3.874 \(\int \frac{1}{(d+e x)^{5/2} \sqrt{c d^2-c e^2 x^2}} \, dx\)

Optimal. Leaf size=150 \[ -\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} \sqrt{c} d^{5/2} e} \]

[Out]

-Sqrt[c*d^2 - c*e^2*x^2]/(4*c*d*e*(d + e*x)^(5/2)) - (3*Sqrt[c*d^2 - c*e^2*x^2])
/(16*c*d^2*e*(d + e*x)^(3/2)) - (3*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt
[c]*Sqrt[d]*Sqrt[d + e*x])])/(16*Sqrt[2]*Sqrt[c]*d^(5/2)*e)

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Rubi [A]  time = 0.218259, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.103 \[ -\frac{3 \sqrt{c d^2-c e^2 x^2}}{16 c d^2 e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{4 c d e (d+e x)^{5/2}}-\frac{3 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{16 \sqrt{2} \sqrt{c} d^{5/2} e} \]

Antiderivative was successfully verified.

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

[Out]

-Sqrt[c*d^2 - c*e^2*x^2]/(4*c*d*e*(d + e*x)^(5/2)) - (3*Sqrt[c*d^2 - c*e^2*x^2])
/(16*c*d^2*e*(d + e*x)^(3/2)) - (3*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt
[c]*Sqrt[d]*Sqrt[d + e*x])])/(16*Sqrt[2]*Sqrt[c]*d^(5/2)*e)

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Rubi in Sympy [A]  time = 21.8761, size = 133, normalized size = 0.89 \[ - \frac{\sqrt{c d^{2} - c e^{2} x^{2}}}{4 c d e \left (d + e x\right )^{\frac{5}{2}}} - \frac{3 \sqrt{c d^{2} - c e^{2} x^{2}}}{16 c d^{2} e \left (d + e x\right )^{\frac{3}{2}}} - \frac{3 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{32 \sqrt{c} d^{\frac{5}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(c*d**2 - c*e**2*x**2)/(4*c*d*e*(d + e*x)**(5/2)) - 3*sqrt(c*d**2 - c*e**2*
x**2)/(16*c*d**2*e*(d + e*x)**(3/2)) - 3*sqrt(2)*atanh(sqrt(2)*sqrt(c*d**2 - c*e
**2*x**2)/(2*sqrt(c)*sqrt(d)*sqrt(d + e*x)))/(32*sqrt(c)*d**(5/2)*e)

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Mathematica [A]  time = 0.14533, size = 140, normalized size = 0.93 \[ \frac{2 \sqrt{d} \sqrt{d+e x} \left (-7 d^2+4 d e x+3 e^2 x^2\right )-3 \sqrt{2} (d+e x)^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{32 d^{5/2} e (d+e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((d + e*x)^(5/2)*Sqrt[c*d^2 - c*e^2*x^2]),x]

[Out]

(2*Sqrt[d]*Sqrt[d + e*x]*(-7*d^2 + 4*d*e*x + 3*e^2*x^2) - 3*Sqrt[2]*(d + e*x)^2*
Sqrt[d^2 - e^2*x^2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])]
)/(32*d^(5/2)*e*(d + e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.024, size = 195, normalized size = 1.3 \[ -{\frac{1}{32\,{d}^{2}ec}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{e}^{2}+6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}+6\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}+14\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/32/(e*x+d)^(5/2)*(-c*(e^2*x^2-d^2))^(1/2)/c*(3*2^(1/2)*arctanh(1/2*(-(e*x-d)*
c)^(1/2)*2^(1/2)/(c*d)^(1/2))*x^2*c*e^2+6*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)
*2^(1/2)/(c*d)^(1/2))*x*c*d*e+3*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(
c*d)^(1/2))*c*d^2+6*x*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)+14*(-(e*x-d)*c)^(1/2)*(c*
d)^(1/2)*d)/(-(e*x-d)*c)^(1/2)/e/d^2/(c*d)^(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*e^2*x^2 + c*d^2)*(e*x + d)^(5/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.225491, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d}{\left (3 \, e x + 7 \, d\right )} \sqrt{e x + d} - 3 \,{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \log \left (-\frac{\sqrt{2}{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} + 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{64 \,{\left (c d^{2} e^{4} x^{3} + 3 \, c d^{3} e^{3} x^{2} + 3 \, c d^{4} e^{2} x + c d^{5} e\right )} \sqrt{c d}}, -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d}{\left (3 \, e x + 7 \, d\right )} \sqrt{e x + d} - 3 \,{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right )\right )}}{32 \,{\left (c d^{2} e^{4} x^{3} + 3 \, c d^{3} e^{3} x^{2} + 3 \, c d^{4} e^{2} x + c d^{5} e\right )} \sqrt{-c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(5/2)),x, algorithm="fricas")

[Out]

[-1/64*sqrt(2)*(2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*(3*e*x + 7*d)*sqrt(
e*x + d) - 3*(c*e^3*x^3 + 3*c*d*e^2*x^2 + 3*c*d^2*e*x + c*d^3)*log(-(sqrt(2)*(e^
2*x^2 - 2*d*e*x - 3*d^2)*sqrt(c*d) + 4*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*d)
/(e^2*x^2 + 2*d*e*x + d^2)))/((c*d^2*e^4*x^3 + 3*c*d^3*e^3*x^2 + 3*c*d^4*e^2*x +
c*d^5*e)*sqrt(c*d)), -1/32*sqrt(2)*(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)
*(3*e*x + 7*d)*sqrt(e*x + d) - 3*(c*e^3*x^3 + 3*c*d*e^2*x^2 + 3*c*d^2*e*x + c*d^
3)*arctan(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*sqrt(e*x + d)/(c*e^2*x^2 -
c*d^2)))/((c*d^2*e^4*x^3 + 3*c*d^3*e^3*x^2 + 3*c*d^4*e^2*x + c*d^5*e)*sqrt(-c*d
))]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )} \left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(sqrt(-c*(-d + e*x)*(d + e*x))*(d + e*x)**(5/2)), x)

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GIAC/XCAS [A]  time = 0.622127, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(-c*e^2*x^2 + c*d^2)*(e*x + d)^(5/2)),x, algorithm="giac")

[Out]

sage0*x