3.1548 \(\int (b+2 c x) (d+e x)^2 \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=195 \[ -\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{32 c^3}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (8 a e+5 b d)+5 b^2 e^2+6 c e x (2 c d-b e)+16 c^2 d^2\right )}{60 c^2}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \]

[Out]

((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(32*c^3) + (2*
(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/5 + ((16*c^2*d^2 + 5*b^2*e^2 - 2*c*e*(5*b*d
+ 8*a*e) + 6*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(60*c^2) - ((b^2 - 4
*a*c)^2*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(64*c^(7/2))

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Rubi [A]  time = 0.702216, antiderivative size = 195, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ -\frac{e \left (b^2-4 a c\right )^2 (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{64 c^{7/2}}+\frac{e \left (b^2-4 a c\right ) (b+2 c x) \sqrt{a+b x+c x^2} (2 c d-b e)}{32 c^3}+\frac{\left (a+b x+c x^2\right )^{3/2} \left (-2 c e (8 a e+5 b d)+5 b^2 e^2+6 c e x (2 c d-b e)+16 c^2 d^2\right )}{60 c^2}+\frac{2}{5} (d+e x)^2 \left (a+b x+c x^2\right )^{3/2} \]

Antiderivative was successfully verified.

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

[Out]

((b^2 - 4*a*c)*e*(2*c*d - b*e)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(32*c^3) + (2*
(d + e*x)^2*(a + b*x + c*x^2)^(3/2))/5 + ((16*c^2*d^2 + 5*b^2*e^2 - 2*c*e*(5*b*d
+ 8*a*e) + 6*c*e*(2*c*d - b*e)*x)*(a + b*x + c*x^2)^(3/2))/(60*c^2) - ((b^2 - 4
*a*c)^2*e*(2*c*d - b*e)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/
(64*c^(7/2))

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Rubi in Sympy [A]  time = 53.6611, size = 190, normalized size = 0.97 \[ \frac{2 \left (d + e x\right )^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5} + \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (- 16 a c e^{2} + 5 b^{2} e^{2} - 10 b c d e + 16 c^{2} d^{2} - 6 c e x \left (b e - 2 c d\right )\right )}{60 c^{2}} - \frac{e \left (b + 2 c x\right ) \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{32 c^{3}} + \frac{e \left (- 4 a c + b^{2}\right )^{2} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{64 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(d + e*x)**2*(a + b*x + c*x**2)**(3/2)/5 + (a + b*x + c*x**2)**(3/2)*(-16*a*c*
e**2 + 5*b**2*e**2 - 10*b*c*d*e + 16*c**2*d**2 - 6*c*e*x*(b*e - 2*c*d))/(60*c**2
) - e*(b + 2*c*x)*(-4*a*c + b**2)*(b*e - 2*c*d)*sqrt(a + b*x + c*x**2)/(32*c**3)
+ e*(-4*a*c + b**2)**2*(b*e - 2*c*d)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)))/(64*c**(7/2))

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Mathematica [A]  time = 0.291214, size = 231, normalized size = 1.18 \[ \frac{\sqrt{a+x (b+c x)} \left (-128 a^2 c^2 e^2+4 a c \left (25 b^2 e^2-2 b c e (25 d+7 e x)+4 c^2 \left (20 d^2+15 d e x+4 e^2 x^2\right )\right )-15 b^4 e^2+10 b^3 c e (3 d+e x)-4 b^2 c^2 e x (5 d+2 e x)+16 b c^3 x \left (20 d^2+25 d e x+9 e^2 x^2\right )+32 c^4 x^2 \left (10 d^2+15 d e x+6 e^2 x^2\right )\right )}{480 c^3}+\frac{e \left (b^2-4 a c\right )^2 (b e-2 c d) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{64 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[a + x*(b + c*x)]*(-15*b^4*e^2 - 128*a^2*c^2*e^2 + 10*b^3*c*e*(3*d + e*x) -
4*b^2*c^2*e*x*(5*d + 2*e*x) + 32*c^4*x^2*(10*d^2 + 15*d*e*x + 6*e^2*x^2) + 16*b
*c^3*x*(20*d^2 + 25*d*e*x + 9*e^2*x^2) + 4*a*c*(25*b^2*e^2 - 2*b*c*e*(25*d + 7*e
*x) + 4*c^2*(20*d^2 + 15*d*e*x + 4*e^2*x^2))))/(480*c^3) + ((b^2 - 4*a*c)^2*e*(-
2*c*d + b*e)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(64*c^(7/2))

