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

Optimal. Leaf size=205 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-6 c x (7 b B-10 A c)-50 A b c+35 b^2 B\right )}{240 c^3}+\frac{\left (b^2-4 a c\right ) \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right )}{128 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]

[Out]

-((7*b^3*B - 10*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x
^2])/(128*c^4) + (B*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + ((35*b^2*B - 50*A*b*c -
32*a*B*c - 6*c*(7*b*B - 10*A*c)*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3) + ((b^2 -
4*a*c)*(7*b^3*B - 10*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(9/2))

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Rubi [A]  time = 0.442857, antiderivative size = 205, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (a+b x+c x^2\right )^{3/2} \left (-32 a B c-6 c x (7 b B-10 A c)-50 A b c+35 b^2 B\right )}{240 c^3}+\frac{\left (b^2-4 a c\right ) \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{256 c^{9/2}}-\frac{(b+2 c x) \sqrt{a+b x+c x^2} \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right )}{128 c^4}+\frac{B x^2 \left (a+b x+c x^2\right )^{3/2}}{5 c} \]

Antiderivative was successfully verified.

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

[Out]

-((7*b^3*B - 10*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*(b + 2*c*x)*Sqrt[a + b*x + c*x
^2])/(128*c^4) + (B*x^2*(a + b*x + c*x^2)^(3/2))/(5*c) + ((35*b^2*B - 50*A*b*c -
32*a*B*c - 6*c*(7*b*B - 10*A*c)*x)*(a + b*x + c*x^2)^(3/2))/(240*c^3) + ((b^2 -
4*a*c)*(7*b^3*B - 10*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*ArcTanh[(b + 2*c*x)/(2*S
qrt[c]*Sqrt[a + b*x + c*x^2])])/(256*c^(9/2))

