3.959 \(\int \frac{A+B x}{x^5 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=231 \[ -\frac{\sqrt{a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{96 a^3 x^2}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}+\frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{9/2}}+\frac{\sqrt{a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{192 a^4 x}-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4} \]

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(4*a*x^4) + ((7*A*b - 8*a*B)*Sqrt[a + b*x + c*x^2])/(
24*a^2*x^3) - ((35*A*b^2 - 40*a*b*B - 36*a*A*c)*Sqrt[a + b*x + c*x^2])/(96*a^3*x
^2) + ((105*A*b^3 - 120*a*b^2*B - 220*a*A*b*c + 128*a^2*B*c)*Sqrt[a + b*x + c*x^
2])/(192*a^4*x) + ((8*a*b*B*(5*b^2 - 12*a*c) - A*(35*b^4 - 120*a*b^2*c + 48*a^2*
c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(9/2))

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Rubi [A]  time = 0.693453, antiderivative size = 231, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 4, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.174 \[ -\frac{\sqrt{a+b x+c x^2} \left (-36 a A c-40 a b B+35 A b^2\right )}{96 a^3 x^2}+\frac{(7 A b-8 a B) \sqrt{a+b x+c x^2}}{24 a^2 x^3}+\frac{\left (8 a b B \left (5 b^2-12 a c\right )-A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{9/2}}+\frac{\sqrt{a+b x+c x^2} \left (128 a^2 B c-220 a A b c-120 a b^2 B+105 A b^3\right )}{192 a^4 x}-\frac{A \sqrt{a+b x+c x^2}}{4 a x^4} \]

Antiderivative was successfully verified.

[In]  Int[(A + B*x)/(x^5*Sqrt[a + b*x + c*x^2]),x]

[Out]

-(A*Sqrt[a + b*x + c*x^2])/(4*a*x^4) + ((7*A*b - 8*a*B)*Sqrt[a + b*x + c*x^2])/(
24*a^2*x^3) - ((35*A*b^2 - 40*a*b*B - 36*a*A*c)*Sqrt[a + b*x + c*x^2])/(96*a^3*x
^2) + ((105*A*b^3 - 120*a*b^2*B - 220*a*A*b*c + 128*a^2*B*c)*Sqrt[a + b*x + c*x^
2])/(192*a^4*x) + ((8*a*b*B*(5*b^2 - 12*a*c) - A*(35*b^4 - 120*a*b^2*c + 48*a^2*
c^2))*ArcTanh[(2*a + b*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(9/2))

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Rubi in Sympy [A]  time = 106.734, size = 230, normalized size = 1. \[ - \frac{A \sqrt{a + b x + c x^{2}}}{4 a x^{4}} + \frac{\left (7 A b - 8 B a\right ) \sqrt{a + b x + c x^{2}}}{24 a^{2} x^{3}} - \frac{\sqrt{a + b x + c x^{2}} \left (- 36 A a c + 35 A b^{2} - 40 B a b\right )}{96 a^{3} x^{2}} + \frac{\sqrt{a + b x + c x^{2}} \left (- 220 A a b c + 105 A b^{3} + 128 B a^{2} c - 120 B a b^{2}\right )}{192 a^{4} x} - \frac{\left (48 A a^{2} c^{2} - 120 A a b^{2} c + 35 A b^{4} + 96 B a^{2} b c - 40 B a b^{3}\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{9}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((B*x+A)/x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

-A*sqrt(a + b*x + c*x**2)/(4*a*x**4) + (7*A*b - 8*B*a)*sqrt(a + b*x + c*x**2)/(2
4*a**2*x**3) - sqrt(a + b*x + c*x**2)*(-36*A*a*c + 35*A*b**2 - 40*B*a*b)/(96*a**
3*x**2) + sqrt(a + b*x + c*x**2)*(-220*A*a*b*c + 105*A*b**3 + 128*B*a**2*c - 120
*B*a*b**2)/(192*a**4*x) - (48*A*a**2*c**2 - 120*A*a*b**2*c + 35*A*b**4 + 96*B*a*
*2*b*c - 40*B*a*b**3)*atanh((2*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(128
*a**(9/2))

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Mathematica [A]  time = 0.332844, size = 219, normalized size = 0.95 \[ \frac{3 \log (x) \left (A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )+8 a b B \left (12 a c-5 b^2\right )\right )-3 \left (A \left (48 a^2 c^2-120 a b^2 c+35 b^4\right )+8 a b B \left (12 a c-5 b^2\right )\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )-\frac{2 \sqrt{a} \sqrt{a+x (b+c x)} \left (16 a^3 (3 A+4 B x)-8 a^2 x (A (7 b+9 c x)+2 B x (5 b+8 c x))+10 a b x^2 (7 A b+22 A c x+12 b B x)-105 A b^3 x^3\right )}{x^4}}{384 a^{9/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(A + B*x)/(x^5*Sqrt[a + b*x + c*x^2]),x]

[Out]

