Integrand size = 19, antiderivative size = 165 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=-\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {3 b^5 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}} \]
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Time = 0.16 (sec) , antiderivative size = 165, 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, 2033, 212} \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\frac {3 b^5 \text {arctanh}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5} \]
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Rule 212
Rule 2033
Rule 2045
Rule 2050
Rubi steps \begin{align*} \text {integral}& = -\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {1}{10} (3 b) \int \frac {\sqrt {a x^2+b x^3}}{x^6} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {1}{80} \left (3 b^2\right ) \int \frac {1}{x^3 \sqrt {a x^2+b x^3}} \, dx \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {b^3 \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx}{32 a} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {\left (3 b^4\right ) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{128 a^2} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}-\frac {\left (3 b^5\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{256 a^3} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {\left (3 b^5\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{128 a^3} \\ & = -\frac {3 b \sqrt {a x^2+b x^3}}{40 x^5}-\frac {b^2 \sqrt {a x^2+b x^3}}{80 a x^4}+\frac {b^3 \sqrt {a x^2+b x^3}}{64 a^2 x^3}-\frac {3 b^4 \sqrt {a x^2+b x^3}}{128 a^3 x^2}-\frac {\left (a x^2+b x^3\right )^{3/2}}{5 x^8}+\frac {3 b^5 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{128 a^{7/2}} \\ \end{align*}
Time = 0.25 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.70 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\frac {\sqrt {x^2 (a+b x)} \left (-\sqrt {a} \sqrt {a+b x} \left (128 a^4+176 a^3 b x+8 a^2 b^2 x^2-10 a b^3 x^3+15 b^4 x^4\right )+15 b^5 x^5 \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{640 a^{7/2} x^6 \sqrt {a+b x}} \]
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Time = 2.87 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.62
method | result | size |
risch | \(-\frac {\left (15 b^{4} x^{4}-10 a \,b^{3} x^{3}+8 a^{2} b^{2} x^{2}+176 a^{3} b x +128 a^{4}\right ) \sqrt {x^{2} \left (b x +a \right )}}{640 x^{6} a^{3}}+\frac {3 b^{5} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {x^{2} \left (b x +a \right )}}{128 a^{\frac {7}{2}} x \sqrt {b x +a}}\) | \(103\) |
default | \(-\frac {\left (b \,x^{3}+a \,x^{2}\right )^{\frac {3}{2}} \left (15 \left (b x +a \right )^{\frac {9}{2}} a^{\frac {7}{2}}-70 \left (b x +a \right )^{\frac {7}{2}} a^{\frac {9}{2}}+128 \left (b x +a \right )^{\frac {5}{2}} a^{\frac {11}{2}}-15 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a^{3} x^{5} b^{5}+70 \left (b x +a \right )^{\frac {3}{2}} a^{\frac {13}{2}}-15 \sqrt {b x +a}\, a^{\frac {15}{2}}\right )}{640 x^{8} \left (b x +a \right )^{\frac {3}{2}} a^{\frac {13}{2}}}\) | \(113\) |
pseudoelliptic | \(-\frac {\frac {693 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{8} x^{8}}{256}+\sqrt {b x +a}\, \left (-\frac {693 \sqrt {a}\, b^{7} x^{7}}{256}+\frac {231 a^{\frac {3}{2}} b^{6} x^{6}}{128}-\frac {231 a^{\frac {5}{2}} b^{5} x^{5}}{160}+\frac {99 a^{\frac {7}{2}} b^{4} x^{4}}{80}-\frac {11 a^{\frac {9}{2}} b^{3} x^{3}}{10}+a^{\frac {11}{2}} b^{2} x^{2}+68 a^{\frac {13}{2}} b x +56 a^{\frac {15}{2}}\right )}{448 a^{\frac {13}{2}} x^{8}}\) | \(116\) |
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Time = 0.28 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.33 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\left [\frac {15 \, \sqrt {a} b^{5} x^{6} \log \left (\frac {b x^{2} + 2 \, a x + 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) - 2 \, {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{1280 \, a^{4} x^{6}}, -\frac {15 \, \sqrt {-a} b^{5} x^{6} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + {\left (15 \, a b^{4} x^{4} - 10 \, a^{2} b^{3} x^{3} + 8 \, a^{3} b^{2} x^{2} + 176 \, a^{4} b x + 128 \, a^{5}\right )} \sqrt {b x^{3} + a x^{2}}}{640 \, a^{4} x^{6}}\right ] \]
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\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int \frac {\left (x^{2} \left (a + b x\right )\right )^{\frac {3}{2}}}{x^{9}}\, dx \]
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\[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int { \frac {{\left (b x^{3} + a x^{2}\right )}^{\frac {3}{2}}}{x^{9}} \,d x } \]
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Time = 0.31 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=-\frac {\frac {15 \, b^{6} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right ) \mathrm {sgn}\left (x\right )}{\sqrt {-a} a^{3}} + \frac {15 \, {\left (b x + a\right )}^{\frac {9}{2}} b^{6} \mathrm {sgn}\left (x\right ) - 70 \, {\left (b x + a\right )}^{\frac {7}{2}} a b^{6} \mathrm {sgn}\left (x\right ) + 128 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} b^{6} \mathrm {sgn}\left (x\right ) + 70 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} b^{6} \mathrm {sgn}\left (x\right ) - 15 \, \sqrt {b x + a} a^{4} b^{6} \mathrm {sgn}\left (x\right )}{a^{3} b^{5} x^{5}}}{640 \, b} \]
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Timed out. \[ \int \frac {\left (a x^2+b x^3\right )^{3/2}}{x^9} \, dx=\int \frac {{\left (b\,x^3+a\,x^2\right )}^{3/2}}{x^9} \,d x \]
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