3.864 \(\int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx\)

Optimal. Leaf size=133 \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.209897, antiderivative size = 133, 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 \[ \frac{3 \sqrt{2} c^{3/2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{e \sqrt{d+e x}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{e (d+e x)^{5/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

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

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Rubi in Sympy [A]  time = 22.2709, size = 119, normalized size = 0.89 \[ \frac{3 \sqrt{2} c^{\frac{3}{2}} \sqrt{d} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{e} - \frac{3 c \sqrt{c d^{2} - c e^{2} x^{2}}}{e \sqrt{d + e x}} - \frac{\left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{e \left (d + e x\right )^{\frac{5}{2}}} \]

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)**(7/2),x)

[Out]

3*sqrt(2)*c**(3/2)*sqrt(d)*atanh(sqrt(2)*sqrt(c*d**2 - c*e**2*x**2)/(2*sqrt(c)*s
qrt(d)*sqrt(d + e*x)))/e - 3*c*sqrt(c*d**2 - c*e**2*x**2)/(e*sqrt(d + e*x)) - (c
*d**2 - c*e**2*x**2)**(3/2)/(e*(d + e*x)**(5/2))

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Mathematica [A]  time = 0.299854, size = 107, normalized size = 0.8 \[ \frac{c \sqrt{c \left (d^2-e^2 x^2\right )} \left (\frac{3 \sqrt{2} \sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\sqrt{d^2-e^2 x^2}}-\frac{2 (2 d+e x)}{(d+e x)^{3/2}}\right )}{e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(7/2),x]

[Out]

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

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Maple [A]  time = 0.03, size = 154, normalized size = 1.2 \[{\frac{c}{e}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xcde+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{2}-2\,xe\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-4\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}d \right ) \left ( ex+d \right ) ^{-{\frac{3}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

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)^(7/2),x)

[Out]

(-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*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^2-2*x*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)-4*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)*d)/(
e*x+d)^(3/2)/(-(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)^(7/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.230334, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, c^{2} e^{2} x^{2} + 4 \, c^{2} d e x - 8 \, c^{2} d^{2} + 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d} c \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} - 2 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{c d} \sqrt{e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \, \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}, \frac{2 \, c^{2} e^{2} x^{2} + 2 \, c^{2} d e x - 4 \, c^{2} d^{2} - 3 \, \sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{-c d} \sqrt{e x + d} c \arctan \left (\frac{\sqrt{2} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-c d}}\right )}{\sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} e}\right ] \]

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)^(7/2),x, algorithm="fricas")

[Out]

[1/2*(4*c^2*e^2*x^2 + 4*c^2*d*e*x - 8*c^2*d^2 + 3*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^
2)*sqrt(c*d)*sqrt(e*x + d)*c*log(-(c*e^2*x^2 - 2*c*d*e*x - 3*c*d^2 - 2*sqrt(2)*s
qrt(-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)))/(sq
rt(-c*e^2*x^2 + c*d^2)*sqrt(e*x + d)*e), (2*c^2*e^2*x^2 + 2*c^2*d*e*x - 4*c^2*d^
2 - 3*sqrt(2)*sqrt(-c*e^2*x^2 + c*d^2)*sqrt(-c*d)*sqrt(e*x + d)*c*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. \[ \int \frac{\left (- c \left (- d + e x\right ) \left (d + e x\right )\right )^{\frac{3}{2}}}{\left (d + e x\right )^{\frac{7}{2}}}\, dx \]

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)**(7/2),x)

[Out]

Integral((-c*(-d + e*x)*(d + e*x))**(3/2)/(d + e*x)**(7/2), x)

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{7}{2}}}\,{d x} \]

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)^(7/2),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(7/2), x)