3.127 \(\int \frac{1}{(a+b x^3)^{3/4} (c+d x^3)^{7/12}} \, dx\)

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

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0173785, 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} \[ \frac{x \left (c+d x^3\right )^{5/12} \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};-\frac{(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{3/4}} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12)),x]

[Out]

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

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 )^{3/4} \left (c+d x^3\right )^{7/12}} \, dx &=\frac{x \left (\frac{c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \left (c+d x^3\right )^{5/12} \, _2F_1\left (\frac{1}{3},\frac{3}{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 )^{3/4}}\\ \end{align*}

Mathematica [A]  time = 0.0212314, size = 86, normalized size = 0.99 \[ \frac{x \left (\frac{b x^3}{a}+1\right )^{3/4} \sqrt [4]{\frac{d x^3}{c}+1} \, _2F_1\left (\frac{1}{3},\frac{3}{4};\frac{4}{3};\frac{(a d-b c) x^3}{a \left (d x^3+c\right )}\right )}{\left (a+b x^3\right )^{3/4} \left (c+d x^3\right )^{7/12}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[1/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12)),x]

[Out]

(x*(1 + (b*x^3)/a)^(3/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[1/3, 3/4, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d
*x^3))])/((a + b*x^3)^(3/4)*(c + d*x^3)^(7/12))

________________________________________________________________________________________

Maple [F]  time = 0.433, size = 0, normalized size = 0. \begin{align*} \int{ \left ( b{x}^{3}+a \right ) ^{-{\frac{3}{4}}} \left ( d{x}^{3}+c \right ) ^{-{\frac{7}{12}}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/(b*x^3+a)^(3/4)/(d*x^3+c)^(7/12),x)

[Out]

int(1/(b*x^3+a)^(3/4)/(d*x^3+c)^(7/12),x)

________________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{7}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(3/4)/(d*x^3+c)^(7/12),x, algorithm="maxima")

[Out]

integrate(1/((b*x^3 + a)^(3/4)*(d*x^3 + c)^(7/12)), x)

________________________________________________________________________________________

Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (b x^{3} + a\right )}^{\frac{1}{4}}{\left (d x^{3} + c\right )}^{\frac{5}{12}}}{b d x^{6} +{\left (b c + a d\right )} x^{3} + a c}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(3/4)/(d*x^3+c)^(7/12),x, algorithm="fricas")

[Out]

integral((b*x^3 + a)^(1/4)*(d*x^3 + c)^(5/12)/(b*d*x^6 + (b*c + a*d)*x^3 + a*c), x)

________________________________________________________________________________________

Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\left (a + b x^{3}\right )^{\frac{3}{4}} \left (c + d x^{3}\right )^{\frac{7}{12}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x**3+a)**(3/4)/(d*x**3+c)**(7/12),x)

[Out]

Integral(1/((a + b*x**3)**(3/4)*(c + d*x**3)**(7/12)), x)

________________________________________________________________________________________

Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{{\left (b x^{3} + a\right )}^{\frac{3}{4}}{\left (d x^{3} + c\right )}^{\frac{7}{12}}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/(b*x^3+a)^(3/4)/(d*x^3+c)^(7/12),x, algorithm="giac")

[Out]

integrate(1/((b*x^3 + a)^(3/4)*(d*x^3 + c)^(7/12)), x)