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Maple [B]  time = 0.017, size = 535, normalized size = 2.7 \[ -{\frac{{b}^{3}x{e}^{2}}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{d{b}^{3}e}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{a{b}^{3}{e}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{{b}^{2}xde}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{4\,a{e}^{2}}{15\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{2\,{d}^{2}}{3} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+dex \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}+{\frac{2\,{e}^{2}{x}^{2}}{5} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{bde}{6\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{abde}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}de}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{{e}^{2}{a}^{2}b}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{b{e}^{2}x}{10\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{{b}^{2}{e}^{2}}{12\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{4}{e}^{2}}{32\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{5}{e}^{2}}{64}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{a{b}^{2}de}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-{\frac{axde}{2}\sqrt{c{x}^{2}+bx+a}}+{\frac{a{b}^{2}{e}^{2}}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{d{b}^{4}e}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{abx{e}^{2}}{4\,c}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*x*e^2+1/16*b^3/c^2*(c*x^2+b*x+a)^(1/2)*d*e-1/8
*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*e^2+1/8*b^2/c*(c*x^2+
b*x+a)^(1/2)*x*d*e-4/15*e^2/c*a*(c*x^2+b*x+a)^(3/2)+2/3*d^2*(c*x^2+b*x+a)^(3/2)+
d*e*x*(c*x^2+b*x+a)^(3/2)+2/5*x^2*e^2*(c*x^2+b*x+a)^(3/2)-1/6*b/c*(c*x^2+b*x+a)^
(3/2)*d*e-1/4*a/c*(c*x^2+b*x+a)^(1/2)*b*d*e-1/2*a^2/c^(1/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*d*e+1/4*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*b*e^2-1/10*x*(c*x^2+b*x+a)^(3/2)/c*b*e^2+1/12*b^2/c^2*(c*x^2+b*x+a)^(3/2
)*e^2-1/32*b^4/c^3*(c*x^2+b*x+a)^(1/2)*e^2+1/64*b^5/c^(7/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))*e^2+1/4*b^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)
^(1/2))*a*d*e-1/2*a*(c*x^2+b*x+a)^(1/2)*x*d*e+1/8*a/c^2*(c*x^2+b*x+a)^(1/2)*b^2*
e^2-1/32*b^4/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d*e+1/4*a/c*(c*
x^2+b*x+a)^(1/2)*x*b*e^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)*(2*c*x + b)*(e*x + d)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.337771, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (192 \, c^{4} e^{2} x^{4} + 320 \, a c^{3} d^{2} + 48 \,{\left (10 \, c^{4} d e + 3 \, b c^{3} e^{2}\right )} x^{3} + 10 \,{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} d e -{\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} e^{2} + 8 \,{\left (40 \, c^{4} d^{2} + 50 \, b c^{3} d e -{\left (b^{2} c^{2} - 8 \, a c^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (160 \, b c^{3} d^{2} - 10 \,{\left (b^{2} c^{2} - 12 \, a c^{3}\right )} d e +{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{1920 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (192 \, c^{4} e^{2} x^{4} + 320 \, a c^{3} d^{2} + 48 \,{\left (10 \, c^{4} d e + 