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

Optimal. Leaf size=449 \[ -\frac{\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac{\left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}}+\frac{\left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^5}+\frac{1}{20} x^2 \left (\frac{a}{c d}-\frac{3 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e} \]

[Out]

((21*c^4*d^8 - 6*a^2*c^2*d^4*e^4 - 8*a^3*c*d^2*e^6 - 7*a^4*e^8)*(c*d^2 + a*e^2 +
2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^4*d^4*e^5) + ((a
/(c*d) - (3*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/20 + (x^3
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - ((105*c^3*d^6 - 21*a*c^2
*d^4*e^2 - 33*a^2*c*d^2*e^4 - 35*a^3*e^6 - 6*c*d*e*(21*c^2*d^4 - 6*a*c*d^2*e^2 -
7*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*c^3*d^3*e^4)
- ((c*d^2 - a*e^2)^3*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e
^6)*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])])/(1024*c^(9/2)*d^(9/2)*e^(11/2))

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Rubi [A]  time = 1.52452, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{\left (-35 a^3 e^6-6 c d e x \left (-7 a^2 e^4-6 a c d^2 e^2+21 c^2 d^4\right )-33 a^2 c d^2 e^4-21 a c^2 d^4 e^2+105 c^3 d^6\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{960 c^3 d^3 e^4}-\frac{\left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{1024 c^{9/2} d^{9/2} e^{11/2}}+\frac{\left (-7 a^4 e^8-8 a^3 c d^2 e^6-6 a^2 c^2 d^4 e^4+21 c^4 d^8\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{512 c^4 d^4 e^5}+\frac{1}{20} x^2 \left (\frac{a}{c d}-\frac{3 d}{e^2}\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}+\frac{x^3 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{6 e} \]

Antiderivative was successfully verified.

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

[Out]

((21*c^4*d^8 - 6*a^2*c^2*d^4*e^4 - 8*a^3*c*d^2*e^6 - 7*a^4*e^8)*(c*d^2 + a*e^2 +
2*c*d*e*x)*Sqrt[a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2])/(512*c^4*d^4*e^5) + ((a
/(c*d) - (3*d)/e^2)*x^2*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/20 + (x^3
*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(6*e) - ((105*c^3*d^6 - 21*a*c^2
*d^4*e^2 - 33*a^2*c*d^2*e^4 - 35*a^3*e^6 - 6*c*d*e*(21*c^2*d^4 - 6*a*c*d^2*e^2 -
7*a^2*e^4)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(960*c^3*d^3*e^4)
- ((c*d^2 - a*e^2)^3*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e
^6)*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])])/(1024*c^(9/2)*d^(9/2)*e^(11/2))

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

[Out]

Timed out

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Mathematica [A]  time = 0.80535, size = 422, normalized size = 0.94 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{32 x^3 \left (\frac{3 a^2 e^2}{c}+14 a d^2-\frac{9 c d^4}{e^2}\right )}{d}+\frac{16 x^2 \left (-7 a^3 e^6+3 a^2 c d^2 e^4-33 a c^2 d^4 e^2+21 c^3 d^6\right )}{c^2 d^2 e^3}-\frac{15 \left (c d^2-a e^2\right )^3 \left (7 a^3 e^6+15 a^2 c d^2 e^4+21 a c^2 d^4 e^2+21 c^3 d^6\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2} e^{11/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{4 x \left (\frac{35 a^4 e^4}{c^3}-\frac{16 a^3 d^2 e^2}{c^2}-\frac{18 a^2 d^4}{c}+\frac{168 a d^6}{e^2}-\frac{105 c d^8}{e^4}\right )}{d^3}+\frac{2 \left (-\frac{105 a^5 e^{10}}{c^4 d^4}+\frac{55 a^4 e^8}{c^3 d^2}+\frac{54 a^3 e^6}{c^2}+\frac{78 a^2 d^2 e^4}{c}-525 a d^4 e^2+315 c d^6\right )}{e^5}+\frac{256 x^4 \left (13 a e^2+c d^2\right )}{e}+2560 c d x^5\right )}{15360} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((2*(315*c*d^6 - 525*a*d^4*e^2 + (78*a^2*d^2*e^4)
/c + (54*a^3*e^6)/c^2 + (55*a^4*e^8)/(c^3*d^2) - (105*a^5*e^10)/(c^4*d^4)))/e^5
+ (4*((-18*a^2*d^4)/c - (105*c*d^8)/e^4 + (168*a*d^6)/e^2 - (16*a^3*d^2*e^2)/c^2
+ (35*a^4*e^4)/c^3)*x)/d^3 + (16*(21*c^3*d^6 - 33*a*c^2*d^4*e^2 + 3*a^2*c*d^2*e
^4 - 7*a^3*e^6)*x^2)/(c^2*d^2*e^3) + (32*(14*a*d^2 - (9*c*d^4)/e^2 + (3*a^2*e^2)
/c)*x^3)/d + (256*(c*d^2 + 13*a*e^2)*x^4)/e + 2560*c*d*x^5 - (15*(c*d^2 - a*e^2)
^3*(21*c^3*d^6 + 21*a*c^2*d^4*e^2 + 15*a^2*c*d^2*e^4 + 7*a^3*e^6)*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^(
9/2)*d^(9/2)*e^(11/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/15360

