3.128 \(\int \frac{1}{(a+b x^3)^{5/4} \sqrt [12]{c+d x^3}} \, dx\)

Optimal. Leaf size=87 \[ \frac{x \left (c+d x^3\right )^{11/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]

[Out]

(x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 5/4, 4/3, -(((b*c - a*d)
*x^3)/(a*(c + d*x^3)))])/(c*(a + b*x^3)^(5/4))

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Rubi [A]  time = 0.0163963, 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} \[ \frac{x \left (c+d x^3\right )^{11/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^(5/4)*(c + d*x^3)^(1/12)),x]

[Out]

(x*((c*(a + b*x^3))/(a*(c + d*x^3)))^(5/4)*(c + d*x^3)^(11/12)*Hypergeometric2F1[1/3, 5/4, 4/3, -(((b*c - a*d)
*x^3)/(a*(c + d*x^3)))])/(c*(a + b*x^3)^(5/4))

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}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx &=\frac{x \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \left (c+d x^3\right )^{11/12} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \left (a+b x^3\right )^{5/4}}\\ \end{align*}

Mathematica [A]  time = 0.0220741, size = 89, normalized size = 1.02 \[ \frac{x \sqrt [4]{\frac{b x^3}{a}+1} \, _2F_1\left (\frac{1}{3},\frac{5}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{a \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3} \sqrt [4]{\frac{d x^3}{c}+1}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(5/4)*(c + d*x^3)^(1/12)),x]

[Out]

(x*(1 + (b*x^3)/a)^(1/4)*Hypergeometric2F1[1/3, 5/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(a*(a + b*x^3
)^(1/4)*(c + d*x^3)^(1/12)*(1 + (d*x^3)/c)^(1/4))

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Maple [F]  time = 0.449, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{3}+a \right ) ^{-{\frac{5}{4}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{1}{12}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(5/4)/(d*x^3+c)^(1/12),x)

[Out]

int(1/(b*x^3+a)^(5/4)/(d*x^3+c)^(1/12),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{5}{4}}{\left (d x^{3} + c\right )}^{\frac{1}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(5/4)/(d*x^3+c)^(1/12),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(5/4)*(d*x^3 + c)^(1/12)), 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^{2} d x^{9} +{\left (b^{2} c + 2 \, a b d\right )} x^{6} +{\left (2 \, a b c + a^{2} d\right )} x^{3} + a^{2} c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(5/4)/(d*x^3+c)^(1/12),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(3/4)*(d*x^3 + c)^(11/12)/(b^2*d*x^9 + (b^2*c + 2*a*b*d)*x^6 + (2*a*b*c + a^2*d)*x^3 + a^
2*c), 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)**(5/4)/(d*x**3+c)**(1/12),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{5}{4}}{\left (d x^{3} + c\right )}^{\frac{1}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(5/4)/(d*x^3+c)^(1/12),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(5/4)*(d*x^3 + c)^(1/12)), x)