3 \, b c^{3} e^{2}\right )} x^{3} + 10 \,{\left (3 \, b^{3} c - 20 \, a b c^{2}\right )} d e -{\left (15 \, b^{4} - 100 \, a b^{2} c + 128 \, a^{2} c^{2}\right )} e^{2} + 8 \,{\left (40 \, c^{4} d^{2} + 50 \, b c^{3} d e -{\left (b^{2} c^{2} - 8 \, a c^{3}\right )} e^{2}\right )} x^{2} + 2 \,{\left (160 \, b c^{3} d^{2} - 10 \,{\left (b^{2} c^{2} - 12 \, a c^{3}\right )} d e +{\left (5 \, b^{3} c - 28 \, a b c^{2}\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 15 \,{\left (2 \,{\left (b^{4} c - 8 \, a b^{2} c^{2} + 16 \, a^{2} c^{3}\right )} d e -{\left (b^{5} - 8 \, a b^{3} c + 16 \, a^{2} b c^{2}\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{960 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/1920*(4*(192*c^4*e^2*x^4 + 320*a*c^3*d^2 + 48*(10*c^4*d*e + 3*b*c^3*e^2)*x^3
+ 10*(3*b^3*c - 20*a*b*c^2)*d*e - (15*b^4 - 100*a*b^2*c + 128*a^2*c^2)*e^2 + 8*(
40*c^4*d^2 + 50*b*c^3*d*e - (b^2*c^2 - 8*a*c^3)*e^2)*x^2 + 2*(160*b*c^3*d^2 - 10
*(b^2*c^2 - 12*a*c^3)*d*e + (5*b^3*c - 28*a*b*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)
*sqrt(c) - 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*a^2*c^3)*d*e - (b^5 - 8*a*b^3*c + 16*
a^2*b*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), 1/960*(2*(192*c^4*e^2*x^4 + 320*a*c^3*d^2
+ 48*(10*c^4*d*e + 3*b*c^3*e^2)*x^3 + 10*(3*b^3*c - 20*a*b*c^2)*d*e - (15*b^4 -
100*a*b^2*c + 128*a^2*c^2)*e^2 + 8*(40*c^4*d^2 + 50*b*c^3*d*e - (b^2*c^2 - 8*a*c
^3)*e^2)*x^2 + 2*(160*b*c^3*d^2 - 10*(b^2*c^2 - 12*a*c^3)*d*e + (5*b^3*c - 28*a*
b*c^2)*e^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 15*(2*(b^4*c - 8*a*b^2*c^2 + 16*
a^2*c^3)*d*e - (b^5 - 8*a*b^3*c + 16*a^2*b*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. \[ \int \left (b + 2 c x\right ) \left (d + e x\right )^{2} \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.286665, size = 416, normalized size = 2.13 \[ \frac{1}{480} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (4 \, c x e^{2} + \frac{10 \, c^{5} d e + 3 \, b c^{4} e^{2}}{c^{4}}\right )} x + \frac{40 \, c^{5} d^{2} + 50 \, b c^{4} d e - b^{2} c^{3} e^{2} + 8 \, a c^{4} e^{2}}{c^{4}}\right )} x + \frac{160 \, b c^{4} d^{2} - 10 \, b^{2} c^{3} d e + 120 \, a c^{4} d e + 5 \, b^{3} c^{2} e^{2} - 28 \, a b c^{3} e^{2}}{c^{4}}\right )} x + \frac{320 \, a c^{4} d^{2} + 30 \, b^{3} c^{2} d e - 200 \, a b c^{3} d e - 15 \, b^{4} c e^{2} + 100 \, a b^{2} c^{2} e^{2} - 128 \, a^{2} c^{3} e^{2}}{c^{4}}\right )} + \frac{{\left (2 \, b^{4} c d e - 16 \, a b^{2} c^{2} d e + 32 \, a^{2} c^{3} d e - b^{5} e^{2} + 8 \, a b^{3} c e^{2} - 16 \, a^{2} b c^{2} e^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{64 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/480*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(4*c*x*e^2 + (10*c^5*d*e + 3*b*c^4*e^2)/c^4
)*x + (40*c^5*d^2 + 50*b*c^4*d*e - b^2*c^3*e^2 + 8*a*c^4*e^2)/c^4)*x + (160*b*c^
4*d^2 - 10*b^2*c^3*d*e + 120*a*c^4*d*e + 5*b^3*c^2*e^2 - 28*a*b*c^3*e^2)/c^4)*x
+ (320*a*c^4*d^2 + 30*b^3*c^2*d*e - 200*a*b*c^3*d*e - 15*b^4*c*e^2 + 100*a*b^2*c
^2*e^2 - 128*a^2*c^3*e^2)/c^4) + 1/64*(2*b^4*c*d*e - 16*a*b^2*c^2*d*e + 32*a^2*c
^3*d*e - b^5*e^2 + 8*a*b^3*c*e^2 - 16*a^2*b*c^2*e^2)*ln(abs(-2*(sqrt(c)*x - sqrt
(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)