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

Optimal. Leaf size=36 \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{11} a c x^{11/2}+\frac{2}{19} c^2 x^{19/2} \]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*c*x^(11/2))/11 + (2*c^2*x^(19/2))/19

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Rubi [A]  time = 0.0082749, antiderivative size = 36, 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} \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{11} a c x^{11/2}+\frac{2}{19} c^2 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[Sqrt[x]*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(3/2))/3 + (4*a*c*x^(11/2))/11 + (2*c^2*x^(19/2))/19

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 \sqrt{x} \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 \sqrt{x}+2 a c x^{9/2}+c^2 x^{17/2}\right ) \, dx\\ &=\frac{2}{3} a^2 x^{3/2}+\frac{4}{11} a c x^{11/2}+\frac{2}{19} c^2 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0075721, size = 30, normalized size = 0.83 \[ \frac{2}{627} x^{3/2} \left (209 a^2+114 a c x^4+33 c^2 x^8\right ) \]

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[x]*(a + c*x^4)^2,x]

[Out]

(2*x^(3/2)*(209*a^2 + 114*a*c*x^4 + 33*c^2*x^8))/627

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{66\,{c}^{2}{x}^{8}+228\,ac{x}^{4}+418\,{a}^{2}}{627}{x}^{{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/627*x^(3/2)*(33*c^2*x^8+114*a*c*x^4+209*a^2)

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Maxima [A]  time = 1.06089, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{11} \, a c x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \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]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)

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Fricas [A]  time = 1.45737, size = 73, normalized size = 2.03 \begin{align*} \frac{2}{627} \,{\left (33 \, c^{2} x^{9} + 114 \, a c x^{5} + 209 \, a^{2} x\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]

2/627*(33*c^2*x^9 + 114*a*c*x^5 + 209*a^2*x)*sqrt(x)

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Sympy [A]  time = 3.06879, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{3}{2}}}{3} + \frac{4 a c x^{\frac{11}{2}}}{11} + \frac{2 c^{2} x^{\frac{19}{2}}}{19} \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*x**(3/2)/3 + 4*a*c*x**(11/2)/11 + 2*c**2*x**(19/2)/19

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Giac [A]  time = 1.13772, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{19} \, c^{2} x^{\frac{19}{2}} + \frac{4}{11} \, a c x^{\frac{11}{2}} + \frac{2}{3} \, a^{2} x^{\frac{3}{2}} \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]

2/19*c^2*x^(19/2) + 4/11*a*c*x^(11/2) + 2/3*a^2*x^(3/2)