Integrand size = 19, antiderivative size = 178 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=-\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}+\frac {21 a^4 \sqrt {b x^{2/3}+a x}}{128 b^4 x^{2/3}}-\frac {21 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^{9/2}} \]
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Time = 0.19 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.211, Rules used = {2045, 2050, 2054, 212} \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=-\frac {21 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {a x+b x^{2/3}}}\right )}{128 b^{9/2}}+\frac {21 a^4 \sqrt {a x+b x^{2/3}}}{128 b^4 x^{2/3}}-\frac {7 a^3 \sqrt {a x+b x^{2/3}}}{64 b^3 x}+\frac {7 a^2 \sqrt {a x+b x^{2/3}}}{80 b^2 x^{4/3}}-\frac {3 a \sqrt {a x+b x^{2/3}}}{40 b x^{5/3}}-\frac {3 \sqrt {a x+b x^{2/3}}}{5 x^2} \]
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Rule 212
Rule 2045
Rule 2050
Rule 2054
Rubi steps \begin{align*} \text {integral}& = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}+\frac {1}{10} a \int \frac {1}{x^2 \sqrt {b x^{2/3}+a x}} \, dx \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}-\frac {\left (7 a^2\right ) \int \frac {1}{x^{5/3} \sqrt {b x^{2/3}+a x}} \, dx}{80 b} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}+\frac {\left (7 a^3\right ) \int \frac {1}{x^{4/3} \sqrt {b x^{2/3}+a x}} \, dx}{96 b^2} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}-\frac {\left (7 a^4\right ) \int \frac {1}{x \sqrt {b x^{2/3}+a x}} \, dx}{128 b^3} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}+\frac {21 a^4 \sqrt {b x^{2/3}+a x}}{128 b^4 x^{2/3}}+\frac {\left (7 a^5\right ) \int \frac {1}{x^{2/3} \sqrt {b x^{2/3}+a x}} \, dx}{256 b^4} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}+\frac {21 a^4 \sqrt {b x^{2/3}+a x}}{128 b^4 x^{2/3}}-\frac {\left (21 a^5\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^4} \\ & = -\frac {3 \sqrt {b x^{2/3}+a x}}{5 x^2}-\frac {3 a \sqrt {b x^{2/3}+a x}}{40 b x^{5/3}}+\frac {7 a^2 \sqrt {b x^{2/3}+a x}}{80 b^2 x^{4/3}}-\frac {7 a^3 \sqrt {b x^{2/3}+a x}}{64 b^3 x}+\frac {21 a^4 \sqrt {b x^{2/3}+a x}}{128 b^4 x^{2/3}}-\frac {21 a^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^{9/2}} \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.63 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\frac {\sqrt {b x^{2/3}+a x} \left (-384 b^4-48 a b^3 \sqrt [3]{x}+56 a^2 b^2 x^{2/3}-70 a^3 b x+105 a^4 x^{4/3}\right )}{640 b^4 x^2}-\frac {21 a^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt [3]{x}}{\sqrt {b x^{2/3}+a x}}\right )}{128 b^{9/2}} \]
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Time = 2.03 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.70
method | result | size |
derivativedivides | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-105 b^{\frac {9}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+490 b^{\frac {11}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-896 b^{\frac {13}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{4} a^{5} x^{\frac {5}{3}}+790 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+105 b^{\frac {17}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{640 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {17}{2}}}\) | \(125\) |
default | \(-\frac {\sqrt {b \,x^{\frac {2}{3}}+a x}\, \left (-105 b^{\frac {9}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {9}{2}}+490 b^{\frac {11}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {7}{2}}-896 b^{\frac {13}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {5}{2}}+105 \,\operatorname {arctanh}\left (\frac {\sqrt {b +a \,x^{\frac {1}{3}}}}{\sqrt {b}}\right ) b^{4} a^{5} x^{\frac {5}{3}}+790 b^{\frac {15}{2}} \left (b +a \,x^{\frac {1}{3}}\right )^{\frac {3}{2}}+105 b^{\frac {17}{2}} \sqrt {b +a \,x^{\frac {1}{3}}}\right )}{640 x^{2} \sqrt {b +a \,x^{\frac {1}{3}}}\, b^{\frac {17}{2}}}\) | \(125\) |
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Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\text {Timed out} \]
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\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{3}}\, dx \]
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\[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int { \frac {\sqrt {a x + b x^{\frac {2}{3}}}}{x^{3}} \,d x } \]
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Time = 0.32 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.71 \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\frac {\frac {105 \, a^{6} \arctan \left (\frac {\sqrt {a x^{\frac {1}{3}} + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{4}} + \frac {105 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {9}{2}} a^{6} - 490 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {7}{2}} a^{6} b + 896 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {5}{2}} a^{6} b^{2} - 790 \, {\left (a x^{\frac {1}{3}} + b\right )}^{\frac {3}{2}} a^{6} b^{3} - 105 \, \sqrt {a x^{\frac {1}{3}} + b} a^{6} b^{4}}{a^{5} b^{4} x^{\frac {5}{3}}}}{640 \, a} \]
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Timed out. \[ \int \frac {\sqrt {b x^{2/3}+a x}}{x^3} \, dx=\int \frac {\sqrt {a\,x+b\,x^{2/3}}}{x^3} \,d x \]
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