3.1897 \(\int (d+e x)^3 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=328 \[ -\frac{7 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac{7 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^4 d^4 e}+\frac{7 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d} \]

[Out]

(7*(c*d^2 - a*e^2)^3*(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^4*d^4*e) + (7*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(48*c^3*d^3) + (7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2))/(40*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2)^(3/2))/(5*c*d) - (7*(c*d^2 - a*e^2)^5*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^(9/2)*d^(9/2)*e^(3/2))

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Rubi [A]  time = 0.732715, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{7 \left (c d^2-a e^2\right )^5 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{256 c^{9/2} d^{9/2} e^{3/2}}+\frac{7 \left (c d^2-a e^2\right )^3 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{128 c^4 d^4 e}+\frac{7 \left (c d^2-a e^2\right )^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{48 c^3 d^3}+\frac{7 (d+e x) \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{40 c^2 d^2}+\frac{(d+e x)^2 \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{5 c d} \]

Antiderivative was successfully verified.

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

[Out]

(7*(c*d^2 - a*e^2)^3*(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^4*d^4*e) + (7*(c*d^2 - a*e^2)^2*(a*d*e + (c*d^2 + a*e^2)*x
+ c*d*e*x^2)^(3/2))/(48*c^3*d^3) + (7*(c*d^2 - a*e^2)*(d + e*x)*(a*d*e + (c*d^2
+ a*e^2)*x + c*d*e*x^2)^(3/2))/(40*c^2*d^2) + ((d + e*x)^2*(a*d*e + (c*d^2 + a*e
^2)*x + c*d*e*x^2)^(3/2))/(5*c*d) - (7*(c*d^2 - a*e^2)^5*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^(9/2)*d^(9/2)*e^(3/2))

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Rubi in Sympy [A]  time = 90.1827, size = 313, normalized size = 0.95 \[ \frac{\left (d + e x\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{5 c d} - \frac{7 \left (d + e x\right ) \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{40 c^{2} d^{2}} + \frac{7 \left (a e^{2} - c d^{2}\right )^{2} \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{48 c^{3} d^{3}} - \frac{7 \left (a e^{2} - c d^{2}\right )^{3} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{128 c^{4} d^{4} e} + \frac{7 \left (a e^{2} - c d^{2}\right )^{5} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{256 c^{\frac{9}{2}} d^{\frac{9}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(5*c*d) - 7*(d +
e*x)*(a*e**2 - c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(40*c**
2*d**2) + 7*(a*e**2 - c*d**2)**2*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/
2)/(48*c**3*d**3) - 7*(a*e**2 - c*d**2)**3*(a*e**2 + c*d**2 + 2*c*d*e*x)*sqrt(a*
d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))/(128*c**4*d**4*e) + 7*(a*e**2 - c*d**2)*
*5*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**(9/2)*d**(9/2)*e**(3/2))

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Mathematica [A]  time = 0.563428, size = 309, normalized size = 0.94 \[ \frac{\sqrt{(d+e x) (a e+c d x)} \left (-\frac{210 a^4 e^7}{c^4 d^4}+\frac{980 a^3 e^5}{c^3 d^2}+\frac{16 e x^2 \left (-7 a^2 e^4+32 a c d^2 e^2+263 c^2 d^4\right )}{c^2 d^2}-\frac{1792 a^2 e^3}{c^2}+\frac{4 x \left (35 a^3 e^6-161 a^2 c d^2 e^4+289 a c^2 d^4 e^2+605 c^3 d^6\right )}{c^3 d^3}-\frac{105 \left (c d^2-a e^2\right )^5 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{9/2} d^{9/2} e^{3/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{96 e^2 x^3 \left (a e^2+31 c d^2\right )}{c d}+\frac{1580 a d^2 e}{c}+\frac{210 d^4}{e}+768 e^3 x^4\right )}{3840} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[(a*e + c*d*x)*(d + e*x)]*((210*d^4)/e + (1580*a*d^2*e)/c - (1792*a^2*e^3)/
c^2 + (980*a^3*e^5)/(c^3*d^2) - (210*a^4*e^7)/(c^4*d^4) + (4*(605*c^3*d^6 + 289*
a*c^2*d^4*e^2 - 161*a^2*c*d^2*e^4 + 35*a^3*e^6)*x)/(c^3*d^3) + (16*e*(263*c^2*d^
4 + 32*a*c*d^2*e^2 - 7*a^2*e^4)*x^2)/(c^2*d^2) + (96*e^2*(31*c*d^2 + a*e^2)*x^3)
/(c*d) + 768*e^3*x^4 - (105*(c*d^2 - a*e^2)^5*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^(3/2)
*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/3840

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Maple [B]  time = 0.013, size = 968, normalized size = 3. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

