Optimal. Leaf size=100 \[ -\frac{a e \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{e \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{3} d x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right ) \]
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Rubi [A] time = 0.154142, antiderivative size = 100, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.278 \[ -\frac{a e \left (a+c x^2\right )^{p+1}}{2 c^2 (p+1)}+\frac{e \left (a+c x^2\right )^{p+2}}{2 c^2 (p+2)}+\frac{1}{3} d x^3 \left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right ) \]
Antiderivative was successfully verified.
[In] Int[x^2*(d + e*x)*(a + c*x^2)^p,x]
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Rubi in Sympy [A] time = 21.4656, size = 82, normalized size = 0.82 \[ - \frac{a e \left (a + c x^{2}\right )^{p + 1}}{2 c^{2} \left (p + 1\right )} + \frac{d x^{3} \left (1 + \frac{c x^{2}}{a}\right )^{- p} \left (a + c x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{- \frac{c x^{2}}{a}} \right )}}{3} + \frac{e \left (a + c x^{2}\right )^{p + 2}}{2 c^{2} \left (p + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**2*(e*x+d)*(c*x**2+a)**p,x)
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Mathematica [A] time = 0.126759, size = 141, normalized size = 1.41 \[ \frac{\left (a+c x^2\right )^p \left (\frac{c x^2}{a}+1\right )^{-p} \left (3 e \left (-a^2 \left (\left (\frac{c x^2}{a}+1\right )^p-1\right )+c^2 (p+1) x^4 \left (\frac{c x^2}{a}+1\right )^p+a c p x^2 \left (\frac{c x^2}{a}+1\right )^p\right )+2 c^2 d \left (p^2+3 p+2\right ) x^3 \, _2F_1\left (\frac{3}{2},-p;\frac{5}{2};-\frac{c x^2}{a}\right )\right )}{6 c^2 (p+1) (p+2)} \]
Antiderivative was successfully verified.
[In] Integrate[x^2*(d + e*x)*(a + c*x^2)^p,x]
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Maple [F] time = 0.059, size = 0, normalized size = 0. \[ \int{x}^{2} \left ( ex+d \right ) \left ( c{x}^{2}+a \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^2*(e*x+d)*(c*x^2+a)^p,x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(c*x^2 + a)^p*x^2,x, algorithm="maxima")
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (e x^{3} + d x^{2}\right )}{\left (c x^{2} + a\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(c*x^2 + a)^p*x^2,x, algorithm="fricas")
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Sympy [A] time = 44.2508, size = 394, normalized size = 3.94 \[ \frac{a^{p} d x^{3}{{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, - p \\ \frac{5}{2} \end{matrix}\middle |{\frac{c x^{2} e^{i \pi }}{a}} \right )}}{3} + e \left (\begin{cases} \frac{a^{p} x^{4}}{4} & \text{for}\: c = 0 \\\frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{a}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} + \frac{c x^{2} \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 a c^{2} + 2 c^{3} x^{2}} & \text{for}\: p = -2 \\- \frac{a \log{\left (- i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} - \frac{a \log{\left (i \sqrt{a} \sqrt{\frac{1}{c}} + x \right )}}{2 c^{2}} + \frac{x^{2}}{2 c} & \text{for}\: p = -1 \\- \frac{a^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{a c p x^{2} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} p x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} + \frac{c^{2} x^{4} \left (a + c x^{2}\right )^{p}}{2 c^{2} p^{2} + 6 c^{2} p + 4 c^{2}} & \text{otherwise} \end{cases}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**2*(e*x+d)*(c*x**2+a)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (e x + d\right )}{\left (c x^{2} + a\right )}^{p} x^{2}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x + d)*(c*x^2 + a)^p*x^2,x, algorithm="giac")
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