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Rubi in Sympy [A]  time = 33.5995, size = 212, normalized size = 1.03 \[ \frac{B x^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{5 c} - \frac{\left (a + b x + c x^{2}\right )^{\frac{3}{2}} \left (8 B a c + \frac{5 b \left (10 A c - 7 B b\right )}{4} - \frac{3 c x \left (10 A c - 7 B b\right )}{2}\right )}{60 c^{3}} - \frac{\left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (8 A a c^{2} - 10 A b^{2} c - 12 B a b c + 7 B b^{3}\right )}{128 c^{4}} + \frac{\left (- 4 a c + b^{2}\right ) \left (8 A a c^{2} - 10 A b^{2} c - 12 B a b c + 7 B b^{3}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{256 c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x**2*(a + b*x + c*x**2)**(3/2)/(5*c) - (a + b*x + c*x**2)**(3/2)*(8*B*a*c + 5*
b*(10*A*c - 7*B*b)/4 - 3*c*x*(10*A*c - 7*B*b)/2)/(60*c**3) - (b + 2*c*x)*sqrt(a
+ b*x + c*x**2)*(8*A*a*c**2 - 10*A*b**2*c - 12*B*a*b*c + 7*B*b**3)/(128*c**4) +
(-4*a*c + b**2)*(8*A*a*c**2 - 10*A*b**2*c - 12*B*a*b*c + 7*B*b**3)*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 = 0.292132, size = 208, normalized size = 1.01 \[ \frac{15 \left (b^2-4 a c\right ) \left (8 a A c^2-12 a b B c-10 A b^2 c+7 b^3 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )-2 \sqrt{c} \sqrt{a+x (b+c x)} \left (-16 c^2 \left (-16 a^2 B+a c x (15 A+8 B x)+6 c^2 x^3 (5 A+4 B x)\right )+4 b^2 c (c x (25 A+14 B x)-115 a B)+8 b c^2 \left (a (65 A+29 B x)-2 c x^2 (5 A+3 B x)\right )-10 b^3 c (15 A+7 B x)+105 b^4 B\right )}{3840 c^{9/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[c]*Sqrt[a + x*(b + c*x)]*(105*b^4*B - 10*b^3*c*(15*A + 7*B*x) - 16*c^2*
(-16*a^2*B + 6*c^2*x^3*(5*A + 4*B*x) + a*c*x*(15*A + 8*B*x)) + 4*b^2*c*(-115*a*B
+ c*x*(25*A + 14*B*x)) + 8*b*c^2*(-2*c*x^2*(5*A + 3*B*x) + a*(65*A + 29*B*x)))
+ 15*(b^2 - 4*a*c)*(7*b^3*B - 10*A*b^2*c - 12*a*b*B*c + 8*a*A*c^2)*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 = 0.013, size = 497, normalized size = 2.4 \[{\frac{Ax}{4\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,Ab}{24\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ax{b}^{2}}{32\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,A{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,a{b}^{2}A}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{5\,A{b}^{4}}{128}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}-{\frac{aAx}{8\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{abA}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{A{a}^{2}}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{B{x}^{2}}{5\,c} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,xBb}{40\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{7\,{b}^{2}B}{48\,{c}^{3}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{7\,Bx{b}^{3}}{64\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{7\,{b}^{4}B}{128\,{c}^{4}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,Ba{b}^{3}}{32}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{7}{2}}}}+{\frac{7\,B{b}^{5}}{256}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{9}{2}}}}+{\frac{3\,Bxab}{16\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,a{b}^{2}B}{32\,{c}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,B{a}^{2}b}{16}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{2\,Ba}{15\,{c}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*A*x*(c*x^2+b*x+a)^(3/2)/c-5/24*A*b/c^2*(c*x^2+b*x+a)^(3/2)+5/32*A*b^2/c^2*(c
*x^2+b*x+a)^(1/2)*x+5/64*A*b^3/c^3*(c*x^2+b*x+a)^(1/2)+3/16*A*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*A*b^4/c^(7/2)*ln((1/2*b+c*x)/c^(1/
2)+(c*x^2+b*x+a)^(1/2))-1/8*A*a/c*(c*x^2+b*x+a)^(1/2)*x-1/16*A*a/c^2*(c*x^2+b*x+
a)^(1/2)*b-1/8*A*a^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+1/5*B*x
^2*(c*x^2+b*x+a)^(3/2)/c-7/40*B*b/c^2*x*(c*x^2+b*x+a)^(3/2)+7/48*B*b^2/c^3*(c*x^
2+b*x+a)^(3/2)-7/64*B*b^3/c^3*(c*x^2+b*x+a)^(1/2)*x-7/128*B*b^4/c^4*(c*x^2+b*x+a
)^(1/2)-5/32*B*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+7/256*B
*b^5/c^(9/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/16*B*b/c^2*a*(c*x^2+b
*x+a)^(1/2)*x+3/32*B*b^2/c^3*a*(c*x^2+b*x+a)^(1/2)+3/16*B*b/c^(5/2)*a^2*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/15*B*a/c^2*(c*x^2+b*x+a)^(3/2)

