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

Optimal. Leaf size=167 \[ -\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{7/2}}-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3} \]

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(3*a*x^3) + ((5*A*b - 6*a*B)*Sqrt[a + b*x + c*x^2])/(
12*a^2*x^2) - ((15*A*b^2 - 18*a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(24*a^3*x
) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[
a]*Sqrt[a + b*x + c*x^2])])/(16*a^(7/2))

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Rubi [A]  time = 0.437258, antiderivative size = 167, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\sqrt{a+b x+c x^2} \left (-16 a A c-18 a b B+15 A b^2\right )}{24 a^3 x}+\frac{(5 A b-6 a B) \sqrt{a+b x+c x^2}}{12 a^2 x^2}+\frac{\left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{16 a^{7/2}}-\frac{A \sqrt{a+b x+c x^2}}{3 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(3*a*x^3) + ((5*A*b - 6*a*B)*Sqrt[a + b*x + c*x^2])/(
12*a^2*x^2) - ((15*A*b^2 - 18*a*b*B - 16*a*A*c)*Sqrt[a + b*x + c*x^2])/(24*a^3*x
) + ((5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*ArcTanh[(2*a + b*x)/(2*Sqrt[
a]*Sqrt[a + b*x + c*x^2])])/(16*a^(7/2))

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Rubi in Sympy [A]  time = 53.9901, size = 162, normalized size = 0.97 \[ - \frac{A \sqrt{a + b x + c x^{2}}}{3 a x^{3}} + \frac{\left (5 A b - 6 B a\right ) \sqrt{a + b x + c x^{2}}}{12 a^{2} x^{2}} - \frac{\sqrt{a + b x + c x^{2}} \left (- 16 A a c + 15 A b^{2} - 18 B a b\right )}{24 a^{3} x} + \frac{\left (- 12 A a b c + 5 A b^{3} + 8 B a^{2} c - 6 B a b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{16 a^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-A*sqrt(a + b*x + c*x**2)/(3*a*x**3) + (5*A*b - 6*B*a)*sqrt(a + b*x + c*x**2)/(1
2*a**2*x**2) - sqrt(a + b*x + c*x**2)*(-16*A*a*c + 15*A*b**2 - 18*B*a*b)/(24*a**
3*x) + (-12*A*a*b*c + 5*A*b**3 + 8*B*a**2*c - 6*B*a*b**2)*atanh((2*a + b*x)/(2*s
qrt(a)*sqrt(a + b*x + c*x**2)))/(16*a**(7/2))

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Mathematica [A]  time = 0.479402, size = 163, normalized size = 0.98 \[ \frac{-\frac{2 \sqrt{a} \sqrt{a+x (b+c x)} \left (4 a^2 (2 A+3 B x)-2 a x (5 A b+8 A c x+9 b B x)+15 A b^2 x^2\right )}{x^3}-3 \log (x) \left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right )+3 \left (8 a^2 B c-12 a A b c-6 a b^2 B+5 A b^3\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{48 a^{7/2}} \]

Antiderivative was successfully verified.

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

[Out]

((-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(15*A*b^2*x^2 + 4*a^2*(2*A + 3*B*x) - 2*a*x*(
5*A*b + 9*b*B*x + 8*A*c*x)))/x^3 - 3*(5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B
*c)*Log[x] + 3*(5*A*b^3 - 6*a*b^2*B - 12*a*A*b*c + 8*a^2*B*c)*Log[2*a + b*x + 2*
Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(48*a^(7/2))

