3.1995 \(\int \frac{1}{(d+e x)^{3/2} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )} \, dx\)

Optimal. Leaf size=120 \[ -\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac{2 c d}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

[Out]

2/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (2*c*d)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]
) - (2*c^(3/2)*d^(3/2)*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 = 0.232387, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}}+\frac{2 c d}{\sqrt{d+e x} \left (c d^2-a e^2\right )^2}+\frac{2}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

2/(3*(c*d^2 - a*e^2)*(d + e*x)^(3/2)) + (2*c*d)/((c*d^2 - a*e^2)^2*Sqrt[d + e*x]
) - (2*c^(3/2)*d^(3/2)*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 = 49.5063, size = 105, normalized size = 0.88 \[ \frac{2 c^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{2 c d}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{2}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

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

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Mathematica [A]  time = 0.339688, size = 111, normalized size = 0.92 \[ \frac{2 \left (c d (4 d+3 e x)-a e^2\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}-\frac{2 c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)),x]

[Out]

(2*(-(a*e^2) + c*d*(4*d + 3*e*x)))/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - (2*c^
(3/2)*d^(3/2)*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 = 0.017, size = 117, normalized size = 1. \[ -{\frac{2}{3\,a{e}^{2}-3\,c{d}^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+2\,{\frac{cd}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ex+d}}}+2\,{\frac{{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2),x)

[Out]

-2/3/(a*e^2-c*d^2)/(e*x+d)^(3/2)+2*c*d/(a*e^2-c*d^2)^2/(e*x+d)^(1/2)+2*c^2*d^2/(
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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.221308, size = 1, normalized size = 0.01 \[ \left [\frac{6 \, c d e x + 8 \, c d^{2} - 2 \, a e^{2} + 3 \,{\left (c d e x + c d^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right )}{3 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}, \frac{2 \,{\left (3 \, c d e x + 4 \, c d^{2} - a e^{2} - 3 \,{\left (c d e x + c d^{2}\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right )\right )}}{3 \,{\left (c^{2} d^{5} - 2 \, a c d^{3} e^{2} + a^{2} d e^{4} +{\left (c^{2} d^{4} e - 2 \, a c d^{2} e^{3} + a^{2} e^{5}\right )} x\right )} \sqrt{e x + d}}\right ] \]

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)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[1/3*(6*c*d*e*x + 8*c*d^2 - 2*a*e^2 + 3*(c*d*e*x + c*d^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)))/((c^2*d^5 - 2*a*c*d^3*e^2 + a^2*d*
e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*x + d)), 2/3*(3*c*d*e*x +
4*c*d^2 - a*e^2 - 3*(c*d*e*x + c*d^2)*sqrt(e*x + d)*sqrt(-c*d/(c*d^2 - a*e^2))*a
rctan(-(c*d^2 - a*e^2)*sqrt(-c*d/(c*d^2 - a*e^2))/(sqrt(e*x + d)*c*d)))/((c^2*d^
5 - 2*a*c*d^3*e^2 + a^2*d*e^4 + (c^2*d^4*e - 2*a*c*d^2*e^3 + a^2*e^5)*x)*sqrt(e*
x + d))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (d + e x\right )^{\frac{5}{2}} \left (a e + c d x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2),x)

[Out]

Integral(1/((d + e*x)**(5/2)*(a*e + c*d*x)), x)

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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)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

Timed out