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

Optimal. Leaf size=51 \[ \frac{6}{13} a^2 c x^{13/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{7} a c^2 x^{21/2}+\frac{2}{29} c^3 x^{29/2} \]

[Out]

(2*a^3*x^(5/2))/5 + (6*a^2*c*x^(13/2))/13 + (2*a*c^2*x^(21/2))/7 + (2*c^3*x^(29/2))/29

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Rubi [A]  time = 0.0125475, antiderivative size = 51, 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} \[ \frac{6}{13} a^2 c x^{13/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{7} a c^2 x^{21/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(3/2)*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(5/2))/5 + (6*a^2*c*x^(13/2))/13 + (2*a*c^2*x^(21/2))/7 + (2*c^3*x^(29/2))/29

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 )^3 \, dx &=\int \left (a^3 x^{3/2}+3 a^2 c x^{11/2}+3 a c^2 x^{19/2}+c^3 x^{27/2}\right ) \, dx\\ &=\frac{2}{5} a^3 x^{5/2}+\frac{6}{13} a^2 c x^{13/2}+\frac{2}{7} a c^2 x^{21/2}+\frac{2}{29} c^3 x^{29/2}\\ \end{align*}

Mathematica [A]  time = 0.0116396, size = 51, normalized size = 1. \[ \frac{6}{13} a^2 c x^{13/2}+\frac{2}{5} a^3 x^{5/2}+\frac{2}{7} a c^2 x^{21/2}+\frac{2}{29} c^3 x^{29/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(3/2)*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(5/2))/5 + (6*a^2*c*x^(13/2))/13 + (2*a*c^2*x^(21/2))/7 + (2*c^3*x^(29/2))/29

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Maple [A]  time = 0.003, size = 38, normalized size = 0.8 \begin{align*}{\frac{910\,{c}^{3}{x}^{12}+3770\,a{c}^{2}{x}^{8}+6090\,{a}^{2}c{x}^{4}+5278\,{a}^{3}}{13195}{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)^3,x)

[Out]

2/13195*x^(5/2)*(455*c^3*x^12+1885*a*c^2*x^8+3045*a^2*c*x^4+2639*a^3)

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Maxima [A]  time = 0.989586, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{2}{7} \, a c^{2} x^{\frac{21}{2}} + \frac{6}{13} \, a^{2} c x^{\frac{13}{2}} + \frac{2}{5} \, a^{3} 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)^3,x, algorithm="maxima")

[Out]

2/29*c^3*x^(29/2) + 2/7*a*c^2*x^(21/2) + 6/13*a^2*c*x^(13/2) + 2/5*a^3*x^(5/2)

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Fricas [A]  time = 1.48921, size = 111, normalized size = 2.18 \begin{align*} \frac{2}{13195} \,{\left (455 \, c^{3} x^{14} + 1885 \, a c^{2} x^{10} + 3045 \, a^{2} c x^{6} + 2639 \, a^{3} 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)^3,x, algorithm="fricas")

[Out]

2/13195*(455*c^3*x^14 + 1885*a*c^2*x^10 + 3045*a^2*c*x^6 + 2639*a^3*x^2)*sqrt(x)

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Sympy [A]  time = 42.6093, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{5}{2}}}{5} + \frac{6 a^{2} c x^{\frac{13}{2}}}{13} + \frac{2 a c^{2} x^{\frac{21}{2}}}{7} + \frac{2 c^{3} x^{\frac{29}{2}}}{29} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**3*x**(5/2)/5 + 6*a**2*c*x**(13/2)/13 + 2*a*c**2*x**(21/2)/7 + 2*c**3*x**(29/2)/29

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Giac [A]  time = 1.10618, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{29} \, c^{3} x^{\frac{29}{2}} + \frac{2}{7} \, a c^{2} x^{\frac{21}{2}} + \frac{6}{13} \, a^{2} c x^{\frac{13}{2}} + \frac{2}{5} \, a^{3} 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)^3,x, algorithm="giac")

[Out]

2/29*c^3*x^(29/2) + 2/7*a*c^2*x^(21/2) + 6/13*a^2*c*x^(13/2) + 2/5*a^3*x^(5/2)