3.2005 \(\int \frac{1}{\sqrt{d+e x} \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^2} \, dx\)

Optimal. Leaf size=158 \[ \frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}-\frac{5 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{5 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2} \]

[Out]

(-5*e)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*
(d + e*x)^(3/2)) - (5*c*d*e)/((c*d^2 - a*e^2)^3*Sqrt[d + e*x]) + (5*c^(3/2)*d^(3
/2)*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e
^2)^(7/2)

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Rubi [A]  time = 0.306301, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ \frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}}-\frac{5 c d e}{\sqrt{d+e x} \left (c d^2-a e^2\right )^3}-\frac{1}{(d+e x)^{3/2} \left (c d^2-a e^2\right ) (a e+c d x)}-\frac{5 e}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2} \]

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)^2),x]

[Out]

(-5*e)/(3*(c*d^2 - a*e^2)^2*(d + e*x)^(3/2)) - 1/((c*d^2 - a*e^2)*(a*e + c*d*x)*
(d + e*x)^(3/2)) - (5*c*d*e)/((c*d^2 - a*e^2)^3*Sqrt[d + e*x]) + (5*c^(3/2)*d^(3
/2)*e*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e
^2)^(7/2)

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Rubi in Sympy [A]  time = 62.7237, size = 141, normalized size = 0.89 \[ \frac{5 c^{\frac{3}{2}} d^{\frac{3}{2}} e \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{\left (a e^{2} - c d^{2}\right )^{\frac{7}{2}}} + \frac{5 c d e}{\sqrt{d + e x} \left (a e^{2} - c d^{2}\right )^{3}} - \frac{5 e}{3 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} + \frac{1}{\left (d + e x\right )^{\frac{3}{2}} \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5*c**(3/2)*d**(3/2)*e*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt(a*e**2 - c*d**2))/
(a*e**2 - c*d**2)**(7/2) + 5*c*d*e/(sqrt(d + e*x)*(a*e**2 - c*d**2)**3) - 5*e/(3
*(d + e*x)**(3/2)*(a*e**2 - c*d**2)**2) + 1/((d + e*x)**(3/2)*(a*e + c*d*x)*(a*e
**2 - c*d**2))

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Mathematica [A]  time = 0.295914, size = 156, normalized size = 0.99 \[ \frac{2 a^2 e^4-2 a c d e^2 (7 d+5 e x)-c^2 d^2 \left (3 d^2+20 d e x+15 e^2 x^2\right )}{3 (d+e x)^{3/2} \left (c d^2-a e^2\right )^3 (a e+c d x)}+\frac{5 c^{3/2} d^{3/2} e \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\left (c d^2-a e^2\right )^{7/2}} \]

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)^2),x]

[Out]

(2*a^2*e^4 - 2*a*c*d*e^2*(7*d + 5*e*x) - c^2*d^2*(3*d^2 + 20*d*e*x + 15*e^2*x^2)
)/(3*(c*d^2 - a*e^2)^3*(a*e + c*d*x)*(d + e*x)^(3/2)) + (5*c^(3/2)*d^(3/2)*e*Arc
Tanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(c*d^2 - a*e^2)^(7/2)

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Maple [A]  time = 0.026, size = 162, normalized size = 1. \[ -{\frac{2\,e}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}} \left ( ex+d \right ) ^{-{\frac{3}{2}}}}+4\,{\frac{dec}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ex+d}}}+{\frac{e{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+5\,{\frac{e{c}^{2}{d}^{2}}{ \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}\arctan \left ({\frac{cd\sqrt{ex+d}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2/3*e/(a*e^2-c*d^2)^2/(e*x+d)^(3/2)+4*e/(a*e^2-c*d^2)^3*c*d/(e*x+d)^(1/2)+e*c^2
*d^2/(a*e^2-c*d^2)^3*(e*x+d)^(1/2)/(c*d*e*x+a*e^2)+5*e*c^2*d^2/(a*e^2-c*d^2)^3/(
(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)^2*sqrt(e*x + d)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.231964, size = 1, normalized size = 0.01 \[ \left [-\frac{30 \, c^{2} d^{2} e^{2} x^{2} + 6 \, c^{2} d^{4} + 28 \, a c d^{2} e^{2} - 4 \, a^{2} e^{4} + 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}} \log \left (\frac{c d e x + 2 \, c d^{2} - a e^{2} - 2 \,{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d} \sqrt{\frac{c d}{c d^{2} - a e^{2}}}}{c d x + a e}\right ) + 20 \,{\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{6 \,{\left (a c^{3} d^{7} e - 3 \, a^{2} c^{2} d^{5} e^{3} + 3 \, a^{3} c d^{3} e^{5} - a^{4} d e^{7} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x^{2} +{\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} x\right )} \sqrt{e x + d}}, -\frac{15 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{4} + 14 \, a c d^{2} e^{2} - 2 \, a^{2} e^{4} - 15 \,{\left (c^{2} d^{2} e^{2} x^{2} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{e x + d} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}} \arctan \left (-\frac{{\left (c d^{2} - a e^{2}\right )} \sqrt{-\frac{c d}{c d^{2} - a e^{2}}}}{\sqrt{e x + d} c d}\right ) + 10 \,{\left (2 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (a c^{3} d^{7} e - 3 \, a^{2} c^{2} d^{5} e^{3} + 3 \, a^{3} c d^{3} e^{5} - a^{4} d e^{7} +{\left (c^{4} d^{7} e - 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} - a^{3} c d e^{7}\right )} x^{2} +{\left (c^{4} d^{8} - 2 \, a c^{3} d^{6} e^{2} + 2 \, a^{3} c d^{2} e^{6} - a^{4} e^{8}\right )} x\right )} \sqrt{e x + d}}\right ] \]

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

[Out]

[-1/6*(30*c^2*d^2*e^2*x^2 + 6*c^2*d^4 + 28*a*c*d^2*e^2 - 4*a^2*e^4 + 15*(c^2*d^2
*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e + a*c*d*e^3)*x)*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)*sqr
t(c*d/(c*d^2 - a*e^2)))/(c*d*x + a*e)) + 20*(2*c^2*d^3*e + a*c*d*e^3)*x)/((a*c^3
*d^7*e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7 + (c^4*d^7*e - 3*a*c^3*
d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x^2 + (c^4*d^8 - 2*a*c^3*d^6*e^2 + 2*
a^3*c*d^2*e^6 - a^4*e^8)*x)*sqrt(e*x + d)), -1/3*(15*c^2*d^2*e^2*x^2 + 3*c^2*d^4
+ 14*a*c*d^2*e^2 - 2*a^2*e^4 - 15*(c^2*d^2*e^2*x^2 + a*c*d^2*e^2 + (c^2*d^3*e +
a*c*d*e^3)*x)*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)) + 10*(2*c^2*d^3*e + a*c*d*e^3)*x
)/((a*c^3*d^7*e - 3*a^2*c^2*d^5*e^3 + 3*a^3*c*d^3*e^5 - a^4*d*e^7 + (c^4*d^7*e -
3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 - a^3*c*d*e^7)*x^2 + (c^4*d^8 - 2*a*c^3*d^6
*e^2 + 2*a^3*c*d^2*e^6 - a^4*e^8)*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(1/(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(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^2*sqrt(e*x + d)),x, algorithm="giac")

[Out]

Timed out