3.2322 \(\int (d+e x)^3 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=248 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b + 2*c*x)*Sqrt
[a + b*x + c*x^2])/(128*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (
e*(192*c^2*d^2 + 35*b^2*e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c*e*(2*c*d - b*e)*x)*
(a + b*x + c*x^2)^(3/2))/(240*c^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 +
7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
c*x^2])])/(256*c^(9/2))

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Rubi [A]  time = 0.642339, antiderivative size = 248, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227 \[ -\frac{\left (b^2-4 a c\right ) (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right )}{128 c^4}+\frac{e \left (a+b x+c x^2\right )^{3/2} \left (-2 c e (16 a e+75 b d)+35 b^2 e^2+42 c e x (2 c d-b e)+192 c^2 d^2\right )}{240 c^3}+\frac{e (d+e x)^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

((2*c*d - b*e)*(16*c^2*d^2 + 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*(b + 2*c*x)*Sqrt
[a + b*x + c*x^2])/(128*c^4) + (e*(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/(5*c) + (
e*(192*c^2*d^2 + 35*b^2*e^2 - 2*c*e*(75*b*d + 16*a*e) + 42*c*e*(2*c*d - b*e)*x)*
(a + b*x + c*x^2)^(3/2))/(240*c^3) - ((b^2 - 4*a*c)*(2*c*d - b*e)*(16*c^2*d^2 +
7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x +
c*x^2])])/(256*c^(9/2))

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Rubi in Sympy [A]  time = 57.0931, size = 253, normalized size = 1.02 \[ \frac{e \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5 c} + \frac{e \left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 8 a c e^{2} + \frac{35 b^{2} e^{2}}{4} - \frac{75 b c d e}{2} + 48 c^{2} d^{2} - \frac{21 c e x \left (b e - 2 c d\right )}{2}\right )}{60 c^{3}} - \frac{\left (b + 2 c x\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}} \left (- 12 a c e^{2} + 7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{128 c^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \left (- 12 a c e^{2} + 7 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)**2*(a + b*x + c*x**2)**(3/2)/(5*c) + e*(a + b*x + c*x**2)**(3/2)*(-8
*a*c*e**2 + 35*b**2*e**2/4 - 75*b*c*d*e/2 + 48*c**2*d**2 - 21*c*e*x*(b*e - 2*c*d
)/2)/(60*c**3) - (b + 2*c*x)*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)*(-12*a*c*e**2
+ 7*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)/(128*c**4) + (-4*a*c + b**2)*(b*e - 2
*c*d)*(-12*a*c*e**2 + 7*b**2*e**2 - 16*b*c*d*e + 16*c**2*d**2)*atanh((b + 2*c*x)
/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(256*c**(9/2))

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Mathematica [A]  time = 0.462088, size = 292, normalized size = 1.18 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (16 c^2 \left (-16 a^2 e^3+a c e \left (120 d^2+45 d e x+8 e^2 x^2\right )+6 c^2 x \left (10 d^3+20 d^2 e x+15 d e^2 x^2+4 e^3 x^3\right )\right )-4 b^2 c e \left (c \left (180 d^2+75 d e x+14 e^2 x^2\right )-115 a e^2\right )+8 b c^2 \left (6 c \left (10 d^3+10 d^2 e x+5 d e^2 x^2+e^3 x^3\right )-a e^2 (195 d+29 e x)\right )-105 b^4 e^3+10 b^3 c e^2 (45 d+7 e x)\right )+15 \left (b^2-4 a c\right ) (b e-2 c d) \left (-4 c e (3 a e+4 b d)+7 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{3840 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(-105*b^4*e^3 + 10*b^3*c*e^2*(45*d + 7*e*x) - 4
*b^2*c*e*(-115*a*e^2 + c*(180*d^2 + 75*d*e*x + 14*e^2*x^2)) + 8*b*c^2*(-(a*e^2*(
195*d + 29*e*x)) + 6*c*(10*d^3 + 10*d^2*e*x + 5*d*e^2*x^2 + e^3*x^3)) + 16*c^2*(
-16*a^2*e^3 + a*c*e*(120*d^2 + 45*d*e*x + 8*e^2*x^2) + 6*c^2*x*(10*d^3 + 20*d^2*
e*x + 15*d*e^2*x^2 + 4*e^3*x^3))) + 15*(b^2 - 4*a*c)*(-2*c*d + b*e)*(16*c^2*d^2
+ 7*b^2*e^2 - 4*c*e*(4*b*d + 3*a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c
*x)]])/(3840*c^(9/2))

