3.453 \(\int \frac{\left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}}{x^4 (d+e x)} \, dx\)

Optimal. Leaf size=211 \[ \frac{\left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d x^3} \]

[Out]

-((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(8*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*d*x^3)
+ ((c*d^2 - a*e^2)^3*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*S
qrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a^(3/2)*d^(5/2)*e^(3/2
))

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Rubi [A]  time = 0.708721, antiderivative size = 211, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{\left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{3/2} d^{5/2} e^{3/2}}-\frac{\left (\frac{c}{a e}-\frac{e}{d^2}\right ) \left (x \left (a e^2+c d^2\right )+2 a d e\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{8 x^2}-\frac{\left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{3 d x^3} \]

Antiderivative was successfully verified.

[In]  Int[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^4*(d + e*x)),x]

[Out]

-((c/(a*e) - e/d^2)*(2*a*d*e + (c*d^2 + a*e^2)*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2])/(8*x^2) - (a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(3*d*x^3)
+ ((c*d^2 - a*e^2)^3*ArcTanh[(2*a*d*e + (c*d^2 + a*e^2)*x)/(2*Sqrt[a]*Sqrt[d]*S
qrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(16*a^(3/2)*d^(5/2)*e^(3/2
))

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Rubi in Sympy [A]  time = 61.1806, size = 194, normalized size = 0.92 \[ \frac{\left (\frac{e}{8 d^{2}} - \frac{c}{8 a e}\right ) \left (2 a d e + x \left (a e^{2} + c d^{2}\right )\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{x^{2}} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{3 d x^{3}} - \frac{\left (a e^{2} - c d^{2}\right )^{3} \operatorname{atanh}{\left (\frac{2 a d e + x \left (a e^{2} + c d^{2}\right )}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{16 a^{\frac{3}{2}} d^{\frac{5}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**4/(e*x+d),x)

[Out]

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

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Mathematica [A]  time = 0.361761, size = 251, normalized size = 1.19 \[ \frac{\sqrt{d+e x} \sqrt{a e+c d x} \left (-2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x} \left (a^2 e^2 \left (8 d^2+2 d e x-3 e^2 x^2\right )+2 a c d^2 e x (7 d+4 e x)+3 c^2 d^4 x^2\right )-3 x^3 \log (x) \left (c d^2-a e^2\right )^3+3 x^3 \left (c d^2-a e^2\right )^3 \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )\right )}{48 a^{3/2} d^{5/2} e^{3/2} x^3 \sqrt{(d+e x) (a e+c d x)}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)/(x^4*(d + e*x)),x]

[Out]

(Sqrt[a*e + c*d*x]*Sqrt[d + e*x]*(-2*Sqrt[a]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*S
qrt[d + e*x]*(3*c^2*d^4*x^2 + 2*a*c*d^2*e*x*(7*d + 4*e*x) + a^2*e^2*(8*d^2 + 2*d
*e*x - 3*e^2*x^2)) - 3*(c*d^2 - a*e^2)^3*x^3*Log[x] + 3*(c*d^2 - a*e^2)^3*x^3*Lo
g[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)]))/(48*a^(3/2)*d^(5/2)*e^(3/2)*x^3*Sqrt[(a*e + c*d*x)*(d + e*x)])

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Maple [B]  time = 0.029, size = 1945, normalized size = 9.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/x^4/(e*x+d),x)

[Out]

