Integrand size = 17, antiderivative size = 149 \[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{a^2}-\frac {5 b^{3/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{9/4} \sqrt {b \sqrt [3]{x}+a x}} \]
[Out]
Time = 0.13 (sec) , antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2043, 2047, 2049, 2036, 335, 226} \[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=-\frac {5 b^{3/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{2 a^{9/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {5 \sqrt {a x+b \sqrt [3]{x}}}{a^2}-\frac {3 x}{a \sqrt {a x+b \sqrt [3]{x}}} \]
[In]
[Out]
Rule 226
Rule 335
Rule 2036
Rule 2043
Rule 2047
Rule 2049
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^5}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {15 \text {Subst}\left (\int \frac {x^2}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a} \\ & = -\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{a^2}-\frac {(5 b) \text {Subst}\left (\int \frac {1}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a^2} \\ & = -\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{a^2}-\frac {\left (5 b \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 )}{2 a^2 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{a^2}-\frac {\left (5 b \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 )}{a^2 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {3 x}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {5 \sqrt {b \sqrt [3]{x}+a x}}{a^2}-\frac {5 b^{3/4} \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 )}{2 a^{9/4} \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 = 10.07 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.55 \[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {\sqrt {b \sqrt [3]{x}+a x} \left (5 b+2 a x^{2/3}-5 b \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {5}{4},-\frac {a x^{2/3}}{b}\right )\right )}{a^2 \left (b+a x^{2/3}\right )} \]
[In]
[Out]
Time = 2.08 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.07
method | result | size |
derivativedivides | \(\frac {3 x^{\frac {1}{3}} b}{a^{2} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 \sqrt {b \,x^{\frac {1}{3}}+a x}}{a^{2}}-\frac {5 b \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{2 a^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(160\) |
default | \(-\frac {5 \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right ) \sqrt {-a b}\, \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, b -6 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {1}{3}} a b -4 x^{\frac {1}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a b -4 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} x}{2 x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right ) a^{3}}\) | \(184\) |
[In]
[Out]
\[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
[In]
[Out]
\[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
\[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {x}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
[In]
[Out]