3.865 \(\int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{9/2}} \, dx\)

Optimal. Leaf size=139 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]

[Out]

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

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Rubi [A]  time = 0.212427, antiderivative size = 139, 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 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{4 \sqrt{2} \sqrt{d} e}+\frac{3 c \sqrt{c d^2-c e^2 x^2}}{4 e (d+e x)^{3/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{2 e (d+e x)^{7/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.2749, size = 124, normalized size = 0.89 \[ - \frac{3 \sqrt{2} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{8 \sqrt{d} e} + \frac{3 c \sqrt{c d^{2} - c e^{2} x^{2}}}{4 e \left (d + e x\right )^{\frac{3}{2}}} - \frac{\left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{2 e \left (d + e x\right )^{\frac{7}{2}}} \]

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)**(9/2),x)

[Out]

-3*sqrt(2)*c**(3/2)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)*sqrt(d)*
sqrt(d + e*x)))/(8*sqrt(d)*e) + 3*c*sqrt(c*d**2 - c*e**2*x**2)/(4*e*(d + e*x)**(
3/2)) - (c*d**2 - c*e**2*x**2)**(3/2)/(2*e*(d + e*x)**(7/2))

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Mathematica [A]  time = 0.246072, size = 109, normalized size = 0.78 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{2 (d+5 e x)}{(d+e x)^{5/2}}-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d} \sqrt{d^2-e^2 x^2}}\right )}{8 e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(9/2),x]

[Out]

(c*Sqrt[c*(d^2 - e^2*x^2)]*((2*(d + 5*e*x))/(d + e*x)^(5/2) - (3*Sqrt[2]*ArcTanh
[Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(Sqrt[d]*Sqrt[d^2 - e^2*x
^2])))/(8*e)

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Maple [A]  time = 0.031, size = 190, normalized size = 1.4 \[ -{\frac{c}{8\,e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{e}^{2}+6\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}-10\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-2\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c*e^2*x^2+c*d^2)^(3/2)/(e*x+d)^(9/2),x)

[Out]

-1/8*(-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
^2-10*x*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)-2*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)*d)/(e*
x+d)^(5/2)/(-(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)^(9/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.22964, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{8 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}, \frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{3} x^{3} + 3 \, c d e^{2} x^{2} + 3 \, c d^{2} e x + c d^{3}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{c}{d}}}\right ) + \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (5 \, c e x + c d\right )} \sqrt{e x + d}}{4 \,{\left (e^{4} x^{3} + 3 \, d e^{3} x^{2} + 3 \, d^{2} e^{2} x + d^{3} e\right )}}\right ] \]

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)^(9/2),x, algorithm="fricas")

[Out]

[1/8*(3*sqrt(1/2)*(c*e^3*x^3 + 3*c*d*e^2*x^2 + 3*c*d^2*e*x + c*d^3)*sqrt(c/d)*lo
g(-(c*e^2*x^2 - 2*c*d*e*x - 3*c*d^2 + 4*sqrt(1/2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(
e*x + d)*d*sqrt(c/d))/(e^2*x^2 + 2*d*e*x + d^2)) + 2*sqrt(-c*e^2*x^2 + c*d^2)*(5
*c*e*x + c*d)*sqrt(e*x + d))/(e^4*x^3 + 3*d*e^3*x^2 + 3*d^2*e^2*x + d^3*e), 1/4*
(3*sqrt(1/2)*(c*e^3*x^3 + 3*c*d*e^2*x^2 + 3*c*d^2*e*x + c*d^3)*sqrt(-c/d)*arctan
(2*sqrt(1/2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)/((e^2*x^2 - d^2)*sqrt(-c/d))
) + sqrt(-c*e^2*x^2 + c*d^2)*(5*c*e*x + c*d)*sqrt(e*x + d))/(e^4*x^3 + 3*d*e^3*x
^2 + 3*d^2*e^2*x + d^3*e)]

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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((-c*e**2*x**2+c*d**2)**(3/2)/(e*x+d)**(9/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{9}{2}}}\,{d x} \]

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)^(9/2),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(9/2), x)