3/16/d*e^4*a*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)
*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)*c+1/16/d^5*e^8*a^3/c*ln((1/2*a*e^2+1/2*c*d^2+
c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(1/2)-1/
16/d^5*e^8*a^3/c*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+
d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)-3/16/d*e^4*a*c*ln((1/2*a*e^2-
1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(
1/2))/(c*d*e)^(1/2)+1/16*d^4/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)*c^3+17/24/d^3/a*e^2*c*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/24*d/a^3/e^2*c^3*(a*d*e+(a*e^2+c*d^2)*x
+c*d*e*x^2)^(3/2)*x+1/12/d/a^2/e^2/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c
-1/3/d^2/a^2/e/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c-1/8*d^2/a^2/e*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c^3-1/8/d*e^2*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2
)*(x+d/e))^(1/2)+1/8/d^3*a*e^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+7/12/d^3/
a/x^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+1/8*d/a*(a*d*e+(a*e^2+c*d^2)*x+c*d
*e*x^2)^(1/2)*c^2+5/24/a^2/e*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^2-1/8/d^5
*e^6*a^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/3/d^4*e^3*(c*d*e*(x+d/e)^2+
(a*e^2-c*d^2)*(x+d/e))^(3/2)+3/8/d^4*e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
+1/4/d^4*e^5*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/8/d^5*e^6*a^2/c
*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+3/16/d^3*e^6*a^2*ln((1/2*a*e^2-1/
2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/
2))/(c*d*e)^(1/2)-1/4/d^2*e^3*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+
1/16*d*e^2*c^2*ln((1/2*a*e^2-1/2*c*d^2+(x+d/e)*c*d*e)/(c*d*e)^(1/2)+(c*d*e*(x+d/
e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2))/(c*d*e)^(1/2)+3/16*a*e^3/(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
)*c+1/24/a^3/e^3/x*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*c^2+3/8/d^2*e^3*(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*c-3/16*d^2*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)*c^2+1/3/
d/a^2*c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x-1/16*d*e^2*ln((1/2*a*e^2+1/2
*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(c*d*e)^(
1/2)*c^2-1/3/d^2/a/e/x^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)-1/24*d^2/a^3/e^
3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c^3-1/8*d^3/a^2/e^2*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*c^3-1/16/d^2*a^2*e^5/(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)+11/24/d^2/a*e*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*c-17/24/d^4/a*e/x*(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(5/2)-1/4/d^4*e^5*a*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x-3/16/d^
3*e^6*a^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(c*d*e)^(1/2)+(a*d*e+(a*e^2+c*d^2)*x+
c*d*e*x^2)^(1/2))/(c*d*e)^(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*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.26075, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} x^{3} \log \left (-\frac{4 \,{\left (2 \, a^{2} d^{2} e^{2} +{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} -{\left (8 \, a^{2} d^{2} e^{2} +{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x^{2} + 8 \,{\left (a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{a d e}}{x^{2}}\right ) + 4 \,{\left (8 \, a^{2} d^{2} e^{2} +{\left (3 \, c^{2} d^{4} + 8 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} x^{2} + 2 \,{\left (7 \, a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{a d e}}{96 \, \sqrt{a d e} a d^{2} e x^{3}}, \frac{3 \,{\left (c^{3} d^{6} - 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )} x^{3} \arctan \left (\frac{{\left (2 \, a d e +{\left (c d^{2} + a e^{2}\right )} x\right )} \sqrt{-a d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} a d e}\right ) - 2 \,{\left (8 \, a^{2} d^{2} e^{2} +{\left (3 \, c^{2} d^{4} + 8 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} x^{2} + 2 \,{\left (7 \, a c d^{3} e + a^{2} d e^{3}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-a d e}}{48 \, \sqrt{-a d e} a d^{2} e x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^4),x, algorithm="fricas")

[Out]

[-1/96*(3*(c^3*d^6 - 3*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*x^3*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) + 4*(8*a^2*d^2*e^2 + (3*c^2*d^4 + 8*a*c*d^2*e^
2 - 3*a^2*e^4)*x^2 + 2*(7*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)*sqrt(a*d*e))/(sqrt(a*d*e)*a*d^2*e*x^3), 1/48*(3*(c^3*d^6 - 3*a*c
^2*d^4*e^2 + 3*a^2*c*d^2*e^4 - a^3*e^6)*x^3*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)) - 2*(8*a^
2*d^2*e^2 + (3*c^2*d^4 + 8*a*c*d^2*e^2 - 3*a^2*e^4)*x^2 + 2*(7*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)*sqrt(-a*d*e))/(sqrt(-a*d*e)
*a*d^2*e*x^3)]

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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((a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2)/x**4/(e*x+d),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 2.14089, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)/((e*x + d)*x^4),x, algorithm="giac")

[Out]

Done