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

Optimal. Leaf size=92 \[ \frac{\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \]

[Out]

-((3*b*B - 4*A*c - 2*B*c*x)*Sqrt[a + b*x + c*x^2])/(4*c^2) + ((3*b^2*B - 4*A*b*c
- 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi [A]  time = 0.119927, antiderivative size = 92, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\left (-4 a B c-4 A b c+3 b^2 B\right ) \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{8 c^{5/2}}-\frac{\sqrt{a+b x+c x^2} (-4 A c+3 b B-2 B c x)}{4 c^2} \]

Antiderivative was successfully verified.

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

[Out]

-((3*b*B - 4*A*c - 2*B*c*x)*Sqrt[a + b*x + c*x^2])/(4*c^2) + ((3*b^2*B - 4*A*b*c
- 4*a*B*c)*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(8*c^(5/2))

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Rubi in Sympy [A]  time = 12.0055, size = 90, normalized size = 0.98 \[ \frac{\sqrt{a + b x + c x^{2}} \left (2 A c - \frac{3 B b}{2} + B c x\right )}{2 c^{2}} + \frac{\left (- 4 A b c - 4 B a c + 3 B b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}}} \]

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]

sqrt(a + b*x + c*x**2)*(2*A*c - 3*B*b/2 + B*c*x)/(2*c**2) + (-4*A*b*c - 4*B*a*c
+ 3*B*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*c**(5/2))

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Mathematica [A]  time = 0.228658, size = 88, normalized size = 0.96 \[ \frac{\sqrt{a+x (b+c x)} (4 A c-3 b B+2 B c x)}{4 c^2}-\frac{\left (4 a B c+4 A b c-3 b^2 B\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{8 c^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-3*b*B + 4*A*c + 2*B*c*x)*Sqrt[a + x*(b + c*x)])/(4*c^2) - ((-3*b^2*B + 4*A*b*
c + 4*a*B*c)*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/(8*c^(5/2))

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Maple [A]  time = 0.009, size = 155, normalized size = 1.7 \[{\frac{A}{c}\sqrt{c{x}^{2}+bx+a}}-{\frac{Ab}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}+{\frac{Bx}{2\,c}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,Bb}{4\,{c}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,{b}^{2}B}{8}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{5}{2}}}}-{\frac{Ba}{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}} \]

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]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.357993, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, B c x - 3 \, B b + 4 \, A c\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \log \left (4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{16 \, c^{\frac{5}{2}}}, \frac{2 \,{\left (2 \, B c x - 3 \, B b + 4 \, A c\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} +{\left (3 \, B b^{2} - 4 \,{\left (B a + A b\right )} c\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{8 \, \sqrt{-c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*B*c*x - 3*B*b + 4*A*c)*sqrt(c*x^2 + b*x + a)*sqrt(c) - (3*B*b^2 - 4*
(B*a + A*b)*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^(5/2), 1/8*(2*(2*B*c*x - 3*B*b + 4*A*c)*sqrt(c*x^2
+ b*x + a)*sqrt(-c) + (3*B*b^2 - 4*(B*a + A*b)*c)*arctan(1/2*(2*c*x + b)*sqrt(-c
)/(sqrt(c*x^2 + b*x + a)*c)))/(sqrt(-c)*c^2)]

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

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 = 0.292862, size = 122, normalized size = 1.33 \[ \frac{1}{4} \, \sqrt{c x^{2} + b x + a}{\left (\frac{2 \, B x}{c} - \frac{3 \, B b - 4 \, A c}{c^{2}}\right )} - \frac{{\left (3 \, B b^{2} - 4 \, B a c - 4 \, A b c\right )}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{8 \, c^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt(c*x^2 + b*x + a)*(2*B*x/c - (3*B*b - 4*A*c)/c^2) - 1/8*(3*B*b^2 - 4*B*a
*c - 4*A*b*c)*ln(abs(-2*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*sqrt(c) - b))/c^(5/2
)