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Maple [A]  time = 0.018, size = 283, normalized size = 1.7 \[ -{\frac{A}{3\,a{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,Ab}{12\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{2}A}{8\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,A{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,Abc}{4}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{2\,Ac}{3\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{B}{2\,a{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Bb}{4\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,{b}^{2}B}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{Bc}{2}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*A*(c*x^2+b*x+a)^(1/2)/a/x^3+5/12*A*b/a^2/x^2*(c*x^2+b*x+a)^(1/2)-5/8*A*b^2/
a^3/x*(c*x^2+b*x+a)^(1/2)+5/16*A*b^3/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)
^(1/2))/x)-3/4*A*b/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+2/3*A
/a^2*c/x*(c*x^2+b*x+a)^(1/2)-1/2*B/a/x^2*(c*x^2+b*x+a)^(1/2)+3/4*B*b/a^2/x*(c*x^
2+b*x+a)^(1/2)-3/8*B*b^2/a^(5/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1
/2*B*c/a^(3/2)*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((B*x + A)/(sqrt(c*x^2 + b*x + a)*x^4),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.37955, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} x^{3} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) - 4 \,{\left (8 \, A a^{2} -{\left (18 \, B a b - 15 \, A b^{2} + 16 \, A a c\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a}}{96 \, a^{\frac{7}{2}} x^{3}}, -\frac{3 \,{\left (6 \, B a b^{2} - 5 \, A b^{3} - 4 \,{\left (2 \, B a^{2} - 3 \, A a b\right )} c\right )} x^{3} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) + 2 \,{\left (8 \, A a^{2} -{\left (18 \, B a b - 15 \, A b^{2} + 16 \, A a c\right )} x^{2} + 2 \,{\left (6 \, B a^{2} - 5 \, A a b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a}}{48 \, \sqrt{-a} a^{3} x^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/96*(3*(6*B*a*b^2 - 5*A*b^3 - 4*(2*B*a^2 - 3*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 - (18*B*a*b - 15*A*b^2 + 16*A*a*c)*x^2 + 2*(6*B*a^2 - 5*A*a*b)*x)*s
qrt(c*x^2 + b*x + a)*sqrt(a))/(a^(7/2)*x^3), -1/48*(3*(6*B*a*b^2 - 5*A*b^3 - 4*(
2*B*a^2 - 3*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 - (18*B*a*b - 15*A*b^2 + 16*A*a*c)*x^2 + 2*(6*B*a^2 - 5*A*a*b)
*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^3*x^3)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.283401, size = 690, normalized size = 4.13 \[ \frac{{\left (6 \, B a b^{2} - 5 \, A b^{3} - 8 \, B a^{2} c + 12 \, A a b c\right )} \arctan \left (-\frac{\sqrt{c} x - \sqrt{c x^{2} + b x + a}}{\sqrt{-a}}\right )}{8 \, \sqrt{-a} a^{3}} - \frac{18 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a b^{2} - 15 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A b^{3} - 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} B a^{2} c + 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{5} A a b c - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} B a^{2} b^{2} + 40 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a b^{3} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{3} A a^{2} b c - 48 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} B a^{3} b \sqrt{c} - 96 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} A a^{3} c^{\frac{3}{2}} + 30 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{3} b^{2} - 33 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{2} b^{3} + 24 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} B a^{4} c - 36 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} A a^{3} b c + 48 \, B a^{4} b \sqrt{c} - 48 \, A a^{3} b^{2} \sqrt{c} + 32 \, A a^{4} c^{\frac{3}{2}}}{24 \,{\left ({\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )}^{2} - a\right )}^{3} a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(6*B*a*b^2 - 5*A*b^3 - 8*B*a^2*c + 12*A*a*b*c)*arctan(-(sqrt(c)*x - sqrt(c*x
^2 + b*x + a))/sqrt(-a))/(sqrt(-a)*a^3) - 1/24*(18*(sqrt(c)*x - sqrt(c*x^2 + b*x
+ a))^5*B*a*b^2 - 15*(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 + 36*(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))^3*B*a^2*b^2 + 40*(sqrt(c)*x - s
qrt(c*x^2 + b*x + a))^3*A*a*b^3 - 96*(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) - 96*(sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2*A*a^3*c^(3/2) + 30*(sqrt(c)*x - sqrt(c*x^2 + b*x + a)
)*B*a^3*b^2 - 33*(sqrt(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 + 48*B*a^4*b*sqrt(c) - 48*A*a^3*b^2*sqrt(c) + 32*A*a^4*c^(3/2))/(((sqrt(c)*x
- sqrt(c*x^2 + b*x + a))^2 - a)^3*a^3)