Optimal. Leaf size=83 \[ \frac{2 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
[Out]
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Rubi [A] time = 0.119389, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{2 (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
[Out]
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Rubi in Sympy [A] time = 28.5659, size = 71, normalized size = 0.86 \[ \frac{2 \left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- 2 p} \left (d + e x\right )^{\frac{3}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{3}{2} \\ \frac{5}{2} \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{3 e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x+d)**(1/2)*(b**2*x**2+2*a*b*x+a**2)**p,x)
[Out]
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Mathematica [A] time = 0.0642225, size = 73, normalized size = 0.88 \[ \frac{2 (d+e x)^{3/2} \left ((a+b x)^2\right )^p \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )}{3 e} \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[d + e*x]*(a^2 + 2*a*b*x + b^2*x^2)^p,x]
[Out]
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Maple [F] time = 0.107, size = 0, normalized size = 0. \[ \int \sqrt{ex+d} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x+d)^(1/2)*(b^2*x^2+2*a*b*x+a^2)^p,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x+d)**(1/2)*(b**2*x**2+2*a*b*x+a**2)**p,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{e x + d}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="giac")
[Out]