3.2052 \(\int \frac{1}{(d+e x)^{5/2} \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2}} \, dx\)

Optimal. Leaf size=207 \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \sqrt{e} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]

[Out]

Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]/(2*(c*d^2 - a*e^2)*(d + e*x)^(5/2))
+ (3*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d^2 - a*e^2)^2*(d +
e*x)^(3/2)) + (3*c^2*d^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])])/(4*Sqrt[e]*(c*d^2 - a*e^2)^(5/2))

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Rubi [A]  time = 0.362584, antiderivative size = 207, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 39, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{3 c^2 d^2 \tan ^{-1}\left (\frac{\sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{\sqrt{d+e x} \sqrt{c d^2-a e^2}}\right )}{4 \sqrt{e} \left (c d^2-a e^2\right )^{5/2}}+\frac{3 c d \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{4 (d+e x)^{3/2} \left (c d^2-a e^2\right )^2}+\frac{\sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{2 (d+e x)^{5/2} \left (c d^2-a e^2\right )} \]

Antiderivative was successfully verified.

[In]  Int[1/((d + e*x)^(5/2)*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]/(2*(c*d^2 - a*e^2)*(d + e*x)^(5/2))
+ (3*c*d*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(4*(c*d^2 - a*e^2)^2*(d +
e*x)^(3/2)) + (3*c^2*d^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])])/(4*Sqrt[e]*(c*d^2 - a*e^2)^(5/2))

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Rubi in Sympy [A]  time = 73.465, size = 190, normalized size = 0.92 \[ - \frac{3 c^{2} d^{2} \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{\sqrt{d + e x} \sqrt{a e^{2} - c d^{2}}} \right )}}{4 \sqrt{e} \left (a e^{2} - c d^{2}\right )^{\frac{5}{2}}} + \frac{3 c d \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{4 \left (d + e x\right )^{\frac{3}{2}} \left (a e^{2} - c d^{2}\right )^{2}} - \frac{\sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{2 \left (d + e x\right )^{\frac{5}{2}} \left (a e^{2} - c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

-3*c**2*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)))/(4*sqrt(e)*(a*e**2 - c*d**2)**(5/2)) + 3*c*d*sq
rt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(4*(d + e*x)**(3/2)*(a*e**2 - c*d**
2)**2) - sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(2*(d + e*x)**(5/2)*(a*e
**2 - c*d**2))

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Mathematica [A]  time = 0.319893, size = 174, normalized size = 0.84 \[ \frac{\sqrt{e} \sqrt{a e^2-c d^2} \left (-2 a^2 e^3+a c d e (5 d+e x)+c^2 d^2 x (5 d+3 e x)\right )-3 c^2 d^2 (d+e x)^2 \sqrt{a e+c d x} \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{a e+c d x}}{\sqrt{a e^2-c d^2}}\right )}{4 \sqrt{e} (d+e x)^{3/2} \left (a e^2-c d^2\right )^{5/2} \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*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]*(-2*a^2*e^3 + a*c*d*e*(5*d + e*x) + c^2*d^2*x*(5
*d + 3*e*x)) - 3*c^2*d^2*Sqrt[a*e + c*d*x]*(d + e*x)^2*ArcTanh[(Sqrt[e]*Sqrt[a*e
+ c*d*x])/Sqrt[-(c*d^2) + a*e^2]])/(4*Sqrt[e]*(-(c*d^2) + a*e^2)^(5/2)*(d + e*x
)^(3/2)*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [A]  time = 0.026, size = 292, normalized size = 1.4 \[ -{\frac{1}{4\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{2}}\sqrt{cde{x}^{2}+a{e}^{2}x+c{d}^{2}x+aed} \left ( 3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){x}^{2}{c}^{2}{d}^{2}{e}^{2}+6\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ) x{c}^{2}{d}^{3}e+3\,{\it Artanh} \left ({\frac{e\sqrt{cdx+ae}}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}} \right ){c}^{2}{d}^{4}-3\,xcde\sqrt{cdx+ae}\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}+2\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}a{e}^{2}-5\,\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}\sqrt{cdx+ae}c{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cdx+ae}}}{\frac{1}{\sqrt{ \left ( a{e}^{2}-c{d}^{2} \right ) e}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(e*x+d)^(5/2)/(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2),x)

[Out]

