3.2003 \(\int \frac{(d+e x)^{3/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=101 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{3/2}}-\frac{\sqrt{d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]

[Out]

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

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Rubi [A]  time = 0.17349, antiderivative size = 101, 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 \[ \frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{3/2}}-\frac{\sqrt{d+e x}}{\left (c d^2-a e^2\right ) (a e+c d x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 41.4782, size = 85, normalized size = 0.84 \[ \frac{\sqrt{d + e x}}{\left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} + \frac{e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\sqrt{c} \sqrt{d} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.169977, size = 102, normalized size = 1.01 \[ \frac{\frac{\sqrt{d+e x}}{a e+c d x}-\frac{e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \sqrt{c d^2-a e^2}}}{a e^2-c d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.011, size = 99, normalized size = 1. \[{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{e}{a{e}^{2}-c{d}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(e*x+d)^(1/2)/(a*e^2-c*d^2)/(c*d*e*x+a*e^2)+e/(a*e^2-c*d^2)/((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)^(3/2)/(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 = 0.22561, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d e x + a e^{2}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right ) + 2 \, \sqrt{c^{2} d^{3} - a c d e^{2}} \sqrt{e x + d}}{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )}}, \frac{{\left (c d e x + a e^{2}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right ) - \sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (a c d^{2} e - a^{2} e^{3} +{\left (c^{2} d^{3} - a c d e^{2}\right )} x\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out