3.1741 \(\int \sqrt{d+e x} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

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]

(2*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[3/2, -2*p, 5/2,
(b*(d + e*x))/(b*d - a*e)])/(3*e*(-((e*(a + b*x))/(b*d - a*e)))^(2*p))

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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]

(2*(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[3/2, -2*p, 5/2,
(b*(d + e*x))/(b*d - a*e)])/(3*e*(-((e*(a + b*x))/(b*d - a*e)))^(2*p))

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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]

2*(e*(a + b*x)/(a*e - b*d))**(-2*p)*(d + e*x)**(3/2)*(a**2 + 2*a*b*x + b**2*x**2
)**p*hyper((-2*p, 3/2), (5/2,), b*(-d - e*x)/(a*e - b*d))/(3*e)

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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]

(2*((a + b*x)^2)^p*(d + e*x)^(3/2)*Hypergeometric2F1[3/2, -2*p, 5/2, (b*(d + e*x
))/(b*d - a*e)])/(3*e*((e*(a + b*x))/(-(b*d) + a*e))^(2*p))

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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]

int((e*x+d)^(1/2)*(b^2*x^2+2*a*b*x+a^2)^p,x)

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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]

integrate(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p, x)

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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]

integral(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p, x)

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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]

Timed 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]

integrate(sqrt(e*x + d)*(b^2*x^2 + 2*a*b*x + a^2)^p, x)