3.401 \(\int x^{3/2} (a+b x^2)^2 (c+d x^2)^2 \, dx\)

Optimal. Leaf size=97 \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (4*b
*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))/21

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Rubi [A]  time = 0.0497726, 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} \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (4*b
*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))/21

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^{3/2} \left (a+b x^2\right )^2 \left (c+d x^2\right )^2 \, dx &=\int \left (a^2 c^2 x^{3/2}+2 a c (b c+a d) x^{7/2}+\left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{11/2}+2 b d (b c+a d) x^{15/2}+b^2 d^2 x^{19/2}\right ) \, dx\\ &=\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{9} a c (b c+a d) x^{9/2}+\frac{2}{13} \left (b^2 c^2+4 a b c d+a^2 d^2\right ) x^{13/2}+\frac{4}{17} b d (b c+a d) x^{17/2}+\frac{2}{21} b^2 d^2 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0316799, size = 97, normalized size = 1. \[ \frac{2}{13} x^{13/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+\frac{2}{5} a^2 c^2 x^{5/2}+\frac{4}{17} b d x^{17/2} (a d+b c)+\frac{4}{9} a c x^{9/2} (a d+b c)+\frac{2}{21} b^2 d^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + b*x^2)^2*(c + d*x^2)^2,x]

[Out]

(2*a^2*c^2*x^(5/2))/5 + (4*a*c*(b*c + a*d)*x^(9/2))/9 + (2*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2))/13 + (4*b
*d*(b*c + a*d)*x^(17/2))/17 + (2*b^2*d^2*x^(21/2))/21

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Maple [A]  time = 0.004, size = 97, normalized size = 1. \begin{align*}{\frac{6630\,{b}^{2}{d}^{2}{x}^{8}+16380\,{x}^{6}ab{d}^{2}+16380\,{x}^{6}{b}^{2}cd+10710\,{x}^{4}{a}^{2}{d}^{2}+42840\,{x}^{4}abcd+10710\,{x}^{4}{b}^{2}{c}^{2}+30940\,{x}^{2}{a}^{2}cd+30940\,a{c}^{2}b{x}^{2}+27846\,{a}^{2}{c}^{2}}{69615}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^2,x)

[Out]

2/69615*x^(5/2)*(3315*b^2*d^2*x^8+8190*a*b*d^2*x^6+8190*b^2*c*d*x^6+5355*a^2*d^2*x^4+21420*a*b*c*d*x^4+5355*b^
2*c^2*x^4+15470*a^2*c*d*x^2+15470*a*b*c^2*x^2+13923*a^2*c^2)

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Maxima [A]  time = 1.10538, size = 115, normalized size = 1.19 \begin{align*} \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{17}{2}} + \frac{2}{13} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} + \frac{4}{9} \,{\left (a b c^{2} + a^{2} c d\right )} x^{\frac{9}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="maxima")

[Out]

2/21*b^2*d^2*x^(21/2) + 4/17*(b^2*c*d + a*b*d^2)*x^(17/2) + 2/13*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^(13/2) + 2/
5*a^2*c^2*x^(5/2) + 4/9*(a*b*c^2 + a^2*c*d)*x^(9/2)

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Fricas [A]  time = 0.884202, size = 220, normalized size = 2.27 \begin{align*} \frac{2}{69615} \,{\left (3315 \, b^{2} d^{2} x^{10} + 8190 \,{\left (b^{2} c d + a b d^{2}\right )} x^{8} + 5355 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{6} + 13923 \, a^{2} c^{2} x^{2} + 15470 \,{\left (a b c^{2} + a^{2} c d\right )} x^{4}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="fricas")

[Out]

2/69615*(3315*b^2*d^2*x^10 + 8190*(b^2*c*d + a*b*d^2)*x^8 + 5355*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^6 + 13923*a
^2*c^2*x^2 + 15470*(a*b*c^2 + a^2*c*d)*x^4)*sqrt(x)

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Sympy [A]  time = 11.3792, size = 136, normalized size = 1.4 \begin{align*} \frac{2 a^{2} c^{2} x^{\frac{5}{2}}}{5} + \frac{4 a^{2} c d x^{\frac{9}{2}}}{9} + \frac{2 a^{2} d^{2} x^{\frac{13}{2}}}{13} + \frac{4 a b c^{2} x^{\frac{9}{2}}}{9} + \frac{8 a b c d x^{\frac{13}{2}}}{13} + \frac{4 a b d^{2} x^{\frac{17}{2}}}{17} + \frac{2 b^{2} c^{2} x^{\frac{13}{2}}}{13} + \frac{4 b^{2} c d x^{\frac{17}{2}}}{17} + \frac{2 b^{2} d^{2} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(3/2)*(b*x**2+a)**2*(d*x**2+c)**2,x)

[Out]

2*a**2*c**2*x**(5/2)/5 + 4*a**2*c*d*x**(9/2)/9 + 2*a**2*d**2*x**(13/2)/13 + 4*a*b*c**2*x**(9/2)/9 + 8*a*b*c*d*
x**(13/2)/13 + 4*a*b*d**2*x**(17/2)/17 + 2*b**2*c**2*x**(13/2)/13 + 4*b**2*c*d*x**(17/2)/17 + 2*b**2*d**2*x**(
21/2)/21

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Giac [A]  time = 1.59996, size = 127, normalized size = 1.31 \begin{align*} \frac{2}{21} \, b^{2} d^{2} x^{\frac{21}{2}} + \frac{4}{17} \, b^{2} c d x^{\frac{17}{2}} + \frac{4}{17} \, a b d^{2} x^{\frac{17}{2}} + \frac{2}{13} \, b^{2} c^{2} x^{\frac{13}{2}} + \frac{8}{13} \, a b c d x^{\frac{13}{2}} + \frac{2}{13} \, a^{2} d^{2} x^{\frac{13}{2}} + \frac{4}{9} \, a b c^{2} x^{\frac{9}{2}} + \frac{4}{9} \, a^{2} c d x^{\frac{9}{2}} + \frac{2}{5} \, a^{2} c^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(3/2)*(b*x^2+a)^2*(d*x^2+c)^2,x, algorithm="giac")

[Out]

2/21*b^2*d^2*x^(21/2) + 4/17*b^2*c*d*x^(17/2) + 4/17*a*b*d^2*x^(17/2) + 2/13*b^2*c^2*x^(13/2) + 8/13*a*b*c*d*x
^(13/2) + 2/13*a^2*d^2*x^(13/2) + 4/9*a*b*c^2*x^(9/2) + 4/9*a^2*c*d*x^(9/2) + 2/5*a^2*c^2*x^(5/2)