3.867 \(\int \frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{(d+e x)^{13/2}} \, dx\)

Optimal. Leaf size=217 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{256 \sqrt{2} d^{5/2} e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]

[Out]

(c*Sqrt[c*d^2 - c*e^2*x^2])/(8*e*(d + e*x)^(7/2)) - (c*Sqrt[c*d^2 - c*e^2*x^2])/
(64*d*e*(d + e*x)^(5/2)) - (3*c*Sqrt[c*d^2 - c*e^2*x^2])/(256*d^2*e*(d + e*x)^(3
/2)) - (c*d^2 - c*e^2*x^2)^(3/2)/(4*e*(d + e*x)^(11/2)) - (3*c^(3/2)*ArcTanh[Sqr
t[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(256*Sqrt[2]*d^(5
/2)*e)

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Rubi [A]  time = 0.375255, antiderivative size = 217, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ -\frac{3 c^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c d^2-c e^2 x^2}}{\sqrt{2} \sqrt{c} \sqrt{d} \sqrt{d+e x}}\right )}{256 \sqrt{2} d^{5/2} e}-\frac{3 c \sqrt{c d^2-c e^2 x^2}}{256 d^2 e (d+e x)^{3/2}}-\frac{c \sqrt{c d^2-c e^2 x^2}}{64 d e (d+e x)^{5/2}}+\frac{c \sqrt{c d^2-c e^2 x^2}}{8 e (d+e x)^{7/2}}-\frac{\left (c d^2-c e^2 x^2\right )^{3/2}}{4 e (d+e x)^{11/2}} \]

Antiderivative was successfully verified.

[In]  Int[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(13/2),x]

[Out]

(c*Sqrt[c*d^2 - c*e^2*x^2])/(8*e*(d + e*x)^(7/2)) - (c*Sqrt[c*d^2 - c*e^2*x^2])/
(64*d*e*(d + e*x)^(5/2)) - (3*c*Sqrt[c*d^2 - c*e^2*x^2])/(256*d^2*e*(d + e*x)^(3
/2)) - (c*d^2 - c*e^2*x^2)^(3/2)/(4*e*(d + e*x)^(11/2)) - (3*c^(3/2)*ArcTanh[Sqr
t[c*d^2 - c*e^2*x^2]/(Sqrt[2]*Sqrt[c]*Sqrt[d]*Sqrt[d + e*x])])/(256*Sqrt[2]*d^(5
/2)*e)

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Rubi in Sympy [A]  time = 38.755, size = 190, normalized size = 0.88 \[ - \frac{3 \sqrt{2} c^{\frac{3}{2}} \operatorname{atanh}{\left (\frac{\sqrt{2} \sqrt{c d^{2} - c e^{2} x^{2}}}{2 \sqrt{c} \sqrt{d} \sqrt{d + e x}} \right )}}{512 d^{\frac{5}{2}} e} + \frac{c \sqrt{c d^{2} - c e^{2} x^{2}}}{8 e \left (d + e x\right )^{\frac{7}{2}}} - \frac{c \sqrt{c d^{2} - c e^{2} x^{2}}}{64 d e \left (d + e x\right )^{\frac{5}{2}}} - \frac{3 c \sqrt{c d^{2} - c e^{2} x^{2}}}{256 d^{2} e \left (d + e x\right )^{\frac{3}{2}}} - \frac{\left (c d^{2} - c e^{2} x^{2}\right )^{\frac{3}{2}}}{4 e \left (d + e x\right )^{\frac{11}{2}}} \]

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)**(13/2),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)))/(512*d**(5/2)*e) + c*sqrt(c*d**2 - c*e**2*x**2)/(8*e*(d + e*x)**
(7/2)) - c*sqrt(c*d**2 - c*e**2*x**2)/(64*d*e*(d + e*x)**(5/2)) - 3*c*sqrt(c*d**
2 - c*e**2*x**2)/(256*d**2*e*(d + e*x)**(3/2)) - (c*d**2 - c*e**2*x**2)**(3/2)/(
4*e*(d + e*x)**(11/2))

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Mathematica [A]  time = 0.365705, size = 145, normalized size = 0.67 \[ \frac{\left (c \left (d^2-e^2 x^2\right )\right )^{3/2} \left (-\frac{3 \sqrt{2} \tanh ^{-1}\left (\frac{\sqrt{d^2-e^2 x^2}}{\sqrt{2} \sqrt{d} \sqrt{d+e x}}\right )}{\left (d^2-e^2 x^2\right )^{3/2}}-\frac{2 \sqrt{d} \left (39 d^3-79 d^2 e x+13 d e^2 x^2+3 e^3 x^3\right )}{(d-e x) (d+e x)^{11/2}}\right )}{512 d^{5/2} e} \]

Antiderivative was successfully verified.

