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

Optimal. Leaf size=352 \[ \frac{\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac{x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]

[Out]

-((c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(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])/(128*c^3*d^3*e^4) + (x^2*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) + ((35*c^2*d^4 - 12*a*c*d^2*e^2
- 15*a^2*e^4 - 6*c*d*e*(7*c*d^2 - 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(240*c^2*d^2*e^3) + ((c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2
+ 3*a^2*e^4)*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])])/(256*c^(7/2)*d^(7/2)*e^(9/2))

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Rubi [A]  time = 0.958964, antiderivative size = 352, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ \frac{\left (-15 a^2 e^4-6 c d e x \left (7 c d^2-3 a e^2\right )-12 a c d^2 e^2+35 c^2 d^4\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{240 c^2 d^2 e^3}+\frac{\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right )^3 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{7/2} d^{7/2} e^{9/2}}-\frac{\left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \left (c d^2-a e^2\right ) \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^3 d^3 e^4}+\frac{x^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 e} \]

Antiderivative was successfully verified.

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

[Out]

-((c*d^2 - a*e^2)*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*(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])/(128*c^3*d^3*e^4) + (x^2*(a*d
*e + (c*d^2 + a*e^2)*x + c*d*e*x^2)^(3/2))/(5*e) + ((35*c^2*d^4 - 12*a*c*d^2*e^2
- 15*a^2*e^4 - 6*c*d*e*(7*c*d^2 - 3*a*e^2)*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*
e*x^2)^(3/2))/(240*c^2*d^2*e^3) + ((c*d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2
+ 3*a^2*e^4)*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])])/(256*c^(7/2)*d^(7/2)*e^(9/2))

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Rubi in Sympy [A]  time = 109.759, size = 354, normalized size = 1.01 \[ \frac{x^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{5 e} - \frac{\left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}} \left (\frac{15 a^{2} e^{4}}{4} + 3 a c d^{2} e^{2} - \frac{35 c^{2} d^{4}}{4} - \frac{3 c d e x \left (3 a e^{2} - 7 c d^{2}\right )}{2}\right )}{60 c^{2} d^{2} e^{3}} + \frac{\left (a e^{2} - c d^{2}\right ) \left (a e^{2} + c d^{2} + 2 c d e x\right ) \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4}\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{128 c^{3} d^{3} e^{4}} - \frac{\left (a e^{2} - c d^{2}\right )^{3} \left (3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 7 c^{2} d^{4}\right ) \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{256 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

x**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(5*e) - (a*d*e + c*d*e*x*
*2 + x*(a*e**2 + c*d**2))**(3/2)*(15*a**2*e**4/4 + 3*a*c*d**2*e**2 - 35*c**2*d**
4/4 - 3*c*d*e*x*(3*a*e**2 - 7*c*d**2)/2)/(60*c**2*d**2*e**3) + (a*e**2 - c*d**2)
*(a*e**2 + c*d**2 + 2*c*d*e*x)*(3*a**2*e**4 + 6*a*c*d**2*e**2 + 7*c**2*d**4)*sqr
t(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(128*c**3*d**3*e**4) - (a*e**2 - c*d
**2)**3*(3*a**2*e**4 + 6*a*c*d**2*e**2 + 7*c**2*d**4)*atanh((a*e**2 + c*d**2 + 2
*c*d*e*x)/(2*sqrt(c)*sqrt(d)*sqrt(e)*sqrt(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**
2))))/(256*c**(7/2)*d**(7/2)*e**(9/2))

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Mathematica [A]  time = 0.5892, size = 320, normalized size = 0.91 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (\frac{90 a^4 e^4}{c^3 d^3}-\frac{60 a^3 e^2}{c^2 d}+\frac{15 \left (c d^2-a e^2\right )^3 \left (3 a^2 e^4+6 a c d^2 e^2+7 c^2 d^4\right ) \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{9/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{16 x^2 \left (\frac{3 a^2 e^2}{c}+12 a d^2-\frac{7 c d^4}{e^2}\right )}{d}-\frac{72 a^2 d}{c}+\frac{4 x \left (-\frac{15 a^3 e^6}{c^2 d^2}+\frac{9 a^2 e^4}{c}-61 a d^2 e^2+35 c d^4\right )}{e^3}+\frac{96 x^3 \left (11 a e^2+c d^2\right )}{e}+\frac{380 a d^3}{e^2}-\frac{210 c d^5}{e^4}+768 c d x^4\right )}{3840} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^2*(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)]*((-72*a^2*d)/c - (210*c*d^5)/e^4 + (380*a*d^3)/e^
2 - (60*a^3*e^2)/(c^2*d) + (90*a^4*e^4)/(c^3*d^3) + (4*(35*c*d^4 - 61*a*d^2*e^2
+ (9*a^2*e^4)/c - (15*a^3*e^6)/(c^2*d^2))*x)/e^3 + (16*(12*a*d^2 - (7*c*d^4)/e^2
+ (3*a^2*e^2)/c)*x^2)/d + (96*(c*d^2 + 11*a*e^2)*x^3)/e + 768*c*d*x^4 + (15*(c*
d^2 - a*e^2)^3*(7*c^2*d^4 + 6*a*c*d^2*e^2 + 3*a^2*e^4)*Log[a*e^2 + 2*Sqrt[c]*Sqr
t[d]*Sqrt[e]*Sqrt[a*e + c*d*x]*Sqrt[d + e*x] + c*d*(d + 2*e*x)])/(c^(7/2)*d^(7/2
)*e^(9/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/3840

