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

Optimal. Leaf size=191 \[ -\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{32 \sqrt{2} c^{3/2} d^{7/2} e}-\frac{5}{16 c d^2 e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}-\frac{1}{4 c d e (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}+\frac{15 \sqrt{d+e x}}{32 c d^3 e \sqrt{c d^2-c e^2 x^2}} \]

[Out]

-1/(4*c*d*e*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2]) - 5/(16*c*d^2*e*Sqrt[d + e*
x]*Sqrt[c*d^2 - c*e^2*x^2]) + (15*Sqrt[d + e*x])/(32*c*d^3*e*Sqrt[c*d^2 - c*e^2*
x^2]) - (15*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*
x])])/(32*Sqrt[2]*c^(3/2)*d^(7/2)*e)

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Rubi [A]  time = 0.304501, antiderivative size = 191, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{15 \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{32 \sqrt{2} c^{3/2} d^{7/2} e}-\frac{5}{16 c d^2 e \sqrt{d+e x} \sqrt{c d^2-c e^2 x^2}}-\frac{1}{4 c d e (d+e x)^{3/2} \sqrt{c d^2-c e^2 x^2}}+\frac{15 \sqrt{d+e x}}{32 c d^3 e \sqrt{c d^2-c e^2 x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-1/(4*c*d*e*(d + e*x)^(3/2)*Sqrt[c*d^2 - c*e^2*x^2]) - 5/(16*c*d^2*e*Sqrt[d + e*
x]*Sqrt[c*d^2 - c*e^2*x^2]) + (15*Sqrt[d + e*x])/(32*c*d^3*e*Sqrt[c*d^2 - c*e^2*
x^2]) - (15*ArcTanh[Sqrt[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*
x])])/(32*Sqrt[2]*c^(3/2)*d^(7/2)*e)

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Rubi in Sympy [A]  time = 31.4912, size = 168, normalized size = 0.88 \[ - \frac{1}{4 c d e \left (d + e x\right )^{\frac{3}{2}} \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{5}{16 c d^{2} e \sqrt{d + e x} \sqrt{c d^{2} - c e^{2} x^{2}}} + \frac{15 \sqrt{d + e x}}{32 c d^{3} e \sqrt{c d^{2} - c e^{2} x^{2}}} - \frac{15 \sqrt{2} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{64 c^{\frac{3}{2}} d^{\frac{7}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/(4*c*d*e*(d + e*x)**(3/2)*sqrt(c*d**2 - c*e**2*x**2)) - 5/(16*c*d**2*e*sqrt(d
+ e*x)*sqrt(c*d**2 - c*e**2*x**2)) + 15*sqrt(d + e*x)/(32*c*d**3*e*sqrt(c*d**2
- c*e**2*x**2)) - 15*sqrt(2)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)
*sqrt(d)*sqrt(d + e*x)))/(64*c**(3/2)*d**(7/2)*e)

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Mathematica [A]  time = 0.1569, size = 143, normalized size = 0.75 \[ \frac{2 \sqrt{d} \sqrt{d+e x} \left (-3 d^2+20 d e x+15 e^2 x^2\right )-15 \sqrt{2} (d+e x)^2 \sqrt{d^2-e^2 x^2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{64 c d^{7/2} e (d+e x)^2 \sqrt{c \left (d^2-e^2 x^2\right )}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[d]*Sqrt[d + e*x]*(-3*d^2 + 20*d*e*x + 15*e^2*x^2) - 15*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
])])/(64*c*d^(7/2)*e*(d + e*x)^2*Sqrt[c*(d^2 - e^2*x^2)])

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Maple [A]  time = 0.033, size = 217, normalized size = 1.1 \[{\frac{1}{64\,{c}^{2} \left ( ex-d \right ) e{d}^{3}}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 15\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}{e}^{2}+30\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xde+15\,\sqrt{- \left ( ex-d \right ) c}\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){d}^{2}-30\,\sqrt{cd}{x}^{2}{e}^{2}-40\,\sqrt{cd}xde+6\,\sqrt{cd}{d}^{2} \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/64/(e*x+d)^(5/2)*(-c*(e^2*x^2-d^2))^(1/2)/c^2*(15*(-(e*x-d)*c)^(1/2)*2^(1/2)*a
rctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*x^2*e^2+30*(-(e*x-d)*c)^(1/2)
*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*x*d*e+15*(-(e*x-d)*
c)^(1/2)*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*d^2-30*(c*d
)^(1/2)*x^2*e^2-40*(c*d)^(1/2)*x*d*e+6*(c*d)^(1/2)*d^2)/(e*x-d)/e/d^3/(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/((-c*e^2*x^2 + c*d^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.235058, size = 1, normalized size = 0.01 \[ \left [-\frac{\sqrt{2}{\left (2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (15 \, e^{2} x^{2} + 20 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} \sqrt{e x + d} - 15 \,{\left (c e^{4} x^{4} + 2 \, c d e^{3} x^{3} - 2 \, c d^{3} e x - c d^{4}\right )} \log \left (-\frac{\sqrt{2}{\left (e^{2} x^{2} - 2 \, d e x - 3 \, d^{2}\right )} \sqrt{c d} + 4 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )\right )}}{128 \,{\left (c^{2} d^{3} e^{5} x^{4} + 2 \, c^{2} d^{4} e^{4} x^{3} - 2 \, c^{2} d^{6} e^{2} x - c^{2} d^{7} e\right )} \sqrt{c d}}, -\frac{\sqrt{2}{\left (\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}}{\left (15 \, e^{2} x^{2} + 20 \, d e x - 3 \, d^{2}\right )} \sqrt{-c d} \sqrt{e x + d} - 15 \,{\left (c e^{4} x^{4} + 2 \, c d e^{3} x^{3} - 2 \, c d^{3} e x - c d^{4}\right )} \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d}}{c e^{2} x^{2} - c d^{2}}\right )\right )}}{64 \,{\left (c^{2} d^{3} e^{5} x^{4} + 2 \, c^{2} d^{4} e^{4} x^{3} - 2 \, c^{2} d^{6} e^{2} x - c^{2} d^{7} e\right )} \sqrt{-c d}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-c*e^2*x^2 + c*d^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="fricas")

[Out]

[-1/128*sqrt(2)*(2*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*(15*e^2*x^2 + 20*d*e*x - 3*d
^2)*sqrt(c*d)*sqrt(e*x + d) - 15*(c*e^4*x^4 + 2*c*d*e^3*x^3 - 2*c*d^3*e*x - c*d^
4)*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^2*d^3*e^5*x^4 + 2*c^2*d^4*e
^4*x^3 - 2*c^2*d^6*e^2*x - c^2*d^7*e)*sqrt(c*d)), -1/64*sqrt(2)*(sqrt(2)*sqrt(-c
*e^2*x^2 + c*d^2)*(15*e^2*x^2 + 20*d*e*x - 3*d^2)*sqrt(-c*d)*sqrt(e*x + d) - 15*
(c*e^4*x^4 + 2*c*d*e^3*x^3 - 2*c*d^3*e*x - c*d^4)*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^2*d^3*e^5*x^4 + 2*c
^2*d^4*e^4*x^3 - 2*c^2*d^6*e^2*x - c^2*d^7*e)*sqrt(-c*d))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((-c*(-d + e*x)*(d + e*x))**(3/2)*(d + e*x)**(3/2)), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 2\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((-c*e^2*x^2 + c*d^2)^(3/2)*(e*x + d)^(3/2)),x, algorithm="giac")

[Out]

[undef, undef, undef, undef, undef, undef, undef, 2]