3.2323 \(\int (d+e x)^2 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=191 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{64 c^3}+\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c} \]

[Out]

((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2
])/(64*c^3) + (5*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(24*c^2) + (e*(d + e*x
)*(a + b*x + c*x^2)^(3/2))/(4*c) - ((b^2 - 4*a*c)*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(4*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^
(7/2))

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Rubi [A]  time = 0.5367, antiderivative size = 191, 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 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right )}{64 c^3}+\frac{5 e \left (a+b x+c x^2\right )^{3/2} (2 c d-b e)}{24 c^2}+\frac{e (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c} \]

Antiderivative was successfully verified.

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

[Out]

((16*c^2*d^2 + 5*b^2*e^2 - 4*c*e*(4*b*d + a*e))*(b + 2*c*x)*Sqrt[a + b*x + c*x^2
])/(64*c^3) + (5*e*(2*c*d - b*e)*(a + b*x + c*x^2)^(3/2))/(24*c^2) + (e*(d + e*x
)*(a + b*x + c*x^2)^(3/2))/(4*c) - ((b^2 - 4*a*c)*(16*c^2*d^2 + 5*b^2*e^2 - 4*c*
e*(4*b*d + a*e))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^
(7/2))

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Rubi in Sympy [A]  time = 39.1516, size = 190, normalized size = 0.99 \[ \frac{e \left (d + e x\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 c} - \frac{5 e \left (b e - 2 c d\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (- 4 a c e^{2} + 5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right )}{64 c^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (- 4 a c e^{2} + 5 b^{2} e^{2} - 16 b c d e + 16 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

e*(d + e*x)*(a + b*x + c*x**2)**(3/2)/(4*c) - 5*e*(b*e - 2*c*d)*(a + b*x + c*x**
2)**(3/2)/(24*c**2) + (b + 2*c*x)*sqrt(a + b*x + c*x**2)*(-4*a*c*e**2 + 5*b**2*e
**2 - 16*b*c*d*e + 16*c**2*d**2)/(64*c**3) - (-4*a*c + b**2)*(-4*a*c*e**2 + 5*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)))/(128*c**(7/2))

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Mathematica [A]  time = 0.282341, size = 192, normalized size = 1.01 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b c \left (2 c \left (6 d^2+4 d e x+e^2 x^2\right )-13 a e^2\right )+8 c^2 \left (a e (16 d+3 e x)+2 c x \left (6 d^2+8 d e x+3 e^2 x^2\right )\right )+15 b^3 e^2-2 b^2 c e (24 d+5 e x)\right )-3 \left (b^2-4 a c\right ) \left (-4 c e (a e+4 b d)+5 b^2 e^2+16 c^2 d^2\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^3*e^2 - 2*b^2*c*e*(24*d + 5*e*x) + 4*b*c*
(-13*a*e^2 + 2*c*(6*d^2 + 4*d*e*x + e^2*x^2)) + 8*c^2*(a*e*(16*d + 3*e*x) + 2*c*
x*(6*d^2 + 8*d*e*x + 3*e^2*x^2))) - 3*(b^2 - 4*a*c)*(16*c^2*d^2 + 5*b^2*e^2 - 4*
c*e*(4*b*d + a*e))*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(384*c^(7/2
))

