3.918 \(\int \frac{(A+B x) \sqrt{a+b x+c x^2}}{x^4} \, dx\)

Optimal. Leaf size=121 \[ -\frac{\left (b^2-4 a c\right ) (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{5/2}}+\frac{(2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{8 a^2 x^2}-\frac{A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3} \]

[Out]

((A*b - 2*a*B)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(8*a^2*x^2) - (A*(a + b*x + c*
x^2)^(3/2))/(3*a*x^3) - ((A*b - 2*a*B)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt
[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(5/2))

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Rubi [A]  time = 0.191524, antiderivative size = 121, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\left (b^2-4 a c\right ) (A b-2 a B) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{5/2}}+\frac{(2 a+b x) (A b-2 a B) \sqrt{a+b x+c x^2}}{8 a^2 x^2}-\frac{A \left (a+b x+c x^2\right )^{3/2}}{3 a x^3} \]

Antiderivative was successfully verified.

[In]  Int[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^4,x]

[Out]

((A*b - 2*a*B)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(8*a^2*x^2) - (A*(a + b*x + c*
x^2)^(3/2))/(3*a*x^3) - ((A*b - 2*a*B)*(b^2 - 4*a*c)*ArcTanh[(2*a + b*x)/(2*Sqrt
[a]*Sqrt[a + b*x + c*x^2])])/(16*a^(5/2))

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Rubi in Sympy [A]  time = 19.3953, size = 110, normalized size = 0.91 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 a x^{3}} + \frac{\left (2 a + b x\right ) \left (A b - 2 B a\right ) \sqrt{a + b x + c x^{2}}}{8 a^{2} x^{2}} - \frac{\left (A b - 2 B a\right ) \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 a^{\frac{5}{2}}} \]

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**4,x)

[Out]

-A*(a + b*x + c*x**2)**(3/2)/(3*a*x**3) + (2*a + b*x)*(A*b - 2*B*a)*sqrt(a + b*x
+ c*x**2)/(8*a**2*x**2) - (A*b - 2*B*a)*(-4*a*c + b**2)*atanh((2*a + b*x)/(2*sq
rt(a)*sqrt(a + b*x + c*x**2)))/(16*a**(5/2))

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Mathematica [A]  time = 0.322838, size = 146, normalized size = 1.21 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (4 a^2 (2 A+3 B x)+2 a x (A (b+4 c x)+3 b B x)-3 A b^2 x^2\right )+3 x^3 \log (x) \left (b^2-4 a c\right ) (A b-2 a B)-3 x^3 \left (b^2-4 a c\right ) (A b-2 a B) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{48 a^{5/2} x^3} \]

Antiderivative was successfully verified.

[In]  Integrate[((A + B*x)*Sqrt[a + b*x + c*x^2])/x^4,x]

[Out]

(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-3*A*b^2*x^2 + 4*a^2*(2*A + 3*B*x) + 2*a*x*(3
*b*B*x + A*(b + 4*c*x))) + 3*(A*b - 2*a*B)*(b^2 - 4*a*c)*x^3*Log[x] - 3*(A*b - 2
*a*B)*(b^2 - 4*a*c)*x^3*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(48*a^
(5/2)*x^3)

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Maple [B]  time = 0.017, size = 386, normalized size = 3.2 \[ -{\frac{A}{3\,a{x}^{3}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Ab}{4\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}A}{8\,{a}^{3}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{A{b}^{3}}{8\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{A{b}^{2}cx}{8\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{Abc}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{Abc}{4}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{2\,a{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{4\,{a}^{2}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}B}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{{b}^{2}B}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{Bxbc}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{Bc}{2\,a}\sqrt{c{x}^{2}+bx+a}}-{\frac{Bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){\frac{1}{\sqrt{a}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)*(c*x^2+b*x+a)^(1/2)/x^4,x)

[Out]

-1/3*A*(c*x^2+b*x+a)^(3/2)/a/x^3+1/4*A*b/a^2/x^2*(c*x^2+b*x+a)^(3/2)-1/8*A*b^2/a
^3/x*(c*x^2+b*x+a)^(3/2)+1/8*A*b^3/a^3*(c*x^2+b*x+a)^(1/2)-1/16*A*b^3/a^(5/2)*ln
((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/8*A*b^2/a^3*c*(c*x^2+b*x+a)^(1/2)*
x-1/4*A*b/a^2*c*(c*x^2+b*x+a)^(1/2)+1/4*A*b/a^(3/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x
^2+b*x+a)^(1/2))/x)-1/2*B/a/x^2*(c*x^2+b*x+a)^(3/2)+1/4*B*b/a^2/x*(c*x^2+b*x+a)^
(3/2)-1/4*B*b^2/a^2*(c*x^2+b*x+a)^(1/2)+1/8*B*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*B*b/a^2*c*(c*x^2+b*x+a)^(1/2)*x+1/2*B/a*c*(c*x^2+b*x
+a)^(1/2)-1/2*B/a^(1/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)

