3.988 \(\int \frac{1}{(c x)^{5/2} (a+b x^2)^{5/4}} \, dx\)

Optimal. Leaf size=55 \[ \frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}} \]

[Out]

2/(a*c*(c*x)^(3/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(3/2))

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Rubi [A]  time = 0.0148584, antiderivative size = 55, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ \frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(5/2)*(a + b*x^2)^(5/4)),x]

[Out]

2/(a*c*(c*x)^(3/2)*(a + b*x^2)^(1/4)) - (8*(a + b*x^2)^(3/4))/(3*a^2*c*(c*x)^(3/2))

Rule 273

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(
a*c*n*(p + 1)), x] + Dist[(m + n*(p + 1) + 1)/(a*n*(p + 1)), Int[(c*x)^m*(a + b*x^n)^(p + 1), x], x] /; FreeQ[
{a, b, c, m, n, p}, x] && ILtQ[Simplify[(m + 1)/n + p + 1], 0] && NeQ[p, -1]

Rule 264

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[((c*x)^(m + 1)*(a + b*x^n)^(p + 1))/(a
*c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[(m + 1)/n + p + 1, 0] && NeQ[m, -1]

Rubi steps

\begin{align*} \int \frac{1}{(c x)^{5/2} \left (a+b x^2\right )^{5/4}} \, dx &=\frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}+\frac{4 \int \frac{1}{(c x)^{5/2} \sqrt [4]{a+b x^2}} \, dx}{a}\\ &=\frac{2}{a c (c x)^{3/2} \sqrt [4]{a+b x^2}}-\frac{8 \left (a+b x^2\right )^{3/4}}{3 a^2 c (c x)^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.0096554, size = 34, normalized size = 0.62 \[ -\frac{2 x \left (a+4 b x^2\right )}{3 a^2 (c x)^{5/2} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(5/2)*(a + b*x^2)^(5/4)),x]

[Out]

(-2*x*(a + 4*b*x^2))/(3*a^2*(c*x)^(5/2)*(a + b*x^2)^(1/4))

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Maple [A]  time = 0.004, size = 29, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 4\,b{x}^{2}+a \right ) }{3\,{a}^{2}} \left ( cx \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(5/2)/(b*x^2+a)^(5/4),x)

[Out]

-2/3*x*(4*b*x^2+a)/(b*x^2+a)^(1/4)/a^2/(c*x)^(5/2)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(b*x^2+a)^(5/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(5/2)), x)

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Fricas [A]  time = 1.53265, size = 105, normalized size = 1.91 \begin{align*} -\frac{2 \,{\left (4 \, b x^{2} + a\right )}{\left (b x^{2} + a\right )}^{\frac{3}{4}} \sqrt{c x}}{3 \,{\left (a^{2} b c^{3} x^{4} + a^{3} c^{3} x^{2}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(b*x^2+a)^(5/4),x, algorithm="fricas")

[Out]

-2/3*(4*b*x^2 + a)*(b*x^2 + a)^(3/4)*sqrt(c*x)/(a^2*b*c^3*x^4 + a^3*c^3*x^2)

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Sympy [A]  time = 128.074, size = 78, normalized size = 1.42 \begin{align*} \frac{\Gamma \left (- \frac{3}{4}\right )}{8 a \sqrt [4]{b} c^{\frac{5}{2}} x^{2} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} + \frac{b^{\frac{3}{4}} \Gamma \left (- \frac{3}{4}\right )}{2 a^{2} c^{\frac{5}{2}} \sqrt [4]{\frac{a}{b x^{2}} + 1} \Gamma \left (\frac{5}{4}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(5/2)/(b*x**2+a)**(5/4),x)

[Out]

gamma(-3/4)/(8*a*b**(1/4)*c**(5/2)*x**2*(a/(b*x**2) + 1)**(1/4)*gamma(5/4)) + b**(3/4)*gamma(-3/4)/(2*a**2*c**
(5/2)*(a/(b*x**2) + 1)**(1/4)*gamma(5/4))

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{5}{4}} \left (c x\right )^{\frac{5}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(5/2)/(b*x^2+a)^(5/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(5/4)*(c*x)^(5/2)), x)