3.1913 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=187 \[ \frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac{3}{4} \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{c} \sqrt{d} e^{5/2}} \]

[Out]

(3*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/4 + (a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(2*e*(d + e*x)) + (3*(c*d^2 - a*e^2)^2*ArcTa
nh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[c]*Sqrt[d]*e^(5/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.352752, antiderivative size = 187, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{2 e (d+e x)}+\frac{3}{4} \left (a-\frac{c d^2}{e^2}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}+\frac{3 \left (c d^2-a e^2\right )^2 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{c} \sqrt{d} e^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(3*(a - (c*d^2)/e^2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/4 + (a*d*e + (
c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(2*e*(d + e*x)) + (3*(c*d^2 - a*e^2)^2*ArcTa
nh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 +
a*e^2)*x + c*d*e*x^2])])/(8*Sqrt[c]*Sqrt[d]*e^(5/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 51.2128, size = 178, normalized size = 0.95 \[ \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )} + \frac{3 \left (a e^{2} - c d^{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 e^{2}} + \frac{3 \left (a e^{2} - c d^{2}\right )^{2} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 \sqrt{c} \sqrt{d} e^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(2*e*(d + e*x)) + 3*(a*e**2 -
c*d**2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*e**2) + 3*(a*e**2 - c*
d**2)**2*atanh((a*e**2 + c*d**2 + 2*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d
*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(8*sqrt(c)*sqrt(d)*e**(5/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.235064, size = 165, normalized size = 0.88 \[ \frac{\frac{3 \sqrt{d+e x} \left (c d^2-a e^2\right )^2 \sqrt{a e+c d x} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{c} \sqrt{d} e^{5/2}}+\frac{2 (d+e x) (a e+c d x) \left (5 a e^2+c d (2 e x-3 d)\right )}{e^2}}{8 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

((2*(a*e + c*d*x)*(d + e*x)*(5*a*e^2 + c*d*(-3*d + 2*e*x)))/e^2 + (3*(c*d^2 - a*
e^2)^2*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqr
t[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(Sqrt[c]*Sqrt[d]*e^(5/2)))/(8*S
qrt[(a*e + c*d*x)*(d + e*x)])

_______________________________________________________________________________________

Maple [B]  time = 0.015, size = 757, normalized size = 4.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/e^2/(a*e^2-c*d^2)/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2)-2/e*
d*c/(a*e^2-c*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)-3/2*e*d*c/(a*e^2
-c*d^2)*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3/4*e^2/(a*e^2-c*d^2)*
a^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+3/8*e^4/(a*e^2-c*d^2)*a^3*ln((
1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*
(d/e+x))^(1/2))/(d*e*c)^(1/2)-9/8*e^2*d^2*c/(a*e^2-c*d^2)*a^2*ln((1/2*a*e^2-1/2*
c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)
)/(d*e*c)^(1/2)+9/8*d^4*c^2/(a*e^2-c*d^2)*a*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x
))/(d*e*c)^(1/2)+(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(1/2)+3/
2/e*d^3*c^2/(a*e^2-c*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x+3/4/e^
2*d^4*c^2/(a*e^2-c*d^2)*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)-3/8/e^2*d^
6*c^3/(a*e^2-c*d^2)*ln((1/2*a*e^2-1/2*c*d^2+c*d*e*(d/e+x))/(d*e*c)^(1/2)+(c*d*(d
/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2))/(d*e*c)^(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((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.260679, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x - 3 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{c d e} + 3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \log \left (4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{16 \, \sqrt{c d e} e^{2}}, \frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (2 \, c d e x - 3 \, c d^{2} + 5 \, a e^{2}\right )} \sqrt{-c d e} + 3 \,{\left (c^{2} d^{4} - 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{8 \, \sqrt{-c d e} e^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x - 3*c*d^2 + 5*a*
e^2)*sqrt(c*d*e) + 3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2*e^4)*log(4*(2*c^2*d^2*e^2*x
+ c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*c^2*d^
2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*d*e^3)*x)*sqr
t(c*d*e)))/(sqrt(c*d*e)*e^2), 1/8*(2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)
*(2*c*d*e*x - 3*c*d^2 + 5*a*e^2)*sqrt(-c*d*e) + 3*(c^2*d^4 - 2*a*c*d^2*e^2 + a^2
*e^4)*arctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*
e + (c*d^2 + a*e^2)*x)*c*d*e)))/(sqrt(-c*d*e)*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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**2,x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError