3.40 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^8} \, dx\)

Optimal. Leaf size=110 \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{7 x^7}+\frac{1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

[Out]

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

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Rubi [A]  time = 0.0671996, 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} \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{7 x^7}+\frac{1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0333246, size = 110, normalized size = 1. \[ 5 a^2 b^2 x^2 (a B+A b)-\frac{a^4 (a B+5 A b)}{4 x^4}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{7 x^7}+\frac{1}{8} b^4 x^8 (5 a B+A b)+a b^3 x^5 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^8,x]

[Out]

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

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Maple [A]  time = 0.007, size = 117, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{11}}{11}}+{\frac{A{x}^{8}{b}^{5}}{8}}+{\frac{5\,B{x}^{8}a{b}^{4}}{8}}+A{x}^{5}a{b}^{4}+2\,B{x}^{5}{a}^{2}{b}^{3}+5\,A{x}^{2}{a}^{2}{b}^{3}+5\,B{x}^{2}{a}^{3}{b}^{2}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{4\,{x}^{4}}}-5\,{\frac{{a}^{3}b \left ( 2\,Ab+Ba \right ) }{x}}-{\frac{A{a}^{5}}{7\,{x}^{7}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^3+a)^5*(B*x^3+A)/x^8,x)

[Out]

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

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Maxima [A]  time = 1.05217, size = 163, normalized size = 1.48 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} + \frac{1}{8} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{8} +{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 5 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{2} - \frac{140 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 4 \, A a^{5} + 7 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="maxima")

[Out]

1/11*B*b^5*x^11 + 1/8*(5*B*a*b^4 + A*b^5)*x^8 + (2*B*a^2*b^3 + A*a*b^4)*x^5 + 5*(B*a^3*b^2 + A*a^2*b^3)*x^2 -
1/28*(140*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 4*A*a^5 + 7*(B*a^5 + 5*A*a^4*b)*x^3)/x^7

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Fricas [A]  time = 1.43642, size = 274, normalized size = 2.49 \begin{align*} \frac{56 \, B b^{5} x^{18} + 77 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 616 \,{\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} - 3080 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 88 \, A a^{5} - 154 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{616 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="fricas")

[Out]

1/616*(56*B*b^5*x^18 + 77*(5*B*a*b^4 + A*b^5)*x^15 + 616*(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 - 3080*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 88*A*a^5 - 154*(B*a^5 + 5*A*a^4*b)*x^3)/x^7

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Sympy [A]  time = 1.22447, size = 126, normalized size = 1.15 \begin{align*} \frac{B b^{5} x^{11}}{11} + x^{8} \left (\frac{A b^{5}}{8} + \frac{5 B a b^{4}}{8}\right ) + x^{5} \left (A a b^{4} + 2 B a^{2} b^{3}\right ) + x^{2} \left (5 A a^{2} b^{3} + 5 B a^{3} b^{2}\right ) - \frac{4 A a^{5} + x^{6} \left (280 A a^{3} b^{2} + 140 B a^{4} b\right ) + x^{3} \left (35 A a^{4} b + 7 B a^{5}\right )}{28 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**8,x)

[Out]

B*b**5*x**11/11 + x**8*(A*b**5/8 + 5*B*a*b**4/8) + x**5*(A*a*b**4 + 2*B*a**2*b**3) + x**2*(5*A*a**2*b**3 + 5*B
*a**3*b**2) - (4*A*a**5 + x**6*(280*A*a**3*b**2 + 140*B*a**4*b) + x**3*(35*A*a**4*b + 7*B*a**5))/(28*x**7)

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Giac [A]  time = 1.15847, size = 171, normalized size = 1.55 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} + \frac{5}{8} \, B a b^{4} x^{8} + \frac{1}{8} \, A b^{5} x^{8} + 2 \, B a^{2} b^{3} x^{5} + A a b^{4} x^{5} + 5 \, B a^{3} b^{2} x^{2} + 5 \, A a^{2} b^{3} x^{2} - \frac{140 \, B a^{4} b x^{6} + 280 \, A a^{3} b^{2} x^{6} + 7 \, B a^{5} x^{3} + 35 \, A a^{4} b x^{3} + 4 \, A a^{5}}{28 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^8,x, algorithm="giac")

[Out]

1/11*B*b^5*x^11 + 5/8*B*a*b^4*x^8 + 1/8*A*b^5*x^8 + 2*B*a^2*b^3*x^5 + A*a*b^4*x^5 + 5*B*a^3*b^2*x^2 + 5*A*a^2*
b^3*x^2 - 1/28*(140*B*a^4*b*x^6 + 280*A*a^3*b^2*x^6 + 7*B*a^5*x^3 + 35*A*a^4*b*x^3 + 4*A*a^5)/x^7