-1/4*(c*d*e*x^2+a*e^2*x+c*d^2*x+a*d*e)^(1/2)*(3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*
e^2-c*d^2)*e)^(1/2))*x^2*c^2*d^2*e^2+6*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2
)*e)^(1/2))*x*c^2*d^3*e+3*arctanh(e*(c*d*x+a*e)^(1/2)/((a*e^2-c*d^2)*e)^(1/2))*c
^2*d^4-3*x*c*d*e*(c*d*x+a*e)^(1/2)*((a*e^2-c*d^2)*e)^(1/2)+2*((a*e^2-c*d^2)*e)^(
1/2)*(c*d*x+a*e)^(1/2)*a*e^2-5*((a*e^2-c*d^2)*e)^(1/2)*(c*d*x+a*e)^(1/2)*c*d^2)/
(e*x+d)^(5/2)/(c*d*x+a*e)^(1/2)/(a*e^2-c*d^2)^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)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.23445, size = 1, normalized size = 0. \[ \left [\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d^{2} e + a e^{3}}{\left (3 \, c d e x + 5 \, c d^{2} - 2 \, a e^{2}\right )} \sqrt{e x + d} + 3 \,{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \log \left (-\frac{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x}{\left (c d^{2} e - a e^{3}\right )} \sqrt{e x + d} +{\left (c d e^{2} x^{2} + 2 \, a e^{3} x - c d^{3} + 2 \, a d e^{2}\right )} \sqrt{-c d^{2} e + a e^{3}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{8 \,{\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} +{\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{-c d^{2} e + a e^{3}}}, \frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}}{\left (3 \, c d e x + 5 \, c d^{2} - 2 \, a e^{2}\right )} \sqrt{e x + d} - 3 \,{\left (c^{2} d^{2} e^{3} x^{3} + 3 \, c^{2} d^{3} e^{2} x^{2} + 3 \, c^{2} d^{4} e x + c^{2} d^{5}\right )} \arctan \left (\frac{\sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d^{2} e - a e^{3}} \sqrt{e x + d}}{c d e^{2} x^{2} + a d e^{2} +{\left (c d^{2} e + a e^{3}\right )} x}\right )}{4 \,{\left (c^{2} d^{7} - 2 \, a c d^{5} e^{2} + a^{2} d^{3} e^{4} +{\left (c^{2} d^{4} e^{3} - 2 \, a c d^{2} e^{5} + a^{2} e^{7}\right )} x^{3} + 3 \,{\left (c^{2} d^{5} e^{2} - 2 \, a c d^{3} e^{4} + a^{2} d e^{6}\right )} x^{2} + 3 \,{\left (c^{2} d^{6} e - 2 \, a c d^{4} e^{3} + a^{2} d^{2} e^{5}\right )} x\right )} \sqrt{c d^{2} e - a e^{3}}}\right ] \]

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)^(5/2)),x, algorithm="fricas")

[Out]

[1/8*(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)*(3*c*
d*e*x + 5*c*d^2 - 2*a*e^2)*sqrt(e*x + d) + 3*(c^2*d^2*e^3*x^3 + 3*c^2*d^3*e^2*x^
2 + 3*c^2*d^4*e*x + c^2*d^5)*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)))/((c^2*d^7 - 2*a*c*d^5*e^2 +
a^2*d^3*e^4 + (c^2*d^4*e^3 - 2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 - 2*
a*c*d^3*e^4 + a^2*d*e^6)*x^2 + 3*(c^2*d^6*e - 2*a*c*d^4*e^3 + a^2*d^2*e^5)*x)*sq
rt(-c*d^2*e + a*e^3)), 1/4*(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d
^2*e - a*e^3)*(3*c*d*e*x + 5*c*d^2 - 2*a*e^2)*sqrt(e*x + d) - 3*(c^2*d^2*e^3*x^3
+ 3*c^2*d^3*e^2*x^2 + 3*c^2*d^4*e*x + c^2*d^5)*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)))/((c^2*d^7 - 2*a*c*d^5*e^2 + a^2*d^3*e^4 + (c^2*d^4*e^3 -
2*a*c*d^2*e^5 + a^2*e^7)*x^3 + 3*(c^2*d^5*e^2 - 2*a*c*d^3*e^4 + a^2*d*e^6)*x^2 +
3*(c^2*d^6*e - 2*a*c*d^4*e^3 + a^2*d^2*e^5)*x)*sqrt(c*d^2*e - a*e^3))]

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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(1/(e*x+d)**(5/2)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(1/2),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.584202, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

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)^(5/2)),x, algorithm="giac")

[Out]

sage0*x