Optimal. Leaf size=133 \[ -\frac{x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (a e^2-c d^2 (2 p+3)\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )}{c (2 p+3)}+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]
[Out]
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Rubi [A] time = 0.193603, antiderivative size = 125, normalized size of antiderivative = 0.94, number of steps used = 4, number of rules used = 4, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.235 \[ x \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (d^2-\frac{a e^2}{2 c p+3 c}\right ) \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+\frac{e (d+e x) \left (a+c x^2\right )^{p+1}}{c (2 p+3)}+\frac{d e (p+2) \left (a+c x^2\right )^{p+1}}{c (p+1) (2 p+3)} \]
Antiderivative was successfully verified.
[In] Int[(d + e*x)^2*(a + c*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 28.8794, size = 107, normalized size = 0.8 \[ \frac{d e \left (a + c x^{2}\right )^{p + 1} \left (p + 2\right )}{c \left (p + 1\right ) \left (2 p + 3\right )} + \frac{e \left (a + c x^{2}\right )^{p + 1} \left (d + e x\right )}{c \left (2 p + 3\right )} - \frac{x \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p} \left (a e^{2} - c d^{2} \left (2 p + 3\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{c \left (2 p + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**2*(c*x**2+a)**p,x)
[Out]
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Mathematica [A] time = 0.16578, size = 133, normalized size = 1. \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 c d^2 (p+1) x \, _2F_1\left (\frac{1}{2},-p;\frac{3}{2};-\frac{c x^2}{a}\right )+e \left (3 d \left (c x^2 \left (\frac{c x^2}{a}+1\right )^p+a \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )\right )+c e (p+1) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )\right )\right )}{3 c (p+1)} \]
Antiderivative was successfully verified.
[In] Integrate[(d + e*x)^2*(a + c*x^2)^p,x]
[Out]
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Maple [F] time = 0.064, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{2} \left ( c{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^2*(c*x^2+a)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{2}{\left (c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(c*x^2 + a)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e^{2} x^{2} + 2 \, d e x + d^{2}\right )}{\left (c x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(c*x^2 + a)^p,x, algorithm="fricas")
[Out]
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Sympy [A] time = 42.4672, size = 97, normalized size = 0.73 \[ a^{p} d^{2} x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, - p \\ \frac{3}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )} + \frac{a^{p} e^{2} x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{3} + 2 d e \left (\begin{cases} \frac{a^{p} x^{2}}{2} & \text{for}\: c = 0 \\\frac{\begin{cases} \frac{\left (a + c x^{2}\right )^{p + 1}}{p + 1} & \text{for}\: p \neq -1 \\\log{\left (a + c x^{2} \right )} & \text{otherwise} \end{cases}}{2 c} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**2*(c*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}^{2}{\left (c x^{2} + a\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)^2*(c*x^2 + a)^p,x, algorithm="giac")
[Out]