3.441 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{x (d+e x)} \, dx\)

Optimal. Leaf size=168 \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{d}} \]

[Out]

(Sqrt[c]*Sqrt[d]*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])])/Sqrt[e] - (Sqrt[a]*Sqrt[e]*ArcTan
h[(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])])/Sqrt[d]

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Rubi [A]  time = 0.554071, antiderivative size = 168, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{e}}-\frac{\sqrt{a} \sqrt{e} \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{\sqrt{d}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[c]*Sqrt[d]*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])])/Sqrt[e] - (Sqrt[a]*Sqrt[e]*ArcTan
h[(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])])/Sqrt[d]

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Rubi in Sympy [A]  time = 49.8846, size = 162, normalized size = 0.96 \[ - \frac{\sqrt{a} \sqrt{e} \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{\sqrt{d}} + \frac{\sqrt{c} \sqrt{d} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{\sqrt{e}} \]

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)**(1/2)/x/(e*x+d),x)

[Out]

-sqrt(a)*sqrt(e)*atanh((2*a*d*e + x*(a*e**2 + c*d**2))/(2*sqrt(a)*sqrt(d)*sqrt(e
)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))))/sqrt(d) + sqrt(c)*sqrt(d)*ata
nh((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))))/sqrt(e)

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Mathematica [A]  time = 0.226393, size = 185, normalized size = 1.1 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-\sqrt{a} e \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )+\sqrt{c} d \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )+\sqrt{a} e \log (x)\right )}{\sqrt{d} \sqrt{e} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 439, normalized size = 2.6 \[{\frac{1}{d}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}+{\frac{a{e}^{2}}{2\,d}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}+{\frac{cd}{2}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}+{\frac{c{d}^{2}}{2}}+cdex \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ){\frac{1}{\sqrt{cde}}}}-{ae\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}-{\frac{1}{d}\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}-{\frac{a{e}^{2}}{2\,d}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ( x+{\frac{d}{e}} \right ) cde \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \right ){\frac{1}{\sqrt{cde}}}}+{\frac{cd}{2}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+ \left ( x+{\frac{d}{e}} \right ) cde \right ){\frac{1}{\sqrt{cde}}}}+\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) } \right ){\frac{1}{\sqrt{cde}}}} \]

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)^(1/2)/x/(e*x+d),x)

[Out]

1/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/2/d*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)*a*e^2+1/
2*d*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-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)-1/d*(c*d*e*(x+d/e)^2+(a*e^
2-c*d^2)*(x+d/e))^(1/2)-1/2/d*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)*a*e^2+1/2*d*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)*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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)/((e*x + d)*x),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.498645, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*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) + 1/2*sqrt(a*e/d)*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), sqrt(-c*d/e)*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)) + 1/2*sqrt(a*e/d)*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), -sqrt(-a*e/d)*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))) + 1/2*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), -sqrt(-a*e/d)*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))) + sqrt(-c*d/e)*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)
)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{x \left (d + e x\right )}\, dx \]

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)**(1/2)/x/(e*x+d),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(x*(d + e*x)), x)

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GIAC/XCAS [A]  time = 2.25542, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x