[In]  Integrate[(c*d^2 - c*e^2*x^2)^(3/2)/(d + e*x)^(13/2),x]

[Out]

((c*(d^2 - e^2*x^2))^(3/2)*((-2*Sqrt[d]*(39*d^3 - 79*d^2*e*x + 13*d*e^2*x^2 + 3*
e^3*x^3))/((d - e*x)*(d + e*x)^(11/2)) - (3*Sqrt[2]*ArcTanh[Sqrt[d^2 - e^2*x^2]/
(Sqrt[2]*Sqrt[d]*Sqrt[d + e*x])])/(d^2 - e^2*x^2)^(3/2)))/(512*d^(5/2)*e)

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Maple [A]  time = 0.034, size = 325, normalized size = 1.5 \[ -{\frac{c}{512\,e{d}^{2}}\sqrt{-c \left ({e}^{2}{x}^{2}-{d}^{2} \right ) } \left ( 3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{4}c{e}^{4}+12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{3}cd{e}^{3}+18\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ){x}^{2}c{d}^{2}{e}^{2}+12\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) xc{d}^{3}e+6\,{x}^{3}{e}^{3}\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}+3\,\sqrt{2}{\it Artanh} \left ( 1/2\,{\frac{\sqrt{- \left ( ex-d \right ) c}\sqrt{2}}{\sqrt{cd}}} \right ) c{d}^{4}+26\,{x}^{2}d{e}^{2}\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}-158\,x{d}^{2}e\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}+78\,\sqrt{- \left ( ex-d \right ) c}\sqrt{cd}{d}^{3} \right ) \left ( ex+d \right ) ^{-{\frac{9}{2}}}{\frac{1}{\sqrt{- \left ( ex-d \right ) c}}}{\frac{1}{\sqrt{cd}}}} \]

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)^(13/2),x)

[Out]

-1/512*(-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^4*c*e^4+12*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*
d)^(1/2))*x^3*c*d*e^3+18*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1
/2))*x^2*c*d^2*e^2+12*2^(1/2)*arctanh(1/2*(-(e*x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2)
)*x*c*d^3*e+6*x^3*e^3*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)+3*2^(1/2)*arctanh(1/2*(-(e*
x-d)*c)^(1/2)*2^(1/2)/(c*d)^(1/2))*c*d^4+26*x^2*d*e^2*(-(e*x-d)*c)^(1/2)*(c*d)^(
1/2)-158*x*d^2*e*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2)+78*(-(e*x-d)*c)^(1/2)*(c*d)^(1/2
)*d^3)/(e*x+d)^(9/2)/(-(e*x-d)*c)^(1/2)/e/d^2/(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)^(13/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.229279, size = 1, normalized size = 0. \[ \left [\frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\right )} \sqrt{\frac{c}{d}} \log \left (-\frac{c e^{2} x^{2} - 2 \, c d e x - 3 \, c d^{2} + 4 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d} d \sqrt{\frac{c}{d}}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \,{\left (3 \, c e^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{512 \,{\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} e\right )}}, \frac{3 \, \sqrt{\frac{1}{2}}{\left (c e^{5} x^{5} + 5 \, c d e^{4} x^{4} + 10 \, c d^{2} e^{3} x^{3} + 10 \, c d^{3} e^{2} x^{2} + 5 \, c d^{4} e x + c d^{5}\right )} \sqrt{-\frac{c}{d}} \arctan \left (\frac{2 \, \sqrt{\frac{1}{2}} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{{\left (e^{2} x^{2} - d^{2}\right )} \sqrt{-\frac{c}{d}}}\right ) -{\left (3 \, c e^{3} x^{3} + 13 \, c d e^{2} x^{2} - 79 \, c d^{2} e x + 39 \, c d^{3}\right )} \sqrt{-c e^{2} x^{2} + c d^{2}} \sqrt{e x + d}}{256 \,{\left (d^{2} e^{6} x^{5} + 5 \, d^{3} e^{5} x^{4} + 10 \, d^{4} e^{4} x^{3} + 10 \, d^{5} e^{3} x^{2} + 5 \, d^{6} e^{2} x + d^{7} e\right )}}\right ] \]

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)^(13/2),x, algorithm="fricas")

[Out]

[1/512*(3*sqrt(1/2)*(c*e^5*x^5 + 5*c*d*e^4*x^4 + 10*c*d^2*e^3*x^3 + 10*c*d^3*e^2
*x^2 + 5*c*d^4*e*x + c*d^5)*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*(3*c*e^3*x^3 + 13*c*d*e^2*x^2 - 79*c*d^2*e*x + 39*c*d^3)*sqrt(-c*e^
2*x^2 + c*d^2)*sqrt(e*x + d))/(d^2*e^6*x^5 + 5*d^3*e^5*x^4 + 10*d^4*e^4*x^3 + 10
*d^5*e^3*x^2 + 5*d^6*e^2*x + d^7*e), 1/256*(3*sqrt(1/2)*(c*e^5*x^5 + 5*c*d*e^4*x
^4 + 10*c*d^2*e^3*x^3 + 10*c*d^3*e^2*x^2 + 5*c*d^4*e*x + c*d^5)*sqrt(-c/d)*arcta
n(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)
)) - (3*c*e^3*x^3 + 13*c*d*e^2*x^2 - 79*c*d^2*e*x + 39*c*d^3)*sqrt(-c*e^2*x^2 +
c*d^2)*sqrt(e*x + d))/(d^2*e^6*x^5 + 5*d^3*e^5*x^4 + 10*d^4*e^4*x^3 + 10*d^5*e^3
*x^2 + 5*d^6*e^2*x + d^7*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)**(13/2),x)

[Out]

Timed out

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (-c e^{2} x^{2} + c d^{2}\right )}^{\frac{3}{2}}}{{\left (e x + d\right )}^{\frac{13}{2}}}\,{d x} \]

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)^(13/2),x, algorithm="giac")

[Out]

integrate((-c*e^2*x^2 + c*d^2)^(3/2)/(e*x + d)^(13/2), x)