3.863 \(\int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{5/2}} \, dx\)

Optimal. Leaf size=136 \[ -\frac{4 \sqrt{2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}+\frac{4 c d \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.212471, antiderivative size = 136, 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 \[ -\frac{4 \sqrt{2} c^{3/2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}+\frac{4 c d \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}+\frac{2 \left (c d^2-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(5/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 23.1196, size = 124, normalized size = 0.91 \[ - \frac{4 \sqrt{2} c^{\frac{3}{2}} d^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{e} + \frac{4 c d \sqrt{c d^{2} - c e^{2} x^{2}}}{e \sqrt{d + e x}} + \frac{2 \left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{3 e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(5/2),x)

[Out]

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

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Mathematica [A]  time = 0.316952, size = 110, normalized size = 0.81 \[ \frac{2 c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{7 d-e x}{\sqrt{d+e x}}-\frac{6 \sqrt{2} d^{3/2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}\right )}{3 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(5/2),x]

[Out]

(2*c*Sqrt[c*(d^2 - e^2*x^2)]*((7*d - e*x)/Sqrt[d + e*x] - (6*Sqrt[2]*d^(3/2)*Arc
Tanh[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/Sqrt[d^2 - e^2*x^2]))
/(3*e)

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Maple [A]  time = 0.022, size = 122, normalized size = 0.9 \[ -{\frac{2\,c}{3\,e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}+xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-7\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ){\frac{1}{\sqrt{ex+d}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(5/2),x)

[Out]

-2/3*(-c*(e^2*x^2-d^2))^(1/2)*c*(6*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2
)/(c*d)^(1/2))*c*d^2+x*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)-7*(-(e*x-d)*c)^(1/2)*(c*
d)^(1/2)*d)/(e*x+d)^(1/2)/(-(e*x-d)*c)^(1/2)/e/(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((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.228795, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (c^{2} e^{3} x^{3} - 7 \, c^{2} d e^{2} x^{2} - c^{2} d^{2} e x + 7 \, c^{2} d^{3} + 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} c d \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}, \frac{2 \,{\left (c^{2} e^{3} x^{3} - 7 \, c^{2} d e^{2} x^{2} - c^{2} d^{2} e x + 7 \, c^{2} d^{3} + 6 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} c d \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d}}\right )\right )}}{3 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="fricas")

[Out]

[2/3*(c^2*e^3*x^3 - 7*c^2*d*e^2*x^2 - c^2*d^2*e*x + 7*c^2*d^3 + 3*sqrt(2)*sqrt(-
c*e^2*x^2 + c*d^2)*sqrt(c*d)*sqrt(e*x + d)*c*d*log(-(c*e^2*x^2 - 2*c*d*e*x - 3*c
*d^2 + 2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(c*d)*sqrt(e*x + d))/(e^2*x^2 + 2*
d*e*x + d^2)))/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e), 2/3*(c^2*e^3*x^3 - 7*
c^2*d*e^2*x^2 - c^2*d^2*e*x + 7*c^2*d^3 + 6*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqr
t(-c*d)*sqrt(e*x + d)*c*d*arctan(sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*
d/((e^2*x^2 - d^2)*sqrt(-c*d))))/(sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{5}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(5/2),x)

[Out]

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)/(d + e*x)**(5/2), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: AttributeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(5/2),x, algorithm="giac")

[Out]

Exception raised: AttributeError