3.2051 \(\int \frac{1}{(d+e x)^{3/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=139 \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\sqrt{e} \left (c d^2-a e^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.222439, antiderivative size = 139, 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 \[ \frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{(d+e x)^{3/2} \left (c d^2-a e^2\right )}+\frac{c d \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{\sqrt{e} \left (c d^2-a e^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

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

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Rubi in Sympy [A]  time = 49.3238, size = 126, normalized size = 0.91 \[ \frac{c d \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{\sqrt{e} \left (a e^{2} - c d^{2}\right )^{\frac{3}{2}}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.283658, size = 136, normalized size = 0.98 \[ \frac{c d (d+e x) \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )-\sqrt{e} \sqrt{a e^2-c d^2} (a e+c d x)}{\sqrt{e} \sqrt{d+e x} \left (a e^2-c d^2\right )^{3/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(3/2)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]),x]

[Out]

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

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Maple [A]  time = 0.028, size = 172, normalized size = 1.2 \[{\frac{1}{a{e}^{2}-c{d}^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ({\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ) xcde+{\it Artanh} \left ({e\sqrt{cdx+ae}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \right ) c{d}^{2}-\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e} \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(3/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

(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))*x*c*d*e+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)^(3/2)/(c*d*x+a*e)^(1/2)/(a*e
^2-c*d^2)/((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(1/(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 = 0.228581, size = 1, normalized size = 0.01 \[ \left [-\frac{{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \log \left (\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} -{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}} \sqrt{e x + d}}{2 \,{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{2} e^{2} - a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt{-c d^{2} e + a e^{3}}}, -\frac{{\left (c d e^{2} x^{2} + 2 \, c d^{2} e x + c d^{3}\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right ) - \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{{\left (c d^{4} - a d^{2} e^{2} +{\left (c d^{2} e^{2} - a e^{4}\right )} x^{2} + 2 \,{\left (c d^{3} e - a d e^{3}\right )} x\right )} \sqrt{c d^{2} e - a e^{3}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(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]

[-1/2*((c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*log((2*sqrt(c*d*e*x^2 + a*d*e + (c*d^
2 + a*e^2)*x)*(c*d^2*e - a*e^3)*sqrt(e*x + d) - (c*d*e^2*x^2 + 2*a*e^3*x - c*d^3
+ 2*a*d*e^2)*sqrt(-c*d^2*e + a*e^3))/(e^2*x^2 + 2*d*e*x + d^2)) - 2*sqrt(c*d*e*
x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d^2*e + a*e^3)*sqrt(e*x + d))/((c*d^4 -
a*d^2*e^2 + (c*d^2*e^2 - a*e^4)*x^2 + 2*(c*d^3*e - a*d*e^3)*x)*sqrt(-c*d^2*e +
a*e^3)), -((c*d*e^2*x^2 + 2*c*d^2*e*x + c*d^3)*arctan(sqrt(c*d*e*x^2 + a*d*e + (
c*d^2 + a*e^2)*x)*sqrt(c*d^2*e - a*e^3)*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*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d^2*e
- a*e^3)*sqrt(e*x + d))/((c*d^4 - a*d^2*e^2 + (c*d^2*e^2 - a*e^4)*x^2 + 2*(c*d^3
*e - a*d*e^3)*x)*sqrt(c*d^2*e - a*e^3))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{\frac{3}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(e*x+d)**(3/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Integral(1/(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(1/(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