3.1740 \(\int (d+e x)^{3/2} \left (a^2+2 a b x+b^2 x^2\right )^p \, dx\)

Optimal. Leaf size=83 \[ \frac{2 (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{5}{2},-2 p;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right )}{5 e} \]

[Out]

(2*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[5/2, -2*p, 7/2,
(b*(d + e*x))/(b*d - a*e)])/(5*e*(-((e*(a + b*x))/(b*d - a*e)))^(2*p))

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Rubi [A]  time = 0.123404, 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 \[ \frac{2 (d+e x)^{5/2} \left (a^2+2 a b x+b^2 x^2\right )^p \left (-\frac{e (a+b x)}{b d-a e}\right )^{-2 p} \, _2F_1\left (\frac{5}{2},-2 p;\frac{7}{2};\frac{b (d+e x)}{b d-a e}\right )}{5 e} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

(2*(d + e*x)^(5/2)*(a^2 + 2*a*b*x + b^2*x^2)^p*Hypergeometric2F1[5/2, -2*p, 7/2,
(b*(d + e*x))/(b*d - a*e)])/(5*e*(-((e*(a + b*x))/(b*d - a*e)))^(2*p))

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Rubi in Sympy [A]  time = 28.0974, size = 71, normalized size = 0.86 \[ \frac{2 \left (\frac{e \left (a + b x\right )}{a e - b d}\right )^{- 2 p} \left (d + e x\right )^{\frac{5}{2}} \left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{p}{{}_{2}F_{1}\left (\begin{matrix} - 2 p, \frac{5}{2} \\ \frac{7}{2} \end{matrix}\middle |{\frac{b \left (- d - e x\right )}{a e - b d}} \right )}}{5 e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(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)**(5/2)*(a**2 + 2*a*b*x + b**2*x**2
)**p*hyper((-2*p, 5/2), (7/2,), b*(-d - e*x)/(a*e - b*d))/(5*e)

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Mathematica [C]  time = 0.564534, size = 202, normalized size = 2.43 \[ \frac{d \sqrt{d+e x} \left ((a+b x)^2\right )^p \left (\frac{9 a e^2 x^2 F_1\left (2;-2 p,-\frac{1}{2};3;-\frac{b x}{a},-\frac{e x}{d}\right )}{6 a d F_1\left (2;-2 p,-\frac{1}{2};3;-\frac{b x}{a},-\frac{e x}{d}\right )+4 b d p x F_1\left (3;1-2 p,-\frac{1}{2};4;-\frac{b x}{a},-\frac{e x}{d}\right )+a e x F_1\left (3;-2 p,\frac{1}{2};4;-\frac{b x}{a},-\frac{e x}{d}\right )}+2 (d+e x) \left (\frac{e (a+b x)}{a e-b d}\right )^{-2 p} \, _2F_1\left (\frac{3}{2},-2 p;\frac{5}{2};\frac{b (d+e x)}{b d-a e}\right )\right )}{3 e} \]

Warning: Unable to verify antiderivative.

[In]  Integrate[(d + e*x)^(3/2)*(a^2 + 2*a*b*x + b^2*x^2)^p,x]

[Out]

(d*((a + b*x)^2)^p*Sqrt[d + e*x]*((9*a*e^2*x^2*AppellF1[2, -2*p, -1/2, 3, -((b*x
)/a), -((e*x)/d)])/(6*a*d*AppellF1[2, -2*p, -1/2, 3, -((b*x)/a), -((e*x)/d)] + 4
*b*d*p*x*AppellF1[3, 1 - 2*p, -1/2, 4, -((b*x)/a), -((e*x)/d)] + a*e*x*AppellF1[
3, -2*p, 1/2, 4, -((b*x)/a), -((e*x)/d)]) + (2*(d + e*x)*Hypergeometric2F1[3/2,
-2*p, 5/2, (b*(d + e*x))/(b*d - a*e)])/((e*(a + b*x))/(-(b*d) + a*e))^(2*p)))/(3
*e)

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Maple [F]  time = 0.107, size = 0, normalized size = 0. \[ \int \left ( ex+d \right ) ^{{\frac{3}{2}}} \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2} \right ) ^{p}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(b^2*x^2+2*a*b*x+a^2)^p,x)

[Out]

int((e*x+d)^(3/2)*(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. \[ \int{\left (e x + d\right )}^{\frac{3}{2}}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)*(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. \[{\rm integral}\left ({\left (e x + d\right )}^{\frac{3}{2}}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="fricas")

[Out]

integral((e*x + d)^(3/2)*(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)**(3/2)*(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. \[ \int{\left (e x + d\right )}^{\frac{3}{2}}{\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(b^2*x^2 + 2*a*b*x + a^2)^p,x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)*(b^2*x^2 + 2*a*b*x + a^2)^p, x)