3.478 \(\int \frac{x^4}{(d+e x) \left (a d e+\left (c d^2+a e^2\right ) x+c d e x^2\right )^{3/2}} \, dx\)
Optimal. Leaf size=438 \[ -\frac{2 x \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac{\left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac{2 d x^3 \left (c d x \left (c d^2-a e^2\right )+a e \left (c d^2-a e^2\right )\right )}{3 e \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
[Out]
(-2*d*x^3*(a*e*(c*d^2 - a*e^2) + c*d*(c*d^2 - a*e^2)*x))/(3*e*(c*d^2 - a*e^2)^2*
(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2)) - (2*x*(a*d*e*(c*d^2 - a*e^2)*(5*
c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e^4) + (c*d^2 - a*e^2)*(5*c^3*d^6 - 9*a*c^2*d^4
*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2]) + ((15*c^3*d^6 - 31*a*c^2*d^4*e^2 + 9*a^2*c*d^2*
e^4 - 9*a^3*e^6)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2*e^3*(c*
d^2 - a*e^2)^3) - ((5*c*d^2 + 3*a*e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sq
rt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^(5/2)*
d^(5/2)*e^(7/2))
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Rubi [A] time = 1.46161, antiderivative size = 438, 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
\[ -\frac{2 x \left (a d e \left (c d^2-a e^2\right ) \left (-3 a^2 e^4-10 a c d^2 e^2+5 c^2 d^4\right )+x \left (c d^2-a e^2\right ) \left (-3 a^3 e^6-a^2 c d^2 e^4-9 a c^2 d^4 e^2+5 c^3 d^6\right )\right )}{3 c d e^2 \left (c d^2-a e^2\right )^4 \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}+\frac{\left (-9 a^3 e^6+9 a^2 c d^2 e^4-31 a c^2 d^4 e^2+15 c^3 d^6\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{3 c^2 d^2 e^3 \left (c d^2-a e^2\right )^3}-\frac{\left (3 a e^2+5 c d^2\right ) \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{2 c^{5/2} d^{5/2} e^{7/2}}-\frac{2 d x^3 (a e+c d x)}{3 e \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Int[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*d*x^3*(a*e + c*d*x))/(3*e*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e
*x^2)^(3/2)) - (2*x*(a*d*e*(c*d^2 - a*e^2)*(5*c^2*d^4 - 10*a*c*d^2*e^2 - 3*a^2*e
^4) + (c*d^2 - a*e^2)*(5*c^3*d^6 - 9*a*c^2*d^4*e^2 - a^2*c*d^2*e^4 - 3*a^3*e^6)*
x))/(3*c*d*e^2*(c*d^2 - a*e^2)^4*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2]) +
((15*c^3*d^6 - 31*a*c^2*d^4*e^2 + 9*a^2*c*d^2*e^4 - 9*a^3*e^6)*Sqrt[a*d*e + (c*d
^2 + a*e^2)*x + c*d*e*x^2])/(3*c^2*d^2*e^3*(c*d^2 - a*e^2)^3) - ((5*c*d^2 + 3*a*
e^2)*ArcTanh[(c*d^2 + a*e^2 + 2*c*d*e*x)/(2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*d*e +
(c*d^2 + a*e^2)*x + c*d*e*x^2])])/(2*c^(5/2)*d^(5/2)*e^(7/2))
