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