Integrand size = 19, antiderivative size = 414 \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=-\frac {418 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{11/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {418 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {209 b^{21/4} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {\frac {b+a x^{2/3}}{\left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right )^2}} \sqrt [6]{x} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}} \]
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Time = 0.39 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 11, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {2043, 2049, 2057, 335, 311, 226, 1210} \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=-\frac {209 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} \operatorname {EllipticF}\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right ),\frac {1}{2}\right )}{221 a^{23/4} \sqrt {a x+b \sqrt [3]{x}}}+\frac {418 b^{21/4} \sqrt [6]{x} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {\frac {a x^{2/3}+b}{\left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right )^2}} E\left (2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [6]{x}}{\sqrt [4]{b}}\right )|\frac {1}{2}\right )}{221 a^{23/4} \sqrt {a x+b \sqrt [3]{x}}}-\frac {418 b^5 \sqrt [3]{x} \left (a x^{2/3}+b\right )}{221 a^{11/2} \left (\sqrt {a} \sqrt [3]{x}+\sqrt {b}\right ) \sqrt {a x+b \sqrt [3]{x}}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {a x+b \sqrt [3]{x}}}{663 a^5}-\frac {2090 b^3 x \sqrt {a x+b \sqrt [3]{x}}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {a x+b \sqrt [3]{x}}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {a x+b \sqrt [3]{x}}}{119 a^2}+\frac {2 x^3 \sqrt {a x+b \sqrt [3]{x}}}{7 a} \]
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Rule 226
Rule 311
Rule 335
Rule 1210
Rule 2043
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = 3 \text {Subst}\left (\int \frac {x^{11}}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right ) \\ & = \frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {(19 b) \text {Subst}\left (\int \frac {x^9}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{7 a} \\ & = -\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {\left (285 b^2\right ) \text {Subst}\left (\int \frac {x^7}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{119 a^2} \\ & = \frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (3135 b^3\right ) \text {Subst}\left (\int \frac {x^5}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{1547 a^3} \\ & = -\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {\left (1045 b^4\right ) \text {Subst}\left (\int \frac {x^3}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{663 a^4} \\ & = \frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (209 b^5\right ) \text {Subst}\left (\int \frac {x}{\sqrt {b x+a x^3}} \, dx,x,\sqrt [3]{x}\right )}{221 a^5} \\ & = \frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (209 b^5 \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 )}{221 a^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (418 b^5 \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 )}{221 a^5 \sqrt {b \sqrt [3]{x}+a x}} \\ & = \frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}-\frac {\left (418 b^{11/2} \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 )}{221 a^{11/2} \sqrt {b \sqrt [3]{x}+a x}}+\frac {\left (418 b^{11/2} \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 )}{221 a^{11/2} \sqrt {b \sqrt [3]{x}+a x}} \\ & = -\frac {418 b^5 \left (b+a x^{2/3}\right ) \sqrt [3]{x}}{221 a^{11/2} \left (\sqrt {b}+\sqrt {a} \sqrt [3]{x}\right ) \sqrt {b \sqrt [3]{x}+a x}}+\frac {418 b^4 \sqrt [3]{x} \sqrt {b \sqrt [3]{x}+a x}}{663 a^5}-\frac {2090 b^3 x \sqrt {b \sqrt [3]{x}+a x}}{4641 a^4}+\frac {570 b^2 x^{5/3} \sqrt {b \sqrt [3]{x}+a x}}{1547 a^3}-\frac {38 b x^{7/3} \sqrt {b \sqrt [3]{x}+a x}}{119 a^2}+\frac {2 x^3 \sqrt {b \sqrt [3]{x}+a x}}{7 a}+\frac {418 b^{21/4} \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 )}{221 a^{23/4} \sqrt {b \sqrt [3]{x}+a x}}-\frac {209 b^{21/4} \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 )}{221 a^{23/4} \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 = 143, normalized size of antiderivative = 0.35 \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\frac {2 \sqrt {b \sqrt [3]{x}+a x} \left (1463 b^5 \sqrt [3]{x}+418 a b^4 x-190 a^2 b^3 x^{5/3}+114 a^3 b^2 x^{7/3}-78 a^4 b x^3+663 a^5 x^{11/3}-1463 b^5 \sqrt {1+\frac {a x^{2/3}}{b}} \sqrt [3]{x} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},\frac {3}{4},\frac {7}{4},-\frac {a x^{2/3}}{b}\right )\right )}{4641 a^5 \left (b+a x^{2/3}\right )} \]
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Time = 3.28 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.63
method | result | size |
default | \(-\frac {-228 x^{\frac {8}{3}} a^{4} b^{2}+156 x^{\frac {10}{3}} a^{5} b +380 a^{3} b^{3} x^{2}+8778 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, E\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-4389 b^{6} \sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (a \,x^{\frac {1}{3}}-\sqrt {-a b}\right )}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, F\left (\sqrt {\frac {a \,x^{\frac {1}{3}}+\sqrt {-a b}}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )-1326 a^{6} x^{4}-2926 x^{\frac {2}{3}} a \,b^{5}-836 x^{\frac {4}{3}} a^{2} b^{4}}{4641 a^{6} \sqrt {x^{\frac {1}{3}} \left (b +a \,x^{\frac {2}{3}}\right )}}\) | \(261\) |
derivativedivides | \(\frac {2 x^{3} \sqrt {b \,x^{\frac {1}{3}}+a x}}{7 a}-\frac {38 b \,x^{\frac {7}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{119 a^{2}}+\frac {570 b^{2} x^{\frac {5}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{1547 a^{3}}-\frac {2090 b^{3} x \sqrt {b \,x^{\frac {1}{3}}+a x}}{4641 a^{4}}+\frac {418 b^{4} x^{\frac {1}{3}} \sqrt {b \,x^{\frac {1}{3}}+a x}}{663 a^{5}}-\frac {209 b^{5} \sqrt {-a b}\, \sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {2 \left (x^{\frac {1}{3}}-\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}\, \sqrt {-\frac {x^{\frac {1}{3}} a}{\sqrt {-a b}}}\, \left (-\frac {2 \sqrt {-a b}\, E\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}+\frac {\sqrt {-a b}\, F\left (\sqrt {\frac {\left (x^{\frac {1}{3}}+\frac {\sqrt {-a b}}{a}\right ) a}{\sqrt {-a b}}}, \frac {\sqrt {2}}{2}\right )}{a}\right )}{221 a^{6} \sqrt {b \,x^{\frac {1}{3}}+a x}}\) | \(276\) |
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\[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{3}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {x^{3}}{\sqrt {a x + b \sqrt [3]{x}}}\, dx \]
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\[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{3}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \]
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\[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int { \frac {x^{3}}{\sqrt {a x + b x^{\frac {1}{3}}}} \,d x } \]
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Timed out. \[ \int \frac {x^3}{\sqrt {b \sqrt [3]{x}+a x}} \, dx=\int \frac {x^3}{\sqrt {a\,x+b\,x^{1/3}}} \,d x \]
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