3.38 \(\int \frac{(a+b x^2)^5 (A+B x^2)}{x^6} \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.0623546, 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} \[ 10 a^2 b^2 x (a B+A b)-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{5 x^5}+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^5*(A + B*x^2))/x^6,x]

[Out]

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

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

Mathematica [A]  time = 0.0346202, size = 111, normalized size = 1. \[ 10 a^2 b^2 x (a B+A b)-\frac{a^4 (a B+5 A b)}{3 x^3}-\frac{5 a^3 b (a B+2 A b)}{x}-\frac{a^5 A}{5 x^5}+\frac{1}{5} b^4 x^5 (5 a B+A b)+\frac{5}{3} a b^3 x^3 (2 a B+A b)+\frac{1}{7} b^5 B x^7 \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^5*(A + B*x^2))/x^6,x]

[Out]

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

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Maple [A]  time = 0.006, size = 113, normalized size = 1. \begin{align*}{\frac{{b}^{5}B{x}^{7}}{7}}+{\frac{A{x}^{5}{b}^{5}}{5}}+B{x}^{5}a{b}^{4}+{\frac{5\,A{x}^{3}a{b}^{4}}{3}}+{\frac{10\,B{x}^{3}{a}^{2}{b}^{3}}{3}}+10\,{a}^{2}{b}^{3}Ax+10\,{a}^{3}{b}^{2}Bx-{\frac{{a}^{4} \left ( 5\,Ab+Ba \right ) }{3\,{x}^{3}}}-{\frac{A{a}^{5}}{5\,{x}^{5}}}-5\,{\frac{{a}^{3}b \left ( 2\,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^6,x)

[Out]

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

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Maxima [A]  time = 0.99986, size = 162, normalized size = 1.46 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + \frac{1}{5} \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + \frac{5}{3} \,{\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{3} + 10 \,{\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x - \frac{3 \, A a^{5} + 75 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 5 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="maxima")

[Out]

1/7*B*b^5*x^7 + 1/5*(5*B*a*b^4 + A*b^5)*x^5 + 5/3*(2*B*a^2*b^3 + A*a*b^4)*x^3 + 10*(B*a^3*b^2 + A*a^2*b^3)*x -
1/15*(3*A*a^5 + 75*(B*a^4*b + 2*A*a^3*b^2)*x^4 + 5*(B*a^5 + 5*A*a^4*b)*x^2)/x^5

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Fricas [A]  time = 1.39156, size = 270, normalized size = 2.43 \begin{align*} \frac{15 \, B b^{5} x^{12} + 21 \,{\left (5 \, B a b^{4} + A b^{5}\right )} x^{10} + 175 \,{\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} - 21 \, A a^{5} - 525 \,{\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} - 35 \,{\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}}{105 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.882236, size = 126, normalized size = 1.14 \begin{align*} \frac{B b^{5} x^{7}}{7} + x^{5} \left (\frac{A b^{5}}{5} + B a b^{4}\right ) + x^{3} \left (\frac{5 A a b^{4}}{3} + \frac{10 B a^{2} b^{3}}{3}\right ) + x \left (10 A a^{2} b^{3} + 10 B a^{3} b^{2}\right ) - \frac{3 A a^{5} + x^{4} \left (150 A a^{3} b^{2} + 75 B a^{4} b\right ) + x^{2} \left (25 A a^{4} b + 5 B a^{5}\right )}{15 x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**5*(B*x**2+A)/x**6,x)

[Out]

B*b**5*x**7/7 + x**5*(A*b**5/5 + B*a*b**4) + x**3*(5*A*a*b**4/3 + 10*B*a**2*b**3/3) + x*(10*A*a**2*b**3 + 10*B
*a**3*b**2) - (3*A*a**5 + x**4*(150*A*a**3*b**2 + 75*B*a**4*b) + x**2*(25*A*a**4*b + 5*B*a**5))/(15*x**5)

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Giac [A]  time = 1.13736, size = 166, normalized size = 1.5 \begin{align*} \frac{1}{7} \, B b^{5} x^{7} + B a b^{4} x^{5} + \frac{1}{5} \, A b^{5} x^{5} + \frac{10}{3} \, B a^{2} b^{3} x^{3} + \frac{5}{3} \, A a b^{4} x^{3} + 10 \, B a^{3} b^{2} x + 10 \, A a^{2} b^{3} x - \frac{75 \, B a^{4} b x^{4} + 150 \, A a^{3} b^{2} x^{4} + 5 \, B a^{5} x^{2} + 25 \, A a^{4} b x^{2} + 3 \, A a^{5}}{15 \, x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^5*(B*x^2+A)/x^6,x, algorithm="giac")

[Out]

1/7*B*b^5*x^7 + B*a*b^4*x^5 + 1/5*A*b^5*x^5 + 10/3*B*a^2*b^3*x^3 + 5/3*A*a*b^4*x^3 + 10*B*a^3*b^2*x + 10*A*a^2
*b^3*x - 1/15*(75*B*a^4*b*x^4 + 150*A*a^3*b^2*x^4 + 5*B*a^5*x^2 + 25*A*a^4*b*x^2 + 3*A*a^5)/x^5