3.2010 \(\int \frac{(d+e x)^{9/2}}{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^3} \, dx\)

Optimal. Leaf size=130 \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2} \]

[Out]

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

_______________________________________________________________________________________

Rubi [A]  time = 0.213665, antiderivative size = 130, 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 \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{3 e \sqrt{d+e x}}{4 c^2 d^2 (a e+c d x)}-\frac{(d+e x)^{3/2}}{2 c d (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Int[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.168, size = 116, normalized size = 0.89 \[ - \frac{\left (d + e x\right )^{\frac{3}{2}}}{2 c d \left (a e + c d x\right )^{2}} - \frac{3 e \sqrt{d + e x}}{4 c^{2} d^{2} \left (a e + c d x\right )} + \frac{3 e^{2} \operatorname{atan}{\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d + e x}}{\sqrt{a e^{2} - c d^{2}}} \right )}}{4 c^{\frac{5}{2}} d^{\frac{5}{2}} \sqrt{a e^{2} - c d^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x+d)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

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

_______________________________________________________________________________________

Mathematica [A]  time = 0.174601, size = 118, normalized size = 0.91 \[ -\frac{3 e^2 \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d} \sqrt{d+e x}}{\sqrt{c d^2-a e^2}}\right )}{4 c^{5/2} d^{5/2} \sqrt{c d^2-a e^2}}-\frac{\sqrt{d+e x} \left (3 a e^2+c d (2 d+5 e x)\right )}{4 c^2 d^2 (a e+c d x)^2} \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^(9/2)/(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^3,x]

[Out]

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

_______________________________________________________________________________________

Maple [A]  time = 0.02, size = 149, normalized size = 1.2 \[ -{\frac{5\,{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}cd} \left ( ex+d \right ) ^{{\frac{3}{2}}}}-{\frac{3\,{e}^{4}a}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}{c}^{2}{d}^{2}}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\, \left ( cdex+a{e}^{2} \right ) ^{2}c}\sqrt{ex+d}}+{\frac{3\,{e}^{2}}{4\,{c}^{2}{d}^{2}}\arctan \left ({cd\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \right ){\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x+d)^(9/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^3,x)

[Out]

-5/4*e^2/(c*d*e*x+a*e^2)^2/c/d*(e*x+d)^(3/2)-3/4*e^4/(c*d*e*x+a*e^2)^2/c^2/d^2*(
e*x+d)^(1/2)*a+3/4*e^2/(c*d*e*x+a*e^2)^2/c*(e*x+d)^(1/2)+3/4*e^2/c^2/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))

_______________________________________________________________________________________

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)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.230155, size = 1, normalized size = 0.01 \[ \left [-\frac{2 \, \sqrt{c^{2} d^{3} - a c d e^{2}}{\left (5 \, c d e x + 2 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d} - 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \log \left (\frac{\sqrt{c^{2} d^{3} - a c d e^{2}}{\left (c d e x + 2 \, c d^{2} - a e^{2}\right )} - 2 \,{\left (c^{2} d^{3} - a c d e^{2}\right )} \sqrt{e x + d}}{c d x + a e}\right )}{8 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )} \sqrt{c^{2} d^{3} - a c d e^{2}}}, -\frac{\sqrt{-c^{2} d^{3} + a c d e^{2}}{\left (5 \, c d e x + 2 \, c d^{2} + 3 \, a e^{2}\right )} \sqrt{e x + d} + 3 \,{\left (c^{2} d^{2} e^{2} x^{2} + 2 \, a c d e^{3} x + a^{2} e^{4}\right )} \arctan \left (-\frac{c d^{2} - a e^{2}}{\sqrt{-c^{2} d^{3} + a c d e^{2}} \sqrt{e x + d}}\right )}{4 \,{\left (c^{4} d^{4} x^{2} + 2 \, a c^{3} d^{3} e x + a^{2} c^{2} d^{2} e^{2}\right )} \sqrt{-c^{2} d^{3} + a c d e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x + d)^(9/2)/(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)*(5*c*d*e*x + 2*c*d^2 + 3*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)))/((c^4*d^4*x^2 + 2*a*c^3*d^3*e*x + a^2*c^2*d^2*e^2)*sqrt(c^2*d^3 - a*c*d*e^2
)), -1/4*(sqrt(-c^2*d^3 + a*c*d*e^2)*(5*c*d*e*x + 2*c*d^2 + 3*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))))/((c^4*d^4*x^2 + 2*a*c^3*d^3*e*x + a^2*c^
2*d^2*e^2)*sqrt(-c^2*d^3 + a*c*d*e^2))]

_______________________________________________________________________________________

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)**(9/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**3,x)

[Out]

Timed out

_______________________________________________________________________________________

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)^(9/2)/(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^3,x, algorithm="giac")

[Out]

Timed out