3.913 \(\int x (A+B x) \sqrt{a+b x+c x^2} \, dx\)

Optimal. Leaf size=144 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]

[Out]

((5*b^2*B - 8*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^3) - ((5
*b*B - 8*A*c - 6*B*c*x)*(a + b*x + c*x^2)^(3/2))/(24*c^2) - ((b^2 - 4*a*c)*(5*b^
2*B - 8*A*b*c - 4*a*B*c)*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 = 0.171423, antiderivative size = 144, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.19 \[ -\frac{\left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{128 c^{7/2}}+\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (-4 a B c-8 A b c+5 b^2 B\right )}{64 c^3}-\frac{\left (a+b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]

Antiderivative was successfully verified.

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

[Out]

((5*b^2*B - 8*A*b*c - 4*a*B*c)*(b + 2*c*x)*Sqrt[a + b*x + c*x^2])/(64*c^3) - ((5
*b*B - 8*A*c - 6*B*c*x)*(a + b*x + c*x^2)^(3/2))/(24*c^2) - ((b^2 - 4*a*c)*(5*b^
2*B - 8*A*b*c - 4*a*B*c)*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 = 15.7786, size = 146, normalized size = 1.01 \[ \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (4 A c - \frac{5 B b}{2} + 3 B c x\right )}{12 c^{2}} + \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (- 8 A b c - 4 B a c + 5 B b^{2}\right )}{64 c^{3}} - \frac{\left (- 4 a c + b^{2}\right ) \left (- 8 A b c - 4 B a c + 5 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

(a + b*x + c*x**2)**(3/2)*(4*A*c - 5*B*b/2 + 3*B*c*x)/(12*c**2) + (b + 2*c*x)*sq
rt(a + b*x + c*x**2)*(-8*A*b*c - 4*B*a*c + 5*B*b**2)/(64*c**3) - (-4*a*c + b**2)
*(-8*A*b*c - 4*B*a*c + 5*B*b**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 = 0.219877, size = 154, normalized size = 1.07 \[ \frac{2 \sqrt{c} \sqrt{a+x (b+c x)} \left (4 b c (2 c x (2 A+B x)-13 a B)+8 c^2 \left (8 a A+3 a B x+8 A c x^2+6 B c x^3\right )-2 b^2 c (12 A+5 B x)+15 b^3 B\right )-3 \left (b^2-4 a c\right ) \left (-4 a B c-8 A b c+5 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{384 c^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(15*b^3*B - 2*b^2*c*(12*A + 5*B*x) + 8*c^2*(8*a
*A + 3*a*B*x + 8*A*c*x^2 + 6*B*c*x^3) + 4*b*c*(-13*a*B + 2*c*x*(2*A + B*x))) - 3
*(b^2 - 4*a*c)*(5*b^2*B - 8*A*b*c - 4*a*B*c)*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 = 0.011, size = 352, normalized size = 2.4 \[{\frac{A}{3\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{Abx}{4\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{{b}^{2}A}{8\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{abA}{4}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{A{b}^{3}}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}+{\frac{Bx}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Bb}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{b}^{2}Bx}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,B{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,a{b}^{2}B}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,{b}^{4}B}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{aBx}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{abB}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{{a}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/3*A*(c*x^2+b*x+a)^(3/2)/c-1/4*A*b/c*(c*x^2+b*x+a)^(1/2)*x-1/8*A*b^2/c^2*(c*x^2
+b*x+a)^(1/2)-1/4*A*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/16
*A*b^3/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/4*B*x*(c*x^2+b*x+a)
^(3/2)/c-5/24*B*b/c^2*(c*x^2+b*x+a)^(3/2)+5/32*B*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x+5
/64*B*b^3/c^3*(c*x^2+b*x+a)^(1/2)+3/16*B*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*B*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2
))-1/8*B*a/c*(c*x^2+b*x+a)^(1/2)*x-1/16*B*a/c^2*(c*x^2+b*x+a)^(1/2)*b-1/8*B*a^2/
c^(3/2)*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)*(B*x + A)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.318547, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 64 \, A a c^{2} + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c - 2 \,{\left (5 \, B b^{2} c - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} + 3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{768 \, c^{\frac{7}{2}}}, \frac{2 \,{\left (48 \, B c^{3} x^{3} + 15 \, B b^{3} + 64 \, A a c^{2} + 8 \,{\left (B b c^{2} + 8 \, A c^{3}\right )} x^{2} - 4 \,{\left (13 \, B a b + 6 \, A b^{2}\right )} c - 2 \,{\left (5 \, B b^{2} c - 4 \,{\left (3 \, B a + 2 \, A b\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} - 3 \,{\left (5 \, B b^{4} + 16 \,{\left (B a^{2} + 2 \, A a b\right )} c^{2} - 8 \,{\left (3 \, B a b^{2} + A b^{3}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{384 \, \sqrt{-c} c^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/768*(4*(48*B*c^3*x^3 + 15*B*b^3 + 64*A*a*c^2 + 8*(B*b*c^2 + 8*A*c^3)*x^2 - 4*
(13*B*a*b + 6*A*b^2)*c - 2*(5*B*b^2*c - 4*(3*B*a + 2*A*b)*c^2)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(c) + 3*(5*B*b^4 + 16*(B*a^2 + 2*A*a*b)*c^2 - 8*(3*B*a*b^2 + A*b^3)*
c)*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*B*c^3*x^3 + 15*B*b^3 + 64*A*a*c^2 + 8*(B*b*
c^2 + 8*A*c^3)*x^2 - 4*(13*B*a*b + 6*A*b^2)*c - 2*(5*B*b^2*c - 4*(3*B*a + 2*A*b)
*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-c) - 3*(5*B*b^4 + 16*(B*a^2 + 2*A*a*b)*c^2
- 8*(3*B*a*b^2 + A*b^3)*c)*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. \[ \int x \left (A + B x\right ) \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(A + B*x)*sqrt(a + b*x + c*x**2), x)

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GIAC/XCAS [A]  time = 0.281998, size = 240, normalized size = 1.67 \[ \frac{1}{192} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \, B x + \frac{B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac{5 \, B b^{2} c - 12 \, B a c^{2} - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac{15 \, B b^{3} - 52 \, B a b c - 24 \, A b^{2} c + 64 \, A a c^{2}}{c^{3}}\right )} + \frac{{\left (5 \, B b^{4} - 24 \, B a b^{2} c - 8 \, A b^{3} c + 16 \, B a^{2} c^{2} + 32 \, A a b c^{2}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{128 \, c^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/192*sqrt(c*x^2 + b*x + a)*(2*(4*(6*B*x + (B*b*c^2 + 8*A*c^3)/c^3)*x - (5*B*b^2
*c - 12*B*a*c^2 - 8*A*b*c^2)/c^3)*x + (15*B*b^3 - 52*B*a*b*c - 24*A*b^2*c + 64*A
*a*c^2)/c^3) + 1/128*(5*B*b^4 - 24*B*a*b^2*c - 8*A*b^3*c + 16*B*a^2*c^2 + 32*A*a
*b*c^2)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(7/2)