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

Optimal. Leaf size=271 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}}+\frac{2 \left (-3 a^3 e^6+7 a^2 c d^2 e^4+a c^2 d^4 e^2+c d e x \left (3 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )+3 c^3 d^6\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(3/2)) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 7*a^2*c*d^2*e^4 - 3*a^3*e^6 + c*d*e*(
3*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2]) - ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt
[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])]/(a^(3/2)*d^(5/
2)*e^(3/2))

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Rubi [A]  time = 1.0034, antiderivative size = 271, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ -\frac{\tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{a^{3/2} d^{5/2} e^{3/2}}+\frac{2 \left (-3 a^3 e^6+7 a^2 c d^2 e^4+a c^2 d^4 e^2+c d e x \left (3 c d^2-a e^2\right ) \left (3 a e^2+c d^2\right )+3 c^3 d^6\right )}{3 a d^2 e \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{2 e (a e+c d x)}{3 d \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2
)^(3/2)) + (2*(3*c^3*d^6 + a*c^2*d^4*e^2 + 7*a^2*c*d^2*e^4 - 3*a^3*e^6 + c*d*e*(
3*c*d^2 - a*e^2)*(c*d^2 + 3*a*e^2)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2]) - ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt
[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])]/(a^(3/2)*d^(5/
2)*e^(3/2))

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Rubi in Sympy [A]  time = 138.43, size = 260, normalized size = 0.96 \[ \frac{2 e \left (a e + c d x\right )}{3 d \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}} + \frac{2 \left (3 a^{3} e^{6} - 7 a^{2} c d^{2} e^{4} - a c^{2} d^{4} e^{2} - 3 c^{3} d^{6} + c d e x \left (a e^{2} - 3 c d^{2}\right ) \left (3 a e^{2} + c d^{2}\right )\right )}{3 a d^{2} e \left (a e^{2} - c d^{2}\right )^{3} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} - \frac{\operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{a^{\frac{3}{2}} d^{\frac{5}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*e*(a*e + c*d*x)/(3*d*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**
2))**(3/2)) + 2*(3*a**3*e**6 - 7*a**2*c*d**2*e**4 - a*c**2*d**4*e**2 - 3*c**3*d*
*6 + c*d*e*x*(a*e**2 - 3*c*d**2)*(3*a*e**2 + c*d**2))/(3*a*d**2*e*(a*e**2 - c*d*
*2)**3*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))) - atanh((2*a*d*e + x*(a*e
**2 + c*d**2))/(2*sqrt(a)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 +
c*d**2))))/(a**(3/2)*d**(5/2)*e**(3/2))

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Mathematica [A]  time = 0.810185, size = 254, normalized size = 0.94 \[ \frac{\frac{2 \sqrt{a} \sqrt{d} \sqrt{e} (a e+c d x) \left (a d e^3 \left (c d^2-a e^2\right ) (a e+c d x)+a e^3 (d+e x) \left (8 c d^2-3 a e^2\right ) (a e+c d x)+3 c^3 d^5 (d+e x)^2\right )}{\left (c d^2-a e^2\right )^3}-3 (d+e x)^{3/2} (a e+c d x)^{3/2} \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )+3 \log (x) (d+e x)^{3/2} (a e+c d x)^{3/2}}{3 a^{3/2} d^{5/2} e^{3/2} ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.02, size = 682, normalized size = 2.5 \[{\frac{1}{a{d}^{2}e}{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-2\,{\frac{x{e}^{2}c}{d \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) \sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}-2\,{\frac{dx{c}^{2}}{a \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) \sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}-{\frac{a{e}^{3}}{{d}^{2} \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-2\,{\frac{ce}{ \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) \sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}-{\frac{{c}^{2}{d}^{2}}{ae \left ( -{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}-{c}^{2}{d}^{4} \right ) }{\frac{1}{\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}}}}}-{\frac{1}{a{d}^{2}e}\ln \left ({\frac{1}{x} \left ( 2\,ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+2\,\sqrt{ade}\sqrt{ade+ \left ( a{e}^{2}+c{d}^{2} \right ) x+cde{x}^{2}} \right ) } \right ){\frac{1}{\sqrt{ade}}}}+{\frac{2}{3\,d \left ( a{e}^{2}-c{d}^{2} \right ) } \left ( x+{\frac{d}{e}} \right ) ^{-1}{\frac{1}{\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{16\,d{c}^{2}{e}^{2}x}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}{\frac{1}{\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{8\,c{e}^{3}a}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}{\frac{1}{\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}}}-{\frac{8\,{c}^{2}{d}^{2}e}{3\, \left ( a{e}^{2}-c{d}^{2} \right ) ^{3}}{\frac{1}{\sqrt{cde \left ( x+{\frac{d}{e}} \right ) ^{2}+ \left ( a{e}^{2}-c{d}^{2} \right ) \left ( x+{\frac{d}{e}} \right ) }}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/d^2/a/e/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-2/d*e^2/(-a^2*e^4+2*a*c*d^2*e^
2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c-2*d/a/(-a^2*e^4+2*a*c*d^2
*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^2-1/d^2*a*e^3/(-a^2*e^
4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-2*e/(-a^2*e^4+2
*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-d^2/a/e/(-a^2*e^
4+2*a*c*d^2*e^2-c^2*d^4)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2-1/d^2/a/e/(
a*d*e)^(1/2)*ln((2*a*d*e+(a*e^2+c*d^2)*x+2*(a*d*e)^(1/2)*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2))/x)+2/3/d/(a*e^2-c*d^2)/(x+d/e)/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*
(x+d/e))^(1/2)-16/3*d*c^2*e^2/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+
d/e))^(1/2)*x-8/3*c*e^3/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^
(1/2)*a-8/3*d^2*c^2*e/(a*e^2-c*d^2)^3/(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1
/2)

