3.1898 \(\int (d+e x)^2 \sqrt{a d e+\left (c d^2+a e^2\right ) x+c d e x^2} \, dx\)

Optimal. Leaf size=268 \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{3/2}}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c^3 d^3 e}+\frac{5 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d} \]

[Out]

(5*(c*d^2 - a*e^2)^2*(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])/(64*c^3*d^3*e) + (5*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2))/(24*c^2*d^2) + ((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(3/2))/(4*c*d) - (5*(c*d^2 - a*e^2)^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])])/(128*c^(7
/2)*d^(7/2)*e^(3/2))

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Rubi [A]  time = 0.482126, antiderivative size = 268, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 37, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.135 \[ -\frac{5 \left (c d^2-a e^2\right )^4 \tanh ^{-1}\left (\frac{a e^2+c d^2+2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}\right )}{128 c^{7/2} d^{7/2} e^{3/2}}+\frac{5 \left (c d^2-a e^2\right )^2 \left (a e^2+c d^2+2 c d e x\right ) \sqrt{x \left (a e^2+c d^2\right )+a d e+c d e x^2}}{64 c^3 d^3 e}+\frac{5 \left (c d^2-a e^2\right ) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{24 c^2 d^2}+\frac{(d+e x) \left (x \left (a e^2+c d^2\right )+a d e+c d e x^2\right )^{3/2}}{4 c d} \]

Antiderivative was successfully verified.

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

[Out]

(5*(c*d^2 - a*e^2)^2*(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])/(64*c^3*d^3*e) + (5*(c*d^2 - a*e^2)*(a*d*e + (c*d^2 + a*e^2)*x + c
*d*e*x^2)^(3/2))/(24*c^2*d^2) + ((d + e*x)*(a*d*e + (c*d^2 + a*e^2)*x + c*d*e*x^
2)^(3/2))/(4*c*d) - (5*(c*d^2 - a*e^2)^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])])/(128*c^(7
/2)*d^(7/2)*e^(3/2))

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Rubi in Sympy [A]  time = 60.5705, size = 255, normalized size = 0.95 \[ \frac{\left (d + e x\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{4 c d} - \frac{5 \left (a e^{2} - c d^{2}\right ) \left (a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )\right )^{\frac{3}{2}}}{24 c^{2} d^{2}} + \frac{5 \left (a e^{2} - c d^{2}\right )^{2} \left (a e^{2} + c d^{2} + 2 c d e x\right ) \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}}{64 c^{3} d^{3} e} - \frac{5 \left (a e^{2} - c d^{2}\right )^{4} \operatorname{atanh}{\left (\frac{a e^{2} + c d^{2} + 2 c d e x}{2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{a d e + c d e x^{2} + x \left (a e^{2} + c d^{2}\right )}} \right )}}{128 c^{\frac{7}{2}} d^{\frac{7}{2}} e^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(d + e*x)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(4*c*d) - 5*(a*e**2
- c*d**2)*(a*d*e + c*d*e*x**2 + x*(a*e**2 + c*d**2))**(3/2)/(24*c**2*d**2) + 5*(
a*e**2 - c*d**2)**2*(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))/(64*c**3*d**3*e) - 5*(a*e**2 - c*d**2)**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))))/(128*c**(7/2)*d**(7/2)*e**(3/2))

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Mathematica [A]  time = 0.442677, size = 237, normalized size = 0.88 \[ \frac{1}{384} \sqrt{(d+e x) (a e+c d x)} \left (\frac{30 a^3 e^5}{c^3 d^3}+4 x \left (-\frac{5 a^2 e^4}{c^2 d^2}+\frac{18 a e^2}{c}+59 d^2\right )-\frac{110 a^2 e^3}{c^2 d}-\frac{15 \left (c d^2-a e^2\right )^4 \log \left (2 \sqrt{c} \sqrt{d} \sqrt{e} \sqrt{d+e x} \sqrt{a e+c d x}+a e^2+c d (d+2 e x)\right )}{c^{7/2} d^{7/2} e^{3/2} \sqrt{d+e x} \sqrt{a e+c d x}}+\frac{16 e x^2 \left (a e^2+17 c d^2\right )}{c d}+\frac{146 a d e}{c}+\frac{30 d^3}{e}+96 e^2 x^3\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(d + e*x)^2*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)]*((30*d^3)/e + (146*a*d*e)/c - (110*a^2*e^3)/(c^2*
d) + (30*a^3*e^5)/(c^3*d^3) + 4*(59*d^2 + (18*a*e^2)/c - (5*a^2*e^4)/(c^2*d^2))*
x + (16*e*(17*c*d^2 + a*e^2)*x^2)/(c*d) + 96*e^2*x^3 - (15*(c*d^2 - a*e^2)^4*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^(7/2)*d^(7/2)*e^(3/2)*Sqrt[a*e + c*d*x]*Sqrt[d + e*x])))/384

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Maple [B]  time = 0.012, size = 730, normalized size = 2.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

5/32*d^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*x+5/64*d^3/e*(a*e*d+(a*e^2+c*d^
2)*x+c*d*e*x^2)^(1/2)-15/64*d*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+11/24/c*(a*e*d+(a*
e^2+c*d^2)*x+c*d*e*x^2)^(3/2)-5/64*d*e/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)
*a+5/32*d^3*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-5/16*e^2/c*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^
2)^(1/2)*x*a+5/64*e^5/d^3/c^3*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^3-5/64*e
^3/d/c^2*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(1/2)*a^2-5/128*d^5/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)+5/32*e^5*a^3/c^2/d*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)+5/32*e^4/d^2/c^2*(a*e*d+(a*e^2
+c*d^2)*x+c*d*e*x^2)^(1/2)*x*a^2-5/128*e^7/d^3/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+1/4
*e*x*(a*e*d+(a*e^2+c*d^2)*x+c*d*e*x^2)^(3/2)/d/c-5/24*e^2/d^2/c^2*(a*e*d+(a*e^2+
c*d^2)*x+c*d*e*x^2)^(3/2)*a

