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

Optimal. Leaf size=108 \[ 2 a^2 b^2 x^5 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{x}+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b
+ 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^5*B*x^11)/11

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Rubi [A]  time = 0.0619118, antiderivative size = 108, 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} \[ 2 a^2 b^2 x^5 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{x}+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b
+ 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (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^2\right )^5 \left (A+B x^2\right )}{x^2} \, dx &=\int \left (a^4 (5 A b+a B)+\frac{a^5 A}{x^2}+5 a^3 b (2 A b+a B) x^2+10 a^2 b^2 (A b+a B) x^4+5 a b^3 (A b+2 a B) x^6+b^4 (A b+5 a B) x^8+b^5 B x^{10}\right ) \, dx\\ &=-\frac{a^5 A}{x}+a^4 (5 A b+a B) x+\frac{5}{3} a^3 b (2 A b+a B) x^3+2 a^2 b^2 (A b+a B) x^5+\frac{5}{7} a b^3 (A b+2 a B) x^7+\frac{1}{9} b^4 (A b+5 a B) x^9+\frac{1}{11} b^5 B x^{11}\\ \end{align*}

Mathematica [A]  time = 0.0294418, size = 108, normalized size = 1. \[ 2 a^2 b^2 x^5 (a B+A b)+\frac{5}{3} a^3 b x^3 (a B+2 A b)+a^4 x (a B+5 A b)-\frac{a^5 A}{x}+\frac{1}{9} b^4 x^9 (5 a B+A b)+\frac{5}{7} a b^3 x^7 (2 a B+A b)+\frac{1}{11} b^5 B x^{11} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-((a^5*A)/x) + a^4*(5*A*b + a*B)*x + (5*a^3*b*(2*A*b + a*B)*x^3)/3 + 2*a^2*b^2*(A*b + a*B)*x^5 + (5*a*b^3*(A*b
+ 2*a*B)*x^7)/7 + (b^4*(A*b + 5*a*B)*x^9)/9 + (b^5*B*x^11)/11

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Maple [A]  time = 0.003, size = 121, normalized size = 1.1 \begin{align*}{\frac{{b}^{5}B{x}^{11}}{11}}+{\frac{A{x}^{9}{b}^{5}}{9}}+{\frac{5\,B{x}^{9}a{b}^{4}}{9}}+{\frac{5\,A{x}^{7}a{b}^{4}}{7}}+{\frac{10\,B{x}^{7}{a}^{2}{b}^{3}}{7}}+2\,A{x}^{5}{a}^{2}{b}^{3}+2\,B{x}^{5}{a}^{3}{b}^{2}+{\frac{10\,A{x}^{3}{a}^{3}{b}^{2}}{3}}+{\frac{5\,B{x}^{3}{a}^{4}b}{3}}+5\,{a}^{4}bAx+{a}^{5}Bx-{\frac{A{a}^{5}}{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^2,x)

[Out]

1/11*b^5*B*x^11+1/9*A*x^9*b^5+5/9*B*x^9*a*b^4+5/7*A*x^7*a*b^4+10/7*B*x^7*a^2*b^3+2*A*x^5*a^2*b^3+2*B*x^5*a^3*b
^2+10/3*A*x^3*a^3*b^2+5/3*B*x^3*a^4*b+5*a^4*b*A*x+a^5*B*x-a^5*A/x

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Maxima [A]  time = 0.980533, size = 157, normalized size = 1.45 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} + \frac{1}{9} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{9} + \frac{5}{7} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{7} + 2 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} - \frac{A a^{5}}{x} + \frac{5}{3} \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} +{\left (B a^{5} + 5 \, A a^{4} b\right )} x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^5*x^11 + 1/9*(5*B*a*b^4 + A*b^5)*x^9 + 5/7*(2*B*a^2*b^3 + A*a*b^4)*x^7 + 2*(B*a^3*b^2 + A*a^2*b^3)*x^
5 - A*a^5/x + 5/3*(B*a^4*b + 2*A*a^3*b^2)*x^3 + (B*a^5 + 5*A*a^4*b)*x

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Fricas [A]  time = 1.39418, size = 271, normalized size = 2.51 \begin{align*} \frac{63 \, B b^{5} x^{12} + 77 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 495 \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{8} + 1386 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{6} - 693 \, A a^{5} + 1155 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 693 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{693 \, x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/693*(63*B*b^5*x^12 + 77*(5*B*a*b^4 + A*b^5)*x^10 + 495*(2*B*a^2*b^3 + A*a*b^4)*x^8 + 1386*(B*a^3*b^2 + A*a^2
*b^3)*x^6 - 693*A*a^5 + 1155*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 693*(B*a^5 + 5*A*a^4*b)*x^2)/x

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Sympy [A]  time = 0.368079, size = 126, normalized size = 1.17 \begin{align*} - \frac{A a^{5}}{x} + \frac{B b^{5} x^{11}}{11} + x^{9} \left (\frac{A b^{5}}{9} + \frac{5 B a b^{4}}{9}\right ) + x^{7} \left (\frac{5 A a b^{4}}{7} + \frac{10 B a^{2} b^{3}}{7}\right ) + x^{5} \left (2 A a^{2} b^{3} + 2 B a^{3} b^{2}\right ) + x^{3} \left (\frac{10 A a^{3} b^{2}}{3} + \frac{5 B a^{4} b}{3}\right ) + x \left (5 A a^{4} b + B a^{5}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-A*a**5/x + B*b**5*x**11/11 + x**9*(A*b**5/9 + 5*B*a*b**4/9) + x**7*(5*A*a*b**4/7 + 10*B*a**2*b**3/7) + x**5*(
2*A*a**2*b**3 + 2*B*a**3*b**2) + x**3*(10*A*a**3*b**2/3 + 5*B*a**4*b/3) + x*(5*A*a**4*b + B*a**5)

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Giac [A]  time = 1.15517, size = 162, normalized size = 1.5 \begin{align*} \frac{1}{11} \, B b^{5} x^{11} + \frac{5}{9} \, B a b^{4} x^{9} + \frac{1}{9} \, A b^{5} x^{9} + \frac{10}{7} \, B a^{2} b^{3} x^{7} + \frac{5}{7} \, A a b^{4} x^{7} + 2 \, B a^{3} b^{2} x^{5} + 2 \, A a^{2} b^{3} x^{5} + \frac{5}{3} \, B a^{4} b x^{3} + \frac{10}{3} \, A a^{3} b^{2} x^{3} + B a^{5} x + 5 \, A a^{4} b x - \frac{A a^{5}}{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/11*B*b^5*x^11 + 5/9*B*a*b^4*x^9 + 1/9*A*b^5*x^9 + 10/7*B*a^2*b^3*x^7 + 5/7*A*a*b^4*x^7 + 2*B*a^3*b^2*x^5 + 2
*A*a^2*b^3*x^5 + 5/3*B*a^4*b*x^3 + 10/3*A*a^3*b^2*x^3 + B*a^5*x + 5*A*a^4*b*x - A*a^5/x