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Maple [B]  time = 0.016, size = 795, normalized size = 3.2 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

3/16*e^3*b/c^2*a*(c*x^2+b*x+a)^(1/2)*x-3/8*d*e^2*a/c*(c*x^2+b*x+a)^(1/2)*x-3/16*
d*e^2*a/c^2*(c*x^2+b*x+a)^(1/2)*b-3/4*d^2*e*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*
x^2+b*x+a)^(1/2))*a-3/4*d^2*e*b/c*(c*x^2+b*x+a)^(1/2)*x+15/32*d*e^2*b^2/c^2*(c*x
^2+b*x+a)^(1/2)*x+9/16*d*e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))*a+7/48*e^3*b^2/c^3*(c*x^2+b*x+a)^(3/2)-7/128*e^3*b^4/c^4*(c*x^2+b*x+a)^(1/2
)+7/256*e^3*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/15*e^3*a/c
^2*(c*x^2+b*x+a)^(3/2)+d^2*e*(c*x^2+b*x+a)^(3/2)/c-5/32*e^3*b^3/c^(7/2)*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/4*d*e^2*x*(c*x^2+b*x+a)^(3/2)/c+15/64*d*
e^2*b^3/c^3*(c*x^2+b*x+a)^(1/2)+3/32*e^3*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)-3/8*d^2*e
*b^2/c^2*(c*x^2+b*x+a)^(1/2)+3/16*e^3*b/c^(5/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^
2+b*x+a)^(1/2))-15/128*d*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1
/2))-3/8*d*e^2*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/2*d^3*(
c*x^2+b*x+a)^(1/2)*x+1/5*e^3*x^2*(c*x^2+b*x+a)^(3/2)/c+1/4*d^3/c*(c*x^2+b*x+a)^(
1/2)*b+1/2*d^3/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*d^3/c^(
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2-5/8*d*e^2*b/c^2*(c*x^2+b*x+
a)^(3/2)-7/40*e^3*b/c^2*x*(c*x^2+b*x+a)^(3/2)+3/16*d^2*e*b^3/c^(5/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-7/64*e^3*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.294337, 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*x^2 + b*x + a)*(e*x + d)^3,x, algorithm="fricas")

[Out]

