3.1051 \(\int \frac{(a+b x^2+c x^4)^2}{\sqrt{x}} \, dx\)

Optimal. Leaf size=62 \[ 2 a^2 \sqrt{x}+\frac{2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{4}{5} a b x^{5/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.022, antiderivative size = 62, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.05, Rules used = {1108} \[ 2 a^2 \sqrt{x}+\frac{2}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{4}{5} a b x^{5/2}+\frac{4}{13} b c x^{13/2}+\frac{2}{17} c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x^2 + c*x^4)^2/Sqrt[x],x]

[Out]

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

Rule 1108

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d*x)^m*(a
+ b*x^2 + c*x^4)^p, x], x] /; FreeQ[{a, b, c, d, m}, x] && IGtQ[p, 0] &&  !IntegerQ[(m + 1)/2]

Rubi steps

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

Mathematica [A]  time = 0.042085, size = 63, normalized size = 1.02 \[ 2 \left (a^2 \sqrt{x}+\frac{1}{9} x^{9/2} \left (2 a c+b^2\right )+\frac{2}{5} a b x^{5/2}+\frac{2}{13} b c x^{13/2}+\frac{1}{17} c^2 x^{17/2}\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x^2 + c*x^4)^2/Sqrt[x],x]

[Out]

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

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Maple [A]  time = 0.046, size = 49, normalized size = 0.8 \begin{align*}{\frac{1170\,{c}^{2}{x}^{8}+3060\,bc{x}^{6}+4420\,{x}^{4}ac+2210\,{b}^{2}{x}^{4}+7956\,ab{x}^{2}+19890\,{a}^{2}}{9945}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+b*x^2+a)^2/x^(1/2),x)

[Out]

2/9945*x^(1/2)*(585*c^2*x^8+1530*b*c*x^6+2210*a*c*x^4+1105*b^2*x^4+3978*a*b*x^2+9945*a^2)

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Maxima [A]  time = 0.964708, size = 65, normalized size = 1.05 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} + \frac{4}{45} \,{\left (5 \, c x^{\frac{9}{2}} + 9 \, b x^{\frac{5}{2}}\right )} a \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(1/2),x, algorithm="maxima")

[Out]

2/17*c^2*x^(17/2) + 4/13*b*c*x^(13/2) + 2/9*b^2*x^(9/2) + 2*a^2*sqrt(x) + 4/45*(5*c*x^(9/2) + 9*b*x^(5/2))*a

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Fricas [A]  time = 1.31232, size = 130, normalized size = 2.1 \begin{align*} \frac{2}{9945} \,{\left (585 \, c^{2} x^{8} + 1530 \, b c x^{6} + 1105 \,{\left (b^{2} + 2 \, a c\right )} x^{4} + 3978 \, a b x^{2} + 9945 \, a^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(1/2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 4.93246, size = 68, normalized size = 1.1 \begin{align*} 2 a^{2} \sqrt{x} + \frac{4 a b x^{\frac{5}{2}}}{5} + \frac{4 a c x^{\frac{9}{2}}}{9} + \frac{2 b^{2} x^{\frac{9}{2}}}{9} + \frac{4 b c x^{\frac{13}{2}}}{13} + \frac{2 c^{2} x^{\frac{17}{2}}}{17} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+b*x**2+a)**2/x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.16505, size = 62, normalized size = 1. \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{13} \, b c x^{\frac{13}{2}} + \frac{2}{9} \, b^{2} x^{\frac{9}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + \frac{4}{5} \, a b x^{\frac{5}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+b*x^2+a)^2/x^(1/2),x, algorithm="giac")

[Out]

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