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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^4,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.343482, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (2 \, B a b^{2} - A b^{3} - 4 \,{\left (2 \, B a^{2} - A a b\right )} c\right )} x^{3} \log \left (-\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} +{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) - 4 \,{\left (8 \, A a^{2} +{\left (6 \, B a b - 3 \, A b^{2} + 8 \, A a c\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a}}{96 \, a^{\frac{5}{2}} x^{3}}, \frac{3 \,{\left (2 \, B a b^{2} - A b^{3} - 4 \,{\left (2 \, B a^{2} - A a b\right )} c\right )} x^{3} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - 2 \,{\left (8 \, A a^{2} +{\left (6 \, B a b - 3 \, A b^{2} + 8 \, A a c\right )} x^{2} + 2 \,{\left (6 \, B a^{2} + A a b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{2} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^4,x, algorithm="fricas")

[Out]

[1/96*(3*(2*B*a*b^2 - A*b^3 - 4*(2*B*a^2 - A*a*b)*c)*x^3*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
*(8*A*a^2 + (6*B*a*b - 3*A*b^2 + 8*A*a*c)*x^2 + 2*(6*B*a^2 + A*a*b)*x)*sqrt(c*x^
2 + b*x + a)*sqrt(a))/(a^(5/2)*x^3), 1/48*(3*(2*B*a*b^2 - A*b^3 - 4*(2*B*a^2 - A
*a*b)*c)*x^3*arctan(1/2*(b*x + 2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 2*(8*A
*a^2 + (6*B*a*b - 3*A*b^2 + 8*A*a*c)*x^2 + 2*(6*B*a^2 + A*a*b)*x)*sqrt(c*x^2 + b
*x + a)*sqrt(-a))/(sqrt(-a)*a^2*x^3)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (A + B x\right ) \sqrt{a + b x + c x^{2}}}{x^{4}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)*(c*x**2+b*x+a)**(1/2)/x**4,x)

[Out]

Integral((A + B*x)*sqrt(a + b*x + c*x**2)/x**4, x)

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GIAC/XCAS [A]  time = 0.281844, size = 707, normalized size = 5.84 \[ -\frac{{\left (2 \, B a b^{2} - A b^{3} - 8 \, B a^{2} c + 4 \, A a b c\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{2}} + \frac{6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a b^{2} - 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A b^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a^{2} c + 12 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A a b c + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} B a^{2} b \sqrt{c} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{4} A a^{2} c^{\frac{3}{2}} + 8 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a b^{3} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt{c} + 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} A a^{2} b^{2} \sqrt{c} - 6 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{3} b^{2} + 3 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{2} b^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{4} c + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{3} b c + 16 \, A a^{4} c^{\frac{3}{2}}}{24 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(sqrt(c*x^2 + b*x + a)*(B*x + A)/x^4,x, algorithm="giac")

[Out]

-1/8*(2*B*a*b^2 - A*b^3 - 8*B*a^2*c + 4*A*a*b*c)*arctan(-(sqrt(c)*x - sqrt(c*x^2
+ b*x + a))/sqrt(-a))/(sqrt(-a)*a^2) + 1/24*(6*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))^5*B*a*b^2 - 3*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*b^3 + 24*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^5*B*a^2*c + 12*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^5*A*a*
b*c + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^4*B*a^2*b*sqrt(c) + 48*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^4*A*a^2*c^(3/2) + 8*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^
3*A*a*b^3 + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))^3*A*a^2*b*c - 48*(sqrt(c)*x -
sqrt(c*x^2 + b*x + a))^2*B*a^3*b*sqrt(c) + 48*(sqrt(c)*x - sqrt(c*x^2 + b*x + a
))^2*A*a^2*b^2*sqrt(c) - 6*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*B*a^3*b^2 + 3*(sq
rt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^2*b^3 - 24*(sqrt(c)*x - sqrt(c*x^2 + b*x +
a))*B*a^4*c + 36*(sqrt(c)*x - sqrt(c*x^2 + b*x + a))*A*a^3*b*c + 16*A*a^4*c^(3/2
))/(((sqrt(c)*x - sqrt(c*x^2 + b*x + a))^2 - a)^3*a^2)