Integrand size = 27, antiderivative size = 107 \[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {2 x^{-3 j/2} (c x)^{3 j/2} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)} \]
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Time = 0.12 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2056, 2055, 2054, 212} \[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\frac {2 x^{-3 j/2} (c x)^{3 j/2} \text {arctanh}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)}-\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}} \]
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Rule 212
Rule 2054
Rule 2055
Rule 2056
Rubi steps \begin{align*} \text {integral}& = \frac {\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac {x^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx}{c} \\ & = -\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {\left (x^{-3 j/2} (c x)^{3 j/2}\right ) \int \frac {x^{-1+\frac {j}{2}}}{\sqrt {a x^j+b x^n}} \, dx}{a c} \\ & = -\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {\left (2 x^{-3 j/2} (c x)^{3 j/2}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a c (j-n)} \\ & = -\frac {2 x^{-j} (c x)^{3 j/2}}{a c (j-n) \sqrt {a x^j+b x^n}}+\frac {2 x^{-3 j/2} (c x)^{3 j/2} \tanh ^{-1}\left (\frac {\sqrt {a} x^{j/2}}{\sqrt {a x^j+b x^n}}\right )}{a^{3/2} c (j-n)} \\ \end{align*}
Time = 0.56 (sec) , antiderivative size = 117, normalized size of antiderivative = 1.09 \[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=-\frac {2 x^{-3 j/2} (c x)^{3 j/2} \left (\sqrt {a} x^{j/2}-\sqrt {b} x^{n/2} \sqrt {1+\frac {a x^{j-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {j-n}{2}}}{\sqrt {b}}\right )\right )}{a^{3/2} c (j-n) \sqrt {a x^j+b x^n}} \]
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\[\int \frac {\left (c x \right )^{-1+\frac {3 j}{2}}}{\left (a \,x^{j}+b \,x^{n}\right )^{\frac {3}{2}}}d x\]
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Exception generated. \[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\int \frac {\left (c x\right )^{\frac {3 j}{2} - 1}}{\left (a x^{j} + b x^{n}\right )^{\frac {3}{2}}}\, dx \]
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\[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}} \,d x } \]
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\[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\int { \frac {\left (c x\right )^{\frac {3}{2} \, j - 1}}{{\left (a x^{j} + b x^{n}\right )}^{\frac {3}{2}}} \,d x } \]
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Timed out. \[ \int \frac {(c x)^{-1+\frac {3 j}{2}}}{\left (a x^j+b x^n\right )^{3/2}} \, dx=\int \frac {{\left (c\,x\right )}^{\frac {3\,j}{2}-1}}{{\left (a\,x^j+b\,x^n\right )}^{3/2}} \,d x \]
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