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

Optimal. Leaf size=256 \[ -\frac{\left (-a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{a} d^{3/2} \sqrt{e}}+c^{3/2} d^{3/2} \sqrt{e} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (x \left (a e^2+5 c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^2} \]

[Out]

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

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Rubi [A]  time = 0.87493, antiderivative size = 256, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\left (-a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{8 \sqrt{a} d^{3/2} \sqrt{e}}+c^{3/2} d^{3/2} \sqrt{e} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )-\frac{\left (x \left (a e^2+5 c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 d x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 105.661, size = 246, normalized size = 0.96 \[ c^{\frac{3}{2}} d^{\frac{3}{2}} \sqrt{e} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )} - \frac{\left (a d e + \frac{x \left (a e^{2} + 5 c d^{2}\right )}{2}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 d x^{2}} + \frac{\left (a^{2} e^{4} - 6 a c d^{2} e^{2} - 3 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{8 \sqrt{a} d^{\frac{3}{2}} \sqrt{e}} \]

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)/x**3/(e*x+d),x)

[Out]

c**(3/2)*d**(3/2)*sqrt(e)*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)))) - (a*d*e + x*(a*e**2 +
5*c*d**2)/2)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*d*x**2) + (a**2*
e**4 - 6*a*c*d**2*e**2 - 3*c**2*d**4)*atanh((2*a*d*e + x*(a*e**2 + c*d**2))/(2*s
qrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(8*sqrt(
a)*d**(3/2)*sqrt(e))

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Mathematica [A]  time = 0.633135, size = 310, normalized size = 1.21 \[ -\frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (x^2 \log (x) \left (a^2 e^4-6 a c d^2 e^2-3 c^2 d^4\right )+x^2 \left (-a^2 e^4+6 a c d^2 e^2+3 c^2 d^4\right ) \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )-8 \sqrt{a} c^{3/2} d^3 e x^2 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )+2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a e (2 d+e x)+5 c d^2 x\right )\right )}{8 \sqrt{a} d^{3/2} \sqrt{e} x^2 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(5*c*d^2*x + a*e*(2*d + e*x)) + (-3*c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e
^4)*x^2*Log[x] + (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2*Log[c*d^2*x + 2*Sqrt[
a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + a*e*(2*d + e*x)] - 8*Sqrt[a
]*c^(3/2)*d^3*e*x^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)]))/(8*Sqrt[a]*d^(3/2)*Sqrt[e]*x^2*Sqrt[(a*e + c*d*x)
*(d + e*x)])

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Maple [B]  time = 0.025, size = 1604, normalized size = 6.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3/8*d^3/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c^2+3/4/d^3/a/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(5/2)-1/4/d^2*a*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/16/d^4*e^7*a^3/c*
ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d
^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-1/16/d^4*e^7*a^3/c*ln((1/2*a*e^2+1/2*c*d^2+c*d
*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-1/4/d
/a^2/e^2/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-3/4/d^2/a*e*c*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(3/2)*x-3/4*d*a*e^2/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)
*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)*c-3/16/d^2*e^5*a^
2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c
*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/16*e^3*a*c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)
*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/
2)+1/4/d*e^2*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/16*d^2*e*c^2*ln
((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2
)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+1/4/d^3*e^4*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x+1/8/d^4*e^5*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+3/16/d^2*e^5*a
^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x
^2)^(1/2))/(c*d*e)^(1/2)-1/2/d*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c+1
/8/d*a^2*e^4/(a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/x)+1/4/a^2/e*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)*x-3/16*a*e^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)*c+1/4*d/a^2/e^2*(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(3/2)*c^2+3/4*d^2/a/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+
17/16*d^2*e*c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)+3/4*d/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*x*c^2-1/2/d^2/a/e/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/8/d^4*e^5*
a^2/c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/4/d^3*e^4*a*(c*d*e*(x+d/e)
^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/3/d^3*e^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+
d/e))^(3/2)+1/8*e*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+7/8*e*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-5/12/d^3*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(3/2)

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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)*x^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.00095, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)*x^3),x, algorithm="fricas")

[Out]

[1/16*(8*sqrt(a*d*e)*sqrt(c*d*e)*c*d^2*x^2*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*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x
+ c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^2*d^3*e + a*c*d*e^3)*x) - (3*c^2*d^4 + 6*a*c
*d^2*e^2 - a^2*e^4)*x^2*log((4*(2*a^2*d^2*e^2 + (a*c*d^3*e + a^2*d*e^3)*x)*sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e
^2 + a^2*e^4)*x^2 + 8*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(a*d*e))/x^2) - 4*sqrt(c*d*
e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (5*c*d^2 + a*e^2)*x)*sqrt(a*d*e))/
(sqrt(a*d*e)*d*x^2), 1/16*(16*sqrt(a*d*e)*sqrt(-c*d*e)*c*d^2*x^2*arctan(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
))) - (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2*log((4*(2*a^2*d^2*e^2 + (a*c*d^3
*e + a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) + (8*a^2*d^2*e^2
+ (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 8*(a*c*d^3*e + a^2*d*e^3)*x)*sqrt(a*
d*e))/x^2) - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (5*c*d^2 +
a*e^2)*x)*sqrt(a*d*e))/(sqrt(a*d*e)*d*x^2), 1/8*(4*sqrt(-a*d*e)*sqrt(c*d*e)*c*d
^2*x^2*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*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*c*d*e*x + c*d^2 + a*e^2)*sqrt(c*d*e) + 8*(c^
2*d^3*e + a*c*d*e^3)*x) - (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2*arctan(1/2*(
2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e
^2)*x)*a*d*e)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e + (5*c*d
^2 + a*e^2)*x)*sqrt(-a*d*e))/(sqrt(-a*d*e)*d*x^2), 1/8*(8*sqrt(-a*d*e)*sqrt(-c*d
*e)*c*d^2*x^2*arctan(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))) - (3*c^2*d^4 + 6*a*c*d^2*e^2 - a^2*e^4)*x^2*arc
tan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*
d^2 + a*e^2)*x)*a*d*e)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d*e
+ (5*c*d^2 + a*e^2)*x)*sqrt(-a*d*e))/(sqrt(-a*d*e)*d*x^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)/x**3/(e*x+d),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)*x^3),x, algorithm="giac")

[Out]

Exception raised: TypeError