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

Optimal. Leaf size=61 \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]

[Out]

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

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Rubi [A]  time = 0.0290469, 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} \[ 2 a^2 c \sqrt{x}+\frac{2}{9} b x^{9/2} (2 a d+b c)+\frac{2}{5} a x^{5/2} (a d+2 b c)+\frac{2}{13} b^2 d x^{13/2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.0309814, size = 53, normalized size = 0.87 \[ \frac{2}{585} \sqrt{x} \left (585 a^2 c+65 b x^4 (2 a d+b c)+117 a x^2 (a d+2 b c)+45 b^2 d x^6\right ) \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*Sqrt[x]*(585*a^2*c + 117*a*(2*b*c + a*d)*x^2 + 65*b*(b*c + 2*a*d)*x^4 + 45*b^2*d*x^6))/585

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Maple [A]  time = 0.005, size = 56, normalized size = 0.9 \begin{align*}{\frac{90\,{b}^{2}d{x}^{6}+260\,{x}^{4}abd+130\,{b}^{2}c{x}^{4}+234\,{x}^{2}{a}^{2}d+468\,abc{x}^{2}+1170\,{a}^{2}c}{585}\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^(1/2),x)

[Out]

2/585*x^(1/2)*(45*b^2*d*x^6+130*a*b*d*x^4+65*b^2*c*x^4+117*a^2*d*x^2+234*a*b*c*x^2+585*a^2*c)

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Maxima [A]  time = 1.1155, size = 69, normalized size = 1.13 \begin{align*} \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \,{\left (b^{2} c + 2 \, a b d\right )} x^{\frac{9}{2}} + 2 \, a^{2} c \sqrt{x} + \frac{2}{5} \,{\left (2 \, a b c + a^{2} 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)/x^(1/2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.908064, size = 131, normalized size = 2.15 \begin{align*} \frac{2}{585} \,{\left (45 \, b^{2} d x^{6} + 65 \,{\left (b^{2} c + 2 \, a b d\right )} x^{4} + 585 \, a^{2} c + 117 \,{\left (2 \, a b c + a^{2} 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)/x^(1/2),x, algorithm="fricas")

[Out]

2/585*(45*b^2*d*x^6 + 65*(b^2*c + 2*a*b*d)*x^4 + 585*a^2*c + 117*(2*a*b*c + a^2*d)*x^2)*sqrt(x)

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Sympy [A]  time = 2.14039, size = 78, normalized size = 1.28 \begin{align*} 2 a^{2} c \sqrt{x} + \frac{2 a^{2} d x^{\frac{5}{2}}}{5} + \frac{4 a b c x^{\frac{5}{2}}}{5} + \frac{4 a b d x^{\frac{9}{2}}}{9} + \frac{2 b^{2} c x^{\frac{9}{2}}}{9} + \frac{2 b^{2} d x^{\frac{13}{2}}}{13} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x**2+a)**2*(d*x**2+c)/x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16651, size = 72, normalized size = 1.18 \begin{align*} \frac{2}{13} \, b^{2} d x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} c x^{\frac{9}{2}} + \frac{4}{9} \, a b d x^{\frac{9}{2}} + \frac{4}{5} \, a b c x^{\frac{5}{2}} + \frac{2}{5} \, a^{2} d x^{\frac{5}{2}} + 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^(1/2),x, algorithm="giac")

[Out]

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