Optimal. Leaf size=168 \[ -\frac{d \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a^2 (p+1)}+\frac{e x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a e^2+3 b d^2 (2 p+1)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{a}-\frac{3 d^2 e \left (a+b x^2\right )^{p+1}}{a x}-\frac{d^3 \left (a+b x^2\right )^{p+1}}{2 a x^2} \]
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Rubi [A] time = 0.2215, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {1807, 764, 266, 65, 246, 245} \[ -\frac{d \left (a+b x^2\right )^{p+1} \left (3 a e^2+b d^2 p\right ) \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a^2 (p+1)}+\frac{e x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (a e^2+3 b d^2 (2 p+1)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{a}-\frac{3 d^2 e \left (a+b x^2\right )^{p+1}}{a x}-\frac{d^3 \left (a+b x^2\right )^{p+1}}{2 a x^2} \]
Antiderivative was successfully verified.
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Rule 1807
Rule 764
Rule 266
Rule 65
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b x^2\right )^p}{x^3} \, dx &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac{\int \frac{\left (a+b x^2\right )^p \left (-6 a d^2 e-2 d \left (3 a e^2+b d^2 p\right ) x-2 a e^3 x^2\right )}{x^2} \, dx}{2 a}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac{3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac{\int \frac{\left (2 a d \left (3 a e^2+b d^2 p\right )+2 a e \left (a e^2+3 b d^2 (1+2 p)\right ) x\right ) \left (a+b x^2\right )^p}{x} \, dx}{2 a^2}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac{3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac{\left (d \left (3 a e^2+b d^2 p\right )\right ) \int \frac{\left (a+b x^2\right )^p}{x} \, dx}{a}+\frac{\left (e \left (a e^2+3 b d^2 (1+2 p)\right )\right ) \int \left (a+b x^2\right )^p \, dx}{a}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac{3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac{\left (d \left (3 a e^2+b d^2 p\right )\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,x^2\right )}{2 a}+\frac{\left (e \left (a e^2+3 b d^2 (1+2 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}{a}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{2 a x^2}-\frac{3 d^2 e \left (a+b x^2\right )^{1+p}}{a x}+\frac{e \left (a e^2+3 b d^2 (1+2 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 )}{a}-\frac{d \left (3 a e^2+b d^2 p\right ) \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^2 (1+p)}\\ \end{align*}
Mathematica [A] time = 0.130246, size = 174, normalized size = 1.04 \[ -\frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (x \left (d \left (a+b x^2\right ) \left (\frac{b x^2}{a}+1\right )^p \left (3 a e^2 \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )-b d^2 \, _2F_1\left (2,p+1;p+2;\frac{b x^2}{a}+1\right )\right )-2 a^2 e^3 (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )\right )+6 a^2 d^2 e (p+1) \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right )\right )}{2 a^2 (p+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.526, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{p}}{{x}^{3}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}{\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F] time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )}{\left (b x^{2} + a\right )}^{p}}{x^{3}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] time = 36.8128, size = 150, normalized size = 0.89 \begin{align*} - \frac{3 a^{p} d^{2} 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} + a^{p} e^{3} 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{b^{p} d^{3} x^{2 p} \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 x^{2} \Gamma \left (2 - p\right )} - \frac{3 b^{p} d e^{2} 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 )} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (e x + d\right )}^{3}{\left (b x^{2} + a\right )}^{p}}{x^{3}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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