3.1914 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{(d+e x)^3} \, dx\)

Optimal. Leaf size=175 \[ -\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^2}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2}-\frac{3 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 e^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.294596, antiderivative size = 175, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.108 \[ -\frac{2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{e (d+e x)^2}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e^2}-\frac{3 \sqrt{c} \sqrt{d} \left (c d^2-a e^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 e^{5/2}} \]

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)^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 48.7799, size = 170, normalized size = 0.97 \[ \frac{3 \sqrt{c} \sqrt{d} \left (a e^{2} - c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 e^{\frac{5}{2}}} + \frac{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e^{2}} - \frac{2 \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{e \left (d + e x\right )^{2}} \]

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)**3,x)

[Out]

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

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Mathematica [A]  time = 0.522503, size = 166, normalized size = 0.95 \[ \frac{1}{2} ((d+e x) (a e+c d x))^{3/2} \left (\frac{3 \sqrt{c} \sqrt{d} \left (a e^2-c d^2\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{e^{5/2} (d+e x)^{3/2} (a e+c d x)^{3/2}}+\frac{2 \left (c d (3 d+e x)-2 a e^2\right )}{e^2 (d+e x)^2 (a e+c d x)}\right ) \]

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)^3,x]

[Out]

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

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Maple [B]  time = 0.017, size = 837, normalized size = 4.8 \[ \text{result too large to display} \]

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)^3,x)

[Out]

-2/e^3/(a*e^2-c*d^2)/(d/e+x)^3*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2)+8/e
^2*d*c/(a*e^2-c*d^2)^2/(d/e+x)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(5/2)-8
/e*d^2*c^2/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(3/2)-6*e*d^2
*c^2/(a*e^2-c*d^2)^2*a*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)*x-3*e^2*d*c
/(a*e^2-c*d^2)^2*a^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d^2)*(d/e+x))^(1/2)+3/2*e^4*d*c/(
a*e^2-c*d^2)^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)-9/2*e^2*d^3*c^2/(a*e^2-c*d^
2)^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)+9/2*d^5*c^3/(a*e^2-c*d^2)^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)+6/e*d^4*c^3/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c
*d^2)*(d/e+x))^(1/2)*x+3/e^2*d^5*c^3/(a*e^2-c*d^2)^2*(c*d*(d/e+x)^2*e+(a*e^2-c*d
^2)*(d/e+x))^(1/2)-3/2/e^2*d^7*c^4/(a*e^2-c*d^2)^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)

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.311795, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} - 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) + 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{4 \,{\left (e^{3} x + d e^{2}\right )}}, -\frac{3 \,{\left (c d^{3} - a d e^{2} +{\left (c d^{2} e - a e^{3}\right )} x\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-\frac{c d}{e}} e}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d e x + 3 \, c d^{2} - 2 \, a e^{2}\right )}}{2 \,{\left (e^{3} x + d e^{2}\right )}}\right ] \]

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)^3,x, algorithm="fricas")

[Out]

[1/4*(3*(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(c*d/e)*log(8*c^2*d^2*e^2*x^
2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 - 4*(2*c*d*e^2*x + c*d^2*e + a*e^3)*sqrt(c
*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d/e) + 8*(c^2*d^3*e + a*c*d*e^3)*x)
+ 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*e*x + 3*c*d^2 - 2*a*e^2))/
(e^3*x + d*e^2), -1/2*(3*(c*d^3 - a*d*e^2 + (c*d^2*e - a*e^3)*x)*sqrt(-c*d/e)*ar
ctan(1/2*(2*c*d*e*x + c*d^2 + a*e^2)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(-c*d/e)*e)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*e*x + 3*
c*d^2 - 2*a*e^2))/(e^3*x + 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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/(e*x+d)**3,x)

[Out]

Timed out

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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)^3,x, algorithm="giac")

[Out]

Exception raised: TypeError