3.1003 \(\int \frac{1}{(c x)^{5/4} \sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=56 \[ -\frac{4 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{c x} \sqrt [4]{a+b x^2}} \]

[Out]

(-4*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, -((b*x^2)/a)])/(c*(c*x)^(1/4)*(a + b*x^2)^(1/4))

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Rubi [A]  time = 0.0170952, antiderivative size = 56, 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 = {365, 364} \[ -\frac{4 \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{c x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, -((b*x^2)/a)])/(c*(c*x)^(1/4)*(a + b*x^2)^(1/4))

Rule 365

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[(a^IntPart[p]*(a + b*x^n)^FracPart[p])
/(1 + (b*x^n)/a)^FracPart[p], Int[(c*x)^m*(1 + (b*x^n)/a)^p, x], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[
p, 0] &&  !(ILtQ[p, 0] || GtQ[a, 0])

Rule 364

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(a^p*(c*x)^(m + 1)*Hypergeometric2F1[-
p, (m + 1)/n, (m + 1)/n + 1, -((b*x^n)/a)])/(c*(m + 1)), x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rubi steps

\begin{align*} \int \frac{1}{(c x)^{5/4} \sqrt [4]{a+b x^2}} \, dx &=\frac{\sqrt [4]{1+\frac{b x^2}{a}} \int \frac{1}{(c x)^{5/4} \sqrt [4]{1+\frac{b x^2}{a}}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=-\frac{4 \sqrt [4]{1+\frac{b x^2}{a}} \, _2F_1\left (-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{c \sqrt [4]{c x} \sqrt [4]{a+b x^2}}\\ \end{align*}

Mathematica [A]  time = 0.0109496, size = 54, normalized size = 0.96 \[ -\frac{4 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (-\frac{1}{8},\frac{1}{4};\frac{7}{8};-\frac{b x^2}{a}\right )}{(c x)^{5/4} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-4*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[-1/8, 1/4, 7/8, -((b*x^2)/a)])/((c*x)^(5/4)*(a + b*x^2)^(1/4))

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Maple [F]  time = 0.034, size = 0, normalized size = 0. \begin{align*} \int{ \left ( cx \right ) ^{-{\frac{5}{4}}}{\frac{1}{\sqrt [4]{b{x}^{2}+a}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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{1}{4}} \left (c x\right )^{\frac{5}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{2} + a\right )}^{\frac{3}{4}} \left (c x\right )^{\frac{3}{4}}}{b c^{2} x^{4} + a c^{2} x^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [C]  time = 10.0957, size = 48, normalized size = 0.86 \begin{align*} \frac{\Gamma \left (- \frac{1}{8}\right ){{}_{2}F_{1}\left (\begin{matrix} - \frac{1}{8}, \frac{1}{4} \\ \frac{7}{8} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} c^{\frac{5}{4}} \sqrt [4]{x} \Gamma \left (\frac{7}{8}\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

gamma(-1/8)*hyper((-1/8, 1/4), (7/8,), b*x**2*exp_polar(I*pi)/a)/(2*a**(1/4)*c**(5/4)*x**(1/4)*gamma(7/8))

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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{1}{4}} \left (c x\right )^{\frac{5}{4}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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