3.725 \(\int x^{3/2} (a+c x^4)^2 \, dx\)

Optimal. Leaf size=36 \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{13} a c x^{13/2}+\frac{2}{21} c^2 x^{21/2} \]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*c*x^(13/2))/13 + (2*c^2*x^(21/2))/21

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Rubi [A]  time = 0.0083541, 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} \[ \frac{2}{5} a^2 x^{5/2}+\frac{4}{13} a c x^{13/2}+\frac{2}{21} c^2 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*c*x^(13/2))/13 + (2*c^2*x^(21/2))/21

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 x^{3/2} \left (a+c x^4\right )^2 \, dx &=\int \left (a^2 x^{3/2}+2 a c x^{11/2}+c^2 x^{19/2}\right ) \, dx\\ &=\frac{2}{5} a^2 x^{5/2}+\frac{4}{13} a c x^{13/2}+\frac{2}{21} c^2 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0075776, size = 30, normalized size = 0.83 \[ \frac{2 x^{5/2} \left (273 a^2+210 a c x^4+65 c^2 x^8\right )}{1365} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + c*x^4)^2,x]

[Out]

(2*x^(5/2)*(273*a^2 + 210*a*c*x^4 + 65*c^2*x^8))/1365

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{130\,{c}^{2}{x}^{8}+420\,ac{x}^{4}+546\,{a}^{2}}{1365}{x}^{{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1365*x^(5/2)*(65*c^2*x^8+210*a*c*x^4+273*a^2)

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Maxima [A]  time = 0.984526, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{13} \, a c x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/13*a*c*x^(13/2) + 2/5*a^2*x^(5/2)

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Fricas [A]  time = 1.41265, size = 78, normalized size = 2.17 \begin{align*} \frac{2}{1365} \,{\left (65 \, c^{2} x^{10} + 210 \, a c x^{6} + 273 \, a^{2} x^{2}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1365*(65*c^2*x^10 + 210*a*c*x^6 + 273*a^2*x^2)*sqrt(x)

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Sympy [A]  time = 13.8, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{5}{2}}}{5} + \frac{4 a c x^{\frac{13}{2}}}{13} + \frac{2 c^{2} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(5/2)/5 + 4*a*c*x**(13/2)/13 + 2*c**2*x**(21/2)/21

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Giac [A]  time = 1.10039, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{21} \, c^{2} x^{\frac{21}{2}} + \frac{4}{13} \, a c x^{\frac{13}{2}} + \frac{2}{5} \, a^{2} x^{\frac{5}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/21*c^2*x^(21/2) + 4/13*a*c*x^(13/2) + 2/5*a^2*x^(5/2)