3.41 \(\int \frac{(a+b x^3)^5 (A+B x^3)}{x^9} \, dx\)

Optimal. Leaf size=113 \[ 10 a^2 b^2 x (a B+A b)-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{a^5 A}{8 x^8}+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

[Out]

-(a^5*A)/(8*x^8) - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + 10*a^2*b^2*(A*b + a*B)*x +
(5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^10)/10

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Rubi [A]  time = 0.0623747, 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} \[ 10 a^2 b^2 x (a B+A b)-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{a^5 A}{8 x^8}+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^3)^5*(A + B*x^3))/x^9,x]

[Out]

-(a^5*A)/(8*x^8) - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + 10*a^2*b^2*(A*b + a*B)*x +
(5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^10)/10

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^9} \, dx &=\int \left (10 a^2 b^2 (A b+a B)+\frac{a^5 A}{x^9}+\frac{a^4 (5 A b+a B)}{x^6}+\frac{5 a^3 b (2 A b+a B)}{x^3}+5 a b^3 (A b+2 a B) x^3+b^4 (A b+5 a B) x^6+b^5 B x^9\right ) \, dx\\ &=-\frac{a^5 A}{8 x^8}-\frac{a^4 (5 A b+a B)}{5 x^5}-\frac{5 a^3 b (2 A b+a B)}{2 x^2}+10 a^2 b^2 (A b+a B) x+\frac{5}{4} a b^3 (A b+2 a B) x^4+\frac{1}{7} b^4 (A b+5 a B) x^7+\frac{1}{10} b^5 B x^{10}\\ \end{align*}

Mathematica [A]  time = 0.0327947, size = 113, normalized size = 1. \[ 10 a^2 b^2 x (a B+A b)-\frac{5 a^3 b (a B+2 A b)}{2 x^2}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{a^5 A}{8 x^8}+\frac{1}{7} b^4 x^7 (5 a B+A b)+\frac{5}{4} a b^3 x^4 (2 a B+A b)+\frac{1}{10} b^5 B x^{10} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^3)^5*(A + B*x^3))/x^9,x]

[Out]

-(a^5*A)/(8*x^8) - (a^4*(5*A*b + a*B))/(5*x^5) - (5*a^3*b*(2*A*b + a*B))/(2*x^2) + 10*a^2*b^2*(A*b + a*B)*x +
(5*a*b^3*(A*b + 2*a*B)*x^4)/4 + (b^4*(A*b + 5*a*B)*x^7)/7 + (b^5*B*x^10)/10

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Maple [A]  time = 0.004, size = 114, normalized size = 1. \begin{align*}{\frac{{b}^{5}B{x}^{10}}{10}}+{\frac{A{x}^{7}{b}^{5}}{7}}+{\frac{5\,B{x}^{7}a{b}^{4}}{7}}+{\frac{5\,A{x}^{4}a{b}^{4}}{4}}+{\frac{5\,B{x}^{4}{a}^{2}{b}^{3}}{2}}+10\,{a}^{2}{b}^{3}Ax+10\,{a}^{3}{b}^{2}Bx-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{A{a}^{5}}{8\,{x}^{8}}}-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{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^9,x)

[Out]

1/10*b^5*B*x^10+1/7*A*x^7*b^5+5/7*B*x^7*a*b^4+5/4*A*x^4*a*b^4+5/2*B*x^4*a^2*b^3+10*a^2*b^3*A*x+10*a^3*b^2*B*x-
1/5*a^4*(5*A*b+B*a)/x^5-1/8*a^5*A/x^8-5/2*a^3*b*(2*A*b+B*a)/x^2

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Maxima [A]  time = 1.22319, size = 162, normalized size = 1.43 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{1}{7} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + \frac{5}{4} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x - \frac{100 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} + 5 \, A a^{5} + 8 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^9,x, algorithm="maxima")

[Out]

1/10*B*b^5*x^10 + 1/7*(5*B*a*b^4 + A*b^5)*x^7 + 5/4*(2*B*a^2*b^3 + A*a*b^4)*x^4 + 10*(B*a^3*b^2 + A*a^2*b^3)*x
- 1/40*(100*(B*a^4*b + 2*A*a^3*b^2)*x^6 + 5*A*a^5 + 8*(B*a^5 + 5*A*a^4*b)*x^3)/x^8

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Fricas [A]  time = 1.42995, size = 271, normalized size = 2.4 \begin{align*} \frac{28 \, B b^{5} x^{18} + 40 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{15} + 350 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{12} + 2800 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{9} - 700 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{6} - 35 \, A a^{5} - 56 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}}{280 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^9,x, algorithm="fricas")

[Out]

1/280*(28*B*b^5*x^18 + 40*(5*B*a*b^4 + A*b^5)*x^15 + 350*(2*B*a^2*b^3 + A*a*b^4)*x^12 + 2800*(B*a^3*b^2 + A*a^
2*b^3)*x^9 - 700*(B*a^4*b + 2*A*a^3*b^2)*x^6 - 35*A*a^5 - 56*(B*a^5 + 5*A*a^4*b)*x^3)/x^8

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Sympy [A]  time = 1.28377, size = 129, normalized size = 1.14 \begin{align*} \frac{B b^{5} x^{10}}{10} + x^{7} \left (\frac{A b^{5}}{7} + \frac{5 B a b^{4}}{7}\right ) + x^{4} \left (\frac{5 A a b^{4}}{4} + \frac{5 B a^{2} b^{3}}{2}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) - \frac{5 A a^{5} + x^{6} \left (200 A a^{3} b^{2} + 100 B a^{4} b\right ) + x^{3} \left (40 A a^{4} b + 8 B a^{5}\right )}{40 x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**3+a)**5*(B*x**3+A)/x**9,x)

[Out]

B*b**5*x**10/10 + x**7*(A*b**5/7 + 5*B*a*b**4/7) + x**4*(5*A*a*b**4/4 + 5*B*a**2*b**3/2) + x*(10*A*a**2*b**3 +
10*B*a**3*b**2) - (5*A*a**5 + x**6*(200*A*a**3*b**2 + 100*B*a**4*b) + x**3*(40*A*a**4*b + 8*B*a**5))/(40*x**8
)

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Giac [A]  time = 1.16013, size = 167, normalized size = 1.48 \begin{align*} \frac{1}{10} \, B b^{5} x^{10} + \frac{5}{7} \, B a b^{4} x^{7} + \frac{1}{7} \, A b^{5} x^{7} + \frac{5}{2} \, B a^{2} b^{3} x^{4} + \frac{5}{4} \, A a b^{4} x^{4} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x - \frac{100 \, B a^{4} b x^{6} + 200 \, A a^{3} b^{2} x^{6} + 8 \, B a^{5} x^{3} + 40 \, A a^{4} b x^{3} + 5 \, A a^{5}}{40 \, x^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^3+a)^5*(B*x^3+A)/x^9,x, algorithm="giac")

[Out]

1/10*B*b^5*x^10 + 5/7*B*a*b^4*x^7 + 1/7*A*b^5*x^7 + 5/2*B*a^2*b^3*x^4 + 5/4*A*a*b^4*x^4 + 10*B*a^3*b^2*x + 10*
A*a^2*b^3*x - 1/40*(100*B*a^4*b*x^6 + 200*A*a^3*b^2*x^6 + 8*B*a^5*x^3 + 40*A*a^4*b*x^3 + 5*A*a^5)/x^8