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

Optimal. Leaf size=34 \[ -\frac{2 a^2}{\sqrt{x}}+\frac{4}{7} a c x^{7/2}+\frac{2}{15} c^2 x^{15/2} \]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*c*x^(7/2))/7 + (2*c^2*x^(15/2))/15

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Rubi [A]  time = 0.0081777, antiderivative size = 34, 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} \[ -\frac{2 a^2}{\sqrt{x}}+\frac{4}{7} a c x^{7/2}+\frac{2}{15} c^2 x^{15/2} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^4)^2/x^(3/2),x]

[Out]

(-2*a^2)/Sqrt[x] + (4*a*c*x^(7/2))/7 + (2*c^2*x^(15/2))/15

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

Mathematica [A]  time = 0.0085726, size = 30, normalized size = 0.88 \[ \frac{2 \left (-105 a^2+30 a c x^4+7 c^2 x^8\right )}{105 \sqrt{x}} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^4)^2/x^(3/2),x]

[Out]

(2*(-105*a^2 + 30*a*c*x^4 + 7*c^2*x^8))/(105*Sqrt[x])

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Maple [A]  time = 0.004, size = 27, normalized size = 0.8 \begin{align*} -{\frac{-14\,{c}^{2}{x}^{8}-60\,ac{x}^{4}+210\,{a}^{2}}{105}{\frac{1}{\sqrt{x}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/105*(-7*c^2*x^8-30*a*c*x^4+105*a^2)/x^(1/2)

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Maxima [A]  time = 0.9709, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{7} \, a c x^{\frac{7}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.46449, size = 68, normalized size = 2. \begin{align*} \frac{2 \,{\left (7 \, c^{2} x^{8} + 30 \, a c x^{4} - 105 \, a^{2}\right )}}{105 \, \sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(7*c^2*x^8 + 30*a*c*x^4 - 105*a^2)/sqrt(x)

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Sympy [A]  time = 7.01912, size = 32, normalized size = 0.94 \begin{align*} - \frac{2 a^{2}}{\sqrt{x}} + \frac{4 a c x^{\frac{7}{2}}}{7} + \frac{2 c^{2} x^{\frac{15}{2}}}{15} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.13659, size = 32, normalized size = 0.94 \begin{align*} \frac{2}{15} \, c^{2} x^{\frac{15}{2}} + \frac{4}{7} \, a c x^{\frac{7}{2}} - \frac{2 \, a^{2}}{\sqrt{x}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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