3.406 \(\int \frac{(a+b x^2)^2 (c+d x^2)^2}{x^{7/2}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]

[Out]

(-2*a^2*c^2)/(5*x^(5/2)) - (4*a*c*(b*c + a*d))/Sqrt[x] + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2))/3 + (4*b*
d*(b*c + a*d)*x^(7/2))/7 + (2*b^2*d^2*x^(11/2))/11

________________________________________________________________________________________

Rubi [A]  time = 0.0471844, 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} \[ \frac{2}{3} x^{3/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )-\frac{2 a^2 c^2}{5 x^{5/2}}+\frac{4}{7} b d x^{7/2} (a d+b c)-\frac{4 a c (a d+b c)}{\sqrt{x}}+\frac{2}{11} b^2 d^2 x^{11/2} \]

Antiderivative was successfully verified.

[In]

Int[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]

[Out]

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

Mathematica [A]  time = 0.0373862, size = 83, normalized size = 0.87 \[ \frac{2 \left (385 x^4 \left (a^2 d^2+4 a b c d+b^2 c^2\right )-231 a^2 c^2+330 b d x^6 (a d+b c)-2310 a c x^2 (a d+b c)+105 b^2 d^2 x^8\right )}{1155 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[((a + b*x^2)^2*(c + d*x^2)^2)/x^(7/2),x]

[Out]

(2*(-231*a^2*c^2 - 2310*a*c*(b*c + a*d)*x^2 + 385*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 330*b*d*(b*c + a*d)*x^
6 + 105*b^2*d^2*x^8))/(1155*x^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.006, size = 97, normalized size = 1. \begin{align*} -{\frac{-210\,{b}^{2}{d}^{2}{x}^{8}-660\,{x}^{6}ab{d}^{2}-660\,{x}^{6}{b}^{2}cd-770\,{x}^{4}{a}^{2}{d}^{2}-3080\,{x}^{4}abcd-770\,{x}^{4}{b}^{2}{c}^{2}+4620\,{x}^{2}{a}^{2}cd+4620\,a{c}^{2}b{x}^{2}+462\,{a}^{2}{c}^{2}}{1155}{x}^{-{\frac{5}{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^(7/2),x)

[Out]

-2/1155*(-105*b^2*d^2*x^8-330*a*b*d^2*x^6-330*b^2*c*d*x^6-385*a^2*d^2*x^4-1540*a*b*c*d*x^4-385*b^2*c^2*x^4+231
0*a^2*c*d*x^2+2310*a*b*c^2*x^2+231*a^2*c^2)/x^(5/2)

________________________________________________________________________________________

Maxima [A]  time = 1.06232, size = 117, normalized size = 1.23 \begin{align*} \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{7}{2}} + \frac{2}{3} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{3}{2}} - \frac{2 \,{\left (a^{2} c^{2} + 10 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="maxima")

[Out]

2/11*b^2*d^2*x^(11/2) + 4/7*(b^2*c*d + a*b*d^2)*x^(7/2) + 2/3*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(3/2) - 2/5*(a
^2*c^2 + 10*(a*b*c^2 + a^2*c*d)*x^2)/x^(5/2)

________________________________________________________________________________________

Fricas [A]  time = 0.75379, size = 204, normalized size = 2.15 \begin{align*} \frac{2 \,{\left (105 \, b^{2} d^{2} x^{8} + 330 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 385 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} - 231 \, a^{2} c^{2} - 2310 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\right )}}{1155 \, x^{\frac{5}{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^(7/2),x, algorithm="fricas")

[Out]

2/1155*(105*b^2*d^2*x^8 + 330*(b^2*c*d + a*b*d^2)*x^6 + 385*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 - 231*a^2*c^2
- 2310*(a*b*c^2 + a^2*c*d)*x^2)/x^(5/2)

________________________________________________________________________________________

Sympy [A]  time = 8.94829, size = 133, normalized size = 1.4 \begin{align*} - \frac{2 a^{2} c^{2}}{5 x^{\frac{5}{2}}} - \frac{4 a^{2} c d}{\sqrt{x}} + \frac{2 a^{2} d^{2} x^{\frac{3}{2}}}{3} - \frac{4 a b c^{2}}{\sqrt{x}} + \frac{8 a b c d x^{\frac{3}{2}}}{3} + \frac{4 a b d^{2} x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c^{2} x^{\frac{3}{2}}}{3} + \frac{4 b^{2} c d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d^{2} x^{\frac{11}{2}}}{11} \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**(7/2),x)

[Out]

-2*a**2*c**2/(5*x**(5/2)) - 4*a**2*c*d/sqrt(x) + 2*a**2*d**2*x**(3/2)/3 - 4*a*b*c**2/sqrt(x) + 8*a*b*c*d*x**(3
/2)/3 + 4*a*b*d**2*x**(7/2)/7 + 2*b**2*c**2*x**(3/2)/3 + 4*b**2*c*d*x**(7/2)/7 + 2*b**2*d**2*x**(11/2)/11

________________________________________________________________________________________

Giac [A]  time = 1.16628, size = 130, normalized size = 1.37 \begin{align*} \frac{2}{11} \, b^{2} d^{2} x^{\frac{11}{2}} + \frac{4}{7} \, b^{2} c d x^{\frac{7}{2}} + \frac{4}{7} \, a b d^{2} x^{\frac{7}{2}} + \frac{2}{3} \, b^{2} c^{2} x^{\frac{3}{2}} + \frac{8}{3} \, a b c d x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d^{2} x^{\frac{3}{2}} - \frac{2 \,{\left (10 \, a b c^{2} x^{2} + 10 \, a^{2} c d x^{2} + a^{2} c^{2}\right )}}{5 \, x^{\frac{5}{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^(7/2),x, algorithm="giac")

[Out]

2/11*b^2*d^2*x^(11/2) + 4/7*b^2*c*d*x^(7/2) + 4/7*a*b*d^2*x^(7/2) + 2/3*b^2*c^2*x^(3/2) + 8/3*a*b*c*d*x^(3/2)
+ 2/3*a^2*d^2*x^(3/2) - 2/5*(10*a*b*c^2*x^2 + 10*a^2*c*d*x^2 + a^2*c^2)/x^(5/2)