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

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

[Out]

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

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Rubi [A]  time = 0.0124189, 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} \[ \frac{6}{5} a^2 c x^{5/2}-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{13} a c^2 x^{13/2}+\frac{2}{21} c^3 x^{21/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/x^(5/2),x]

[Out]

(-2*a^3)/(3*x^(3/2)) + (6*a^2*c*x^(5/2))/5 + (6*a*c^2*x^(13/2))/13 + (2*c^3*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 \frac{\left (a+c x^4\right )^3}{x^{5/2}} \, dx &=\int \left (\frac{a^3}{x^{5/2}}+3 a^2 c x^{3/2}+3 a c^2 x^{11/2}+c^3 x^{19/2}\right ) \, dx\\ &=-\frac{2 a^3}{3 x^{3/2}}+\frac{6}{5} a^2 c x^{5/2}+\frac{6}{13} a c^2 x^{13/2}+\frac{2}{21} c^3 x^{21/2}\\ \end{align*}

Mathematica [A]  time = 0.0117046, size = 41, normalized size = 0.8 \[ \frac{2 \left (819 a^2 c x^4-455 a^3+315 a c^2 x^8+65 c^3 x^{12}\right )}{1365 x^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/x^(5/2),x]

[Out]

(2*(-455*a^3 + 819*a^2*c*x^4 + 315*a*c^2*x^8 + 65*c^3*x^12))/(1365*x^(3/2))

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Maple [A]  time = 0.004, size = 38, normalized size = 0.8 \begin{align*} -{\frac{-130\,{c}^{3}{x}^{12}-630\,a{c}^{2}{x}^{8}-1638\,{a}^{2}c{x}^{4}+910\,{a}^{3}}{1365}{x}^{-{\frac{3}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/1365*(-65*c^3*x^12-315*a*c^2*x^8-819*a^2*c*x^4+455*a^3)/x^(3/2)

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Maxima [A]  time = 1.07083, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{13} \, a c^{2} x^{\frac{13}{2}} + \frac{6}{5} \, a^{2} c x^{\frac{5}{2}} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.43207, size = 97, normalized size = 1.9 \begin{align*} \frac{2 \,{\left (65 \, c^{3} x^{12} + 315 \, a c^{2} x^{8} + 819 \, a^{2} c x^{4} - 455 \, a^{3}\right )}}{1365 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/1365*(65*c^3*x^12 + 315*a*c^2*x^8 + 819*a^2*c*x^4 - 455*a^3)/x^(3/2)

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Sympy [A]  time = 21.7077, size = 49, normalized size = 0.96 \begin{align*} - \frac{2 a^{3}}{3 x^{\frac{3}{2}}} + \frac{6 a^{2} c x^{\frac{5}{2}}}{5} + \frac{6 a c^{2} x^{\frac{13}{2}}}{13} + \frac{2 c^{3} x^{\frac{21}{2}}}{21} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.11344, size = 47, normalized size = 0.92 \begin{align*} \frac{2}{21} \, c^{3} x^{\frac{21}{2}} + \frac{6}{13} \, a c^{2} x^{\frac{13}{2}} + \frac{6}{5} \, a^{2} c x^{\frac{5}{2}} - \frac{2 \, a^{3}}{3 \, x^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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