3.857 \(\int \frac{\sqrt{c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=141 \[ \frac{\sqrt{c d^2-c e^2 x^2}}{8 d e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{2 e (d+e x)^{5/2}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{8 \sqrt{2} d^{3/2} e} \]

[Out]

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

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Rubi [A]  time = 0.223877, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\sqrt{c d^2-c e^2 x^2}}{8 d e (d+e x)^{3/2}}-\frac{\sqrt{c d^2-c e^2 x^2}}{2 e (d+e x)^{5/2}}+\frac{\sqrt{c} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{8 \sqrt{2} d^{3/2} e} \]

Antiderivative was successfully verified.

[In]  Int[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 21.8421, size = 121, normalized size = 0.86 \[ \frac{\sqrt{2} \sqrt{c} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{16 d^{\frac{3}{2}} e} - \frac{\sqrt{c d^{2} - c e^{2} x^{2}}}{2 e \left (d + e x\right )^{\frac{5}{2}}} + \frac{\sqrt{c d^{2} - c e^{2} x^{2}}}{8 d e \left (d + e x\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((-c*e**2*x**2+c*d**2)**(1/2)/(e*x+d)**(7/2),x)

[Out]

sqrt(2)*sqrt(c)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)*sqrt(d)*sqrt
(d + e*x)))/(16*d**(3/2)*e) - sqrt(c*d**2 - c*e**2*x**2)/(2*e*(d + e*x)**(5/2))
+ sqrt(c*d**2 - c*e**2*x**2)/(8*d*e*(d + e*x)**(3/2))

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Mathematica [A]  time = 0.196036, size = 111, normalized size = 0.79 \[ \frac{\sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{\sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{d^{3/2} \sqrt{d^2-e^2 x^2}}+\frac{2 e x-6 d}{d (d+e x)^{5/2}}\right )}{16 e} \]

Antiderivative was successfully verified.

[In]  Integrate[Sqrt[c*d^2 - c*e^2*x^2]/(d + e*x)^(7/2),x]

[Out]

(Sqrt[c*(d^2 - e^2*x^2)]*((-6*d + 2*e*x)/(d*(d + e*x)^(5/2)) + (Sqrt[2]*ArcTanh[
Sqrt[d^2 - e^2*x^2]/(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(d^(3/2)*Sqrt[d^2 - e^2*x^
2])))/(16*e)

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Maple [A]  time = 0.029, size = 190, normalized size = 1.4 \[{\frac{1}{16\,de}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( \sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ){x}^{2}c{e}^{2}+2\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+\sqrt{2}{\it Artanh} \left ({\frac{\sqrt{2}}{2}\sqrt{- \left ( ex-d \right ) c}{\frac{1}{\sqrt{cd}}}} \right ) c{d}^{2}+2\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-6\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{5}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((-c*e^2*x^2+c*d^2)^(1/2)/(e*x+d)^(7/2),x)

[Out]

1/16*(-c*(e^2*x^2-d^2))^(1/2)*(2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c
*d)^(1/2))*x^2*c*e^2+2*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2
))*x*c*d*e+2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*c*d^2+2*x
*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)-6*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)*d)/(e*x+d)^(5
/2)/(-(e*x-d)*c)^(1/2)/e/d/(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(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.236838, size = 1, normalized size = 0.01 \[ \left [\frac{\sqrt{\frac{1}{2}}{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) + 2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}{\left (e x - 3 \, d\right )}}{16 \,{\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}}, -\frac{\sqrt{\frac{1}{2}}{\left (e^{3} x^{3} + 3 \, d e^{2} x^{2} + 3 \, d^{2} e x + d^{3}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{c}{d}}}\right ) - \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}{\left (e x - 3 \, d\right )}}{8 \,{\left (d e^{4} x^{3} + 3 \, d^{2} e^{3} x^{2} + 3 \, d^{3} e^{2} x + d^{4} e\right )}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(7/2),x, algorithm="fricas")

[Out]

[1/16*(sqrt(1/2)*(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + d^3)*sqrt(c/d)*log(-(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)*sqrt(e*x +
d)*(e*x - 3*d))/(d*e^4*x^3 + 3*d^2*e^3*x^2 + 3*d^3*e^2*x + d^4*e), -1/8*(sqrt(1/
2)*(e^3*x^3 + 3*d*e^2*x^2 + 3*d^2*e*x + 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)*sqrt(e*x + d)*(e*x - 3*d))/(d*e^4*x^3 + 3*d^2*e^3*x^2 + 3*d^3*e^2*x +
d^4*e)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{- c \left (- d + e x\right ) \left (d + e x\right )}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((-c*e**2*x**2+c*d**2)**(1/2)/(e*x+d)**(7/2),x)

[Out]

Integral(sqrt(-c*(-d + e*x)*(d + e*x))/(d + e*x)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{-c e^{2} x^{2} + c d^{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(7/2),x, algorithm="giac")

[Out]

integrate(sqrt(-c*e^2*x^2 + c*d^2)/(e*x + d)^(7/2), x)