3.2004 \(\int \frac{\sqrt{d+e x}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=128 \[ -\frac{3 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}-\frac{1}{\sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)}+\frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

[Out]

(-3*e)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*Sqrt
[d + e*x]) + (3*Sqrt[c]*Sqrt[d]*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c
*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(5/2)

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Rubi [A]  time = 0.216524, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{3 e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}-\frac{1}{\sqrt{d+e x} \left (c d^2-a e^2\right ) (a e+c d x)}+\frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-3*e)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*Sqrt
[d + e*x]) + (3*Sqrt[c]*Sqrt[d]*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c
*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(5/2)

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Rubi in Sympy [A]  time = 50.8267, size = 112, normalized size = 0.88 \[ - \frac{3 \sqrt{c} \sqrt{d} e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} - \frac{3 e}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{\sqrt{d + e x} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*sqrt(c)*sqrt(d)*e*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/(
a*e**2 - c*d**2)**(5/2) - 3*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)**2) + 1/(sqrt(d +
e*x)*(a*e + c*d*x)*(a*e**2 - c*d**2))

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Mathematica [A]  time = 0.269767, size = 118, normalized size = 0.92 \[ \frac{3 \sqrt{c} \sqrt{d} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}-\frac{2 a e^2+c d (d+3 e x)}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2 (a e+c d x)} \]

Antiderivative was successfully verified.

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

[Out]

-((2*a*e^2 + c*d*(d + 3*e*x))/((c*d^2 - a*e^2)^2*(a*e + c*d*x)*Sqrt[d + e*x])) +
(3*Sqrt[c]*Sqrt[d]*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2
]])/(c*d^2 - a*e^2)^(5/2)

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Maple [A]  time = 0.026, size = 129, normalized size = 1. \[ -2\,{\frac{e}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}-{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}-3\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*e/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)-e*c*d/(a*e^2-c*d^2)^2*(e*x+d)^(1/2)/(c*d*e*x+
a*e^2)-3*e*c*d/(a*e^2-c*d^2)^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(sqrt(e*x + d)/(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 = 0.227678, size = 1, normalized size = 0.01 \[ \left [-\frac{6 \, c d e x + 2 \, c d^{2} + 4 \, a e^{2} - 3 \,{\left (c d e x + a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right )}{2 \,{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{e x + d}}, -\frac{3 \, c d e x + c d^{2} + 2 \, a e^{2} - 3 \,{\left (c d e x + a e^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right )}{{\left (a c^{2} d^{4} e - 2 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} +{\left (c^{3} d^{5} - 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} x\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/2*(6*c*d*e*x + 2*c*d^2 + 4*a*e^2 - 3*(c*d*e*x + a*e^2)*sqrt(e*x + d)*sqrt(c*
d/(c*d^2 - a*e^2))*log((c*d*e*x + 2*c*d^2 - a*e^2 + 2*(c*d^2 - a*e^2)*sqrt(e*x +
d)*sqrt(c*d/(c*d^2 - a*e^2)))/(c*d*x + a*e)))/((a*c^2*d^4*e - 2*a^2*c*d^2*e^3 +
a^3*e^5 + (c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e^4)*x)*sqrt(e*x + d)), -(3*c*d*
e*x + c*d^2 + 2*a*e^2 - 3*(c*d*e*x + a*e^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e
^2))*arctan(-(c*d^2 - a*e^2)*sqrt(-c*d/(c*d^2 - a*e^2))/(sqrt(e*x + d)*c*d)))/((
a*c^2*d^4*e - 2*a^2*c*d^2*e^3 + a^3*e^5 + (c^3*d^5 - 2*a*c^2*d^3*e^2 + a^2*c*d*e
^4)*x)*sqrt(e*x + d))]

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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)**(1/2)/(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(sqrt(e*x + d)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2,x, algorithm="giac")

[Out]

Timed out