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate(1/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)*x), x)

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Fricas [A]  time = 0.72245, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/6*(4*(3*c^3*d^7 + 9*a^2*c*d^3*e^4 - 4*a^3*d*e^6 + (3*c^3*d^5*e^2 + 8*a*c^2*d^
3*e^4 - 3*a^2*c*d*e^6)*x^2 + (6*c^3*d^6*e + 9*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 -
3*a^3*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(a*d*e) + 3*(a*c^3
*d^8*e - 3*a^2*c^2*d^6*e^3 + 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*
c^3*d^5*e^4 + 3*a^2*c^2*d^3*e^6 - a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5*a*c^3*d^6*
e^3 + 3*a^2*c^2*d^4*e^5 + a^3*c*d^2*e^7 - a^4*e^9)*x^2 + (c^4*d^9 - a*c^3*d^7*e^
2 - 3*a^2*c^2*d^5*e^4 + 5*a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)*log(-(4*(2*a^2*d^2*e^2
+ (a*c*d^3*e + a^2*d*e^3)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x) - (8*a
^2*d^2*e^2 + (c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4)*x^2 + 8*(a*c*d^3*e + a^2*d*e^3)
*x)*sqrt(a*d*e))/x^2))/((a^2*c^3*d^10*e^2 - 3*a^3*c^2*d^8*e^4 + 3*a^4*c*d^6*e^6
- a^5*d^4*e^8 + (a*c^4*d^9*e^3 - 3*a^2*c^3*d^7*e^5 + 3*a^3*c^2*d^5*e^7 - a^4*c*d
^3*e^9)*x^3 + (2*a*c^4*d^10*e^2 - 5*a^2*c^3*d^8*e^4 + 3*a^3*c^2*d^6*e^6 + a^4*c*
d^4*e^8 - a^5*d^2*e^10)*x^2 + (a*c^4*d^11*e - a^2*c^3*d^9*e^3 - 3*a^3*c^2*d^7*e^
5 + 5*a^4*c*d^5*e^7 - 2*a^5*d^3*e^9)*x)*sqrt(a*d*e)), 1/3*(2*(3*c^3*d^7 + 9*a^2*
c*d^3*e^4 - 4*a^3*d*e^6 + (3*c^3*d^5*e^2 + 8*a*c^2*d^3*e^4 - 3*a^2*c*d*e^6)*x^2
+ (6*c^3*d^6*e + 9*a*c^2*d^4*e^3 + 4*a^2*c*d^2*e^5 - 3*a^3*e^7)*x)*sqrt(c*d*e*x^
2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a*d*e) - 3*(a*c^3*d^8*e - 3*a^2*c^2*d^6*e^3
+ 3*a^3*c*d^4*e^5 - a^4*d^2*e^7 + (c^4*d^7*e^2 - 3*a*c^3*d^5*e^4 + 3*a^2*c^2*d^
3*e^6 - a^3*c*d*e^8)*x^3 + (2*c^4*d^8*e - 5*a*c^3*d^6*e^3 + 3*a^2*c^2*d^4*e^5 +
a^3*c*d^2*e^7 - a^4*e^9)*x^2 + (c^4*d^9 - a*c^3*d^7*e^2 - 3*a^2*c^2*d^5*e^4 + 5*
a^3*c*d^3*e^6 - 2*a^4*d*e^8)*x)*arctan(1/2*(2*a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-a
*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*a*d*e)))/((a^2*c^3*d^10*e^2 -
3*a^3*c^2*d^8*e^4 + 3*a^4*c*d^6*e^6 - a^5*d^4*e^8 + (a*c^4*d^9*e^3 - 3*a^2*c^3*
d^7*e^5 + 3*a^3*c^2*d^5*e^7 - a^4*c*d^3*e^9)*x^3 + (2*a*c^4*d^10*e^2 - 5*a^2*c^3
*d^8*e^4 + 3*a^3*c^2*d^6*e^6 + a^4*c*d^4*e^8 - a^5*d^2*e^10)*x^2 + (a*c^4*d^11*e
- a^2*c^3*d^9*e^3 - 3*a^3*c^2*d^7*e^5 + 5*a^4*c*d^5*e^7 - 2*a^5*d^3*e^9)*x)*sqr
t(-a*d*e))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[undef, undef, undef, 1]