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

Optimal. Leaf size=36 \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{15} a c x^{15/2}+\frac{2}{23} c^2 x^{23/2} \]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*c*x^(15/2))/15 + (2*c^2*x^(23/2))/23

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Rubi [A]  time = 0.0086518, 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} \[ \frac{2}{7} a^2 x^{7/2}+\frac{4}{15} a c x^{15/2}+\frac{2}{23} c^2 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[x^(5/2)*(a + c*x^4)^2,x]

[Out]

(2*a^2*x^(7/2))/7 + (4*a*c*x^(15/2))/15 + (2*c^2*x^(23/2))/23

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

Mathematica [A]  time = 0.0083621, size = 30, normalized size = 0.83 \[ \frac{2 x^{7/2} \left (345 a^2+322 a c x^4+105 c^2 x^8\right )}{2415} \]

Antiderivative was successfully verified.

[In]

Integrate[x^(5/2)*(a + c*x^4)^2,x]

[Out]

(2*x^(7/2)*(345*a^2 + 322*a*c*x^4 + 105*c^2*x^8))/2415

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Maple [A]  time = 0.005, size = 27, normalized size = 0.8 \begin{align*}{\frac{210\,{c}^{2}{x}^{8}+644\,ac{x}^{4}+690\,{a}^{2}}{2415}{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)^2,x)

[Out]

2/2415*x^(7/2)*(105*c^2*x^8+322*a*c*x^4+345*a^2)

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Maxima [A]  time = 1.01983, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} 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)^2,x, algorithm="maxima")

[Out]

2/23*c^2*x^(23/2) + 4/15*a*c*x^(15/2) + 2/7*a^2*x^(7/2)

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Fricas [A]  time = 1.48837, size = 80, normalized size = 2.22 \begin{align*} \frac{2}{2415} \,{\left (105 \, c^{2} x^{11} + 322 \, a c x^{7} + 345 \, a^{2} 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)^2,x, algorithm="fricas")

[Out]

2/2415*(105*c^2*x^11 + 322*a*c*x^7 + 345*a^2*x^3)*sqrt(x)

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Sympy [A]  time = 25.6285, size = 34, normalized size = 0.94 \begin{align*} \frac{2 a^{2} x^{\frac{7}{2}}}{7} + \frac{4 a c x^{\frac{15}{2}}}{15} + \frac{2 c^{2} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*a**2*x**(7/2)/7 + 4*a*c*x**(15/2)/15 + 2*c**2*x**(23/2)/23

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Giac [A]  time = 1.11313, size = 32, normalized size = 0.89 \begin{align*} \frac{2}{23} \, c^{2} x^{\frac{23}{2}} + \frac{4}{15} \, a c x^{\frac{15}{2}} + \frac{2}{7} \, a^{2} 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)^2,x, algorithm="giac")

[Out]

2/23*c^2*x^(23/2) + 4/15*a*c*x^(15/2) + 2/7*a^2*x^(7/2)