3.405 \(\int \frac{(a+b x^2)^2 (c+d x^2)^2}{x^{5/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]

[Out]

(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2))/5 + (4*b*d*
(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*x^(13/2))/13

________________________________________________________________________________________

Rubi [A]  time = 0.0466474, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.042, Rules used = {448} \[ \frac{2}{5} x^{5/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{3 x^{3/2}}+\frac{4}{9} b d x^{9/2} (a d+b c)+4 a c \sqrt{x} (a d+b c)+\frac{2}{13} b^2 d^2 x^{13/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]

[Out]

(-2*a^2*c^2)/(3*x^(3/2)) + 4*a*c*(b*c + a*d)*Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2))/5 + (4*b*d*
(b*c + a*d)*x^(9/2))/9 + (2*b^2*d^2*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 )^2 \left (c+d x^2\right )^2}{x^{5/2}} \, dx &=\int \left (\frac{a^2 c^2}{x^{5/2}}+\frac{2 a c (b c+a d)}{\sqrt{x}}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{3/2}+2 b d (b c+a d) x^{7/2}+b^2 d^2 x^{11/2}\right ) \, dx\\ &=-\frac{2 a^2 c^2}{3 x^{3/2}}+4 a c (b c+a d) \sqrt{x}+\frac{2}{5} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{5/2}+\frac{4}{9} b d (b c+a d) x^{9/2}+\frac{2}{13} b^2 d^2 x^{13/2}\\ \end{align*}

Mathematica [A]  time = 0.0318285, size = 83, normalized size = 0.87 \[ \frac{2 \left (117 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-195 a^2 c^2+130 b d x^6 (a d+b c)+1170 a c x^2 (a d+b c)+45 b^2 d^2 x^8\right )}{585 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(5/2),x]

[Out]

(2*(-195*a^2*c^2 + 1170*a*c*(b*c + a*d)*x^2 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 130*b*d*(b*c + a*d)*x^
6 + 45*b^2*d^2*x^8))/(585*x^(3/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 97, normalized size = 1. \begin{align*} -{\frac{-90\,{b}^{2}{d}^{2}{x}^{8}-260\,{x}^{6}ab{d}^{2}-260\,{x}^{6}{b}^{2}cd-234\,{x}^{4}{a}^{2}{d}^{2}-936\,{x}^{4}abcd-234\,{x}^{4}{b}^{2}{c}^{2}-2340\,{x}^{2}{a}^{2}cd-2340\,a{c}^{2}b{x}^{2}+390\,{a}^{2}{c}^{2}}{585}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x^2+a)^2*(d*x^2+c)^2/x^(5/2),x)

[Out]

-2/585*(-45*b^2*d^2*x^8-130*a*b*d^2*x^6-130*b^2*c*d*x^6-117*a^2*d^2*x^4-468*a*b*c*d*x^4-117*b^2*c^2*x^4-1170*a
^2*c*d*x^2-1170*a*b*c^2*x^2+195*a^2*c^2)/x^(3/2)

________________________________________________________________________________________

Maxima [A]  time = 1.02923, size = 115, normalized size = 1.21 \begin{align*} \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{9}{2}} + \frac{2}{5} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{5}{2}} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} + 4 \,{\left (a b c^{2} + a^{2} c d\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*(b^2*c*d + a*b*d^2)*x^(9/2) + 2/5*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(5/2) - 2/3*a^
2*c^2/x^(3/2) + 4*(a*b*c^2 + a^2*c*d)*sqrt(x)

________________________________________________________________________________________

Fricas [A]  time = 0.802958, size = 201, normalized size = 2.12 \begin{align*} \frac{2 \,{\left (45 \, b^{2} d^{2} x^{8} + 130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 117 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 195 \, a^{2} c^{2} + 1170 \,{\left (a b c^{2} + a^{2} c d\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)^2*(d*x^2+c)^2/x^(5/2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d^2*x^8 + 130*(b^2*c*d + a*b*d^2)*x^6 + 117*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 195*a^2*c^2 +
1170*(a*b*c^2 + a^2*c*d)*x^2)/x^(3/2)

________________________________________________________________________________________

Sympy [A]  time = 6.6326, size = 133, normalized size = 1.4 \begin{align*} - \frac{2 a^{2} c^{2}}{3 x^{\frac{3}{2}}} + 4 a^{2} c d \sqrt{x} + \frac{2 a^{2} d^{2} x^{\frac{5}{2}}}{5} + 4 a b c^{2} \sqrt{x} + \frac{8 a b c d x^{\frac{5}{2}}}{5} + \frac{4 a b d^{2} x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 b^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d^{2} x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)**2/x**(5/2),x)

[Out]

-2*a**2*c**2/(3*x**(3/2)) + 4*a**2*c*d*sqrt(x) + 2*a**2*d**2*x**(5/2)/5 + 4*a*b*c**2*sqrt(x) + 8*a*b*c*d*x**(5
/2)/5 + 4*a*b*d**2*x**(9/2)/9 + 2*b**2*c**2*x**(5/2)/5 + 4*b**2*c*d*x**(9/2)/9 + 2*b**2*d**2*x**(13/2)/13

________________________________________________________________________________________

Giac [A]  time = 1.15176, size = 127, normalized size = 1.34 \begin{align*} \frac{2}{13} \, b^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, b^{2} c d x^{\frac{9}{2}} + \frac{4}{9} \, a b d^{2} x^{\frac{9}{2}} + \frac{2}{5} \, b^{2} c^{2} x^{\frac{5}{2}} + \frac{8}{5} \, a b c d x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d^{2} x^{\frac{5}{2}} + 4 \, a b c^{2} \sqrt{x} + 4 \, a^{2} c d \sqrt{x} - \frac{2 \, a^{2} c^{2}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/13*b^2*d^2*x^(13/2) + 4/9*b^2*c*d*x^(9/2) + 4/9*a*b*d^2*x^(9/2) + 2/5*b^2*c^2*x^(5/2) + 8/5*a*b*c*d*x^(5/2)
+ 2/5*a^2*d^2*x^(5/2) + 4*a*b*c^2*sqrt(x) + 4*a^2*c*d*sqrt(x) - 2/3*a^2*c^2/x^(3/2)