Integrand size = 23, antiderivative size = 155 \[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}+\frac {60 a x \sqrt [4]{a+b x^3}}{133 c^2 \left (c+d x^3\right )^{7/12}}+\frac {45 a^2 x \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \left (c+d x^3\right )^{5/12} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (c+d x^3\right )}\right )}{133 c^3 \left (a+b x^3\right )^{3/4}} \]
4/19*x*(b*x^3+a)^(5/4)/c/(d*x^3+c)^(19/12)+60/133*a*x*(b*x^3+a)^(1/4)/c^2/ (d*x^3+c)^(7/12)+45/133*a^2*x*(c*(b*x^3+a)/a/(d*x^3+c))^(3/4)*(d*x^3+c)^(5 /12)*hypergeom([1/3, 3/4],[4/3],-(-a*d+b*c)*x^3/a/(d*x^3+c))/c^3/(b*x^3+a) ^(3/4)
Time = 5.46 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.58 \[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\frac {a x \sqrt [4]{a+b x^3} \sqrt [4]{1+\frac {d x^3}{c}} \operatorname {Hypergeometric2F1}\left (-\frac {5}{4},\frac {1}{3},\frac {4}{3},\frac {(-b c+a d) x^3}{a \left (c+d x^3\right )}\right )}{c^2 \sqrt [4]{1+\frac {b x^3}{a}} \left (c+d x^3\right )^{7/12}} \]
(a*x*(a + b*x^3)^(1/4)*(1 + (d*x^3)/c)^(1/4)*Hypergeometric2F1[-5/4, 1/3, 4/3, ((-(b*c) + a*d)*x^3)/(a*(c + d*x^3))])/(c^2*(1 + (b*x^3)/a)^(1/4)*(c + d*x^3)^(7/12))
Time = 0.25 (sec) , antiderivative size = 161, normalized size of antiderivative = 1.04, number of steps used = 3, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.130, Rules used = {903, 903, 905}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {15 a \int \frac {\sqrt [4]{b x^3+a}}{\left (d x^3+c\right )^{19/12}}dx}{19 c}+\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}\) |
\(\Big \downarrow \) 903 |
\(\displaystyle \frac {15 a \left (\frac {3 a \int \frac {1}{\left (b x^3+a\right )^{3/4} \left (d x^3+c\right )^{7/12}}dx}{7 c}+\frac {4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}\right )}{19 c}+\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}\) |
\(\Big \downarrow \) 905 |
\(\displaystyle \frac {15 a \left (\frac {3 a x \left (c+d x^3\right )^{5/12} \left (\frac {c \left (a+b x^3\right )}{a \left (c+d x^3\right )}\right )^{3/4} \operatorname {Hypergeometric2F1}\left (\frac {1}{3},\frac {3}{4},\frac {4}{3},-\frac {(b c-a d) x^3}{a \left (d x^3+c\right )}\right )}{7 c^2 \left (a+b x^3\right )^{3/4}}+\frac {4 x \sqrt [4]{a+b x^3}}{7 c \left (c+d x^3\right )^{7/12}}\right )}{19 c}+\frac {4 x \left (a+b x^3\right )^{5/4}}{19 c \left (c+d x^3\right )^{19/12}}\) |
(4*x*(a + b*x^3)^(5/4))/(19*c*(c + d*x^3)^(19/12)) + (15*a*((4*x*(a + b*x^ 3)^(1/4))/(7*c*(c + d*x^3)^(7/12)) + (3*a*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)))])/(7*c^2*(a + b*x^3)^(3/4))))/(19*c)
3.2.23.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Simp[(-x)*(a + b*x^n)^(p + 1)*((c + d*x^n)^q/(a*n*(p + 1))), x] - Simp[ c*(q/(a*(p + 1))) Int[(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 1), x], x] /; FreeQ[{a, b, c, d, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[n*(p + q + 1) + 1, 0] && GtQ[q, 0] && NeQ[p, -1]
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]
\[\int \frac {\left (b \,x^{3}+a \right )^{\frac {5}{4}}}{\left (d \,x^{3}+c \right )^{\frac {31}{12}}}d x\]
\[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{4}}}{{\left (d x^{3} + c\right )}^{\frac {31}{12}}} \,d x } \]
integral((b*x^3 + a)^(5/4)*(d*x^3 + c)^(5/12)/(d^3*x^9 + 3*c*d^2*x^6 + 3*c ^2*d*x^3 + c^3), x)
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\text {Timed out} \]
\[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{4}}}{{\left (d x^{3} + c\right )}^{\frac {31}{12}}} \,d x } \]
\[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\int { \frac {{\left (b x^{3} + a\right )}^{\frac {5}{4}}}{{\left (d x^{3} + c\right )}^{\frac {31}{12}}} \,d x } \]
Timed out. \[ \int \frac {\left (a+b x^3\right )^{5/4}}{\left (c+d x^3\right )^{31/12}} \, dx=\int \frac {{\left (b\,x^3+a\right )}^{5/4}}{{\left (d\,x^3+c\right )}^{31/12}} \,d x \]