3.403 \(\int \frac{(a+b x^2)^2 (c+d x^2)^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=95 \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.0459258, 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} \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0313195, size = 95, normalized size = 1. \[ \frac{2}{9} x^{9/2} \left (a^2 d^2+4 a b c d+b^2 c^2\right )+2 a^2 c^2 \sqrt{x}+\frac{4}{13} b d x^{13/2} (a d+b c)+\frac{4}{5} a c x^{5/2} (a d+b c)+\frac{2}{17} b^2 d^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.005, size = 97, normalized size = 1. \begin{align*}{\frac{1170\,{b}^{2}{d}^{2}{x}^{8}+3060\,{x}^{6}ab{d}^{2}+3060\,{x}^{6}{b}^{2}cd+2210\,{x}^{4}{a}^{2}{d}^{2}+8840\,{x}^{4}abcd+2210\,{x}^{4}{b}^{2}{c}^{2}+7956\,{x}^{2}{a}^{2}cd+7956\,a{c}^{2}b{x}^{2}+19890\,{a}^{2}{c}^{2}}{9945}\sqrt{x}} \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^(1/2),x)

[Out]

2/9945*x^(1/2)*(585*b^2*d^2*x^8+1530*a*b*d^2*x^6+1530*b^2*c*d*x^6+1105*a^2*d^2*x^4+4420*a*b*c*d*x^4+1105*b^2*c
^2*x^4+3978*a^2*c*d*x^2+3978*a*b*c^2*x^2+9945*a^2*c^2)

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Maxima [A]  time = 1.02062, size = 115, normalized size = 1.21 \begin{align*} \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \,{\left (b^{2} c d + a b d^{2}\right )} x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{\frac{9}{2}} + 2 \, a^{2} c^{2} \sqrt{x} + \frac{4}{5} \,{\left (a b c^{2} + a^{2} c d\right )} 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^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.70194, size = 208, normalized size = 2.19 \begin{align*} \frac{2}{9945} \,{\left (585 \, b^{2} d^{2} x^{8} + 1530 \,{\left (b^{2} c d + a b d^{2}\right )} x^{6} + 1105 \,{\left (b^{2} c^{2} + 4 \, a b c d + a^{2} d^{2}\right )} x^{4} + 9945 \, a^{2} c^{2} + 3978 \,{\left (a b c^{2} + a^{2} c d\right )} x^{2}\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^(1/2),x, algorithm="fricas")

[Out]

2/9945*(585*b^2*d^2*x^8 + 1530*(b^2*c*d + a*b*d^2)*x^6 + 1105*(b^2*c^2 + 4*a*b*c*d + a^2*d^2)*x^4 + 9945*a^2*c
^2 + 3978*(a*b*c^2 + a^2*c*d)*x^2)*sqrt(x)

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Sympy [A]  time = 4.98539, size = 134, normalized size = 1.41 \begin{align*} 2 a^{2} c^{2} \sqrt{x} + \frac{4 a^{2} c d x^{\frac{5}{2}}}{5} + \frac{2 a^{2} d^{2} x^{\frac{9}{2}}}{9} + \frac{4 a b c^{2} x^{\frac{5}{2}}}{5} + \frac{8 a b c d x^{\frac{9}{2}}}{9} + \frac{4 a b d^{2} x^{\frac{13}{2}}}{13} + \frac{2 b^{2} c^{2} x^{\frac{9}{2}}}{9} + \frac{4 b^{2} c d x^{\frac{13}{2}}}{13} + \frac{2 b^{2} d^{2} x^{\frac{17}{2}}}{17} \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**(1/2),x)

[Out]

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

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Giac [A]  time = 1.17831, size = 127, normalized size = 1.34 \begin{align*} \frac{2}{17} \, b^{2} d^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b^{2} c d x^{\frac{13}{2}} + \frac{4}{13} \, a b d^{2} x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c^{2} x^{\frac{9}{2}} + \frac{8}{9} \, a b c d x^{\frac{9}{2}} + \frac{2}{9} \, a^{2} d^{2} x^{\frac{9}{2}} + \frac{4}{5} \, a b c^{2} x^{\frac{5}{2}} + \frac{4}{5} \, a^{2} c d x^{\frac{5}{2}} + 2 \, a^{2} c^{2} \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^(1/2),x, algorithm="giac")

[Out]

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