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Maple [B]  time = 0.034, size = 1883, normalized size = 4.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*d^6/e^5*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+29/192/c^2/d*(a*d*e+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^2+43/96*d^2/e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*x-7/256*e/c^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3+21/512*d^4/
e^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a-43/512*d^6/e^5*c*(a*d*e+(a*e^2+c*d
^2)*x+c*d*e*x^2)^(1/2)+1/16*d^2*e*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
^6/e^3*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)-1/32*e^2/c^2*a^3/d*(a*d*e+(a*e^2+c*
d^2)*x+c*d*e*x^2)^(1/2)*x+7/96*e/c^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)
*x*a^2-3/512*e^5/c^3/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^5-7/256*e^4/c^3/d^3*(a*d*e+(a*e
^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^4+7/1024*e^7/c^4/d^4*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^6-
17/256*d^2*e/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)*a^3-75/512*d^6/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)*
a-3/16*d^4/e*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^5/e^4*c*(c*d*e*(x+d/e)^
2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x-1/16*d^8/e^5*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)-7/60/e/c^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)*a+7/192*e^2/c^3/d^3*(
a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^3-7/512*e^5/c^4/d^4*(a*d*e+(a*e^2+c*d^2
)*x+c*d*e*x^2)^(1/2)*a^5-15/512*e^3/c^3/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*a^4+11/48/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+1/6/e^2*x*(a*d*e+(
a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d+65/192*d/e^2/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(3/2)*a-3/128*d/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+1/4*d^3/e^2
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-43/256*d^5/e^4*c*(a*d*e+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)*x+29/256*d^2/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a
^2-3/1024*e^3/c^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^4+177/1024*d^4/e*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+43/1024*d^8/e^5*c^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)-1/8*d^2/e*a^2/c*(c*d*e*(x+d/e)^2+
(a*e^2-c*d^2)*(x+d/e))^(1/2)-1/4*d^3/e^2*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e
))^(1/2)*x-1/3*d^3/e^4*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/2)-19/60/e^3/c
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)+43/192*d^3/e^4*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(3/2)

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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)*x^3/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.390298, size = 1, normalized size = 0. \[ \text{result too large to display} \]

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)*x^3/(e*x + d),x, algorithm="fricas")

[Out]

[1/30720*(4*(1280*c^5*d^5*e^5*x^5 + 315*c^5*d^10 - 525*a*c^4*d^8*e^2 + 78*a^2*c^
3*d^6*e^4 + 54*a^3*c^2*d^4*e^6 + 55*a^4*c*d^2*e^8 - 105*a^5*e^10 + 128*(c^5*d^6*
e^4 + 13*a*c^4*d^4*e^6)*x^4 - 16*(9*c^5*d^7*e^3 - 14*a*c^4*d^5*e^5 - 3*a^2*c^3*d
^3*e^7)*x^3 + 8*(21*c^5*d^8*e^2 - 33*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - 7*a^3*c
^2*d^2*e^8)*x^2 - 2*(105*c^5*d^9*e - 168*a*c^4*d^7*e^3 + 18*a^2*c^3*d^5*e^5 + 16
*a^3*c^2*d^3*e^7 - 35*a^4*c*d*e^9)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x
)*sqrt(c*d*e) - 15*(21*c^6*d^12 - 42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^3
*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8 + 6*a^5*c*d^2*e^10 - 7*a^6*e^12)*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)))/(sqrt(c*d*e)*c^4*d^4*e^5), 1/15360*(2*(1280*c^5*d^5*e^5*x^5
+ 315*c^5*d^10 - 525*a*c^4*d^8*e^2 + 78*a^2*c^3*d^6*e^4 + 54*a^3*c^2*d^4*e^6 + 5
5*a^4*c*d^2*e^8 - 105*a^5*e^10 + 128*(c^5*d^6*e^4 + 13*a*c^4*d^4*e^6)*x^4 - 16*(
9*c^5*d^7*e^3 - 14*a*c^4*d^5*e^5 - 3*a^2*c^3*d^3*e^7)*x^3 + 8*(21*c^5*d^8*e^2 -
33*a*c^4*d^6*e^4 + 3*a^2*c^3*d^4*e^6 - 7*a^3*c^2*d^2*e^8)*x^2 - 2*(105*c^5*d^9*e
- 168*a*c^4*d^7*e^3 + 18*a^2*c^3*d^5*e^5 + 16*a^3*c^2*d^3*e^7 - 35*a^4*c*d*e^9)
*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 15*(21*c^6*d^12 -
42*a*c^5*d^10*e^2 + 15*a^2*c^4*d^8*e^4 + 4*a^3*c^3*d^6*e^6 + 3*a^4*c^2*d^4*e^8
+ 6*a^5*c*d^2*e^10 - 7*a^6*e^12)*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)))/(sqrt(-c*d*e)*c^4*d^4
*e^5)]

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

[Out]

Timed out

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

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)*x^3/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError