3.737 \(\int \frac{(a+c x^4)^3}{x^{7/2}} \, dx\)

Optimal. Leaf size=49 \[ 2 a^2 c x^{3/2}-\frac{2 a^3}{5 x^{5/2}}+\frac{6}{11} a c^2 x^{11/2}+\frac{2}{19} c^3 x^{19/2} \]

[Out]

(-2*a^3)/(5*x^(5/2)) + 2*a^2*c*x^(3/2) + (6*a*c^2*x^(11/2))/11 + (2*c^3*x^(19/2))/19

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Rubi [A]  time = 0.0115815, 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} \[ 2 a^2 c x^{3/2}-\frac{2 a^3}{5 x^{5/2}}+\frac{6}{11} a c^2 x^{11/2}+\frac{2}{19} c^3 x^{19/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^3/x^(7/2),x]

[Out]

(-2*a^3)/(5*x^(5/2)) + 2*a^2*c*x^(3/2) + (6*a*c^2*x^(11/2))/11 + (2*c^3*x^(19/2))/19

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^{7/2}} \, dx &=\int \left (\frac{a^3}{x^{7/2}}+3 a^2 c \sqrt{x}+3 a c^2 x^{9/2}+c^3 x^{17/2}\right ) \, dx\\ &=-\frac{2 a^3}{5 x^{5/2}}+2 a^2 c x^{3/2}+\frac{6}{11} a c^2 x^{11/2}+\frac{2}{19} c^3 x^{19/2}\\ \end{align*}

Mathematica [A]  time = 0.0126796, size = 41, normalized size = 0.84 \[ \frac{2 \left (1045 a^2 c x^4-209 a^3+285 a c^2 x^8+55 c^3 x^{12}\right )}{1045 x^{5/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^3/x^(7/2),x]

[Out]

(2*(-209*a^3 + 1045*a^2*c*x^4 + 285*a*c^2*x^8 + 55*c^3*x^12))/(1045*x^(5/2))

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Maple [A]  time = 0.006, size = 38, normalized size = 0.8 \begin{align*} -{\frac{-110\,{c}^{3}{x}^{12}-570\,a{c}^{2}{x}^{8}-2090\,{a}^{2}c{x}^{4}+418\,{a}^{3}}{1045}{x}^{-{\frac{5}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^4+a)^3/x^(7/2),x)

[Out]

-2/1045*(-55*c^3*x^12-285*a*c^2*x^8-1045*a^2*c*x^4+209*a^3)/x^(5/2)

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Maxima [A]  time = 0.971887, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{19} \, c^{3} x^{\frac{19}{2}} + \frac{6}{11} \, a c^{2} x^{\frac{11}{2}} + 2 \, a^{2} c x^{\frac{3}{2}} - \frac{2 \, a^{3}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(7/2),x, algorithm="maxima")

[Out]

2/19*c^3*x^(19/2) + 6/11*a*c^2*x^(11/2) + 2*a^2*c*x^(3/2) - 2/5*a^3/x^(5/2)

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Fricas [A]  time = 1.43578, size = 99, normalized size = 2.02 \begin{align*} \frac{2 \,{\left (55 \, c^{3} x^{12} + 285 \, a c^{2} x^{8} + 1045 \, a^{2} c x^{4} - 209 \, a^{3}\right )}}{1045 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(7/2),x, algorithm="fricas")

[Out]

2/1045*(55*c^3*x^12 + 285*a*c^2*x^8 + 1045*a^2*c*x^4 - 209*a^3)/x^(5/2)

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Sympy [A]  time = 27.6456, size = 48, normalized size = 0.98 \begin{align*} - \frac{2 a^{3}}{5 x^{\frac{5}{2}}} + 2 a^{2} c x^{\frac{3}{2}} + \frac{6 a c^{2} x^{\frac{11}{2}}}{11} + \frac{2 c^{3} x^{\frac{19}{2}}}{19} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**4+a)**3/x**(7/2),x)

[Out]

-2*a**3/(5*x**(5/2)) + 2*a**2*c*x**(3/2) + 6*a*c**2*x**(11/2)/11 + 2*c**3*x**(19/2)/19

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Giac [A]  time = 1.09356, size = 47, normalized size = 0.96 \begin{align*} \frac{2}{19} \, c^{3} x^{\frac{19}{2}} + \frac{6}{11} \, a c^{2} x^{\frac{11}{2}} + 2 \, a^{2} c x^{\frac{3}{2}} - \frac{2 \, a^{3}}{5 \, x^{\frac{5}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^4+a)^3/x^(7/2),x, algorithm="giac")

[Out]

2/19*c^3*x^(19/2) + 6/11*a*c^2*x^(11/2) + 2*a^2*c*x^(3/2) - 2/5*a^3/x^(5/2)