Integrand size = 20, antiderivative size = 171 \[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\frac {3 d e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {e^3 x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}-\frac {e \left (a e^2-3 b d^2 (3+2 p)\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )}{b (3+2 p)}-\frac {d^3 \left (a+b x^2\right )^{1+p} \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )}{2 a (1+p)} \]
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Time = 0.10 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.96, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {1666, 457, 81, 67, 396, 252, 251} \[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=-\frac {d^3 \left (a+b x^2\right )^{p+1} \operatorname {Hypergeometric2F1}\left (1,p+1,p+2,\frac {b x^2}{a}+1\right )}{2 a (p+1)}+e x \left (a+b x^2\right )^p \left (\frac {b x^2}{a}+1\right )^{-p} \left (3 d^2-\frac {a e^2}{2 b p+3 b}\right ) \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )+\frac {3 d e^2 \left (a+b x^2\right )^{p+1}}{2 b (p+1)}+\frac {e^3 x \left (a+b x^2\right )^{p+1}}{b (2 p+3)} \]
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Rule 67
Rule 81
Rule 251
Rule 252
Rule 396
Rule 457
Rule 1666
Rubi steps \begin{align*} \text {integral}& = \int \frac {\left (a+b x^2\right )^p \left (d^3+3 d e^2 x^2\right )}{x} \, dx+\int \left (a+b x^2\right )^p \left (3 d^2 e+e^3 x^2\right ) \, dx \\ & = \frac {e^3 x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+\frac {1}{2} \text {Subst}\left (\int \frac {(a+b x)^p \left (d^3+3 d e^2 x\right )}{x} \, dx,x,x^2\right )+\left (e \left (3 d^2-\frac {a e^2}{3 b+2 b p}\right )\right ) \int \left (a+b x^2\right )^p \, dx \\ & = \frac {3 d e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {e^3 x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+\frac {1}{2} d^3 \text {Subst}\left (\int \frac {(a+b x)^p}{x} \, dx,x,x^2\right )+\left (e \left (3 d^2-\frac {a e^2}{3 b+2 b p}\right ) \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p}\right ) \int \left (1+\frac {b x^2}{a}\right )^p \, dx \\ & = \frac {3 d e^2 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}+\frac {e^3 x \left (a+b x^2\right )^{1+p}}{b (3+2 p)}+e \left (3 d^2-\frac {a e^2}{3 b+2 b p}\right ) x \left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \, _2F_1\left (\frac {1}{2},-p;\frac {3}{2};-\frac {b x^2}{a}\right )-\frac {d^3 \left (a+b x^2\right )^{1+p} \, _2F_1\left (1,1+p;2+p;1+\frac {b x^2}{a}\right )}{2 a (1+p)} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.99 \[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\frac {\left (a+b x^2\right )^p \left (1+\frac {b x^2}{a}\right )^{-p} \left (18 a b d^2 e (1+p) x \operatorname {Hypergeometric2F1}\left (\frac {1}{2},-p,\frac {3}{2},-\frac {b x^2}{a}\right )-3 b d^3 \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p \operatorname {Hypergeometric2F1}\left (1,1+p,2+p,1+\frac {b x^2}{a}\right )+a e^2 \left (9 d \left (a+b x^2\right ) \left (1+\frac {b x^2}{a}\right )^p+2 b e (1+p) x^3 \operatorname {Hypergeometric2F1}\left (\frac {3}{2},-p,\frac {5}{2},-\frac {b x^2}{a}\right )\right )\right )}{6 a b (1+p)} \]
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\[\int \frac {\left (e x +d \right )^{3} \left (b \,x^{2}+a \right )^{p}}{x}d x\]
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\[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{x} \,d x } \]
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Time = 6.36 (sec) , antiderivative size = 144, normalized size of antiderivative = 0.84 \[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=3 a^{p} d^{2} e x {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, - p \\ \frac {3}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )} + \frac {a^{p} e^{3} x^{3} {{}_{2}F_{1}\left (\begin {matrix} \frac {3}{2}, - p \\ \frac {5}{2} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{3} - \frac {b^{p} d^{3} x^{2 p} \Gamma \left (- p\right ) {{}_{2}F_{1}\left (\begin {matrix} - p, - p \\ 1 - p \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{2 \Gamma \left (1 - p\right )} + 3 d e^{2} \left (\begin {cases} \frac {a^{p} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\begin {cases} \frac {\left (a + b x^{2}\right )^{p + 1}}{p + 1} & \text {for}\: p \neq -1 \\\log {\left (a + b x^{2} \right )} & \text {otherwise} \end {cases}}{2 b} & \text {otherwise} \end {cases}\right ) \]
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\[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{x} \,d x } \]
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\[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\int { \frac {{\left (e x + d\right )}^{3} {\left (b x^{2} + a\right )}^{p}}{x} \,d x } \]
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Timed out. \[ \int \frac {(d+e x)^3 \left (a+b x^2\right )^p}{x} \, dx=\int \frac {{\left (b\,x^2+a\right )}^p\,{\left (d+e\,x\right )}^3}{x} \,d x \]
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