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Maple [B]  time = 0.013, size = 484, normalized size = 2.5 \[{\frac{{d}^{2}x}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{{d}^{2}b}{4\,c}\sqrt{c{x}^{2}+bx+a}}+{\frac{a{d}^{2}}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}-{\frac{{b}^{2}{d}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{{e}^{2}x}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,b{e}^{2}}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}{e}^{2}x}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,{b}^{3}{e}^{2}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}{e}^{2}a}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,{e}^{2}{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{a{e}^{2}x}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{a{e}^{2}b}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{2\,de}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bdex}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}de}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{abde}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{de{b}^{3}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*d^2*(c*x^2+b*x+a)^(1/2)*x+1/4*d^2/c*(c*x^2+b*x+a)^(1/2)*b+1/2*d^2/c^(1/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*b^2+1/4*e^2*x*(c*x^2+b*x+a)^(3/2)/c-5/24*e^2*b/c^2*(c*x^
2+b*x+a)^(3/2)+5/32*e^2*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x+5/64*e^2*b^3/c^3*(c*x^2+b*
x+a)^(1/2)+3/16*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-5/
128*e^2*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/8*e^2*a/c*(c*x
^2+b*x+a)^(1/2)*x-1/16*e^2*a/c^2*(c*x^2+b*x+a)^(1/2)*b-1/8*e^2*a^2/c^(3/2)*ln((1
/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+2/3*d*e*(c*x^2+b*x+a)^(3/2)/c-1/2*d*e*b/c
*(c*x^2+b*x+a)^(1/2)*x-1/4*d*e*b^2/c^2*(c*x^2+b*x+a)^(1/2)-1/2*d*e*b/c^(3/2)*ln(
(1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/8*d*e*b^3/c^(5/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.25505, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, c^{3} e^{2} x^{3} + 48 \, b c^{2} d^{2} - 16 \,{\left (3 \, b^{2} c - 8 \, a c^{2}\right )} d e +{\left (15 \, b^{3} - 52 \, a b c\right )} e^{2} + 8 \,{\left (16 \, c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{3} d^{2} + 16 \, b c^{2} d e -{\left (5 \, b^{2} c - 12 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d e +{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{2}\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, c^{3} e^{2} x^{3} + 48 \, b c^{2} d^{2} - 16 \,{\left (3 \, b^{2} c - 8 \, a c^{2}\right )} d e +{\left (15 \, b^{3} - 52 \, a b c\right )} e^{2} + 8 \,{\left (16 \, c^{3} d e + b c^{2} e^{2}\right )} x^{2} + 2 \,{\left (48 \, c^{3} d^{2} + 16 \, b c^{2} d e -{\left (5 \, b^{2} c - 12 \, a c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (16 \,{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} d^{2} - 16 \,{\left (b^{3} c - 4 \, a b c^{2}\right )} d e +{\left (5 \, b^{4} - 24 \, a b^{2} c + 16 \, a^{2} c^{2}\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*c^3*e^2*x^3 + 48*b*c^2*d^2 - 16*(3*b^2*c - 8*a*c^2)*d*e + (15*b^3
- 52*a*b*c)*e^2 + 8*(16*c^3*d*e + b*c^2*e^2)*x^2 + 2*(48*c^3*d^2 + 16*b*c^2*d*e
- (5*b^2*c - 12*a*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(c) + 3*(16*(b^2*c^2 -
4*a*c^3)*d^2 - 16*(b^3*c - 4*a*b*c^2)*d*e + (5*b^4 - 24*a*b^2*c + 16*a^2*c^2)*e^
2)*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)*sqrt(c)))/c^(7/2), 1/384*(2*(48*c^3*e^2*x^3 + 48*b*c^2*d^2 - 16*(3*b^2*c -
8*a*c^2)*d*e + (15*b^3 - 52*a*b*c)*e^2 + 8*(16*c^3*d*e + b*c^2*e^2)*x^2 + 2*(48*
c^3*d^2 + 16*b*c^2*d*e - (5*b^2*c - 12*a*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt
(-c) - 3*(16*(b^2*c^2 - 4*a*c^3)*d^2 - 16*(b^3*c - 4*a*b*c^2)*d*e + (5*b^4 - 24*
a*b^2*c + 16*a^2*c^2)*e^2)*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a
)*c)))/(sqrt(-c)*c^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.222764, size = 317, normalized size = 1.66 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, x e^{2} + \frac{16 \, c^{3} d e + b c^{2} e^{2}}{c^{3}}\right )} x + \frac{48 \, c^{3} d^{2} + 16 \, b c^{2} d e - 5 \, b^{2} c e^{2} + 12 \, a c^{2} e^{2}}{c^{3}}\right )} x + \frac{48 \, b c^{2} d^{2} - 48 \, b^{2} c d e + 128 \, a c^{2} d e + 15 \, b^{3} e^{2} - 52 \, a b c e^{2}}{c^{3}}\right )} + \frac{{\left (16 \, b^{2} c^{2} d^{2} - 64 \, a c^{3} d^{2} - 16 \, b^{3} c d e + 64 \, a b c^{2} d e + 5 \, b^{4} e^{2} - 24 \, a b^{2} c e^{2} + 16 \, a^{2} c^{2} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*x*e^2 + (16*c^3*d*e + b*c^2*e^2)/c^3)*x + (
48*c^3*d^2 + 16*b*c^2*d*e - 5*b^2*c*e^2 + 12*a*c^2*e^2)/c^3)*x + (48*b*c^2*d^2 -
48*b^2*c*d*e + 128*a*c^2*d*e + 15*b^3*e^2 - 52*a*b*c*e^2)/c^3) + 1/128*(16*b^2*
c^2*d^2 - 64*a*c^3*d^2 - 16*b^3*c*d*e + 64*a*b*c^2*d*e + 5*b^4*e^2 - 24*a*b^2*c*
e^2 + 16*a^2*c^2*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b)
)/c^(7/2)