3.40 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^8} \, dx\)

Optimal. Leaf size=111 \[ -\frac{10 a^2 b^2 (a B+A b)}{x}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{a^5 A}{7 x^7}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

[Out]

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

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Rubi [A]  time = 0.0617063, antiderivative size = 111, 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} \[ -\frac{10 a^2 b^2 (a B+A b)}{x}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{a^5 A}{7 x^7}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^8,x]

[Out]

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

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

Mathematica [A]  time = 0.0365976, size = 111, normalized size = 1. \[ -\frac{10 a^2 b^2 (a B+A b)}{x}-\frac{a^4 (a B+5 A b)}{5 x^5}-\frac{5 a^3 b (a B+2 A b)}{3 x^3}-\frac{a^5 A}{7 x^7}+\frac{1}{3} b^4 x^3 (5 a B+A b)+5 a b^3 x (2 a B+A b)+\frac{1}{5} b^5 B x^5 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^8,x]

[Out]

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

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Maple [A]  time = 0.006, size = 108, normalized size = 1. \begin{align*}{\frac{{b}^{5}B{x}^{5}}{5}}+{\frac{A{x}^{3}{b}^{5}}{3}}+{\frac{5\,B{x}^{3}a{b}^{4}}{3}}+5\,a{b}^{4}Ax+10\,{a}^{2}{b}^{3}Bx-{\frac{5\,{a}^{3}b \left ( 2\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{5\,{x}^{5}}}-{\frac{A{a}^{5}}{7\,{x}^{7}}}-10\,{\frac{{b}^{2}{a}^{2} \left ( Ab+Ba \right ) }{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^5*(B*x^2+A)/x^8,x)

[Out]

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

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Maxima [A]  time = 1.05277, size = 162, normalized size = 1.46 \begin{align*} \frac{1}{5} \, B b^{5} x^{5} + \frac{1}{3} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{3} + 5 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x - \frac{1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} + 15 \, A a^{5} + 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 21 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^8,x, algorithm="maxima")

[Out]

1/5*B*b^5*x^5 + 1/3*(5*B*a*b^4 + A*b^5)*x^3 + 5*(2*B*a^2*b^3 + A*a*b^4)*x - 1/105*(1050*(B*a^3*b^2 + A*a^2*b^3
)*x^6 + 15*A*a^5 + 175*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 21*(B*a^5 + 5*A*a^4*b)*x^2)/x^7

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Fricas [A]  time = 1.37307, size = 270, normalized size = 2.43 \begin{align*} \frac{21 \, B b^{5} x^{12} + 35 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 525 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} - 1050 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 15 \, A a^{5} - 175 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 21 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^8,x, algorithm="fricas")

[Out]

1/105*(21*B*b^5*x^12 + 35*(5*B*a*b^4 + A*b^5)*x^10 + 525*(2*B*a^2*b^3 + A*a*b^4)*x^8 - 1050*(B*a^3*b^2 + A*a^2
*b^3)*x^6 - 15*A*a^5 - 175*(B*a^4*b + 2*A*a^3*b^2)*x^4 - 21*(B*a^5 + 5*A*a^4*b)*x^2)/x^7

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Sympy [A]  time = 1.69062, size = 126, normalized size = 1.14 \begin{align*} \frac{B b^{5} x^{5}}{5} + x^{3} \left (\frac{A b^{5}}{3} + \frac{5 B a b^{4}}{3}\right ) + x \left (5 A a b^{4} + 10 B a^{2} b^{3}\right ) - \frac{15 A a^{5} + x^{6} \left (1050 A a^{2} b^{3} + 1050 B a^{3} b^{2}\right ) + x^{4} \left (350 A a^{3} b^{2} + 175 B a^{4} b\right ) + x^{2} \left (105 A a^{4} b + 21 B a^{5}\right )}{105 x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**8,x)

[Out]

B*b**5*x**5/5 + x**3*(A*b**5/3 + 5*B*a*b**4/3) + x*(5*A*a*b**4 + 10*B*a**2*b**3) - (15*A*a**5 + x**6*(1050*A*a
**2*b**3 + 1050*B*a**3*b**2) + x**4*(350*A*a**3*b**2 + 175*B*a**4*b) + x**2*(105*A*a**4*b + 21*B*a**5))/(105*x
**7)

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Giac [A]  time = 1.1622, size = 167, normalized size = 1.5 \begin{align*} \frac{1}{5} \, B b^{5} x^{5} + \frac{5}{3} \, B a b^{4} x^{3} + \frac{1}{3} \, A b^{5} x^{3} + 10 \, B a^{2} b^{3} x + 5 \, A a b^{4} x - \frac{1050 \, B a^{3} b^{2} x^{6} + 1050 \, A a^{2} b^{3} x^{6} + 175 \, B a^{4} b x^{4} + 350 \, A a^{3} b^{2} x^{4} + 21 \, B a^{5} x^{2} + 105 \, A a^{4} b x^{2} + 15 \, A a^{5}}{105 \, x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^8,x, algorithm="giac")

[Out]

1/5*B*b^5*x^5 + 5/3*B*a*b^4*x^3 + 1/3*A*b^5*x^3 + 10*B*a^2*b^3*x + 5*A*a*b^4*x - 1/105*(1050*B*a^3*b^2*x^6 + 1
050*A*a^2*b^3*x^6 + 175*B*a^4*b*x^4 + 350*A*a^3*b^2*x^4 + 21*B*a^5*x^2 + 105*A*a^4*b*x^2 + 15*A*a^5)/x^7