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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)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.24586, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{6} + 73 \, a c^{2} d^{4} e^{2} - 55 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 8 \,{\left (17 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (59 \, c^{3} d^{5} e + 18 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{c d e} + 15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \log \left (-4 \,{\left (2 \, c^{2} d^{2} e^{2} x + c^{2} d^{3} e + a c d e^{3}\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} +{\left (8 \, c^{2} d^{2} e^{2} x^{2} + c^{2} d^{4} + 6 \, a c d^{2} e^{2} + a^{2} e^{4} + 8 \,{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \sqrt{c d e}\right )}{768 \, \sqrt{c d e} c^{3} d^{3} e}, \frac{2 \,{\left (48 \, c^{3} d^{3} e^{3} x^{3} + 15 \, c^{3} d^{6} + 73 \, a c^{2} d^{4} e^{2} - 55 \, a^{2} c d^{2} e^{4} + 15 \, a^{3} e^{6} + 8 \,{\left (17 \, c^{3} d^{4} e^{2} + a c^{2} d^{2} e^{4}\right )} x^{2} + 2 \,{\left (59 \, c^{3} d^{5} e + 18 \, a c^{2} d^{3} e^{3} - 5 \, a^{2} c d e^{5}\right )} x\right )} \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} \sqrt{-c d e} - 15 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \arctan \left (\frac{{\left (2 \, c d e x + c d^{2} + a e^{2}\right )} \sqrt{-c d e}}{2 \, \sqrt{c d e x^{2} + a d e +{\left (c d^{2} + a e^{2}\right )} x} c d e}\right )}{384 \, \sqrt{-c d e} c^{3} d^{3} e}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

[1/768*(4*(48*c^3*d^3*e^3*x^3 + 15*c^3*d^6 + 73*a*c^2*d^4*e^2 - 55*a^2*c*d^2*e^4
+ 15*a^3*e^6 + 8*(17*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 2*(59*c^3*d^5*e + 18*a*
c^2*d^3*e^3 - 5*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt
(c*d*e) + 15*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 +
a^4*e^8)*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), 1/384*(2*(4
8*c^3*d^3*e^3*x^3 + 15*c^3*d^6 + 73*a*c^2*d^4*e^2 - 55*a^2*c*d^2*e^4 + 15*a^3*e^
6 + 8*(17*c^3*d^4*e^2 + a*c^2*d^2*e^4)*x^2 + 2*(59*c^3*d^5*e + 18*a*c^2*d^3*e^3
- 5*a^2*c*d*e^5)*x)*sqrt(c*d*e*x^2 + a*d*e + (c*d^2 + a*e^2)*x)*sqrt(-c*d*e) - 1
5*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*ar
ctan(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)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \sqrt{\left (d + e x\right ) \left (a e + c d x\right )} \left (d + e x\right )^{2}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt((d + e*x)*(a*e + c*d*x))*(d + e*x)**2, x)

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GIAC/XCAS [A]  time = 0.255081, size = 402, normalized size = 1.5 \[ \frac{1}{192} \, \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}{\left (2 \,{\left (4 \,{\left (6 \, x e^{2} + \frac{{\left (17 \, c^{3} d^{4} e^{4} + a c^{2} d^{2} e^{6}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x + \frac{{\left (59 \, c^{3} d^{5} e^{3} + 18 \, a c^{2} d^{3} e^{5} - 5 \, a^{2} c d e^{7}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} x + \frac{{\left (15 \, c^{3} d^{6} e^{2} + 73 \, a c^{2} d^{4} e^{4} - 55 \, a^{2} c d^{2} e^{6} + 15 \, a^{3} e^{8}\right )} e^{\left (-3\right )}}{c^{3} d^{3}}\right )} + \frac{5 \,{\left (c^{4} d^{8} - 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} - 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}\right )} \sqrt{c d} e^{\left (-\frac{3}{2}\right )}{\rm ln}\left ({\left | -\sqrt{c d} c d^{2} e^{\frac{1}{2}} - 2 \,{\left (\sqrt{c d} x e^{\frac{1}{2}} - \sqrt{c d x^{2} e + c d^{2} x + a x e^{2} + a d e}\right )} c d e - \sqrt{c d} a e^{\frac{5}{2}} \right |}\right )}{128 \, c^{4} d^{4}} \]

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)^2,x, algorithm="giac")

[Out]

1/192*sqrt(c*d*x^2*e + c*d^2*x + a*x*e^2 + a*d*e)*(2*(4*(6*x*e^2 + (17*c^3*d^4*e
^4 + a*c^2*d^2*e^6)*e^(-3)/(c^3*d^3))*x + (59*c^3*d^5*e^3 + 18*a*c^2*d^3*e^5 - 5
*a^2*c*d*e^7)*e^(-3)/(c^3*d^3))*x + (15*c^3*d^6*e^2 + 73*a*c^2*d^4*e^4 - 55*a^2*
c*d^2*e^6 + 15*a^3*e^8)*e^(-3)/(c^3*d^3)) + 5/128*(c^4*d^8 - 4*a*c^3*d^6*e^2 + 6
*a^2*c^2*d^4*e^4 - 4*a^3*c*d^2*e^6 + a^4*e^8)*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^4*d^4)