3.486 \(\int \frac{1}{x^4 (d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=664 \[ \frac{2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}+\frac{5 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}-\frac{\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}-\frac{2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

[Out]

(-2*e*(a*e + c*d*x))/(3*d*(c*d^2 - a*e^2)*x^3*(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 + 13*a^2*c*d^2*e^4 - 9*a^3*e^6 + c*
d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*x^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((7*c^3*d^6 - 3*a*c^2*d^4*e^2 +
33*a^2*c*d^2*e^4 - 21*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*
a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^3) + ((35*c^4*d^8 - 16*a*c^3*d^6*e^2 - 18*a^2*c^
2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x^2) - ((105*c^5*d^10 - 55*a*c^4*d^8
*e^2 - 54*a^2*c^3*d^6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e^8 - 315*a^5*e^1
0)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*a^4*d^5*e^4*(c*d^2 - a*e^2)^
3*x) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*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])])/(16*a^(9/2)*d^(11/2)*e^(9/2))

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Rubi [A]  time = 2.7994, antiderivative size = 664, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{2 \left (-9 a^3 e^6+c d e x \left (-9 a^2 e^4+14 a c d^2 e^2+3 c^2 d^4\right )+13 a^2 c d^2 e^4+a c^2 d^4 e^2+3 c^3 d^6\right )}{3 a d^2 e x^3 \left (c d^2-a e^2\right )^3 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}-\frac{\left (-21 a^3 e^6+33 a^2 c d^2 e^4-3 a c^2 d^4 e^2+7 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 a^2 d^3 e^2 x^3 \left (c d^2-a e^2\right )^3}+\frac{5 \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) \tanh ^{-1}\left (\frac{x \left (a e^2+c d^2\right )+2 a d e}{2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{16 a^{9/2} d^{11/2} e^{9/2}}+\frac{\left (-105 a^4 e^8+168 a^3 c d^2 e^6-18 a^2 c^2 d^4 e^4-16 a c^3 d^6 e^2+35 c^4 d^8\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{12 a^3 d^4 e^3 x^2 \left (c d^2-a e^2\right )^3}-\frac{\left (-315 a^5 e^{10}+525 a^4 c d^2 e^8-78 a^3 c^2 d^4 e^6-54 a^2 c^3 d^6 e^4-55 a c^4 d^8 e^2+105 c^5 d^{10}\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{24 a^4 d^5 e^4 x \left (c d^2-a e^2\right )^3}-\frac{2 e (a e+c d x)}{3 d x^3 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Int[1/(x^4*(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)*x^3*(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 + 13*a^2*c*d^2*e^4 - 9*a^3*e^6 + c*
d*e*(3*c^2*d^4 + 14*a*c*d^2*e^2 - 9*a^2*e^4)*x))/(3*a*d^2*e*(c*d^2 - a*e^2)^3*x^
3*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) - ((7*c^3*d^6 - 3*a*c^2*d^4*e^2 +
33*a^2*c*d^2*e^4 - 21*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*
a^2*d^3*e^2*(c*d^2 - a*e^2)^3*x^3) + ((35*c^4*d^8 - 16*a*c^3*d^6*e^2 - 18*a^2*c^
2*d^4*e^4 + 168*a^3*c*d^2*e^6 - 105*a^4*e^8)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*
d*e*x^2])/(12*a^3*d^4*e^3*(c*d^2 - a*e^2)^3*x^2) - ((105*c^5*d^10 - 55*a*c^4*d^8
*e^2 - 54*a^2*c^3*d^6*e^4 - 78*a^3*c^2*d^4*e^6 + 525*a^4*c*d^2*e^8 - 315*a^5*e^1
0)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(24*a^4*d^5*e^4*(c*d^2 - a*e^2)^
3*x) + (5*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*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])])/(16*a^(9/2)*d^(11/2)*e^(9/2))

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Rubi in Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)

[Out]

