3.1902 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^2} \, dx\)

Optimal. Leaf size=124 \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{3/2}}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)} \]

[Out]

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

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Rubi [A]  time = 0.165692, antiderivative size = 124, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.081 \[ \frac{\sqrt{c} \sqrt{d} \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{e^{3/2}}-\frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e (d+e x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 26.1133, size = 117, normalized size = 0.94 \[ \frac{\sqrt{c} \sqrt{d} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{e^{\frac{3}{2}}} - \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e \left (d + e x\right )} \]

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

[Out]

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

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Mathematica [A]  time = 0.257589, size = 125, normalized size = 1.01 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{\sqrt{c} \sqrt{d} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{\sqrt{d+e x} \sqrt{a e+c d x}}-\frac{2 \sqrt{e}}{d+e x}\right )}{e^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 317, normalized size = 2.6 \[ -2\,{\frac{1}{{e}^{2} \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( cd \left ({\frac{d}{e}}+x \right ) ^{2}e+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) \right ) ^{3/2} \left ({\frac{d}{e}}+x \right ) ^{-2}}+2\,{\frac{cd}{e \left ( a{e}^{2}-c{d}^{2} \right ) }\sqrt{cd \left ({\frac{d}{e}}+x \right ) ^{2}e+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) }}+{\frac{acde}{a{e}^{2}-c{d}^{2}}\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+cde \left ({\frac{d}{e}}+x \right ) \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{cd \left ({\frac{d}{e}}+x \right ) ^{2}e+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \right ){\frac{1}{\sqrt{dec}}}}-{\frac{{c}^{2}{d}^{3}}{e \left ( a{e}^{2}-c{d}^{2} \right ) }\ln \left ({1 \left ({\frac{a{e}^{2}}{2}}-{\frac{c{d}^{2}}{2}}+cde \left ({\frac{d}{e}}+x \right ) \right ){\frac{1}{\sqrt{dec}}}}+\sqrt{cd \left ({\frac{d}{e}}+x \right ) ^{2}e+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ({\frac{d}{e}}+x \right ) } \right ){\frac{1}{\sqrt{dec}}}} \]

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)^(1/2)/(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))^(3/2)+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))^(1/2)+e*d*c/(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)*a-1/e*d^3*c^2/(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)

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.283907, size = 1, normalized size = 0.01 \[ \left [\frac{{\left (e x + d\right )} \sqrt{\frac{c d}{e}} \log \left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 4 \,{\left (2 \, c d e^{2} x + c d^{2} e + a e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{\frac{c d}{e}} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right ) - 4 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{2 \,{\left (e^{2} x + d e\right )}}, \frac{{\left (e x + d\right )} \sqrt{-\frac{c d}{e}} \arctan \left (\frac{2 \, c d e x + c d^{2} + a e^{2}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-\frac{c d}{e}} e}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}}{e^{2} x + d e}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{2}}\, dx \]

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

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**2, x)

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

[Out]

Exception raised: TypeError