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

Optimal. Leaf size=34 \[ 2 a^2 \sqrt{x}+\frac{4}{9} a c x^{9/2}+\frac{2}{17} c^2 x^{17/2} \]

[Out]

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

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Rubi [A]  time = 0.0081666, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.067, Rules used = {270} \[ 2 a^2 \sqrt{x}+\frac{4}{9} a c x^{9/2}+\frac{2}{17} c^2 x^{17/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/Sqrt[x],x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0075751, size = 30, normalized size = 0.88 \[ \frac{2}{153} \sqrt{x} \left (153 a^2+34 a c x^4+9 c^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(153*a^2 + 34*a*c*x^4 + 9*c^2*x^8))/153

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Maple [A]  time = 0.003, size = 27, normalized size = 0.8 \begin{align*}{\frac{18\,{c}^{2}{x}^{8}+68\,ac{x}^{4}+306\,{a}^{2}}{153}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153*x^(1/2)*(9*c^2*x^8+34*a*c*x^4+153*a^2)

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Maxima [A]  time = 0.983175, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.45164, size = 68, normalized size = 2. \begin{align*} \frac{2}{153} \,{\left (9 \, c^{2} x^{8} + 34 \, a c x^{4} + 153 \, a^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/153*(9*c^2*x^8 + 34*a*c*x^4 + 153*a^2)*sqrt(x)

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Sympy [A]  time = 5.00596, size = 32, normalized size = 0.94 \begin{align*} 2 a^{2} \sqrt{x} + \frac{4 a c x^{\frac{9}{2}}}{9} + \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+a)**2/x**(1/2),x)

[Out]

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

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Giac [A]  time = 1.10926, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{17} \, c^{2} x^{\frac{17}{2}} + \frac{4}{9} \, a c x^{\frac{9}{2}} + 2 \, a^{2} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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