3.1994 \(\int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

Optimal. Leaf size=91 \[ \frac{2}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.195403, antiderivative size = 91, 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 \[ \frac{2}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 39.47, size = 80, normalized size = 0.88 \[ - \frac{2 \sqrt{c} \sqrt{d} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} - \frac{2}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.122084, size = 91, normalized size = 1. \[ \frac{2}{\sqrt{d+e x} \left (c d^2-a e^2\right )}-\frac{2 \sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.015, size = 88, normalized size = 1. \[ -2\,{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) }-2\,{\frac{1}{ \left ( a{e}^{2}-c{d}^{2} \right ) \sqrt{ex+d}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.219177, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} + 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) - 2}{{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}, -\frac{2 \,{\left (\sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right ) - 1\right )}}{{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.72894, size = 287, normalized size = 3.15 \[ \frac{2 c d \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{c d}{a e^{2} - c d^{2}}} \sqrt{d + e x}} \right )}}{\sqrt{\frac{c d}{a e^{2} - c d^{2}}} \left (a e^{2} - c d^{2}\right )} & \text{for}\: \frac{c d}{a e^{2} - c d^{2}} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{c d}{a e^{2} - c d^{2}}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c d}{a e^{2} - c d^{2}}} \left (a e^{2} - c d^{2}\right )} & \text{for}\: \frac{1}{d + e x} > - \frac{c d}{a e^{2} - c d^{2}} \wedge \frac{c d}{a e^{2} - c d^{2}} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{c d}{a e^{2} - c d^{2}}} \sqrt{d + e x}} \right )}}{\sqrt{- \frac{c d}{a e^{2} - c d^{2}}} \left (a e^{2} - c d^{2}\right )} & \text{for}\: \frac{c d}{a e^{2} - c d^{2}} < 0 \wedge \frac{1}{d + e x} < - \frac{c d}{a e^{2} - c d^{2}} \end{cases}\right )}{a e^{2} - c d^{2}} - \frac{2}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*d*Piecewise((atan(1/(sqrt(c*d/(a*e**2 - c*d**2))*sqrt(d + e*x)))/(sqrt(c*d/(
a*e**2 - c*d**2))*(a*e**2 - c*d**2)), c*d/(a*e**2 - c*d**2) > 0), (-acoth(1/(sqr
t(-c*d/(a*e**2 - c*d**2))*sqrt(d + e*x)))/(sqrt(-c*d/(a*e**2 - c*d**2))*(a*e**2
- c*d**2)), (c*d/(a*e**2 - c*d**2) < 0) & (1/(d + e*x) > -c*d/(a*e**2 - c*d**2))
), (-atanh(1/(sqrt(-c*d/(a*e**2 - c*d**2))*sqrt(d + e*x)))/(sqrt(-c*d/(a*e**2 -
c*d**2))*(a*e**2 - c*d**2)), (c*d/(a*e**2 - c*d**2) < 0) & (1/(d + e*x) < -c*d/(
a*e**2 - c*d**2))))/(a*e**2 - c*d**2) - 2/(sqrt(d + e*x)*(a*e**2 - c*d**2))

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

[Out]

Timed out