Integrand size = 20, antiderivative size = 141 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=-\frac {a \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (-\frac {2}{3},-\frac {3}{2},-\frac {3}{2},\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \]
[Out]
Time = 0.08 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {1399, 524} \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=-\frac {a \sqrt {a+b x^3+c x^6} \operatorname {AppellF1}\left (-\frac {2}{3},-\frac {3}{2},-\frac {3}{2},\frac {1}{3},-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}+1} \sqrt {\frac {2 c x^3}{\sqrt {b^2-4 a c}+b}+1}} \]
[In]
[Out]
Rule 524
Rule 1399
Rubi steps \begin{align*} \text {integral}& = \frac {\left (a \sqrt {a+b x^3+c x^6}\right ) \int \frac {\left (1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}\right )^{3/2} \left (1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )^{3/2}}{x^3} \, dx}{\sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \\ & = -\frac {a \sqrt {a+b x^3+c x^6} F_1\left (-\frac {2}{3};-\frac {3}{2},-\frac {3}{2};\frac {1}{3};-\frac {2 c x^3}{b-\sqrt {b^2-4 a c}},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}\right )}{2 x^2 \sqrt {1+\frac {2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {1+\frac {2 c x^3}{b+\sqrt {b^2-4 a c}}}} \\ \end{align*}
Leaf count is larger than twice the leaf count of optimal. \(379\) vs. \(2(141)=282\).
Time = 10.35 (sec) , antiderivative size = 379, normalized size of antiderivative = 2.69 \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\frac {8 \left (-28 a^2-11 a b x^3+17 b^2 x^6-20 a c x^6+25 b c x^9+8 c^2 x^{12}\right )+648 a b x^3 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {1}{3},\frac {1}{2},\frac {1}{2},\frac {4}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )+27 \left (b^2+8 a c\right ) x^6 \sqrt {\frac {b-\sqrt {b^2-4 a c}+2 c x^3}{b-\sqrt {b^2-4 a c}}} \sqrt {\frac {b+\sqrt {b^2-4 a c}+2 c x^3}{b+\sqrt {b^2-4 a c}}} \operatorname {AppellF1}\left (\frac {4}{3},\frac {1}{2},\frac {1}{2},\frac {7}{3},-\frac {2 c x^3}{b+\sqrt {b^2-4 a c}},\frac {2 c x^3}{-b+\sqrt {b^2-4 a c}}\right )}{448 x^2 \sqrt {a+b x^3+c x^6}} \]
[In]
[Out]
\[\int \frac {\left (c \,x^{6}+b \,x^{3}+a \right )^{\frac {3}{2}}}{x^{3}}d x\]
[In]
[Out]
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\int \frac {\left (a + b x^{3} + c x^{6}\right )^{\frac {3}{2}}}{x^{3}}\, dx \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\int { \frac {{\left (c x^{6} + b x^{3} + a\right )}^{\frac {3}{2}}}{x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a+b x^3+c x^6\right )^{3/2}}{x^3} \, dx=\int \frac {{\left (c\,x^6+b\,x^3+a\right )}^{3/2}}{x^3} \,d x \]
[In]
[Out]