3.731 \(\int x^{5/2} (a+c x^4)^3 \, dx\)

Optimal. Leaf size=51 \[ \frac{2}{5} a^2 c x^{15/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} a c^2 x^{23/2}+\frac{2}{31} c^3 x^{31/2} \]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*c*x^(15/2))/5 + (6*a*c^2*x^(23/2))/23 + (2*c^3*x^(31/2))/31

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Rubi [A]  time = 0.0127232, 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} \[ \frac{2}{5} a^2 c x^{15/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} a c^2 x^{23/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*c*x^(15/2))/5 + (6*a*c^2*x^(23/2))/23 + (2*c^3*x^(31/2))/31

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^{5/2} \left (a+c x^4\right )^3 \, dx &=\int \left (a^3 x^{5/2}+3 a^2 c x^{13/2}+3 a c^2 x^{21/2}+c^3 x^{29/2}\right ) \, dx\\ &=\frac{2}{7} a^3 x^{7/2}+\frac{2}{5} a^2 c x^{15/2}+\frac{6}{23} a c^2 x^{23/2}+\frac{2}{31} c^3 x^{31/2}\\ \end{align*}

Mathematica [A]  time = 0.0123016, size = 51, normalized size = 1. \[ \frac{2}{5} a^2 c x^{15/2}+\frac{2}{7} a^3 x^{7/2}+\frac{6}{23} a c^2 x^{23/2}+\frac{2}{31} c^3 x^{31/2} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + c*x^4)^3,x]

[Out]

(2*a^3*x^(7/2))/7 + (2*a^2*c*x^(15/2))/5 + (6*a*c^2*x^(23/2))/23 + (2*c^3*x^(31/2))/31

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Maple [A]  time = 0.005, size = 38, normalized size = 0.8 \begin{align*}{\frac{1610\,{c}^{3}{x}^{12}+6510\,a{c}^{2}{x}^{8}+9982\,{a}^{2}c{x}^{4}+7130\,{a}^{3}}{24955}{x}^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^(5/2)*(c*x^4+a)^3,x)

[Out]

2/24955*x^(7/2)*(805*c^3*x^12+3255*a*c^2*x^8+4991*a^2*c*x^4+3565*a^3)

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Maxima [A]  time = 0.975635, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{6}{23} \, a c^{2} x^{\frac{23}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{15}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a)^3,x, algorithm="maxima")

[Out]

2/31*c^3*x^(31/2) + 6/23*a*c^2*x^(23/2) + 2/5*a^2*c*x^(15/2) + 2/7*a^3*x^(7/2)

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Fricas [A]  time = 1.48204, size = 111, normalized size = 2.18 \begin{align*} \frac{2}{24955} \,{\left (805 \, c^{3} x^{15} + 3255 \, a c^{2} x^{11} + 4991 \, a^{2} c x^{7} + 3565 \, a^{3} x^{3}\right )} \sqrt{x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a)^3,x, algorithm="fricas")

[Out]

2/24955*(805*c^3*x^15 + 3255*a*c^2*x^11 + 4991*a^2*c*x^7 + 3565*a^3*x^3)*sqrt(x)

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Sympy [A]  time = 61.7511, size = 49, normalized size = 0.96 \begin{align*} \frac{2 a^{3} x^{\frac{7}{2}}}{7} + \frac{2 a^{2} c x^{\frac{15}{2}}}{5} + \frac{6 a c^{2} x^{\frac{23}{2}}}{23} + \frac{2 c^{3} x^{\frac{31}{2}}}{31} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**(5/2)*(c*x**4+a)**3,x)

[Out]

2*a**3*x**(7/2)/7 + 2*a**2*c*x**(15/2)/5 + 6*a*c**2*x**(23/2)/23 + 2*c**3*x**(31/2)/31

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Giac [A]  time = 1.09888, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{31} \, c^{3} x^{\frac{31}{2}} + \frac{6}{23} \, a c^{2} x^{\frac{23}{2}} + \frac{2}{5} \, a^{2} c x^{\frac{15}{2}} + \frac{2}{7} \, a^{3} x^{\frac{7}{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^(5/2)*(c*x^4+a)^3,x, algorithm="giac")

[Out]

2/31*c^3*x^(31/2) + 6/23*a*c^2*x^(23/2) + 2/5*a^2*c*x^(15/2) + 2/7*a^3*x^(7/2)