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

Optimal. Leaf size=97 \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (4
*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23

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Rubi [A]  time = 0.0459392, antiderivative size = 97, 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} \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (4
*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23

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 x^{5/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{5/2}+2 a c (b c+a d) x^{9/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+2 b d (b c+a d) x^{17/2}+b^2 d^2 x^{21/2}\right ) \, dx\\ &=\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{11} a c (b c+a d) x^{11/2}+\frac{2}{15} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{15/2}+\frac{4}{19} b d (b c+a d) x^{19/2}+\frac{2}{23} b^2 d^2 x^{23/2}\\ \end{align*}

Mathematica [A]  time = 0.0330486, size = 97, normalized size = 1. \[ \frac{2}{15} x^{15/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{7} a^2 c^2 x^{7/2}+\frac{4}{19} b d x^{19/2} (a d+b c)+\frac{4}{11} a c x^{11/2} (a d+b c)+\frac{2}{23} b^2 d^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*a^2*c^2*x^(7/2))/7 + (4*a*c*(b*c + a*d)*x^(11/2))/11 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2))/15 + (4
*b*d*(b*c + a*d)*x^(19/2))/19 + (2*b^2*d^2*x^(23/2))/23

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Maple [A]  time = 0.006, size = 97, normalized size = 1. \begin{align*}{\frac{43890\,{b}^{2}{d}^{2}{x}^{8}+106260\,{x}^{6}ab{d}^{2}+106260\,{x}^{6}{b}^{2}cd+67298\,{x}^{4}{a}^{2}{d}^{2}+269192\,{x}^{4}abcd+67298\,{x}^{4}{b}^{2}{c}^{2}+183540\,{x}^{2}{a}^{2}cd+183540\,a{c}^{2}b{x}^{2}+144210\,{a}^{2}{c}^{2}}{504735}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/504735*x^(7/2)*(21945*b^2*d^2*x^8+53130*a*b*d^2*x^6+53130*b^2*c*d*x^6+33649*a^2*d^2*x^4+134596*a*b*c*d*x^4+3
3649*b^2*c^2*x^4+91770*a^2*c*d*x^2+91770*a*b*c^2*x^2+72105*a^2*c^2)

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Maxima [A]  time = 1.04838, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{19}{2}} + \frac{2}{15} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} + \frac{4}{11} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{11}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*(b^2*c*d + a*b*d^2)*x^(19/2) + 2/15*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(15/2) + 2/
7*a^2*c^2*x^(7/2) + 4/11*(a*b*c^2 + a^2*c*d)*x^(11/2)

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Fricas [A]  time = 0.74764, size = 225, normalized size = 2.32 \begin{align*} \frac{2}{504735} \,{\left (21945 \, b^{2} d^{2} x^{11} + 53130 \,{\left (b^{2} c d + a b d^{2}\right )} x^{9} + 33649 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{7} + 72105 \, a^{2} c^{2} x^{3} + 91770 \,{\left (a b c^{2} + a^{2} c d\right )} x^{5}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/504735*(21945*b^2*d^2*x^11 + 53130*(b^2*c*d + a*b*d^2)*x^9 + 33649*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^7 + 721
05*a^2*c^2*x^3 + 91770*(a*b*c^2 + a^2*c*d)*x^5)*sqrt(x)

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Sympy [A]  time = 20.5765, size = 136, normalized size = 1.4 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{7}{2}}}{7} + \frac{4 a^{2} c d x^{\frac{11}{2}}}{11} + \frac{2 a^{2} d^{2} x^{\frac{15}{2}}}{15} + \frac{4 a b c^{2} x^{\frac{11}{2}}}{11} + \frac{8 a b c d x^{\frac{15}{2}}}{15} + \frac{4 a b d^{2} x^{\frac{19}{2}}}{19} + \frac{2 b^{2} c^{2} x^{\frac{15}{2}}}{15} + \frac{4 b^{2} c d x^{\frac{19}{2}}}{19} + \frac{2 b^{2} d^{2} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*c**2*x**(7/2)/7 + 4*a**2*c*d*x**(11/2)/11 + 2*a**2*d**2*x**(15/2)/15 + 4*a*b*c**2*x**(11/2)/11 + 8*a*b*
c*d*x**(15/2)/15 + 4*a*b*d**2*x**(19/2)/19 + 2*b**2*c**2*x**(15/2)/15 + 4*b**2*c*d*x**(19/2)/19 + 2*b**2*d**2*
x**(23/2)/23

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Giac [A]  time = 1.14413, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{23} \, b^{2} d^{2} x^{\frac{23}{2}} + \frac{4}{19} \, b^{2} c d x^{\frac{19}{2}} + \frac{4}{19} \, a b d^{2} x^{\frac{19}{2}} + \frac{2}{15} \, b^{2} c^{2} x^{\frac{15}{2}} + \frac{8}{15} \, a b c d x^{\frac{15}{2}} + \frac{2}{15} \, a^{2} d^{2} x^{\frac{15}{2}} + \frac{4}{11} \, a b c^{2} x^{\frac{11}{2}} + \frac{4}{11} \, a^{2} c d x^{\frac{11}{2}} + \frac{2}{7} \, a^{2} c^{2} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/23*b^2*d^2*x^(23/2) + 4/19*b^2*c*d*x^(19/2) + 4/19*a*b*d^2*x^(19/2) + 2/15*b^2*c^2*x^(15/2) + 8/15*a*b*c*d*x
^(15/2) + 2/15*a^2*d^2*x^(15/2) + 4/11*a*b*c^2*x^(11/2) + 4/11*a^2*c*d*x^(11/2) + 2/7*a^2*c^2*x^(7/2)