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Maple [B]  time = 0.022, size = 1560, normalized size = 4.4 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*d^2/e*a*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)*x+1/8*d*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*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)-1/8*d^5/e^4*c*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(1/2)+9/128/e^4*c*d^5
*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-1/4/e/c*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(3/2)*a-3/64/e^2*d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+1/5/e^2*(a*d*e
+(a*e^2+c*d^2)*x+c*d*e*x^2)^(5/2)/c/d-3/8/e^2*d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2
)^(3/2)*x-21/128*d^3*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^2-3/32/c*d*(a*d*e+(a*e^2+c*d^2)*x+c
*d*e*x^2)^(1/2)*a^2+3/256*e^4/c^2/d*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+9/128*e^2/c*d*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+33/256/e^2*c*d^5*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-1/16*d*e^2*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^5/e^2*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)+3/64*e^3/c^2/d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3-3/256*e^6/c^
3/d^3*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+1/16*d^7/e^4*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)-15/64/e*d^2*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-1/8/c/d*(a*d*e+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(3/2)*x*a+3/64*e^2/c^2/d*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^
(1/2)*a^3-9/256/e^4*c^2*d^7*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*e/c^2/d^2*(a*d*e+(a*e^2+c
*d^2)*x+c*d*e*x^2)^(3/2)*a^2+9/64/e^3*c*d^4*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1
/2)*x+3/128*e^4/c^3/d^3*(a*d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^4+3/64*e/c*(a*
d*e+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-1/4*d^4/e^3*c*(c*d*e*(x+d/e)^2+(a*e^2
-c*d^2)*(x+d/e))^(1/2)*x+1/3*d^2/e^3*(c*d*e*(x+d/e)^2+(a*e^2-c*d^2)*(x+d/e))^(3/
2)-3/16/e^3*d^2*(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^2/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.330387, 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^2/(e*x + d),x, algorithm="fricas")

[Out]

[1/7680*(4*(384*c^4*d^4*e^4*x^4 - 105*c^4*d^8 + 190*a*c^3*d^6*e^2 - 36*a^2*c^2*d
^4*e^4 - 30*a^3*c*d^2*e^6 + 45*a^4*e^8 + 48*(c^4*d^5*e^3 + 11*a*c^3*d^3*e^5)*x^3
- 8*(7*c^4*d^6*e^2 - 12*a*c^3*d^4*e^4 - 3*a^2*c^2*d^2*e^6)*x^2 + 2*(35*c^4*d^7*
e - 61*a*c^3*d^5*e^3 + 9*a^2*c^2*d^3*e^5 - 15*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a
*d*e + (c*d^2 + a*e^2)*x)*sqrt(c*d*e) - 15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^
2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*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^3*d^3*e^4), 1/3840*(2*(384*c^4*d^4*e^4*x^4 -
105*c^4*d^8 + 190*a*c^3*d^6*e^2 - 36*a^2*c^2*d^4*e^4 - 30*a^3*c*d^2*e^6 + 45*a^
4*e^8 + 48*(c^4*d^5*e^3 + 11*a*c^3*d^3*e^5)*x^3 - 8*(7*c^4*d^6*e^2 - 12*a*c^3*d^
4*e^4 - 3*a^2*c^2*d^2*e^6)*x^2 + 2*(35*c^4*d^7*e - 61*a*c^3*d^5*e^3 + 9*a^2*c^2*
d^3*e^5 - 15*a^3*c*d*e^7)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c
*d*e) + 15*(7*c^5*d^10 - 15*a*c^4*d^8*e^2 + 6*a^2*c^3*d^6*e^4 + 2*a^3*c^2*d^4*e^
6 + 3*a^4*c*d^2*e^8 - 3*a^5*e^10)*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^3*d^
3*e^4)]

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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**2*(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^2/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError