Integrand size = 39, antiderivative size = 566 \[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} (d+e x)^{2/3} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{c d^2-a e^2} \sqrt [3]{1+\frac {e x}{d}}+c^{2/3} d^{4/3} \left (1+\frac {e x}{d}\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}}} \]
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Time = 0.49 (sec) , antiderivative size = 566, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.128, Rules used = {693, 691, 52, 65, 231} \[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {3^{3/4} (d+e x)^{2/3} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1} \sqrt [3]{c d^2-a e^2}+c^{2/3} d^{4/3} \left (\frac {e x}{d}+1\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )^2}} \operatorname {EllipticF}\left (\arccos \left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}}\right ),\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{\frac {e x}{d}+1}\right )^2}}}+\frac {3 (d+e x)^{2/3} (a e+c d x)}{2 c d \sqrt {x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]
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Rule 52
Rule 65
Rule 231
Rule 691
Rule 693
Rubi steps \begin{align*} \text {integral}& = \frac {(d+e x)^{2/3} \int \frac {\left (1+\frac {e x}{d}\right )^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx}{\left (1+\frac {e x}{d}\right )^{2/3}} \\ & = \frac {\left (\sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac {\sqrt [6]{1+\frac {e x}{d}}}{\sqrt {a d e+c d^2 x}} \, dx}{\sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (\left (1-\frac {a e^2}{c d^2}\right ) \sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \int \frac {1}{\sqrt {a d e+c d^2 x} \left (1+\frac {e x}{d}\right )^{5/6}} \, dx}{4 \sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {\left (3 d \left (1-\frac {a e^2}{c d^2}\right ) \sqrt {a d e+c d^2 x} (d+e x)^{2/3}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {-\frac {c d^3}{e}+a d e+\frac {c d^3 x^6}{e}}} \, dx,x,\sqrt [6]{1+\frac {e x}{d}}\right )}{2 e \sqrt [6]{1+\frac {e x}{d}} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \\ & = \frac {3 (a e+c d x) (d+e x)^{2/3}}{2 c d \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}+\frac {3^{3/4} \left (c d^2-a e^2\right )^{2/3} \sqrt {a d e+c d^2 x} (d+e x)^{2/3} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right ) \sqrt {\frac {\left (c d^2-a e^2\right )^{2/3}+\sqrt [3]{c} d^{2/3} \sqrt [3]{c d^2-a e^2} \sqrt [3]{1+\frac {e x}{d}}+c^{2/3} d^{4/3} \left (1+\frac {e x}{d}\right )^{2/3}}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}} F\left (\cos ^{-1}\left (\frac {\sqrt [3]{c d^2-a e^2}-\left (1-\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}{\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{4 c d e \sqrt {d (a e+c d x)} \sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \sqrt {-\frac {\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}} \left (\sqrt [3]{c d^2-a e^2}-\sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )}{\left (\sqrt [3]{c d^2-a e^2}-\left (1+\sqrt {3}\right ) \sqrt [3]{c} d^{2/3} \sqrt [3]{1+\frac {e x}{d}}\right )^2}}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.05 (sec) , antiderivative size = 95, normalized size of antiderivative = 0.17 \[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\frac {2 \sqrt {(a e+c d x) (d+e x)} \operatorname {Hypergeometric2F1}\left (-\frac {1}{6},\frac {1}{2},\frac {3}{2},\frac {e (a e+c d x)}{-c d^2+a e^2}\right )}{c d \sqrt [3]{d+e x} \sqrt [6]{\frac {c d (d+e x)}{c d^2-a e^2}}} \]
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\[\int \frac {\left (e x +d \right )^{\frac {2}{3}}}{\sqrt {a d e +\left (e^{2} a +c \,d^{2}\right ) x +c d e \,x^{2}}}d x\]
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\[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {2}{3}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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\[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {\left (d + e x\right )^{\frac {2}{3}}}{\sqrt {\left (d + e x\right ) \left (a e + c d x\right )}}\, dx \]
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\[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {2}{3}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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\[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int { \frac {{\left (e x + d\right )}^{\frac {2}{3}}}{\sqrt {c d e x^{2} + a d e + {\left (c d^{2} + a e^{2}\right )} x}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^{2/3}}{\sqrt {a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx=\int \frac {{\left (d+e\,x\right )}^{2/3}}{\sqrt {c\,d\,e\,x^2+\left (c\,d^2+a\,e^2\right )\,x+a\,d\,e}} \,d x \]
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