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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(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 = 1.16008, size = 261, normalized size = 0.6 \[ \frac{\frac{2 (d+e x)^2 (a e+c d x)^2 \left (-\frac{6 a^4 e^7}{c^2 \left (c d^2-a e^2\right )^3 (a e+c d x)}-\frac{2 d^6}{(d+e x)^2 \left (c d^2-a e^2\right )^2}+\frac{2 d^5 \left (7 c d^2-12 a e^2\right )}{(d+e x) \left (c d^2-a e^2\right )^3}+\frac{3}{c^2}\right )}{3 d^2 e^3}-\frac{(d+e x)^{3/2} \left (3 a e^2+5 c d^2\right ) (a e+c d x)^{3/2} \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{5/2} d^{5/2} e^{7/2}}}{2 ((d+e x) (a e+c d x))^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[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*(a*e + c*d*x)^2*(d + e*x)^2*(3/c^2 - (6*a^4*e^7)/(c^2*(c*d^2 - a*e^2)^3*(a*e
+ c*d*x)) - (2*d^6)/((c*d^2 - a*e^2)^2*(d + e*x)^2) + (2*d^5*(7*c*d^2 - 12*a*e^
2))/((c*d^2 - a*e^2)^3*(d + e*x))))/(3*d^2*e^3) - ((5*c*d^2 + 3*a*e^2)*(a*e + c*
d*x)^(3/2)*(d + e*x)^(3/2)*Log[a*e^2 + 2*Sqrt[c]*Sqrt[d]*Sqrt[e]*Sqrt[a*e + c*d*
x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(5/2)*d^(5/2)*e^(7/2)))/(2*((a*e + c*d*x
)*(d + e*x))^(3/2))
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Maple [B] time = 0.018, size = 1266, normalized size = 2.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4/(e*x+d)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2),x)
[Out]
-3/2*e^3/c^2/d^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*a^3+5/2/e^3/c*x/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/4/c^3/d^3/
(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-9/4/e^4/c*d/(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)-5/2/e^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)+16/3*d^6/e^3*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+8/3*d^5/e^2*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+3/2/e/c^2/d^2*x/(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)*a-3/2*e/c/(-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*a^2-9/2/e*d^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*a-9/2/e^3*c*d^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)*x-3/4*e^4/c^3/d^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)*a^4-3/2*e^2/c^2/d/(-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)*a^3-3/2/e/c^
2/d^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)*a-2/3*d^4/e^5/(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)+8/3*d^7/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*d^3/e^4*(2*c*d*e*x+a*e^2+c*d^2)/(4*a*c*d^2*e^2-(a*
e^2+c*d^2)^2)/(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)+1/e^2*x^2/c/d/(a*d*e+(a*e^
2+c*d^2)*x+c*d*e*x^2)^(1/2)-3/c*d/(-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)*a^2-9/2/e^2*d^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)*a-9/4/e^4*c*d^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)
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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(x^4/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 1.6216, size = 1, normalized size = 0. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)),x, algorithm="fricas")
[Out]
[1/12*(4*(15*a*c^3*d^8*e - 31*a^2*c^2*d^6*e^3 + 9*a^3*c*d^4*e^5 - 9*a^4*d^2*e^7
+ 3*(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 + (20*
c^4*d^8*e - 39*a*c^3*d^6*e^3 + 9*a^2*c^2*d^4*e^5 + 3*a^3*c*d^2*e^7 - 9*a^4*e^9)*
