Integrand size = 21, antiderivative size = 122 \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {2 a^{3/2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}} \]
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Time = 0.12 (sec) , antiderivative size = 122, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {2053, 2056, 2054, 212} \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\frac {2 a^{3/2} \sqrt {x} \text {arctanh}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}}-\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}} \]
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Rule 212
Rule 2053
Rule 2054
Rule 2056
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {a \int \frac {\sqrt {a x+b x^n}}{(c x)^{3/2}} \, dx}{c} \\ & = -\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {a^2 \int \frac {1}{\sqrt {c x} \sqrt {a x+b x^n}} \, dx}{c^2} \\ & = -\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {\left (a^2 \sqrt {x}\right ) \int \frac {1}{\sqrt {x} \sqrt {a x+b x^n}} \, dx}{c^2 \sqrt {c x}} \\ & = -\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {\left (2 a^2 \sqrt {x}\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}} \\ & = -\frac {2 a \sqrt {a x+b x^n}}{c^2 (1-n) \sqrt {c x}}-\frac {2 \left (a x+b x^n\right )^{3/2}}{3 c (1-n) (c x)^{3/2}}+\frac {2 a^{3/2} \sqrt {x} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {x}}{\sqrt {a x+b x^n}}\right )}{c^2 (1-n) \sqrt {c x}} \\ \end{align*}
Time = 1.62 (sec) , antiderivative size = 120, normalized size of antiderivative = 0.98 \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\frac {x \left (8 a^2 x^2+2 b^2 x^{2 n}+10 a b x^{1+n}-6 a^{3/2} \sqrt {b} x^{\frac {3+n}{2}} \sqrt {1+\frac {a x^{1-n}}{b}} \text {arcsinh}\left (\frac {\sqrt {a} x^{\frac {1}{2}-\frac {n}{2}}}{\sqrt {b}}\right )\right )}{3 (-1+n) (c x)^{5/2} \sqrt {a x+b x^n}} \]
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\[\int \frac {\left (a x +b \,x^{n}\right )^{\frac {3}{2}}}{\left (c x \right )^{\frac {5}{2}}}d x\]
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Exception generated. \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\text {Exception raised: TypeError} \]
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\[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\int \frac {\left (a x + b x^{n}\right )^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\int { \frac {{\left (a x + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\int { \frac {{\left (a x + b x^{n}\right )}^{\frac {3}{2}}}{\left (c x\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {\left (a x+b x^n\right )^{3/2}}{(c x)^{5/2}} \, dx=\int \frac {{\left (b\,x^n+a\,x\right )}^{3/2}}{{\left (c\,x\right )}^{5/2}} \,d x \]
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