Integrand size = 18, antiderivative size = 50 \[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^3 x (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {b x}{a}\right )}{a^4 c^2 (1+n) \sqrt {c x^2}} \]
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Time = 0.01 (sec) , antiderivative size = 50, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {15, 67} \[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^3 x (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (4,n+1,n+2,\frac {b x}{a}+1\right )}{a^4 c^2 (n+1) \sqrt {c x^2}} \]
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Rule 15
Rule 67
Rubi steps \begin{align*} \text {integral}& = \frac {x \int \frac {(a+b x)^n}{x^4} \, dx}{c^2 \sqrt {c x^2}} \\ & = \frac {b^3 x (a+b x)^{1+n} \, _2F_1\left (4,1+n;2+n;1+\frac {b x}{a}\right )}{a^4 c^2 (1+n) \sqrt {c x^2}} \\ \end{align*}
Time = 0.00 (sec) , antiderivative size = 49, normalized size of antiderivative = 0.98 \[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\frac {b^3 x^5 (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (4,1+n,2+n,1+\frac {b x}{a}\right )}{a^4 (1+n) \left (c x^2\right )^{5/2}} \]
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\[\int \frac {x \left (b x +a \right )^{n}}{\left (c \,x^{2}\right )^{\frac {5}{2}}}d x\]
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\[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\int \frac {x \left (a + b x\right )^{n}}{\left (c x^{2}\right )^{\frac {5}{2}}}\, dx \]
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\[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac {5}{2}}} \,d x } \]
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\[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\int { \frac {{\left (b x + a\right )}^{n} x}{\left (c x^{2}\right )^{\frac {5}{2}}} \,d x } \]
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Timed out. \[ \int \frac {x (a+b x)^n}{\left (c x^2\right )^{5/2}} \, dx=\int \frac {x\,{\left (a+b\,x\right )}^n}{{\left (c\,x^2\right )}^{5/2}} \,d x \]
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