Integrand size = 20, antiderivative size = 48 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\frac {b c \sqrt {c x^2} (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n) x} \]
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Time = 0.01 (sec) , antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {15, 67} \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\frac {b c \sqrt {c x^2} (a+b x)^{n+1} \operatorname {Hypergeometric2F1}\left (2,n+1,n+2,\frac {b x}{a}+1\right )}{a^2 (n+1) x} \]
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Rule 15
Rule 67
Rubi steps \begin{align*} \text {integral}& = \frac {\left (c \sqrt {c x^2}\right ) \int \frac {(a+b x)^n}{x^2} \, dx}{x} \\ & = \frac {b c \sqrt {c x^2} (a+b x)^{1+n} \, _2F_1\left (2,1+n;2+n;1+\frac {b x}{a}\right )}{a^2 (1+n) x} \\ \end{align*}
Time = 0.01 (sec) , antiderivative size = 47, normalized size of antiderivative = 0.98 \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\frac {b \left (c x^2\right )^{3/2} (a+b x)^{1+n} \operatorname {Hypergeometric2F1}\left (2,1+n,2+n,1+\frac {b x}{a}\right )}{a^2 (1+n) x^3} \]
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\[\int \frac {\left (c \,x^{2}\right )^{\frac {3}{2}} \left (b x +a \right )^{n}}{x^{5}}d x\]
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\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{5}} \,d x } \]
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\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\int \frac {\left (c x^{2}\right )^{\frac {3}{2}} \left (a + b x\right )^{n}}{x^{5}}\, dx \]
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\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{5}} \,d x } \]
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\[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\int { \frac {\left (c x^{2}\right )^{\frac {3}{2}} {\left (b x + a\right )}^{n}}{x^{5}} \,d x } \]
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Timed out. \[ \int \frac {\left (c x^2\right )^{3/2} (a+b x)^n}{x^5} \, dx=\int \frac {{\left (c\,x^2\right )}^{3/2}\,{\left (a+b\,x\right )}^n}{x^5} \,d x \]
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