25/48*d/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)+7/64*d^3*(a*e*d+(a*e^2+c*d^2)*
x+c*d*e*x^2)^(1/2)*x+7/48*e^4/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a^
2+1/5*e^2*x^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/c-7/15*e^2/d/c^2*(a*e*d+
(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)*a+35/128*e^5/c^2*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*
x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^3+7/12
8*d^4/e*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)-35/128*d^2*e^3/c*ln((1/2*a*e^2+1
/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)
^(1/2)*a^2-35/256*e^7/d^2/c^3*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*
e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)*a^4-21/64*e^2*d/c*(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a-7/128*e^7/d^4/c^4*(a*e*d+(a*e^2+c*d^2)*x+c*d*e
*x^2)^(1/2)*a^4-7/64*d^2*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a+35/256*d^
4*e*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*
x^2)^(1/2))/(d*e*c)^(1/2)*a-7/256*d^6/e*c*ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*
c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2))/(d*e*c)^(1/2)+23/40*e/c*x*(a*e
*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-7/40*e^3/d^2/c^2*x*(a*e*d+(a*e^2+c*d^2)*x+c*
d*e*x^2)^(3/2)*a-7/64*e^6/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^3+
21/64*e^4/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2+7/256*e^9/d^4/c^4*
ln((1/2*a*e^2+1/2*c*d^2+c*d*e*x)/(d*e*c)^(1/2)+(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)
^(1/2))/(d*e*c)^(1/2)*a^5+7/64*e^5/d^2/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/
2)*a^3

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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(sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*(e*x + d)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.259121, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*c^4*d^4*e^4*x^4 + 105*c^4*d^8 + 790*a*c^3*d^6*e^2 - 896*a^2*c^2*
d^4*e^4 + 490*a^3*c*d^2*e^6 - 105*a^4*e^8 + 48*(31*c^4*d^5*e^3 + a*c^3*d^3*e^5)*
x^3 + 8*(263*c^4*d^6*e^2 + 32*a*c^3*d^4*e^4 - 7*a^2*c^2*d^2*e^6)*x^2 + 2*(605*c^
4*d^7*e + 289*a*c^3*d^5*e^3 - 161*a^2*c^2*d^3*e^5 + 35*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) + 105*(c^5*d^10 - 5*a*c^4*d^8*e^2
+ 10*a^2*c^3*d^6*e^4 - 10*a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - 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^4*d^4*e), 1/3840*(2*(384*c^4*d^4*e^4*
x^4 + 105*c^4*d^8 + 790*a*c^3*d^6*e^2 - 896*a^2*c^2*d^4*e^4 + 490*a^3*c*d^2*e^6
- 105*a^4*e^8 + 48*(31*c^4*d^5*e^3 + a*c^3*d^3*e^5)*x^3 + 8*(263*c^4*d^6*e^2 + 3
2*a*c^3*d^4*e^4 - 7*a^2*c^2*d^2*e^6)*x^2 + 2*(605*c^4*d^7*e + 289*a*c^3*d^5*e^3
- 161*a^2*c^2*d^3*e^5 + 35*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) - 105*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*
a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - 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^4*d^4*e)]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.25286, size = 521, normalized size = 1.59 \[ \frac{1}{1920} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x e^{3} + \frac{{\left (31 \, c^{4} d^{5} e^{6} + a c^{3} d^{3} e^{8}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (263 \, c^{4} d^{6} e^{5} + 32 \, a c^{3} d^{4} e^{7} - 7 \, a^{2} c^{2} d^{2} e^{9}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (605 \, c^{4} d^{7} e^{4} + 289 \, a c^{3} d^{5} e^{6} - 161 \, a^{2} c^{2} d^{3} e^{8} + 35 \, a^{3} c d e^{10}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} x + \frac{{\left (105 \, c^{4} d^{8} e^{3} + 790 \, a c^{3} d^{6} e^{5} - 896 \, a^{2} c^{2} d^{4} e^{7} + 490 \, a^{3} c d^{2} e^{9} - 105 \, a^{4} e^{11}\right )} e^{\left (-4\right )}}{c^{4} d^{4}}\right )} + \frac{7 \,{\left (c^{5} d^{10} - 5 \, a c^{4} d^{8} e^{2} + 10 \, a^{2} c^{3} d^{6} e^{4} - 10 \, a^{3} c^{2} d^{4} e^{6} + 5 \, a^{4} c d^{2} e^{8} - a^{5} e^{10}\right )} \sqrt{c d} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{256 \, c^{5} d^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(6*(8*x*e^3 + (31*c^4*d
^5*e^6 + a*c^3*d^3*e^8)*e^(-4)/(c^4*d^4))*x + (263*c^4*d^6*e^5 + 32*a*c^3*d^4*e^
7 - 7*a^2*c^2*d^2*e^9)*e^(-4)/(c^4*d^4))*x + (605*c^4*d^7*e^4 + 289*a*c^3*d^5*e^
6 - 161*a^2*c^2*d^3*e^8 + 35*a^3*c*d*e^10)*e^(-4)/(c^4*d^4))*x + (105*c^4*d^8*e^
3 + 790*a*c^3*d^6*e^5 - 896*a^2*c^2*d^4*e^7 + 490*a^3*c*d^2*e^9 - 105*a^4*e^11)*
e^(-4)/(c^4*d^4)) + 7/256*(c^5*d^10 - 5*a*c^4*d^8*e^2 + 10*a^2*c^3*d^6*e^4 - 10*
a^3*c^2*d^4*e^6 + 5*a^4*c*d^2*e^8 - a^5*e^10)*sqrt(c*d)*e^(-3/2)*ln(abs(-sqrt(c*
d)*c*d^2*e^(1/2) - 2*(sqrt(c*d)*x*e^(1/2) - sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 +
a*d*e))*c*d*e - sqrt(c*d)*a*e^(5/2)))/(c^5*d^5)