Integrand size = 19, antiderivative size = 246 \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {663 a^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{154 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \]
[Out]
Time = 0.25 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2043, 2048, 2050, 2036, 335, 226} \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {663 a^{15/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{154 b^{21/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {663 a^3 \sqrt {a x+b \sqrt [3]{x}}}{77 b^5 x^{2/3}}-\frac {1989 a^2 \sqrt {a x+b \sqrt [3]{x}}}{385 b^4 x^{4/3}}+\frac {221 a \sqrt {a x+b \sqrt [3]{x}}}{55 b^3 x^2}-\frac {17 \sqrt {a x+b \sqrt [3]{x}}}{5 b^2 x^{8/3}}+\frac {3}{b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}} \]
[In]
[Out]
Rule 226
Rule 335
Rule 2036
Rule 2043
Rule 2048
Rule 2050
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {1}{x^7 \left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}+\frac {51 \text {Subst}\left (\int \frac {1}{x^8 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 b} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}-\frac {(221 a) \text {Subst}\left (\int \frac {1}{x^6 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 b^2} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}+\frac {\left (1989 a^2\right ) \text {Subst}\left (\int \frac {1}{x^4 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{110 b^3} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}-\frac {\left (1989 a^3\right ) \text {Subst}\left (\int \frac {1}{x^2 \sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^4} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {x} \sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{154 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {\left (663 a^4 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{77 b^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {3}{b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}-\frac {17 \sqrt {b \sqrt [3]{x}+a x}}{5 b^2 x^{8/3}}+\frac {221 a \sqrt {b \sqrt [3]{x}+a x}}{55 b^3 x^2}-\frac {1989 a^2 \sqrt {b \sqrt [3]{x}+a x}}{385 b^4 x^{4/3}}+\frac {663 a^3 \sqrt {b \sqrt [3]{x}+a x}}{77 b^5 x^{2/3}}+\frac {663 a^{15/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{154 b^{21/4} \sqrt {b \sqrt [3]{x}+a x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 10.06 (sec) , antiderivative size = 64, normalized size of antiderivative = 0.26 \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {2 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (-\frac {15}{4},\frac {3}{2},-\frac {11}{4},-\frac {a x^{2/3}}{b}\right )}{5 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}} \]
[In]
[Out]
Time = 5.33 (sec) , antiderivative size = 231, normalized size of antiderivative = 0.94
method | result | size |
derivativedivides | \(-\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{5 b^{2} x^{\frac {8}{3}}}+\frac {56 a \sqrt {b \,x^{\frac {1}{3}}+a x}}{55 b^{3} x^{2}}-\frac {834 a^{2} \sqrt {b \,x^{\frac {1}{3}}+a x}}{385 b^{4} x^{\frac {4}{3}}}+\frac {432 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{77 b^{5} x^{\frac {2}{3}}}+\frac {3 x^{\frac {1}{3}} a^{4}}{b^{5} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {663 a^{3} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{154 b^{5} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(231\) |
default | \(\frac {3315 x^{\frac {14}{3}} \sqrt {-a b}\, \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3}-884 x^{\frac {11}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b^{2}+2652 x^{\frac {13}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} b +476 x^{3} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{3}+2310 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{5} a^{4}+4320 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{4} x^{5}-308 x^{\frac {7}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b^{4}}{770 b^{5} x^{5} \left (b +a \,x^{\frac {2}{3}}\right )}\) | \(261\) |
[In]
[Out]
\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^{3} \left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {1}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}} x^{3}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{x^3 \left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {1}{x^3\,{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
[In]
[Out]