3.917 \(\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{x^3} \, dx\)

Optimal. Leaf size=133 \[ \frac{\left (-4 a A c-4 a b B+A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac{\sqrt{a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \]

[Out]

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

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Rubi [A]  time = 0.296011, antiderivative size = 133, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\left (4 a b B-A \left (b^2-4 a c\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{8 a^{3/2}}-\frac{\sqrt{a+b x+c x^2} (x (4 a B+A b)+2 a A)}{4 a x^2}+B \sqrt{c} \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^3,x]

[Out]

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

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Rubi in Sympy [A]  time = 43.9966, size = 124, normalized size = 0.93 \[ B \sqrt{c} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )} - \frac{\left (A a + x \left (\frac{A b}{2} + 2 B a\right )\right ) \sqrt{a + b x + c x^{2}}}{2 a x^{2}} + \frac{\left (- 4 A a c + A b^{2} - 4 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{8 a^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**3,x)

[Out]

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

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Mathematica [A]  time = 0.354291, size = 159, normalized size = 1.2 \[ \frac{x^2 \log (x) \left (4 a A c+4 a b B-A b^2\right )+x^2 \left (A \left (b^2-4 a c\right )-4 a b B\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-2 \sqrt{a} \left (\sqrt{a+x (b+c x)} (2 a (A+2 B x)+A b x)-4 a B \sqrt{c} x^2 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )\right )}{8 a^{3/2} x^2} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^3,x]

[Out]

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

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Maple [B]  time = 0.015, size = 304, normalized size = 2.3 \[ -{\frac{A}{2\,a{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{4\,{a}^{2}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}A}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}A}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{Abcx}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{Ac}{2\,a}\sqrt{c{x}^{2}+bx+a}}-{\frac{Ac}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}-{\frac{B}{ax} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{a}\sqrt{c{x}^{2}+bx+a}}-{\frac{Bb}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}}+{\frac{Bcx}{a}\sqrt{c{x}^{2}+bx+a}}+B\sqrt{c}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^3,x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.663825, size = 1, normalized size = 0.01 \[ \left [\frac{8 \, B a^{\frac{3}{2}} \sqrt{c} x^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) +{\left (4 \, B a b - A b^{2} + 4 \, A a c\right )} x^{2} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{a}}{16 \, a^{\frac{3}{2}} x^{2}}, \frac{16 \, B a^{\frac{3}{2}} \sqrt{-c} x^{2} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) +{\left (4 \, B a b - A b^{2} + 4 \, A a c\right )} x^{2} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{a}}{16 \, a^{\frac{3}{2}} x^{2}}, \frac{4 \, B \sqrt{-a} a \sqrt{c} x^{2} \log \left (-8 \, c^{2} x^{2} - 8 \, b c x - b^{2} - 4 \, \sqrt{c x^{2} + b x + a}{\left (2 \, c x + b\right )} \sqrt{c} - 4 \, a c\right ) -{\left (4 \, B a b - A b^{2} + 4 \, A a c\right )} x^{2} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - 2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{-a}}{8 \, \sqrt{-a} a x^{2}}, \frac{8 \, B \sqrt{-a} a \sqrt{-c} x^{2} \arctan \left (\frac{2 \, c x + b}{2 \, \sqrt{c x^{2} + b x + a} \sqrt{-c}}\right ) -{\left (4 \, B a b - A b^{2} + 4 \, A a c\right )} x^{2} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - 2 \, \sqrt{c x^{2} + b x + a}{\left (2 \, A a +{\left (4 \, B a + A b\right )} x\right )} \sqrt{-a}}{8 \, \sqrt{-a} a x^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^3,x, algorithm="fricas")

[Out]

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

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{a + b x + c x^{2}}}{x^{3}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**3,x)

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**3, x)

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GIAC/XCAS [A]  time = 0.617222, size = 4, normalized size = 0.03 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^3,x, algorithm="giac")

[Out]

sage0*x