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

Optimal. Leaf size=26 \[ -\frac {2 \sqrt [4]{a+b x^2}}{a c \sqrt {c x}} \]

[Out]

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

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Rubi [A]  time = 0.01, antiderivative size = 26, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {264} \[ -\frac {2 \sqrt [4]{a+b x^2}}{a c \sqrt {c x}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(a + b*x^2)^(1/4))/(a*c*Sqrt[c*x])

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

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Mathematica [A]  time = 0.01, size = 24, normalized size = 0.92 \[ -\frac {2 x \sqrt [4]{a+b x^2}}{a (c x)^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 0.97, size = 25, normalized size = 0.96 \[ -\frac {2 \, {\left (b x^{2} + a\right )}^{\frac {1}{4}} \sqrt {c x}}{a c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [A]  time = 0.00, size = 21, normalized size = 0.81 \[ -\frac {2 \left (b \,x^{2}+a \right )^{\frac {1}{4}} x}{\left (c x \right )^{\frac {3}{2}} a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {1}{{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {3}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 4.90, size = 22, normalized size = 0.85 \[ -\frac {2\,{\left (b\,x^2+a\right )}^{1/4}}{a\,c\,\sqrt {c\,x}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.92, size = 36, normalized size = 1.38 \[ \frac {\sqrt [4]{b} \sqrt [4]{\frac {a}{b x^{2}} + 1} \Gamma \left (- \frac {1}{4}\right )}{2 a c^{\frac {3}{2}} \Gamma \left (\frac {3}{4}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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