3.53 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{21}} \, dx\)

Optimal. Leaf size=117 \[ -\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{a^5 A}{20 x^{20}}-\frac{5 a b^3 (2 a B+A b)}{8 x^8}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{b^5 B}{2 x^2} \]

[Out]

-(a^5*A)/(20*x^20) - (a^4*(5*A*b + a*B))/(17*x^17) - (5*a^3*b*(2*A*b + a*B))/(14*x^14) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(8*x^8) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(2*x^2)

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Rubi [A]  time = 0.0588298, antiderivative size = 117, 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} \[ -\frac{10 a^2 b^2 (a B+A b)}{11 x^{11}}-\frac{a^4 (a B+5 A b)}{17 x^{17}}-\frac{5 a^3 b (a B+2 A b)}{14 x^{14}}-\frac{a^5 A}{20 x^{20}}-\frac{5 a b^3 (2 a B+A b)}{8 x^8}-\frac{b^4 (5 a B+A b)}{5 x^5}-\frac{b^5 B}{2 x^2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

-(a^5*A)/(20*x^20) - (a^4*(5*A*b + a*B))/(17*x^17) - (5*a^3*b*(2*A*b + a*B))/(14*x^14) - (10*a^2*b^2*(A*b + a*
B))/(11*x^11) - (5*a*b^3*(A*b + 2*a*B))/(8*x^8) - (b^4*(A*b + 5*a*B))/(5*x^5) - (b^5*B)/(2*x^2)

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^{21}} \, dx &=\int \left (\frac{a^5 A}{x^{21}}+\frac{a^4 (5 A b+a B)}{x^{18}}+\frac{5 a^3 b (2 A b+a B)}{x^{15}}+\frac{10 a^2 b^2 (A b+a B)}{x^{12}}+\frac{5 a b^3 (A b+2 a B)}{x^9}+\frac{b^4 (A b+5 a B)}{x^6}+\frac{b^5 B}{x^3}\right ) \, dx\\ &=-\frac{a^5 A}{20 x^{20}}-\frac{a^4 (5 A b+a B)}{17 x^{17}}-\frac{5 a^3 b (2 A b+a B)}{14 x^{14}}-\frac{10 a^2 b^2 (A b+a B)}{11 x^{11}}-\frac{5 a b^3 (A b+2 a B)}{8 x^8}-\frac{b^4 (A b+5 a B)}{5 x^5}-\frac{b^5 B}{2 x^2}\\ \end{align*}

Mathematica [A]  time = 0.0305845, size = 121, normalized size = 1.03 \[ -\frac{5950 a^2 b^3 x^9 \left (8 A+11 B x^3\right )+3400 a^3 b^2 x^6 \left (11 A+14 B x^3\right )+1100 a^4 b x^3 \left (14 A+17 B x^3\right )+154 a^5 \left (17 A+20 B x^3\right )+6545 a b^4 x^{12} \left (5 A+8 B x^3\right )+5236 b^5 x^{15} \left (2 A+5 B x^3\right )}{52360 x^{20}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^21,x]

[Out]

-(5236*b^5*x^15*(2*A + 5*B*x^3) + 6545*a*b^4*x^12*(5*A + 8*B*x^3) + 5950*a^2*b^3*x^9*(8*A + 11*B*x^3) + 3400*a
^3*b^2*x^6*(11*A + 14*B*x^3) + 1100*a^4*b*x^3*(14*A + 17*B*x^3) + 154*a^5*(17*A + 20*B*x^3))/(52360*x^20)

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Maple [A]  time = 0.007, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{20\,{x}^{20}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{17\,{x}^{17}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{14\,{x}^{14}}}-{\frac{10\,{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{11\,{x}^{11}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{8\,{x}^{8}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{5\,{x}^{5}}}-{\frac{B{b}^{5}}{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^21,x)

[Out]

-1/20*a^5*A/x^20-1/17*a^4*(5*A*b+B*a)/x^17-5/14*a^3*b*(2*A*b+B*a)/x^14-10/11*a^2*b^2*(A*b+B*a)/x^11-5/8*a*b^3*
(A*b+2*B*a)/x^8-1/5*b^4*(A*b+5*B*a)/x^5-1/2*b^5*B/x^2

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Maxima [A]  time = 1.1166, size = 163, normalized size = 1.39 \begin{align*} -\frac{26180 \, B b^{5} x^{18} + 10472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="maxima")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 47600*(B*a^
3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

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Fricas [A]  time = 1.40634, size = 297, normalized size = 2.54 \begin{align*} -\frac{26180 \, B b^{5} x^{18} + 10472 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 32725 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 47600 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 18700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 2618 \, A a^{5} + 3080 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{52360 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="fricas")

[Out]

-1/52360*(26180*B*b^5*x^18 + 10472*(5*B*a*b^4 + A*b^5)*x^15 + 32725*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 47600*(B*a^
3*b^2 + A*a^2*b^3)*x^9 + 18700*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 2618*A*a^5 + 3080*(B*a^5 + 5*A*a^4*b)*x^3)/x^20

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**21,x)

[Out]

Timed out

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Giac [A]  time = 1.17573, size = 171, normalized size = 1.46 \begin{align*} -\frac{26180 \, B b^{5} x^{18} + 52360 \, B a b^{4} x^{15} + 10472 \, A b^{5} x^{15} + 65450 \, B a^{2} b^{3} x^{12} + 32725 \, A a b^{4} x^{12} + 47600 \, B a^{3} b^{2} x^{9} + 47600 \, A a^{2} b^{3} x^{9} + 18700 \, B a^{4} b x^{6} + 37400 \, A a^{3} b^{2} x^{6} + 3080 \, B a^{5} x^{3} + 15400 \, A a^{4} b x^{3} + 2618 \, A a^{5}}{52360 \, x^{20}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^21,x, algorithm="giac")

[Out]

-1/52360*(26180*B*b^5*x^18 + 52360*B*a*b^4*x^15 + 10472*A*b^5*x^15 + 65450*B*a^2*b^3*x^12 + 32725*A*a*b^4*x^12
+ 47600*B*a^3*b^2*x^9 + 47600*A*a^2*b^3*x^9 + 18700*B*a^4*b*x^6 + 37400*A*a^3*b^2*x^6 + 3080*B*a^5*x^3 + 1540
0*A*a^4*b*x^3 + 2618*A*a^5)/x^20