3.2021 \(\int \frac{\sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}}{(d+e x)^{3/2}} \, dx\)

Optimal. Leaf size=128 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.271059, antiderivative size = 128, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{2 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{e \sqrt{d+e x}}-\frac{2 \sqrt{c d^2-a e^2} \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{e^{3/2}} \]

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)^(3/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 54.8991, size = 117, normalized size = 0.91 \[ \frac{2 \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{e \sqrt{d + e x}} - \frac{2 \sqrt{a e^{2} - c d^{2}} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{e^{\frac{3}{2}}} \]

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)**(1/2)/(e*x+d)**(3/2),x)

[Out]

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

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Mathematica [A]  time = 0.133381, size = 118, normalized size = 0.92 \[ \frac{2 \sqrt{d+e x} \sqrt{a e+c d x} \left (\sqrt{e} \sqrt{a e+c d x}-\sqrt{a e^2-c d^2} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )\right )}{e^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.03, size = 163, normalized size = 1.3 \[ -2\,{\frac{\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed}}{\sqrt{ex+d}\sqrt{cdx+ae}e\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}} \left ({\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) a{e}^{2}-{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) c{d}^{2}-\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e} \right ) } \]

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)^(3/2),x)

[Out]

-2*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-
c*d^2)*e)^(1/2))*a*e^2-arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c*d^
2-(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2))/(e*x+d)^(1/2)/(c*d*x+a*e)^(1/2)/e/(
(a*e^2-c*d^2)*e)^(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)^(3/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230019, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c d e x^{2} + 2 \, a d e + \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{-\frac{c d^{2} - a e^{2}}{e}} \log \left (-\frac{c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2} - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e \sqrt{-\frac{c d^{2} - a e^{2}}{e}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \,{\left (c d^{2} + a e^{2}\right )} x}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e}, \frac{2 \,{\left (c d e x^{2} + a d e - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} \sqrt{\frac{c d^{2} - a e^{2}}{e}} \arctan \left (-\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} - a e^{2}\right )} \sqrt{e x + d}}{{\left (c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x\right )} \sqrt{\frac{c d^{2} - a e^{2}}{e}}}\right ) +{\left (c d^{2} + a e^{2}\right )} x\right )}}{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{e x + d} e}\right ] \]

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)^(3/2),x, algorithm="fricas")

[Out]

[(2*c*d*e*x^2 + 2*a*d*e + sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x +
d)*sqrt(-(c*d^2 - a*e^2)/e)*log(-(c*d*e^2*x^2 + 2*a*e^3*x - c*d^3 + 2*a*d*e^2 -
2*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e*sqrt(-(c*d^2 - a*
e^2)/e))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*(c*d^2 + a*e^2)*x)/(sqrt(c*d*e*x^2 + a*d
*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e), 2*(c*d*e*x^2 + a*d*e - sqrt(c*d*e*x^2
+ a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*sqrt((c*d^2 - a*e^2)/e)*arctan(-sqrt(
c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(c*d^2 - a*e^2)*sqrt(e*x + d)/((c*d*e^2*x
^2 + a*d*e^2 + (c*d^2*e + a*e^3)*x)*sqrt((c*d^2 - a*e^2)/e))) + (c*d^2 + a*e^2)*
x)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(e*x + d)*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )}}{\left (d + e x\right )^{\frac{3}{2}}}\, dx \]

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)**(3/2),x)

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))/(d + e*x)**(3/2), x)

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GIAC/XCAS [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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)^(3/2),x, algorithm="giac")

[Out]

Timed out