3.1001 \(\int \frac {1}{\sqrt [4]{c x} \sqrt [4]{a+b x^2}} \, dx\)

Optimal. Leaf size=58 \[ \frac {4 (c x)^{3/4} \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {3}{8};\frac {11}{8};-\frac {b x^2}{a}\right )}{3 c \sqrt [4]{a+b x^2}} \]

[Out]

4/3*(c*x)^(3/4)*(1+b*x^2/a)^(1/4)*hypergeom([1/4, 3/8],[11/8],-b*x^2/a)/c/(b*x^2+a)^(1/4)

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Rubi [A]  time = 0.02, antiderivative size = 58, normalized size of antiderivative = 1.00, 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} \[ \frac {4 (c x)^{3/4} \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {3}{8};\frac {11}{8};-\frac {b x^2}{a}\right )}{3 c \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*(c*x)^(3/4)*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/8, 11/8, -((b*x^2)/a)])/(3*c*(a + b*x^2)^(1/4))

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])

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])

Rubi steps

\begin {align*} \int \frac {1}{\sqrt [4]{c x} \sqrt [4]{a+b x^2}} \, dx &=\frac {\sqrt [4]{1+\frac {b x^2}{a}} \int \frac {1}{\sqrt [4]{c x} \sqrt [4]{1+\frac {b x^2}{a}}} \, dx}{\sqrt [4]{a+b x^2}}\\ &=\frac {4 (c x)^{3/4} \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (\frac {1}{4},\frac {3}{8};\frac {11}{8};-\frac {b x^2}{a}\right )}{3 c \sqrt [4]{a+b x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.01, size = 56, normalized size = 0.97 \[ \frac {4 x \sqrt [4]{\frac {b x^2}{a}+1} \, _2F_1\left (\frac {1}{4},\frac {3}{8};\frac {11}{8};-\frac {b x^2}{a}\right )}{3 \sqrt [4]{c x} \sqrt [4]{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(4*x*(1 + (b*x^2)/a)^(1/4)*Hypergeometric2F1[1/4, 3/8, 11/8, -((b*x^2)/a)])/(3*(c*x)^(1/4)*(a + b*x^2)^(1/4))

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fricas [F]  time = 0.72, size = 0, normalized size = 0.00 \[ {\rm integral}\left (\frac {{\left (b x^{2} + a\right )}^{\frac {3}{4}} \left (c x\right )^{\frac {3}{4}}}{b c x^{3} + a c x}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 0.29, size = 0, normalized size = 0.00 \[ \int \frac {1}{\left (c x \right )^{\frac {1}{4}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \[ \int \frac {1}{{\left (c\,x\right )}^{1/4}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [C]  time = 1.30, size = 44, normalized size = 0.76 \[ \frac {x^{\frac {3}{4}} \Gamma \left (\frac {3}{8}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {3}{8} \\ \frac {11}{8} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 \sqrt [4]{a} \sqrt [4]{c} \Gamma \left (\frac {11}{8}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**(3/4)*gamma(3/8)*hyper((1/4, 3/8), (11/8,), b*x**2*exp_polar(I*pi)/a)/(2*a**(1/4)*c**(1/4)*gamma(11/8))

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