3.2012 \(\int \frac{(d+e x)^{5/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=146 \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]

[Out]

-Sqrt[d + e*x]/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^2) + (3*e*Sqrt[d + e*x])/(4*(c*d
^2 - a*e^2)^2*(a*e + c*d*x)) - (3*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sq
rt[c*d^2 - a*e^2]])/(4*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(5/2))

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Rubi [A]  time = 0.249161, antiderivative size = 146, 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 \[ \frac{3 e \sqrt{d+e x}}{4 \left (c d^2-a e^2\right )^2 (a e+c d x)}-\frac{\sqrt{d+e x}}{2 \left (c d^2-a e^2\right ) (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

-Sqrt[d + e*x]/(2*(c*d^2 - a*e^2)*(a*e + c*d*x)^2) + (3*e*Sqrt[d + e*x])/(4*(c*d
^2 - a*e^2)^2*(a*e + c*d*x)) - (3*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sq
rt[c*d^2 - a*e^2]])/(4*Sqrt[c]*Sqrt[d]*(c*d^2 - a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 53.8161, size = 128, normalized size = 0.88 \[ \frac{3 e \sqrt{d + e x}}{4 \left (a e + c d x\right ) \left (a e^{2} - c d^{2}\right )^{2}} + \frac{\sqrt{d + e x}}{2 \left (a e + c d x\right )^{2} \left (a e^{2} - c d^{2}\right )} + \frac{3 e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 \sqrt{c} \sqrt{d} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

3*e*sqrt(d + e*x)/(4*(a*e + c*d*x)*(a*e**2 - c*d**2)**2) + sqrt(d + e*x)/(2*(a*e
+ c*d*x)**2*(a*e**2 - c*d**2)) + 3*e**2*atan(sqrt(c)*sqrt(d)*sqrt(d + e*x)/sqrt
(a*e**2 - c*d**2))/(4*sqrt(c)*sqrt(d)*(a*e**2 - c*d**2)**(5/2))

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Mathematica [A]  time = 0.350775, size = 125, normalized size = 0.86 \[ \frac{1}{4} \left (\frac{\sqrt{d+e x} \left (5 a e^2+c d (3 e x-2 d)\right )}{\left (c d^2-a e^2\right )^2 (a e+c d x)^2}-\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{\sqrt{c} \sqrt{d} \left (c d^2-a e^2\right )^{5/2}}\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(5/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

((Sqrt[d + e*x]*(5*a*e^2 + c*d*(-2*d + 3*e*x)))/((c*d^2 - a*e^2)^2*(a*e + c*d*x)
^2) - (3*e^2*ArcTanh[(Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])/Sqrt[c*d^2 - a*e^2]])/(Sqrt
[c]*Sqrt[d]*(c*d^2 - a*e^2)^(5/2)))/4

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Maple [A]  time = 0.014, size = 144, normalized size = 1. \[{\frac{{e}^{2}}{ \left ( 2\,a{e}^{2}-2\,c{d}^{2} \right ) \left ( cdex+a{e}^{2} \right ) ^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2} \left ( cdex+a{e}^{2} \right ) }\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230514, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (3 \, c d e x - 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{e x + d} + 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right )}{8 \,{\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )} \sqrt{c^{2} d^{3} - a c d e^{2}}}, \frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (3 \, c d e x - 2 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{e x + d} - 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right )}{4 \,{\left (a^{2} c^{2} d^{4} e^{2} - 2 \, a^{3} c d^{2} e^{4} + a^{4} e^{6} +{\left (c^{4} d^{6} - 2 \, a c^{3} d^{4} e^{2} + a^{2} c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (a c^{3} d^{5} e - 2 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x\right )} \sqrt{-c^{2} d^{3} + a c d e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="fricas")

[Out]

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

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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)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,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)^(5/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out