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

Optimal. Leaf size=240 \[ -\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (a e-c d x)}{x}+\frac{\sqrt{c} \sqrt{d} \left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.856779, antiderivative size = 240, 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 \[ -\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2} (a e-c d x)}{x}+\frac{\sqrt{c} \sqrt{d} \left (3 a e^2+c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \left (a e^2+3 c d^2\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 \sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 83.9138, size = 228, normalized size = 0.95 \[ - \frac{\sqrt{a} \sqrt{e} \left (a e^{2} + 3 c d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 \sqrt{d}} + \frac{\sqrt{c} \sqrt{d} \left (3 a e^{2} + c d^{2}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{2 \sqrt{e}} - \frac{\left (a e - c d x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{x} \]

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

[Out]

-sqrt(a)*sqrt(e)*(a*e**2 + 3*c*d**2)*atanh((2*a*d*e + x*(a*e**2 + c*d**2))/(2*sq
rt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/(2*sqrt(d
)) + sqrt(c)*sqrt(d)*(3*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*sqrt
(e)) - (a*e - c*d*x)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/x

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Mathematica [A]  time = 0.561272, size = 356, normalized size = 1.48 \[ \frac{\sqrt{a} \sqrt{e} \log (x) \left (a e^2+3 c d^2\right ) \sqrt{(d+e x) (a e+c d x)}}{2 \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x}}-\frac{\sqrt{a} \sqrt{e} \left (a e^2+3 c d^2\right ) \sqrt{(d+e x) (a e+c d x)} \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+2 a d e+a e^2 x+c d^2 x\right )}{2 \sqrt{d} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{\sqrt{c} \sqrt{d} \left (3 a e^2+c d^2\right ) \sqrt{(d+e x) (a e+c d x)} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d^2+2 c d e x\right )}{2 \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}}+\sqrt{(d+e x) (a e+c d x)} \left (c d-\frac{a e}{x}\right ) \]

Antiderivative was successfully verified.

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

[Out]

(c*d - (a*e)/x)*Sqrt[(a*e + c*d*x)*(d + e*x)] + (Sqrt[a]*Sqrt[e]*(3*c*d^2 + a*e^
2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Log[x])/(2*Sqrt[d]*Sqrt[a*e + c*d*x]*Sqrt[d + e
*x]) - (Sqrt[a]*Sqrt[e]*(3*c*d^2 + a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Log[2*a*
d*e + c*d^2*x + a*e^2*x + 2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e
*x]])/(2*Sqrt[d]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]) + (Sqrt[c]*Sqrt[d]*(c*d^2 + 3*
a*e^2)*Sqrt[(a*e + c*d*x)*(d + e*x)]*Log[c*d^2 + a*e^2 + 2*c*d*e*x + 2*Sqrt[c]*S
qrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x]])/(2*Sqrt[e]*Sqrt[a*e + c*d*x]*Sq
rt[d + e*x])

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Maple [B]  time = 0.023, size = 1310, normalized size = 5.5 \[ \text{result too large to display} \]

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^2/(e*x+d),x)

[Out]

7/16*d^3*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)-1/2*a^2*e^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)+1/a/e*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c+5/4*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)*x*c+1/d*a*e^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/4*e*c*(c*d*e*(x+d/e)^
2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/16*d^3*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)+
1/16*e^6/d^3*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)-1/16*e^6/d^3*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)-3/16*e^2*d*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)+27/16*d*a*e^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)*c-3/2*d^2*a*e/(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-1/4*e^3/d^2*a*(a*d*e+(a
*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-1/8*e^4/d^3*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2)+3/16*e^4/d*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/d*a^2*e^4*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/d^2/a/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/4*e^3/d^2
*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/8*e^4/d^3*a^2/c*(c*d*e*(x+d
/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+1/d*c/a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/
2)*x+1/3*e/d^2*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-1/8*d*c*(c*d*e*(x+d
/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+2/3/d^2*e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(
3/2)+17/8*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c

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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^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.70358, 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^2),x, algorithm="fricas")

[Out]

[1/4*((c*d^2 + 3*a*e^2)*sqrt(c*d/e)*x*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) + (3*c*d^2 + a*e^2)*s
qrt(a*e/d)*x*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sq
rt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d^2*e + (c*d^3 + a*d*e^2)*x)*sqrt
(a*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(c*d*x - a*e))/x, 1/4*(2*(c*d^2 + 3*a*e^2)*sqrt(-c*d/e)*x*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)*e)) + (3*c*d^2 + a*e^2)*sqrt(a*e/d)*x*log((8*a^2*d^2*e^2 + (c^2*d^4 + 6*a
*c*d^2*e^2 + a^2*e^4)*x^2 - 4*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(2*a*d
^2*e + (c*d^3 + a*d*e^2)*x)*sqrt(a*e/d) + 8*(a*c*d^3*e + a^2*d*e^3)*x)/x^2) + 4*
sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d*x - a*e))/x, -1/4*(2*(3*c*d^2 +
a*e^2)*sqrt(-a*e/d)*x*arctan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)/(sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*d*sqrt(-a*e/d))) - (c*d^2 + 3*a*e^2)*sqrt(c*d/e)*x*
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*x - a*
e))/x, -1/2*((3*c*d^2 + a*e^2)*sqrt(-a*e/d)*x*arctan(1/2*(2*a*d*e + (c*d^2 + a*e
^2)*x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*d*sqrt(-a*e/d))) - (c*d^2 +
3*a*e^2)*sqrt(-c*d/e)*x*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)*e)) - 2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2
+ a*e^2)*x)*(c*d*x - a*e))/x]

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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**2/(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^2),x, algorithm="giac")

[Out]

Exception raised: TypeError