\(\int \frac {1}{(a+b x^3)^{5/4} \sqrt [12]{c+d x^3}} \, dx\) [128]

Optimal result

Integrand size = 23, antiderivative size = 87 \[ \int \frac {1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx=\frac {x \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \left (c+d x^3\right )^{11/12} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{4},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {388} \[ \int \frac {1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx=\frac {x \left (c+d x^3\right )^{11/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{5/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{4},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{c \left (a+b x^3\right )^{5/4}} \]

[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 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[x*((a + b*x^n)^p/(c*(c*((a
+ b*x^n)/(a*(c + d*x^n))))^p*(c + d*x^n)^(1/n + p)))*Hypergeometric2F1[1/n, -p, 1 + 1/n, (-(b*c - a*d))*(x^n/(
a*(c + d*x^n)))], 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*} \text {integral}& = \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] (warning: unable to verify)

Time = 3.49 (sec) , antiderivative size = 89, normalized size of antiderivative = 1.02 \[ \int \frac {1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx=\frac {x \sqrt [4]{1+\frac {b x^3}{a}} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {5}{4},\frac {4}{3},\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{a \sqrt [4]{a+b x^3} \sqrt [12]{c+d x^3} \sqrt [4]{1+\frac {d x^3}{c}}} \]

[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))

Maple [F]

\[\int \frac {1}{\left (b \,x^{3}+a \right )^{\frac {5}{4}} \left (d \,x^{3}+c \right )^{\frac {1}{12}}}d x\]

[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)

Fricas [F]

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

[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)

Sympy [F]

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

[In]

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

[Out]

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

Maxima [F]

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

[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)

Giac [F]

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

[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)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b x^3\right )^{5/4} \sqrt [12]{c+d x^3}} \, dx=\int \frac {1}{{\left (b\,x^3+a\right )}^{5/4}\,{\left (d\,x^3+c\right )}^{1/12}} \,d x \]

[In]

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

[Out]

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