3.126 \(\int \frac{1}{\sqrt [4]{a+b x^3} (c+d x^3)^{13/12}} \, dx\)

Optimal. Leaf size=87 \[ \frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]

[Out]

(x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/4)*Hypergeometric2F1[1/4, 1/3, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)
))])/(c*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12))

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Rubi [A]  time = 0.0168134, antiderivative size = 87, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.043, Rules used = {380} \[ \frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12)),x]

[Out]

(x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(1/4)*Hypergeometric2F1[1/4, 1/3, 4/3, -(((b*c - a*d)*x^3)/(a*(c + d*x^3)
))])/(c*(a + b*x^3)^(1/4)*(c + d*x^3)^(1/12))

Rule 380

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(x*(a + b*x^n)^p*Hypergeome
tric2F1[1/n, -p, 1 + 1/n, -(((b*c - a*d)*x^n)/(a*(c + d*x^n)))])/(c*((c*(a + b*x^n))/(a*(c + d*x^n)))^p*(c + d
*x^n)^(1/n + p)), x] /; FreeQ[{a, b, c, d, n, p, q}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0]

Rubi steps

\begin{align*} \int \frac{1}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \, dx &=\frac{x \sqrt [4]{\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3}}\\ \end{align*}

Mathematica [A]  time = 0.0287124, size = 86, normalized size = 0.99 \[ \frac{x \sqrt [4]{\frac{b x^3}{a}+1} \left (\frac{d x^3}{c}+1\right )^{3/4} \, _2F_1\left (\frac{1}{4},\frac{1}{3};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\sqrt [4]{a+b x^3} \left (c+d x^3\right )^{13/12}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12)),x]

[Out]

(x*(1 + (b*x^3)/a)^(1/4)*(1 + (d*x^3)/c)^(3/4)*Hypergeometric2F1[1/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d
*x^3))])/((a + b*x^3)^(1/4)*(c + d*x^3)^(13/12))

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Maple [F]  time = 0.446, size = 0, normalized size = 0. \begin{align*} \int{{\frac{1}{\sqrt [4]{b{x}^{3}+a}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{13}{12}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(1/4)/(d*x^3+c)^(13/12),x)

[Out]

int(1/(b*x^3+a)^(1/4)/(d*x^3+c)^(13/12),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{1}{4}}{\left (d x^{3} + c\right )}^{\frac{13}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(1/4)/(d*x^3+c)^(13/12),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(1/4)*(d*x^3 + c)^(13/12)), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{11}{12}}}{b d^{2} x^{9} +{\left (2 \, b c d + a d^{2}\right )} x^{6} +{\left (b c^{2} + 2 \, a c d\right )} x^{3} + a c^{2}}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(1/4)/(d*x^3+c)^(13/12),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(3/4)*(d*x^3 + c)^(11/12)/(b*d^2*x^9 + (2*b*c*d + a*d^2)*x^6 + (b*c^2 + 2*a*c*d)*x^3 + a*
c^2), x)

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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/(b*x**3+a)**(1/4)/(d*x**3+c)**(13/12),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^{3} + a\right )}^{\frac{1}{4}}{\left (d x^{3} + c\right )}^{\frac{13}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(1/4)/(d*x^3+c)^(13/12),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(1/4)*(d*x^3 + c)^(13/12)), x)