Timed out

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Mathematica [A]  time = 2.5358, size = 429, normalized size = 0.65 \[ \frac{-\frac{15 \log (x) (d+e x)^{3/2} \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) (a e+c d x)^{3/2}}{a^{9/2}}+\frac{15 (d+e x)^{3/2} \left (21 a^3 e^6+21 a^2 c d^2 e^4+15 a c^2 d^4 e^2+7 c^3 d^6\right ) (a e+c d x)^{3/2} \log \left (2 \sqrt{a} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e (2 d+e x)+c d^2 x\right )}{a^{9/2}}+2 \sqrt{d} \sqrt{e} (d+e x)^2 (a e+c d x)^2 \left (\frac{\frac{48 c^6 d^{11}}{\left (a e^2-c d^2\right )^3 (a e+c d x)}-\frac{57 c^2 d^4}{x}}{a^4}+\frac{2 c d^2 e (11 d-58 e x)}{a^3 x^2}+\frac{e^2 \left (-8 d^2+34 d e x-123 e^2 x^2\right )}{a^2 x^3}+\frac{192 a e^{11}}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{16 d e^9 \left (a e^2-c d (18 d+17 e x)\right )}{(d+e x)^2 \left (c d^2-a e^2\right )^3}\right )}{48 d^{11/2} e^{9/2} ((d+e x) (a e+c d x))^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[1/(x^4*(d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)),x]

[Out]

(2*Sqrt[d]*Sqrt[e]*(a*e + c*d*x)^2*(d + e*x)^2*((2*c*d^2*e*(11*d - 58*e*x))/(a^3
*x^2) + (192*a*e^11)/((c*d^2 - a*e^2)^3*(d + e*x)) + (e^2*(-8*d^2 + 34*d*e*x - 1
23*e^2*x^2))/(a^2*x^3) + ((-57*c^2*d^4)/x + (48*c^6*d^11)/((-(c*d^2) + a*e^2)^3*
(a*e + c*d*x)))/a^4 + (16*d*e^9*(a*e^2 - c*d*(18*d + 17*e*x)))/((c*d^2 - a*e^2)^
3*(d + e*x)^2)) - (15*(7*c^3*d^6 + 15*a*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*
e^6)*(a*e + c*d*x)^(3/2)*(d + e*x)^(3/2)*Log[x])/a^(9/2) + (15*(7*c^3*d^6 + 15*a
*c^2*d^4*e^2 + 21*a^2*c*d^2*e^4 + 21*a^3*e^6)*(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)])/a^(9/2))/(48*d^(11/2)*e^(9/2)*((a*e + c*d*x)*(d + e*x))^(3/2))

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Maple [B]  time = 0.03, size = 1705, normalized size = 2.6 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)

[Out]

8/3/d^3*e^6*c/(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+16
/3/d^2*e^5*c^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-1
/6/a^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
)*x*c^3+105/8/d^4*e^5/(-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+7/12/d/a^2/e^2/x^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c-
43/24/d/a*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)*c^2+155/48*d^3/a^3/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)*c^4+35/16*d^5/a^4/e^4/(-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^5+75/16/d/a^3/e^2/(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+35/16*d/a^4/e^4/(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^3-17/6/d^2/a^2/e/x/(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*c-105/16/d^5/a*e^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2
)-105/16/d^3/a^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c+13/12/d^3/a/x^2/(a*d*
e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+35/8*d^4/a^4/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)*x*c^5-41/12/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)*x*c^2+25/12*d^2/a^3/
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)*x*c^4
+105/16/d^5/a*e^2/(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)-1/3/d^2/a/e/x^3/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)-35/24/a^3/e^3/x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^2+23/2
4*d/a^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)
*c^3+105/16/d^3/a^2/(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+233/48/d^3*e^4/(-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-89/24/d^4/a*e/x/(a*d*e+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-75/16/d/a^3/e^2/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*c^2-35/16*d/a^4/e^4/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*c^3+105/16/d^5
*a*e^6/(-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)+
8/3/d*e^4*c^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)-2/3/
d^4*e^3/(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)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x\right )}^{\frac{3}{2}}{\left (e x + d\right )} x^{4}}\,{d x} \]

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^4),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^4), x)

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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^4),x, algorithm="fricas")

[Out]

Timed out

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

[Out]

Timed out

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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^4),x, algorithm="giac")

[Out]

[undef, undef, undef, 1]