Integrand size = 18, antiderivative size = 92 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=-\frac {e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x}+\frac {b d \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)} \]
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Time = 0.04 (sec) , antiderivative size = 92, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {778, 272, 67, 372, 371} \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\frac {b d \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (2,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a^2 (p+1)}-\frac {e \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x} \]
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Rule 67
Rule 272
Rule 371
Rule 372
Rule 778
Rubi steps \begin{align*} \text {integral}& = d \int \frac {\left (a+b x^2\right )^p}{x^3} \, dx+e \int \frac {\left (a+b x^2\right )^p}{x^2} \, dx \\ & = \frac {1}{2} d \text {Subst}\left (\int \frac {(a+b x)^p}{x^2} \, dx,x,x^2\right )+\left (e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \frac {\left (1+\frac {b x^2}{a}\right )^p}{x^2} \, dx \\ & = -\frac {e \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (-\frac {1}{2},-p;\frac {1}{2};-\frac {b x^2}{a}\right )}{x}+\frac {b d \left (a+b x^2\right )^{1+p} \, _2F_1\left (2,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a^2 (1+p)} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.97 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\frac {1}{2} \left (a+b x^2\right )^p \left (-\frac {2 e \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (-\frac {1}{2},-p,\frac {1}{2},-\frac {b x^2}{a}\right )}{x}+\frac {b d \left (a+b x^2\right ) \operatorname {Hypergeometric2F1}\left (2,1+p,2+p,1+\frac {b x^2}{a}\right )}{a^2 (1+p)}\right ) \]
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\[\int \frac {\left (e x +d \right ) \left (b \,x^{2}+a \right )^{p}}{x^{3}}d x\]
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\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}} \,d x } \]
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Result contains complex when optimal does not.
Time = 5.50 (sec) , antiderivative size = 70, normalized size of antiderivative = 0.76 \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=- \frac {a^{p} e {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{2}, - p \\ \frac {1}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{x} - \frac {b^{p} d x^{2 p - 2} \Gamma \left (1 - p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, 1 - p \\ 2 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (2 - p\right )} \]
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\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}} \,d x } \]
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\[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\int { \frac {{\left (e x + d\right )} {\left (b x^{2} + a\right )}^{p}}{x^{3}} \,d x } \]
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Timed out. \[ \int \frac {(d+e x) \left (a+b x^2\right )^p}{x^3} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,\left (d+e\,x\right )}{x^3} \,d x \]
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