3.52 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^{20}} \, dx\)

Optimal. Leaf size=113 \[ -\frac{a^2 b^2 (a B+A b)}{x^{10}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^5 A}{19 x^{19}}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{b^5 B}{x} \]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (a^2*b^2*(A*b + a*B))
/x^10 - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b^4*(A*b + 5*a*B))/(4*x^4) - (b^5*B)/x

________________________________________________________________________________________

Rubi [A]  time = 0.0643632, 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} \[ -\frac{a^2 b^2 (a B+A b)}{x^{10}}-\frac{a^4 (a B+5 A b)}{16 x^{16}}-\frac{5 a^3 b (a B+2 A b)}{13 x^{13}}-\frac{a^5 A}{19 x^{19}}-\frac{5 a b^3 (2 a B+A b)}{7 x^7}-\frac{b^4 (5 a B+A b)}{4 x^4}-\frac{b^5 B}{x} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^20,x]

[Out]

-(a^5*A)/(19*x^19) - (a^4*(5*A*b + a*B))/(16*x^16) - (5*a^3*b*(2*A*b + a*B))/(13*x^13) - (a^2*b^2*(A*b + a*B))
/x^10 - (5*a*b^3*(A*b + 2*a*B))/(7*x^7) - (b^4*(A*b + 5*a*B))/(4*x^4) - (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^{20}} \, dx &=\int \left (\frac{a^5 A}{x^{20}}+\frac{a^4 (5 A b+a B)}{x^{17}}+\frac{5 a^3 b (2 A b+a B)}{x^{14}}+\frac{10 a^2 b^2 (A b+a B)}{x^{11}}+\frac{5 a b^3 (A b+2 a B)}{x^8}+\frac{b^4 (A b+5 a B)}{x^5}+\frac{b^5 B}{x^2}\right ) \, dx\\ &=-\frac{a^5 A}{19 x^{19}}-\frac{a^4 (5 A b+a B)}{16 x^{16}}-\frac{5 a^3 b (2 A b+a B)}{13 x^{13}}-\frac{a^2 b^2 (A b+a B)}{x^{10}}-\frac{5 a b^3 (A b+2 a B)}{7 x^7}-\frac{b^4 (A b+5 a B)}{4 x^4}-\frac{b^5 B}{x}\\ \end{align*}

Mathematica [A]  time = 0.0295196, size = 119, normalized size = 1.05 \[ -\frac{3952 a^2 b^3 x^9 \left (7 A+10 B x^3\right )+2128 a^3 b^2 x^6 \left (10 A+13 B x^3\right )+665 a^4 b x^3 \left (13 A+16 B x^3\right )+91 a^5 \left (16 A+19 B x^3\right )+4940 a b^4 x^{12} \left (4 A+7 B x^3\right )+6916 b^5 x^{15} \left (A+4 B x^3\right )}{27664 x^{19}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^20,x]

[Out]

-(6916*b^5*x^15*(A + 4*B*x^3) + 4940*a*b^4*x^12*(4*A + 7*B*x^3) + 3952*a^2*b^3*x^9*(7*A + 10*B*x^3) + 2128*a^3
*b^2*x^6*(10*A + 13*B*x^3) + 665*a^4*b*x^3*(13*A + 16*B*x^3) + 91*a^5*(16*A + 19*B*x^3))/(27664*x^19)

________________________________________________________________________________________

Maple [A]  time = 0.008, size = 104, normalized size = 0.9 \begin{align*} -{\frac{A{a}^{5}}{19\,{x}^{19}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{16\,{x}^{16}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{13\,{x}^{13}}}-{\frac{{a}^{2}{b}^{2} \left ( Ab+Ba \right ) }{{x}^{10}}}-{\frac{5\,a{b}^{3} \left ( Ab+2\,Ba \right ) }{7\,{x}^{7}}}-{\frac{{b}^{4} \left ( Ab+5\,Ba \right ) }{4\,{x}^{4}}}-{\frac{B{b}^{5}}{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^20,x)

[Out]

-1/19*a^5*A/x^19-1/16*a^4*(5*A*b+B*a)/x^16-5/13*a^3*b*(2*A*b+B*a)/x^13-a^2*b^2*(A*b+B*a)/x^10-5/7*a*b^3*(A*b+2
*B*a)/x^7-1/4*b^4*(A*b+5*B*a)/x^4-b^5*B/x

________________________________________________________________________________________

Maxima [A]  time = 1.12837, size = 163, normalized size = 1.44 \begin{align*} -\frac{27664 \, B b^{5} x^{18} + 6916 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 19760 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 27664 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 10640 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 1456 \, A a^{5} + 1729 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{27664 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^20,x, algorithm="maxima")

[Out]

-1/27664*(27664*B*b^5*x^18 + 6916*(5*B*a*b^4 + A*b^5)*x^15 + 19760*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 27664*(B*a^3
*b^2 + A*a^2*b^3)*x^9 + 10640*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 1456*A*a^5 + 1729*(B*a^5 + 5*A*a^4*b)*x^3)/x^19

________________________________________________________________________________________

Fricas [A]  time = 1.41254, size = 296, normalized size = 2.62 \begin{align*} -\frac{27664 \, B b^{5} x^{18} + 6916 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 19760 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 27664 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} + 10640 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 1456 \, A a^{5} + 1729 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{27664 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^20,x, algorithm="fricas")

[Out]

-1/27664*(27664*B*b^5*x^18 + 6916*(5*B*a*b^4 + A*b^5)*x^15 + 19760*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 27664*(B*a^3
*b^2 + A*a^2*b^3)*x^9 + 10640*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 1456*A*a^5 + 1729*(B*a^5 + 5*A*a^4*b)*x^3)/x^19

________________________________________________________________________________________

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**20,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [A]  time = 1.2132, size = 171, normalized size = 1.51 \begin{align*} -\frac{27664 \, B b^{5} x^{18} + 34580 \, B a b^{4} x^{15} + 6916 \, A b^{5} x^{15} + 39520 \, B a^{2} b^{3} x^{12} + 19760 \, A a b^{4} x^{12} + 27664 \, B a^{3} b^{2} x^{9} + 27664 \, A a^{2} b^{3} x^{9} + 10640 \, B a^{4} b x^{6} + 21280 \, A a^{3} b^{2} x^{6} + 1729 \, B a^{5} x^{3} + 8645 \, A a^{4} b x^{3} + 1456 \, A a^{5}}{27664 \, x^{19}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^20,x, algorithm="giac")

[Out]

-1/27664*(27664*B*b^5*x^18 + 34580*B*a*b^4*x^15 + 6916*A*b^5*x^15 + 39520*B*a^2*b^3*x^12 + 19760*A*a*b^4*x^12
+ 27664*B*a^3*b^2*x^9 + 27664*A*a^2*b^3*x^9 + 10640*B*a^4*b*x^6 + 21280*A*a^3*b^2*x^6 + 1729*B*a^5*x^3 + 8645*
A*a^4*b*x^3 + 1456*A*a^5)/x^19