[1/7680*(4*(384*c^4*e^3*x^4 + 480*b*c^3*d^3 - 240*(3*b^2*c^2 - 8*a*c^3)*d^2*e +
30*(15*b^3*c - 52*a*b*c^2)*d*e^2 - (105*b^4 - 460*a*b^2*c + 256*a^2*c^2)*e^3 + 4
8*(30*c^4*d*e^2 + b*c^3*e^3)*x^3 + 8*(240*c^4*d^2*e + 30*b*c^3*d*e^2 - (7*b^2*c^
2 - 16*a*c^3)*e^3)*x^2 + 2*(480*c^4*d^3 + 240*b*c^3*d^2*e - 30*(5*b^2*c^2 - 12*a
*c^3)*d*e^2 + (35*b^3*c - 116*a*b*c^2)*e^3)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) - 1
5*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e + 6*(5*b^4*c - 24
*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*c^2)*e^3)*log(-4
*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqr
t(c)))/c^(9/2), 1/3840*(2*(384*c^4*e^3*x^4 + 480*b*c^3*d^3 - 240*(3*b^2*c^2 - 8*
a*c^3)*d^2*e + 30*(15*b^3*c - 52*a*b*c^2)*d*e^2 - (105*b^4 - 460*a*b^2*c + 256*a
^2*c^2)*e^3 + 48*(30*c^4*d*e^2 + b*c^3*e^3)*x^3 + 8*(240*c^4*d^2*e + 30*b*c^3*d*
e^2 - (7*b^2*c^2 - 16*a*c^3)*e^3)*x^2 + 2*(480*c^4*d^3 + 240*b*c^3*d^2*e - 30*(5
*b^2*c^2 - 12*a*c^3)*d*e^2 + (35*b^3*c - 116*a*b*c^2)*e^3)*x)*sqrt(c*x^2 + b*x +
a)*sqrt(-c) - 15*(32*(b^2*c^3 - 4*a*c^4)*d^3 - 48*(b^3*c^2 - 4*a*b*c^3)*d^2*e +
6*(5*b^4*c - 24*a*b^2*c^2 + 16*a^2*c^3)*d*e^2 - (7*b^5 - 40*a*b^3*c + 48*a^2*b*
c^2)*e^3)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*
c^4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \left (d + e x\right )^{3} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((d + e*x)**3*sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.220314, size = 513, normalized size = 2.07 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, x e^{3} + \frac{30 \, c^{4} d e^{2} + b c^{3} e^{3}}{c^{4}}\right )} x + \frac{240 \, c^{4} d^{2} e + 30 \, b c^{3} d e^{2} - 7 \, b^{2} c^{2} e^{3} + 16 \, a c^{3} e^{3}}{c^{4}}\right )} x + \frac{480 \, c^{4} d^{3} + 240 \, b c^{3} d^{2} e - 150 \, b^{2} c^{2} d e^{2} + 360 \, a c^{3} d e^{2} + 35 \, b^{3} c e^{3} - 116 \, a b c^{2} e^{3}}{c^{4}}\right )} x + \frac{480 \, b c^{3} d^{3} - 720 \, b^{2} c^{2} d^{2} e + 1920 \, a c^{3} d^{2} e + 450 \, b^{3} c d e^{2} - 1560 \, a b c^{2} d e^{2} - 105 \, b^{4} e^{3} + 460 \, a b^{2} c e^{3} - 256 \, a^{2} c^{2} e^{3}}{c^{4}}\right )} + \frac{{\left (32 \, b^{2} c^{3} d^{3} - 128 \, a c^{4} d^{3} - 48 \, b^{3} c^{2} d^{2} e + 192 \, a b c^{3} d^{2} e + 30 \, b^{4} c d e^{2} - 144 \, a b^{2} c^{2} d e^{2} + 96 \, a^{2} c^{3} d e^{2} - 7 \, b^{5} e^{3} + 40 \, a b^{3} c e^{3} - 48 \, a^{2} b c^{2} e^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*x*e^3 + (30*c^4*d*e^2 + b*c^3*e^3)/c^4)
*x + (240*c^4*d^2*e + 30*b*c^3*d*e^2 - 7*b^2*c^2*e^3 + 16*a*c^3*e^3)/c^4)*x + (4
80*c^4*d^3 + 240*b*c^3*d^2*e - 150*b^2*c^2*d*e^2 + 360*a*c^3*d*e^2 + 35*b^3*c*e^
3 - 116*a*b*c^2*e^3)/c^4)*x + (480*b*c^3*d^3 - 720*b^2*c^2*d^2*e + 1920*a*c^3*d^
2*e + 450*b^3*c*d*e^2 - 1560*a*b*c^2*d*e^2 - 105*b^4*e^3 + 460*a*b^2*c*e^3 - 256
*a^2*c^2*e^3)/c^4) + 1/256*(32*b^2*c^3*d^3 - 128*a*c^4*d^3 - 48*b^3*c^2*d^2*e +
192*a*b*c^3*d^2*e + 30*b^4*c*d*e^2 - 144*a*b^2*c^2*d*e^2 + 96*a^2*c^3*d*e^2 - 7*
b^5*e^3 + 40*a*b^3*c*e^3 - 48*a^2*b*c^2*e^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 +
b*x + a))*sqrt(c) - b))/c^(9/2)