Integrand size = 19, antiderivative size = 349 \[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {77 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}-\frac {77 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 b^{9/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{10 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Time = 0.29 (sec) , antiderivative size = 349, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2043, 2047, 2049, 2057, 335, 311, 226, 1210} \[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {77 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{10 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 b^{9/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{15/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {77 b^2 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{5 a^{7/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}-\frac {77 b \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{15 a^3}+\frac {11 x \sqrt {a x+b \sqrt [3]{x}}}{3 a^2}-\frac {3 x^2}{a \sqrt {a x+b \sqrt [3]{x}}} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2047
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^8}{\left (b x+a x^3\right )^{3/2}} \, dx,x,\sqrt [3]{x}\right ) \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {33 \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{2 a} \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}-\frac {(77 b) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{6 a^2} \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}+\frac {\left (77 b^2\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{10 a^3} \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}+\frac {\left (77 b^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {\sqrt {x}}{\sqrt {b+a x^2}} \, dx,x,\sqrt [3]{x}\right )}{10 a^3 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}+\frac {\left (77 b^2 \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {x^2}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 a^3 \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}+\frac {\left (77 b^{5/2} \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 )}{5 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}}-\frac {\left (77 b^{5/2} \sqrt {b+a x^{2/3}} \sqrt [6]{x}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt {a} x^2}{\sqrt {b}}}{\sqrt {b+a x^4}} \, dx,x,\sqrt [6]{x}\right )}{5 a^{7/2} \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {77 b^2 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{5 a^{7/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}-\frac {3 x^2}{a \sqrt {b \sqrt [3]{x}+a x}}-\frac {77 b \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{15 a^3}+\frac {11 x \sqrt {b \sqrt [3]{x}+a x}}{3 a^2}-\frac {77 b^{9/4} \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} E\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{5 a^{15/4} \sqrt {b \sqrt [3]{x}+a x}}+\frac {77 b^{9/4} \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 )}{10 a^{15/4} \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 = 10.09 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.27 \[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\frac {2 x^{2/3} \left (77 b^2-11 a b x^{2/3}+5 a^2 x^{4/3}-77 b^2 \sqrt {1+\frac {a x^{2/3}}{b}} \operatorname {Hypergeometric2F1}\left (\frac {3}{4},\frac {3}{2},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )\right )}{15 a^3 \sqrt {b \sqrt [3]{x}+a x}} \]
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Time = 2.15 (sec) , antiderivative size = 237, normalized size of antiderivative = 0.68
method | result | size |
derivativedivides | \(-\frac {3 x^{\frac {2}{3}} b^{2}}{a^{3} \sqrt {\left (x^{\frac {2}{3}}+\frac {b}{a}\right ) x^{\frac {1}{3}} a}}+\frac {2 x \sqrt {b \,x^{\frac {1}{3}}+a x}}{3 a^{2}}-\frac {32 b \,x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{15 a^{3}}+\frac {77 b^{2} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{10 a^{4} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(237\) |
default | \(-\frac {-462 b^{3} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, E\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+231 b^{3} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )+90 \sqrt {b \,x^{\frac {1}{3}}+a x}\, x^{\frac {2}{3}} a \,b^{2}+64 x^{\frac {2}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a \,b^{2}+44 x^{\frac {4}{3}} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{2} b -20 \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\, a^{3} x^{2}}{30 a^{4} x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}\) | \(312\) |
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\[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x^{2}}{\left (a x + b \sqrt [3]{x}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int { \frac {x^{2}}{{\left (a x + b x^{\frac {1}{3}}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x^2}{\left (b \sqrt [3]{x}+a x\right )^{3/2}} \, dx=\int \frac {x^2}{{\left (a\,x+b\,x^{1/3}\right )}^{3/2}} \,d x \]
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