3.734 \(\int \frac{(a+c x^4)^3}{\sqrt{x}} \, dx\)

Optimal. Leaf size=49 \[ \frac{2}{3} a^2 c x^{9/2}+2 a^3 \sqrt{x}+\frac{6}{17} a c^2 x^{17/2}+\frac{2}{25} c^3 x^{25/2} \]

[Out]

2*a^3*Sqrt[x] + (2*a^2*c*x^(9/2))/3 + (6*a*c^2*x^(17/2))/17 + (2*c^3*x^(25/2))/25

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Rubi [A]  time = 0.0120546, antiderivative size = 49, 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} \[ \frac{2}{3} a^2 c x^{9/2}+2 a^3 \sqrt{x}+\frac{6}{17} a c^2 x^{17/2}+\frac{2}{25} c^3 x^{25/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/Sqrt[x],x]

[Out]

2*a^3*Sqrt[x] + (2*a^2*c*x^(9/2))/3 + (6*a*c^2*x^(17/2))/17 + (2*c^3*x^(25/2))/25

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 )^3}{\sqrt{x}} \, dx &=\int \left (\frac{a^3}{\sqrt{x}}+3 a^2 c x^{7/2}+3 a c^2 x^{15/2}+c^3 x^{23/2}\right ) \, dx\\ &=2 a^3 \sqrt{x}+\frac{2}{3} a^2 c x^{9/2}+\frac{6}{17} a c^2 x^{17/2}+\frac{2}{25} c^3 x^{25/2}\\ \end{align*}

Mathematica [A]  time = 0.0106029, size = 41, normalized size = 0.84 \[ \frac{2 \sqrt{x} \left (425 a^2 c x^4+1275 a^3+225 a c^2 x^8+51 c^3 x^{12}\right )}{1275} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/Sqrt[x],x]

[Out]

(2*Sqrt[x]*(1275*a^3 + 425*a^2*c*x^4 + 225*a*c^2*x^8 + 51*c^3*x^12))/1275

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{102\,{c}^{3}{x}^{12}+450\,a{c}^{2}{x}^{8}+850\,{a}^{2}c{x}^{4}+2550\,{a}^{3}}{1275}\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^3/x^(1/2),x)

[Out]

2/1275*x^(1/2)*(51*c^3*x^12+225*a*c^2*x^8+425*a^2*c*x^4+1275*a^3)

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Maxima [A]  time = 0.98161, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{6}{17} \, a c^{2} x^{\frac{17}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{9}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(1/2),x, algorithm="maxima")

[Out]

2/25*c^3*x^(25/2) + 6/17*a*c^2*x^(17/2) + 2/3*a^2*c*x^(9/2) + 2*a^3*sqrt(x)

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Fricas [A]  time = 1.44936, size = 99, normalized size = 2.02 \begin{align*} \frac{2}{1275} \,{\left (51 \, c^{3} x^{12} + 225 \, a c^{2} x^{8} + 425 \, a^{2} c x^{4} + 1275 \, a^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(1/2),x, algorithm="fricas")

[Out]

2/1275*(51*c^3*x^12 + 225*a*c^2*x^8 + 425*a^2*c*x^4 + 1275*a^3)*sqrt(x)

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Sympy [A]  time = 22.8643, size = 48, normalized size = 0.98 \begin{align*} 2 a^{3} \sqrt{x} + \frac{2 a^{2} c x^{\frac{9}{2}}}{3} + \frac{6 a c^{2} x^{\frac{17}{2}}}{17} + \frac{2 c^{3} x^{\frac{25}{2}}}{25} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**3/x**(1/2),x)

[Out]

2*a**3*sqrt(x) + 2*a**2*c*x**(9/2)/3 + 6*a*c**2*x**(17/2)/17 + 2*c**3*x**(25/2)/25

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Giac [A]  time = 1.0754, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{25} \, c^{3} x^{\frac{25}{2}} + \frac{6}{17} \, a c^{2} x^{\frac{17}{2}} + \frac{2}{3} \, a^{2} c x^{\frac{9}{2}} + 2 \, a^{3} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(1/2),x, algorithm="giac")

[Out]

2/25*c^3*x^(25/2) + 6/17*a*c^2*x^(17/2) + 2/3*a^2*c*x^(9/2) + 2*a^3*sqrt(x)