Optimal. Leaf size=159 \[ \frac{d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{a}-\frac{3 d^2 e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a (p+1)}-\frac{d^3 \left (a+b x^2\right )^{p+1}}{a x}+\frac{e^3 \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
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Rubi [A] time = 0.189275, antiderivative size = 159, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 8, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {1807, 1652, 446, 80, 65, 12, 246, 245} \[ \frac{d x \left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (3 a e^2+b d^2 (2 p+1)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )}{a}-\frac{3 d^2 e \left (a+b x^2\right )^{p+1} \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )}{2 a (p+1)}-\frac{d^3 \left (a+b x^2\right )^{p+1}}{a x}+\frac{e^3 \left (a+b x^2\right )^{p+1}}{2 b (p+1)} \]
Antiderivative was successfully verified.
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Rule 1807
Rule 1652
Rule 446
Rule 80
Rule 65
Rule 12
Rule 246
Rule 245
Rubi steps
\begin{align*} \int \frac{(d+e x)^3 \left (a+b x^2\right )^p}{x^2} \, dx &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{a x}-\frac{\int \frac{\left (a+b x^2\right )^p \left (-3 a d^2 e-d \left (3 a e^2+b d^2 (1+2 p)\right ) x-a e^3 x^2\right )}{x} \, dx}{a}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{a x}+\frac{\int d \left (3 a e^2+b d^2 (1+2 p)\right ) \left (a+b x^2\right )^p \, dx}{a}-\frac{\int \frac{\left (a+b x^2\right )^p \left (-3 a d^2 e-a e^3 x^2\right )}{x} \, dx}{a}\\ &=-\frac{d^3 \left (a+b x^2\right )^{1+p}}{a x}-\frac{\operatorname{Subst}\left (\int \frac{(a+b x)^p \left (-3 a d^2 e-a e^3 x\right )}{x} \, dx,x,x^2\right )}{2 a}+\frac{\left (d \left (3 a e^2+b d^2 (1+2 p)\right )\right ) \int \left (a+b x^2\right )^p \, dx}{a}\\ &=\frac{e^3 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}-\frac{d^3 \left (a+b x^2\right )^{1+p}}{a x}+\frac{1}{2} \left (3 d^2 e\right ) \operatorname{Subst}\left (\int \frac{(a+b x)^p}{x} \, dx,x,x^2\right )+\frac{\left (d \left (3 a e^2+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{e^3 \left (a+b x^2\right )^{1+p}}{2 b (1+p)}-\frac{d^3 \left (a+b x^2\right )^{1+p}}{a x}+\frac{d \left (3 a e^2+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{3 d^2 e \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*}
Mathematica [A] time = 0.123825, size = 154, normalized size = 0.97 \[ \frac{\left (a+b x^2\right )^p \left (\frac{b x^2}{a}+1\right )^{-p} \left (e x \left (\left (a+b x^2\right ) \left (\frac{b x^2}{a}+1\right )^p \left (a e^2-3 b d^2 \, _2F_1\left (1,p+1;p+2;\frac{b x^2}{a}+1\right )\right )+6 a b d e (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{b x^2}{a}\right )\right )-2 a b d^3 (p+1) \, _2F_1\left (-\frac{1}{2},-p;\frac{1}{2};-\frac{b x^2}{a}\right )\right )}{2 a b (p+1) x} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.515, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( ex+d \right ) ^{3} \left ( b{x}^{2}+a \right ) ^{p}}{{x}^{2}}}\, 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^{2}}\,{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^{2}}, x\right ) \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A] time = 29.6593, size = 143, normalized size = 0.9 \begin{align*} - \frac{a^{p} d^{3}{{}_{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} + 3 a^{p} d e^{2} 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{3 b^{p} d^{2} e 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 )} + e^{3} \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 ) \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^{2}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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