x^2 + (15*c^4*d^9 - 11*a*c^3*d^7*e^2 - 33*a^2*c^2*d^5*e^4 + 15*a^3*c*d^3*e^6 - 1
8*a^4*d*e^8)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) + 3*(5*a
*c^4*d^10*e - 12*a^2*c^3*d^8*e^3 + 6*a^3*c^2*d^6*e^5 + 4*a^4*c*d^4*e^7 - 3*a^5*d
^2*e^9 + (5*c^5*d^9*e^2 - 12*a*c^4*d^7*e^4 + 6*a^2*c^3*d^5*e^6 + 4*a^3*c^2*d^3*e
^8 - 3*a^4*c*d*e^10)*x^3 + (10*c^5*d^10*e - 19*a*c^4*d^8*e^3 + 14*a^3*c^2*d^4*e^
7 - 2*a^4*c*d^2*e^9 - 3*a^5*e^11)*x^2 + (5*c^5*d^11 - 2*a*c^4*d^9*e^2 - 18*a^2*c
^3*d^7*e^4 + 16*a^3*c^2*d^5*e^6 + 5*a^4*c*d^3*e^8 - 6*a^5*d*e^10)*x)*log(-4*(2*c
^2*d^2*e^2*x + c^2*d^3*e + a*c*d*e^3)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
) + (8*c^2*d^2*e^2*x^2 + c^2*d^4 + 6*a*c*d^2*e^2 + a^2*e^4 + 8*(c^2*d^3*e + a*c*
d*e^3)*x)*sqrt(c*d*e)))/((a*c^5*d^10*e^4 - 3*a^2*c^4*d^8*e^6 + 3*a^3*c^3*d^6*e^8
- a^4*c^2*d^4*e^10 + (c^6*d^9*e^5 - 3*a*c^5*d^7*e^7 + 3*a^2*c^4*d^5*e^9 - a^3*c
^3*d^3*e^11)*x^3 + (2*c^6*d^10*e^4 - 5*a*c^5*d^8*e^6 + 3*a^2*c^4*d^6*e^8 + a^3*c
^3*d^4*e^10 - a^4*c^2*d^2*e^12)*x^2 + (c^6*d^11*e^3 - a*c^5*d^9*e^5 - 3*a^2*c^4*
d^7*e^7 + 5*a^3*c^3*d^5*e^9 - 2*a^4*c^2*d^3*e^11)*x)*sqrt(c*d*e)), 1/6*(2*(15*a*
c^3*d^8*e - 31*a^2*c^2*d^6*e^3 + 9*a^3*c*d^4*e^5 - 9*a^4*d^2*e^7 + 3*(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 + (20*c^4*d^8*e - 39*
a*c^3*d^6*e^3 + 9*a^2*c^2*d^4*e^5 + 3*a^3*c*d^2*e^7 - 9*a^4*e^9)*x^2 + (15*c^4*d
^9 - 11*a*c^3*d^7*e^2 - 33*a^2*c^2*d^5*e^4 + 15*a^3*c*d^3*e^6 - 18*a^4*d*e^8)*x)
*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 3*(5*a*c^4*d^10*e -
12*a^2*c^3*d^8*e^3 + 6*a^3*c^2*d^6*e^5 + 4*a^4*c*d^4*e^7 - 3*a^5*d^2*e^9 + (5*c^
5*d^9*e^2 - 12*a*c^4*d^7*e^4 + 6*a^2*c^3*d^5*e^6 + 4*a^3*c^2*d^3*e^8 - 3*a^4*c*d
*e^10)*x^3 + (10*c^5*d^10*e - 19*a*c^4*d^8*e^3 + 14*a^3*c^2*d^4*e^7 - 2*a^4*c*d^
2*e^9 - 3*a^5*e^11)*x^2 + (5*c^5*d^11 - 2*a*c^4*d^9*e^2 - 18*a^2*c^3*d^7*e^4 + 1
6*a^3*c^2*d^5*e^6 + 5*a^4*c*d^3*e^8 - 6*a^5*d*e^10)*x)*arctan(1/2*(2*c*d*e*x + c
*d^2 + a*e^2)*sqrt(-c*d*e)/(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*c*d*e)))
/((a*c^5*d^10*e^4 - 3*a^2*c^4*d^8*e^6 + 3*a^3*c^3*d^6*e^8 - a^4*c^2*d^4*e^10 + (
c^6*d^9*e^5 - 3*a*c^5*d^7*e^7 + 3*a^2*c^4*d^5*e^9 - a^3*c^3*d^3*e^11)*x^3 + (2*c
^6*d^10*e^4 - 5*a*c^5*d^8*e^6 + 3*a^2*c^4*d^6*e^8 + a^3*c^3*d^4*e^10 - a^4*c^2*d
^2*e^12)*x^2 + (c^6*d^11*e^3 - a*c^5*d^9*e^5 - 3*a^2*c^4*d^7*e^7 + 5*a^3*c^3*d^5
*e^9 - 2*a^4*c^2*d^3*e^11)*x)*sqrt(-c*d*e))]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{4}}{\left (\left (d + e x\right ) \left (a e + c d x\right )\right )^{\frac{3}{2}} \left (d + e x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**4/(e*x+d)/(a*d*e+(a*e**2+c*d**2)*x+c*d*e*x**2)**(3/2),x)
[Out]
Integral(x**4/(((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. \[ \left [\mathit{undef}, \mathit{undef}, \mathit{undef}, \mathit{undef}, 1\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^4/((c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)^(3/2)*(e*x + d)),x, algorithm="giac")
[Out]
[undef, undef, undef, undef, 1]