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

Optimal. Leaf size=61 \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]

[Out]

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

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Rubi [A]  time = 0.030651, antiderivative size = 61, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.045, Rules used = {448} \[ -\frac{2 a^2 c}{\sqrt{x}}+\frac{2}{7} b x^{7/2} (2 a d+b c)+\frac{2}{3} a x^{3/2} (a d+2 b c)+\frac{2}{11} b^2 d x^{11/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Mathematica [A]  time = 0.0212652, size = 60, normalized size = 0.98 \[ \frac{-154 a^2 \left (3 c-d x^2\right )+44 a b x^2 \left (7 c+3 d x^2\right )+6 b^2 x^4 \left (11 c+7 d x^2\right )}{231 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-154*a^2*(3*c - d*x^2) + 44*a*b*x^2*(7*c + 3*d*x^2) + 6*b^2*x^4*(11*c + 7*d*x^2))/(231*Sqrt[x])

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Maple [A]  time = 0.003, size = 56, normalized size = 0.9 \begin{align*} -{\frac{-42\,{b}^{2}d{x}^{6}-132\,{x}^{4}abd-66\,{b}^{2}c{x}^{4}-154\,{x}^{2}{a}^{2}d-308\,abc{x}^{2}+462\,{a}^{2}c}{231}{\frac{1}{\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)/x^(3/2),x)

[Out]

-2/231*(-21*b^2*d*x^6-66*a*b*d*x^4-33*b^2*c*x^4-77*a^2*d*x^2-154*a*b*c*x^2+231*a^2*c)/x^(1/2)

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Maxima [A]  time = 1.07836, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{7}{2}} - \frac{2 \, a^{2} c}{\sqrt{x}} + \frac{2}{3} \,{\left (2 \, a b c + a^{2} d\right )} 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)/x^(3/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.628379, size = 130, normalized size = 2.13 \begin{align*} \frac{2 \,{\left (21 \, b^{2} d x^{6} + 33 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} - 231 \, a^{2} c + 77 \,{\left (2 \, a b c + a^{2} d\right )} x^{2}\right )}}{231 \, \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)/x^(3/2),x, algorithm="fricas")

[Out]

2/231*(21*b^2*d*x^6 + 33*(b^2*c + 2*a*b*d)*x^4 - 231*a^2*c + 77*(2*a*b*c + a^2*d)*x^2)/sqrt(x)

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Sympy [A]  time = 2.46512, size = 78, normalized size = 1.28 \begin{align*} - \frac{2 a^{2} c}{\sqrt{x}} + \frac{2 a^{2} d x^{\frac{3}{2}}}{3} + \frac{4 a b c x^{\frac{3}{2}}}{3} + \frac{4 a b d x^{\frac{7}{2}}}{7} + \frac{2 b^{2} c x^{\frac{7}{2}}}{7} + \frac{2 b^{2} d 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)/x**(3/2),x)

[Out]

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

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Giac [A]  time = 1.17261, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{11} \, b^{2} d x^{\frac{11}{2}} + \frac{2}{7} \, b^{2} c x^{\frac{7}{2}} + \frac{4}{7} \, a b d x^{\frac{7}{2}} + \frac{4}{3} \, a b c x^{\frac{3}{2}} + \frac{2}{3} \, a^{2} d x^{\frac{3}{2}} - \frac{2 \, a^{2} c}{\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)/x^(3/2),x, algorithm="giac")

[Out]

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