Optimal. Leaf size=81 \[ \frac{2 \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{1}{2},-2 p;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{e} \]
[Out]
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Rubi [A] time = 0.120251, antiderivative size = 81, 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 \[ \frac{2 \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{1}{2},-2 p;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{e} \]
Antiderivative was successfully verified.
[In] Int[(a^2 + 2*a*b*x + b^2*x^2)^p/Sqrt[d + e*x],x]
[Out]
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Rubi in Sympy [A] time = 28.5869, size = 70, normalized size = 0.86 \[ \frac{2 \left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- 2 p} \sqrt{d + e x} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{e} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b**2*x**2+2*a*b*x+a**2)**p/(e*x+d)**(1/2),x)
[Out]
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Mathematica [A] time = 0.0505688, size = 71, normalized size = 0.88 \[ \frac{2 \sqrt{d+e x} \left ((a+b x)^2\right )^p \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (\frac{1}{2},-2 p;\frac{3}{2};\frac{b (d+e x)}{b d-a e}\right )}{e} \]
Antiderivative was successfully verified.
[In] Integrate[(a^2 + 2*a*b*x + b^2*x^2)^p/Sqrt[d + e*x],x]
[Out]
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Maple [F] time = 0.112, size = 0, normalized size = 0. \[ \int{ \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}{\frac{1}{\sqrt{ex+d}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b^2*x^2+2*a*b*x+a^2)^p/(e*x+d)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{\sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{\sqrt{e x + d}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (\left (a + b x\right )^{2}\right )^{p}}{\sqrt{d + e x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b**2*x**2+2*a*b*x+a**2)**p/(e*x+d)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}}{\sqrt{e x + d}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b^2*x^2 + 2*a*b*x + a^2)^p/sqrt(e*x + d),x, algorithm="giac")
[Out]