3.974 \(\int \frac{1}{(c x)^{11/2} (a+b x^2)^{3/4}} \, dx\)

Optimal. Leaf size=83 \[ -\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]

[Out]

(-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) + (16*(a + b*x^2)^(5/4))/(5*a^2*c*(c*x)^(9/2)) - (64*(a + b*x^2)^(9/4
))/(45*a^3*c*(c*x)^(9/2))

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Rubi [A]  time = 0.024621, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.105, Rules used = {273, 264} \[ -\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}} \]

Antiderivative was successfully verified.

[In]

Int[1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*(a + b*x^2)^(1/4))/(a*c*(c*x)^(9/2)) + (16*(a + b*x^2)^(5/4))/(5*a^2*c*(c*x)^(9/2)) - (64*(a + b*x^2)^(9/4
))/(45*a^3*c*(c*x)^(9/2))

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)^{11/2} \left (a+b x^2\right )^{3/4}} \, dx &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}-\frac{8 \int \frac{\sqrt [4]{a+b x^2}}{(c x)^{11/2}} \, dx}{a}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}+\frac{32 \int \frac{\left (a+b x^2\right )^{5/4}}{(c x)^{11/2}} \, dx}{5 a^2}\\ &=-\frac{2 \sqrt [4]{a+b x^2}}{a c (c x)^{9/2}}+\frac{16 \left (a+b x^2\right )^{5/4}}{5 a^2 c (c x)^{9/2}}-\frac{64 \left (a+b x^2\right )^{9/4}}{45 a^3 c (c x)^{9/2}}\\ \end{align*}

Mathematica [A]  time = 0.0233931, size = 52, normalized size = 0.63 \[ -\frac{2 \sqrt{c x} \sqrt [4]{a+b x^2} \left (5 a^2-8 a b x^2+32 b^2 x^4\right )}{45 a^3 c^6 x^5} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((c*x)^(11/2)*(a + b*x^2)^(3/4)),x]

[Out]

(-2*Sqrt[c*x]*(a + b*x^2)^(1/4)*(5*a^2 - 8*a*b*x^2 + 32*b^2*x^4))/(45*a^3*c^6*x^5)

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Maple [A]  time = 0.006, size = 42, normalized size = 0.5 \begin{align*} -{\frac{2\,x \left ( 32\,{b}^{2}{x}^{4}-8\,ab{x}^{2}+5\,{a}^{2} \right ) }{45\,{a}^{3}}\sqrt [4]{b{x}^{2}+a} \left ( cx \right ) ^{-{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x)

[Out]

-2/45*x*(b*x^2+a)^(1/4)*(32*b^2*x^4-8*a*b*x^2+5*a^2)/a^3/(c*x)^(11/2)

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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{3}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="maxima")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)

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Fricas [A]  time = 1.29528, size = 111, normalized size = 1.34 \begin{align*} -\frac{2 \,{\left (32 \, b^{2} x^{4} - 8 \, a b x^{2} + 5 \, a^{2}\right )}{\left (b x^{2} + a\right )}^{\frac{1}{4}} \sqrt{c x}}{45 \, a^{3} c^{6} x^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="fricas")

[Out]

-2/45*(32*b^2*x^4 - 8*a*b*x^2 + 5*a^2)*(b*x^2 + a)^(1/4)*sqrt(c*x)/(a^3*c^6*x^5)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)**(11/2)/(b*x**2+a)**(3/4),x)

[Out]

Timed out

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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{3}{4}} \left (c x\right )^{\frac{11}{2}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(c*x)^(11/2)/(b*x^2+a)^(3/4),x, algorithm="giac")

[Out]

integrate(1/((b*x^2 + a)^(3/4)*(c*x)^(11/2)), x)