3.2002 \(\int \frac{(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=94 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]

[Out]

-(Sqrt[d + e*x]/(c*d*(a*e + c*d*x))) - (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]
)/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2])

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Rubi [A]  time = 0.155438, antiderivative size = 94, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]/(c*d*(a*e + c*d*x))) - (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]
)/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2])

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Rubi in Sympy [A]  time = 40.4508, size = 78, normalized size = 0.83 \[ - \frac{\sqrt{d + e x}}{c d \left (a e + c d x\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{a e^{2} - c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

-sqrt(d + e*x)/(c*d*(a*e + c*d*x)) + e*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a
*e**2 - c*d**2))/(c**(3/2)*d**(3/2)*sqrt(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.158628, size = 94, normalized size = 1. \[ -\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{c^{3/2} d^{3/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x}}{c d (a e+c d x)} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^2,x]

[Out]

-(Sqrt[d + e*x]/(c*d*(a*e + c*d*x))) - (e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x]
)/Sqrt[c*d^2 - a*e^2]])/(c^(3/2)*d^(3/2)*Sqrt[c*d^2 - a*e^2])

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Maple [A]  time = 0.017, size = 84, normalized size = 0.9 \[ -{\frac{e}{cd \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{e}{cd}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^2,x)

[Out]

-e/d/c*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+e/d/c/((a*e^2-c*d^2)*c*d)^(1/2)*arctan(c*d*
(e*x+d)^(1/2)/((a*e^2-c*d^2)*c*d)^(1/2))

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.224263, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (c d e x + a e^{2}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right ) - 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c^{2} d^{2} x + a c d e\right )}}, -\frac{{\left (c d e x + a e^{2}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right ) + \sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (c^{2} d^{2} x + a c d e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="fricas")

[Out]

[1/2*((c*d*e*x + a*e^2)*log((sqrt(c^2*d^3 - a*c*d*e^2)*(c*d*e*x + 2*c*d^2 - a*e^
2) - 2*(c^2*d^3 - a*c*d*e^2)*sqrt(e*x + d))/(c*d*x + a*e)) - 2*sqrt(c^2*d^3 - a*
c*d*e^2)*sqrt(e*x + d))/(sqrt(c^2*d^3 - a*c*d*e^2)*(c^2*d^2*x + a*c*d*e)), -((c*
d*e*x + a*e^2)*arctan(-(c*d^2 - a*e^2)/(sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d)
)) + sqrt(-c^2*d^3 + a*c*d*e^2)*sqrt(e*x + d))/(sqrt(-c^2*d^3 + a*c*d*e^2)*(c^2*
d^2*x + a*c*d*e))]

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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)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**2,x)

[Out]

Timed out

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GIAC/XCAS [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)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Timed out