3.365 \(\int \frac{(a+b x^2)^3 (A+B x^2)}{x^{5/2}} \, dx\)

Optimal. Leaf size=83 \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{13} b^3 B x^{13/2} \]

[Out]

(-2*a^3*A)/(3*x^(3/2)) + 2*a^2*(3*A*b + a*B)*Sqrt[x] + (6*a*b*(A*b + a*B)*x^(5/2))/5 + (2*b^2*(A*b + 3*a*B)*x^
(9/2))/9 + (2*b^3*B*x^(13/2))/13

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Rubi [A]  time = 0.0432517, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ 2 a^2 \sqrt{x} (a B+3 A b)-\frac{2 a^3 A}{3 x^{3/2}}+\frac{2}{9} b^2 x^{9/2} (3 a B+A b)+\frac{6}{5} a b x^{5/2} (a B+A b)+\frac{2}{13} b^3 B x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]

[Out]

(-2*a^3*A)/(3*x^(3/2)) + 2*a^2*(3*A*b + a*B)*Sqrt[x] + (6*a*b*(A*b + a*B)*x^(5/2))/5 + (2*b^2*(A*b + 3*a*B)*x^
(9/2))/9 + (2*b^3*B*x^(13/2))/13

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 )^3 \left (A+B x^2\right )}{x^{5/2}} \, dx &=\int \left (\frac{a^3 A}{x^{5/2}}+\frac{a^2 (3 A b+a B)}{\sqrt{x}}+3 a b (A b+a B) x^{3/2}+b^2 (A b+3 a B) x^{7/2}+b^3 B x^{11/2}\right ) \, dx\\ &=-\frac{2 a^3 A}{3 x^{3/2}}+2 a^2 (3 A b+a B) \sqrt{x}+\frac{6}{5} a b (A b+a B) x^{5/2}+\frac{2}{9} b^2 (A b+3 a B) x^{9/2}+\frac{2}{13} b^3 B x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0261891, size = 78, normalized size = 0.94 \[ \frac{702 a^2 b x^2 \left (5 A+B x^2\right )-390 a^3 \left (A-3 B x^2\right )+78 a b^2 x^4 \left (9 A+5 B x^2\right )+10 b^3 x^6 \left (13 A+9 B x^2\right )}{585 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^3*(A + B*x^2))/x^(5/2),x]

[Out]

(-390*a^3*(A - 3*B*x^2) + 702*a^2*b*x^2*(5*A + B*x^2) + 78*a*b^2*x^4*(9*A + 5*B*x^2) + 10*b^3*x^6*(13*A + 9*B*
x^2))/(585*x^(3/2))

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Maple [A]  time = 0.006, size = 80, normalized size = 1. \begin{align*} -{\frac{-90\,{b}^{3}B{x}^{8}-130\,{x}^{6}{b}^{3}A-390\,{x}^{6}a{b}^{2}B-702\,{x}^{4}a{b}^{2}A-702\,{x}^{4}{a}^{2}bB-3510\,A{a}^{2}b{x}^{2}-1170\,B{a}^{3}{x}^{2}+390\,{a}^{3}A}{585}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x)

[Out]

-2/585*(-45*B*b^3*x^8-65*A*b^3*x^6-195*B*a*b^2*x^6-351*A*a*b^2*x^4-351*B*a^2*b*x^4-1755*A*a^2*b*x^2-585*B*a^3*
x^2+195*A*a^3)/x^(3/2)

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Maxima [A]  time = 1.032, size = 99, normalized size = 1.19 \begin{align*} \frac{2}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{\frac{9}{2}} + \frac{6}{5} \,{\left (B a^{2} b + A a b^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, A a^{3}}{3 \, x^{\frac{3}{2}}} + 2 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="maxima")

[Out]

2/13*B*b^3*x^(13/2) + 2/9*(3*B*a*b^2 + A*b^3)*x^(9/2) + 6/5*(B*a^2*b + A*a*b^2)*x^(5/2) - 2/3*A*a^3/x^(3/2) +
2*(B*a^3 + 3*A*a^2*b)*sqrt(x)

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Fricas [A]  time = 0.864668, size = 177, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (45 \, B b^{3} x^{8} + 65 \,{\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 351 \,{\left (B a^{2} b + A a b^{2}\right )} x^{4} - 195 \, A a^{3} + 585 \,{\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )}}{585 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*B*b^3*x^8 + 65*(3*B*a*b^2 + A*b^3)*x^6 + 351*(B*a^2*b + A*a*b^2)*x^4 - 195*A*a^3 + 585*(B*a^3 + 3*A*
a^2*b)*x^2)/x^(3/2)

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Sympy [A]  time = 6.68447, size = 110, normalized size = 1.33 \begin{align*} - \frac{2 A a^{3}}{3 x^{\frac{3}{2}}} + 6 A a^{2} b \sqrt{x} + \frac{6 A a b^{2} x^{\frac{5}{2}}}{5} + \frac{2 A b^{3} x^{\frac{9}{2}}}{9} + 2 B a^{3} \sqrt{x} + \frac{6 B a^{2} b x^{\frac{5}{2}}}{5} + \frac{2 B a b^{2} x^{\frac{9}{2}}}{3} + \frac{2 B b^{3} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**3*(B*x**2+A)/x**(5/2),x)

[Out]

-2*A*a**3/(3*x**(3/2)) + 6*A*a**2*b*sqrt(x) + 6*A*a*b**2*x**(5/2)/5 + 2*A*b**3*x**(9/2)/9 + 2*B*a**3*sqrt(x) +
6*B*a**2*b*x**(5/2)/5 + 2*B*a*b**2*x**(9/2)/3 + 2*B*b**3*x**(13/2)/13

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Giac [A]  time = 1.13398, size = 104, normalized size = 1.25 \begin{align*} \frac{2}{13} \, B b^{3} x^{\frac{13}{2}} + \frac{2}{3} \, B a b^{2} x^{\frac{9}{2}} + \frac{2}{9} \, A b^{3} x^{\frac{9}{2}} + \frac{6}{5} \, B a^{2} b x^{\frac{5}{2}} + \frac{6}{5} \, A a b^{2} x^{\frac{5}{2}} + 2 \, B a^{3} \sqrt{x} + 6 \, A a^{2} b \sqrt{x} - \frac{2 \, A a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x^2+a)^3*(B*x^2+A)/x^(5/2),x, algorithm="giac")

[Out]

2/13*B*b^3*x^(13/2) + 2/3*B*a*b^2*x^(9/2) + 2/9*A*b^3*x^(9/2) + 6/5*B*a^2*b*x^(5/2) + 6/5*A*a*b^2*x^(5/2) + 2*
B*a^3*sqrt(x) + 6*A*a^2*b*sqrt(x) - 2/3*A*a^3/x^(3/2)