3.782 \(\int \frac{(d+e x)^{3/2} (f+g x)^n}{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=213 \[ \frac{\sqrt{d+e x} (f+g x)^{n+1} (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \, _2F_1\left (1,n+\frac{3}{2};n+2;\frac{c d (f+g x)}{c d f-a e g}\right )}{c d g (n+1) (2 n+3) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (c d f-a e g)}+\frac{2 e (f+g x)^{n+1} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt{d+e x}} \]

[Out]

(2*e*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*g*(3 +
2*n)*Sqrt[d + e*x]) + ((2*a*e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e + c*
d*x)*Sqrt[d + e*x]*(f + g*x)^(1 + n)*Hypergeometric2F1[1, 3/2 + n, 2 + n, (c*d*(
f + g*x))/(c*d*f - a*e*g)])/(c*d*g*(c*d*f - a*e*g)*(1 + n)*(3 + 2*n)*Sqrt[a*d*e
+ (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi [A]  time = 0.773666, antiderivative size = 222, normalized size of antiderivative = 1.04, number of steps used = 4, number of rules used = 4, integrand size = 46, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.087 \[ \frac{2 e (f+g x)^{n+1} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{c d g (2 n+3) \sqrt{d+e x}}-\frac{2 \sqrt{d+e x} (f+g x)^n (a e+c d x) \left (2 a e^2 g (n+1)+c d (e f-d g (2 n+3))\right ) \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};-\frac{g (a e+c d x)}{c d f-a e g}\right )}{c^2 d^2 g (2 n+3) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}} \]

Antiderivative was successfully verified.

[In]  Int[((d + e*x)^(3/2)*(f + g*x)^n)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*e*(f + g*x)^(1 + n)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(c*d*g*(3 +
2*n)*Sqrt[d + e*x]) - (2*(2*a*e^2*g*(1 + n) + c*d*(e*f - d*g*(3 + 2*n)))*(a*e +
c*d*x)*Sqrt[d + e*x]*(f + g*x)^n*Hypergeometric2F1[1/2, -n, 3/2, -((g*(a*e + c*d
*x))/(c*d*f - a*e*g))])/(c^2*d^2*g*(3 + 2*n)*((c*d*(f + g*x))/(c*d*f - a*e*g))^n
*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])

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Rubi in Sympy [A]  time = 89.9942, size = 202, normalized size = 0.95 \[ \frac{2 e \left (f + g x\right )^{n + 1} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{c d g \sqrt{d + e x} \left (2 n + 3\right )} - \frac{2 \left (\frac{c d \left (- f - g x\right )}{a e g - c d f}\right )^{- n} \left (f + g x\right )^{n} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )} \left (- c d^{2} g \left (4 n + 5\right ) + c d e f + 2 g \left (n + 1\right ) \left (a e^{2} + c d^{2}\right )\right ){{}_{2}F_{1}\left (\begin{matrix} - n, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{g \left (a e + c d x\right )}{a e g - c d f}} \right )}}{c^{2} d^{2} g \sqrt{d + e x} \left (2 n + 3\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(3/2)*(g*x+f)**n/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

2*e*(f + g*x)**(n + 1)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(c*d*g*sqr
t(d + e*x)*(2*n + 3)) - 2*(c*d*(-f - g*x)/(a*e*g - c*d*f))**(-n)*(f + g*x)**n*sq
rt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))*(-c*d**2*g*(4*n + 5) + c*d*e*f + 2*
g*(n + 1)*(a*e**2 + c*d**2))*hyper((-n, 1/2), (3/2,), g*(a*e + c*d*x)/(a*e*g - c
*d*f))/(c**2*d**2*g*sqrt(d + e*x)*(2*n + 3))

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Mathematica [A]  time = 0.28842, size = 157, normalized size = 0.74 \[ \frac{2 (f+g x)^n \sqrt{(d+e x) (a e+c d x)} \left (\frac{c d (f+g x)}{c d f-a e g}\right )^{-n} \left (3 \left (c d^2-a e^2\right ) \, _2F_1\left (\frac{1}{2},-n;\frac{3}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )+e (a e+c d x) \, _2F_1\left (\frac{3}{2},-n;\frac{5}{2};\frac{g (a e+c d x)}{a e g-c d f}\right )\right )}{3 c^2 d^2 \sqrt{d+e x}} \]

Antiderivative was successfully verified.

[In]  Integrate[((d + e*x)^(3/2)*(f + g*x)^n)/Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2],x]

[Out]

(2*Sqrt[(a*e + c*d*x)*(d + e*x)]*(f + g*x)^n*(3*(c*d^2 - a*e^2)*Hypergeometric2F
1[1/2, -n, 3/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)] + e*(a*e + c*d*x)*Hypergeo
metric2F1[3/2, -n, 5/2, (g*(a*e + c*d*x))/(-(c*d*f) + a*e*g)]))/(3*c^2*d^2*Sqrt[
d + e*x]*((c*d*(f + g*x))/(c*d*f - a*e*g))^n)

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Maple [F]  time = 0.112, size = 0, normalized size = 0. \[ \int{ \left ( gx+f \right ) ^{n} \left ( ex+d \right ) ^{{\frac{3}{2}}}{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(3/2)*(g*x+f)^n/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

int((e*x+d)^(3/2)*(g*x+f)^n/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{n}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="maxima")

[Out]

integrate((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
), x)

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Fricas [F]  time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{n}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="fricas")

[Out]

integral((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
, 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)*(g*x+f)**n/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (e x + d\right )}^{\frac{3}{2}}{\left (g x + f\right )}^{n}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x),x, algorithm="giac")

[Out]

integrate((e*x + d)^(3/2)*(g*x + f)^n/sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
), x)