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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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.350985, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, B c^{4} x^{4} - 105 \, B b^{4} + 48 \,{\left (B b c^{3} + 10 \, A c^{4}\right )} x^{3} - 8 \,{\left (32 \, B a^{2} + 65 \, A a b\right )} c^{2} - 8 \,{\left (7 \, B b^{2} c^{2} - 2 \,{\left (8 \, B a + 5 \, A b\right )} c^{3}\right )} x^{2} + 10 \,{\left (46 \, B a b^{2} + 15 \, A b^{3}\right )} c + 2 \,{\left (35 \, B b^{3} c + 120 \, A a c^{3} - 2 \,{\left (58 \, B a b + 25 \, A b^{2}\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} - 15 \,{\left (7 \, B b^{5} - 32 \, A a^{2} c^{3} + 48 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 10 \,{\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{7680 \, c^{\frac{9}{2}}}, \frac{2 \,{\left (384 \, B c^{4} x^{4} - 105 \, B b^{4} + 48 \,{\left (B b c^{3} + 10 \, A c^{4}\right )} x^{3} - 8 \,{\left (32 \, B a^{2} + 65 \, A a b\right )} c^{2} - 8 \,{\left (7 \, B b^{2} c^{2} - 2 \,{\left (8 \, B a + 5 \, A b\right )} c^{3}\right )} x^{2} + 10 \,{\left (46 \, B a b^{2} + 15 \, A b^{3}\right )} c + 2 \,{\left (35 \, B b^{3} c + 120 \, A a c^{3} - 2 \,{\left (58 \, B a b + 25 \, A b^{2}\right )} c^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} + 15 \,{\left (7 \, B b^{5} - 32 \, A a^{2} c^{3} + 48 \,{\left (B a^{2} b + A a b^{2}\right )} c^{2} - 10 \,{\left (4 \, B a b^{3} + A b^{4}\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{3840 \, \sqrt{-c} c^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/7680*(4*(384*B*c^4*x^4 - 105*B*b^4 + 48*(B*b*c^3 + 10*A*c^4)*x^3 - 8*(32*B*a^
2 + 65*A*a*b)*c^2 - 8*(7*B*b^2*c^2 - 2*(8*B*a + 5*A*b)*c^3)*x^2 + 10*(46*B*a*b^2
+ 15*A*b^3)*c + 2*(35*B*b^3*c + 120*A*a*c^3 - 2*(58*B*a*b + 25*A*b^2)*c^2)*x)*s
qrt(c*x^2 + b*x + a)*sqrt(c) - 15*(7*B*b^5 - 32*A*a^2*c^3 + 48*(B*a^2*b + A*a*b^
2)*c^2 - 10*(4*B*a*b^3 + A*b^4)*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^(9/2), 1/3840*(2*(384*B*c^4*x^4
- 105*B*b^4 + 48*(B*b*c^3 + 10*A*c^4)*x^3 - 8*(32*B*a^2 + 65*A*a*b)*c^2 - 8*(7*
B*b^2*c^2 - 2*(8*B*a + 5*A*b)*c^3)*x^2 + 10*(46*B*a*b^2 + 15*A*b^3)*c + 2*(35*B*
b^3*c + 120*A*a*c^3 - 2*(58*B*a*b + 25*A*b^2)*c^2)*x)*sqrt(c*x^2 + b*x + a)*sqrt
(-c) + 15*(7*B*b^5 - 32*A*a^2*c^3 + 48*(B*a^2*b + A*a*b^2)*c^2 - 10*(4*B*a*b^3 +
A*b^4)*c)*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. \[ \int x^{2} \left (A + B x\right ) \sqrt{a + b x + c x^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.287359, size = 331, normalized size = 1.61 \[ \frac{1}{1920} \, \sqrt{c x^{2} + b x + a}{\left (2 \,{\left (4 \,{\left (6 \,{\left (8 \, B x + \frac{B b c^{3} + 10 \, A c^{4}}{c^{4}}\right )} x - \frac{7 \, B b^{2} c^{2} - 16 \, B a c^{3} - 10 \, A b c^{3}}{c^{4}}\right )} x + \frac{35 \, B b^{3} c - 116 \, B a b c^{2} - 50 \, A b^{2} c^{2} + 120 \, A a c^{3}}{c^{4}}\right )} x - \frac{105 \, B b^{4} - 460 \, B a b^{2} c - 150 \, A b^{3} c + 256 \, B a^{2} c^{2} + 520 \, A a b c^{2}}{c^{4}}\right )} - \frac{{\left (7 \, B b^{5} - 40 \, B a b^{3} c - 10 \, A b^{4} c + 48 \, B a^{2} b c^{2} + 48 \, A a b^{2} c^{2} - 32 \, A a^{2} c^{3}\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{256 \, c^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/1920*sqrt(c*x^2 + b*x + a)*(2*(4*(6*(8*B*x + (B*b*c^3 + 10*A*c^4)/c^4)*x - (7*
B*b^2*c^2 - 16*B*a*c^3 - 10*A*b*c^3)/c^4)*x + (35*B*b^3*c - 116*B*a*b*c^2 - 50*A
*b^2*c^2 + 120*A*a*c^3)/c^4)*x - (105*B*b^4 - 460*B*a*b^2*c - 150*A*b^3*c + 256*
B*a^2*c^2 + 520*A*a*b*c^2)/c^4) - 1/256*(7*B*b^5 - 40*B*a*b^3*c - 10*A*b^4*c + 4
8*B*a^2*b*c^2 + 48*A*a*b^2*c^2 - 32*A*a^2*c^3)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))*sqrt(c) - b))/c^(9/2)