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

Optimal. Leaf size=49 \[ \frac{6}{7} a^2 c x^{7/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} a c^2 x^{15/2}+\frac{2}{23} c^3 x^{23/2} \]

[Out]

(-2*a^3)/Sqrt[x] + (6*a^2*c*x^(7/2))/7 + (2*a*c^2*x^(15/2))/5 + (2*c^3*x^(23/2))/23

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Rubi [A]  time = 0.0127762, antiderivative size = 49, 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} \[ \frac{6}{7} a^2 c x^{7/2}-\frac{2 a^3}{\sqrt{x}}+\frac{2}{5} a c^2 x^{15/2}+\frac{2}{23} c^3 x^{23/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/x^(3/2),x]

[Out]

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

Mathematica [A]  time = 0.0118718, size = 41, normalized size = 0.84 \[ \frac{2 \left (345 a^2 c x^4-805 a^3+161 a c^2 x^8+35 c^3 x^{12}\right )}{805 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/x^(3/2),x]

[Out]

(2*(-805*a^3 + 345*a^2*c*x^4 + 161*a*c^2*x^8 + 35*c^3*x^12))/(805*Sqrt[x])

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Maple [A]  time = 0.007, size = 38, normalized size = 0.8 \begin{align*} -{\frac{-70\,{c}^{3}{x}^{12}-322\,a{c}^{2}{x}^{8}-690\,{a}^{2}c{x}^{4}+1610\,{a}^{3}}{805}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/805*(-35*c^3*x^12-161*a*c^2*x^8-345*a^2*c*x^4+805*a^3)/x^(1/2)

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Maxima [A]  time = 1.01055, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.51151, size = 96, normalized size = 1.96 \begin{align*} \frac{2 \,{\left (35 \, c^{3} x^{12} + 161 \, a c^{2} x^{8} + 345 \, a^{2} c x^{4} - 805 \, a^{3}\right )}}{805 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/805*(35*c^3*x^12 + 161*a*c^2*x^8 + 345*a^2*c*x^4 - 805*a^3)/sqrt(x)

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Sympy [A]  time = 21.3535, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{\sqrt{x}} + \frac{6 a^{2} c x^{\frac{7}{2}}}{7} + \frac{2 a c^{2} x^{\frac{15}{2}}}{5} + \frac{2 c^{3} x^{\frac{23}{2}}}{23} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.0988, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{23} \, c^{3} x^{\frac{23}{2}} + \frac{2}{5} \, a c^{2} x^{\frac{15}{2}} + \frac{6}{7} \, a^{2} c x^{\frac{7}{2}} - \frac{2 \, a^{3}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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