))/(4*x^8) - (a*b^3*(A*b + 2*a*B))/x^5 - (b^4*(A*b + 5*a*B))/(2*x^2) + b^5*B*x

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Maple [A]  time = 0.007, size = 101, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{17\,{x}^{17}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{4\,{x}^{8}}}-{\frac{a{b}^{3} \left ( Ab+2\,Ba \right ) }{{x}^{5}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{2\,{x}^{2}}}+{b}^{5}Bx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^18,x)

[Out]

-1/17*a^5*A/x^17-1/14*a^4*(5*A*b+B*a)/x^14-5/11*a^3*b*(2*A*b+B*a)/x^11-5/4*a^2*b^2*(A*b+B*a)/x^8-a*b^3*(A*b+2*
B*a)/x^5-1/2*b^4*(A*b+5*B*a)/x^2+b^5*B*x

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Maxima [A]  time = 1.31328, size = 161, normalized size = 1.46 \begin{align*} B b^{5} x - \frac{2618 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 5236 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 6545 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 2380 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 308 \, A a^{5} + 374 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="maxima")

[Out]

B*b^5*x - 1/5236*(2618*(5*B*a*b^4 + A*b^5)*x^15 + 5236*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 6545*(B*a^3*b^2 + A*a^2*
b^3)*x^9 + 2380*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 308*A*a^5 + 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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Fricas [A]  time = 1.40694, size = 285, normalized size = 2.59 \begin{align*} \frac{5236 \, B b^{5} x^{18} - 2618 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} - 5236 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} - 6545 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 2380 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 308 \, A a^{5} - 374 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{5236 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="fricas")

[Out]

1/5236*(5236*B*b^5*x^18 - 2618*(5*B*a*b^4 + A*b^5)*x^15 - 5236*(2*B*a^2*b^3 + A*a*b^4)*x^12 - 6545*(B*a^3*b^2
+ A*a^2*b^3)*x^9 - 2380*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 308*A*a^5 - 374*(B*a^5 + 5*A*a^4*b)*x^3)/x^17

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Sympy [A]  time = 125.843, size = 122, normalized size = 1.11 \begin{align*} B b^{5} x - \frac{308 A a^{5} + x^{15} \left (2618 A b^{5} + 13090 B a b^{4}\right ) + x^{12} \left (5236 A a b^{4} + 10472 B a^{2} b^{3}\right ) + x^{9} \left (6545 A a^{2} b^{3} + 6545 B a^{3} b^{2}\right ) + x^{6} \left (4760 A a^{3} b^{2} + 2380 B a^{4} b\right ) + x^{3} \left (1870 A a^{4} b + 374 B a^{5}\right )}{5236 x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**18,x)

[Out]

B*b**5*x - (308*A*a**5 + x**15*(2618*A*b**5 + 13090*B*a*b**4) + x**12*(5236*A*a*b**4 + 10472*B*a**2*b**3) + x*
*9*(6545*A*a**2*b**3 + 6545*B*a**3*b**2) + x**6*(4760*A*a**3*b**2 + 2380*B*a**4*b) + x**3*(1870*A*a**4*b + 374
*B*a**5))/(5236*x**17)

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Giac [A]  time = 1.19891, size = 169, normalized size = 1.54 \begin{align*} B b^{5} x - \frac{13090 \, B a b^{4} x^{15} + 2618 \, A b^{5} x^{15} + 10472 \, B a^{2} b^{3} x^{12} + 5236 \, A a b^{4} x^{12} + 6545 \, B a^{3} b^{2} x^{9} + 6545 \, A a^{2} b^{3} x^{9} + 2380 \, B a^{4} b x^{6} + 4760 \, A a^{3} b^{2} x^{6} + 374 \, B a^{5} x^{3} + 1870 \, A a^{4} b x^{3} + 308 \, A a^{5}}{5236 \, x^{17}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^18,x, algorithm="giac")

[Out]

B*b^5*x - 1/5236*(13090*B*a*b^4*x^15 + 2618*A*b^5*x^15 + 10472*B*a^2*b^3*x^12 + 5236*A*a*b^4*x^12 + 6545*B*a^3
*b^2*x^9 + 6545*A*a^2*b^3*x^9 + 2380*B*a^4*b*x^6 + 4760*A*a^3*b^2*x^6 + 374*B*a^5*x^3 + 1870*A*a^4*b*x^3 + 308
*A*a^5)/x^17