((-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(-105*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) + 10*a
*b*x^2*(7*A*b + 12*b*B*x + 22*A*c*x) - 8*a^2*x*(2*B*x*(5*b + 8*c*x) + A*(7*b + 9
*c*x))))/x^4 + 3*(8*a*b*B*(-5*b^2 + 12*a*c) + A*(35*b^4 - 120*a*b^2*c + 48*a^2*c
^2))*Log[x] - 3*(8*a*b*B*(-5*b^2 + 12*a*c) + A*(35*b^4 - 120*a*b^2*c + 48*a^2*c^
2))*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])/(384*a^(9/2))

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Maple [B]  time = 0.019, size = 417, normalized size = 1.8 \[ -{\frac{A}{4\,a{x}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{7\,Ab}{24\,{a}^{2}{x}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,{b}^{2}A}{96\,{a}^{3}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{35\,A{b}^{3}}{64\,{a}^{4}x}\sqrt{c{x}^{2}+bx+a}}-{\frac{35\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{9}{2}}}}+{\frac{15\,A{b}^{2}c}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{55\,Abc}{48\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{3\,Ac}{8\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}-{\frac{B}{3\,a{x}^{3}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,Bb}{12\,{a}^{2}{x}^{2}}\sqrt{c{x}^{2}+bx+a}}-{\frac{5\,{b}^{2}B}{8\,{a}^{3}x}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{3\,Bbc}{4}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{2\,Bc}{3\,{a}^{2}x}\sqrt{c{x}^{2}+bx+a}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((B*x+A)/x^5/(c*x^2+b*x+a)^(1/2),x)

[Out]

-1/4*A*(c*x^2+b*x+a)^(1/2)/a/x^4+7/24*A*b/a^2/x^3*(c*x^2+b*x+a)^(1/2)-35/96*A*b^
2/a^3/x^2*(c*x^2+b*x+a)^(1/2)+35/64*A*b^3/a^4/x*(c*x^2+b*x+a)^(1/2)-35/128*A*b^4
/a^(9/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+15/16*A*b^2/a^(7/2)*c*ln(
(2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)-55/48*A*b/a^3*c/x*(c*x^2+b*x+a)^(1/2)
+3/8*A/a^2*c/x^2*(c*x^2+b*x+a)^(1/2)-3/8*A/a^(5/2)*c^2*ln((2*a+b*x+2*a^(1/2)*(c*
x^2+b*x+a)^(1/2))/x)-1/3*B/a/x^3*(c*x^2+b*x+a)^(1/2)+5/12*B*b/a^2/x^2*(c*x^2+b*x
+a)^(1/2)-5/8*B*b^2/a^3/x*(c*x^2+b*x+a)^(1/2)+5/16*B*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*B*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*B/a^2*c/x*(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)/(sqrt(c*x^2 + b*x + a)*x^5),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.408436, size = 1, normalized size = 0. \[ \left [-\frac{3 \,{\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \,{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} x^{4} \log \left (\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \,{\left (48 \, A a^{3} +{\left (120 \, B a b^{2} - 105 \, A b^{3} - 4 \,{\left (32 \, B a^{2} - 55 \, A a b\right )} c\right )} x^{3} - 2 \,{\left (40 \, B a^{2} b - 35 \, A a b^{2} + 36 \, A a^{2} c\right )} x^{2} + 8 \,{\left (8 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a}}{768 \, a^{\frac{9}{2}} x^{4}}, \frac{3 \,{\left (40 \, B a b^{3} - 35 \, A b^{4} - 48 \, A a^{2} c^{2} - 24 \,{\left (4 \, B a^{2} b - 5 \, A a b^{2}\right )} c\right )} x^{4} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) - 2 \,{\left (48 \, A a^{3} +{\left (120 \, B a b^{2} - 105 \, A b^{3} - 4 \,{\left (32 \, B a^{2} - 55 \, A a b\right )} c\right )} x^{3} - 2 \,{\left (40 \, B a^{2} b - 35 \, A a b^{2} + 36 \, A a^{2} c\right )} x^{2} + 8 \,{\left (8 \, B a^{3} - 7 \, A a^{2} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a}}{384 \, \sqrt{-a} a^{4} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*x^5),x, algorithm="fricas")

[Out]

[-1/768*(3*(40*B*a*b^3 - 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - 5*A*a*b^2)*c)
*x^4*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*(48*A*a^3 + (120*B*a*b^2 - 105*A*b^3 - 4*(32*B*a^2 -
55*A*a*b)*c)*x^3 - 2*(40*B*a^2*b - 35*A*a*b^2 + 36*A*a^2*c)*x^2 + 8*(8*B*a^3 -
7*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(9/2)*x^4), 1/384*(3*(40*B*a*b^3
- 35*A*b^4 - 48*A*a^2*c^2 - 24*(4*B*a^2*b - 5*A*a*b^2)*c)*x^4*arctan(1/2*(b*x +
2*a)*sqrt(-a)/(sqrt(c*x^2 + b*x + a)*a)) - 2*(48*A*a^3 + (120*B*a*b^2 - 105*A*b
^3 - 4*(32*B*a^2 - 55*A*a*b)*c)*x^3 - 2*(40*B*a^2*b - 35*A*a*b^2 + 36*A*a^2*c)*x
^2 + 8*(8*B*a^3 - 7*A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^4*x^
4)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{A + B x}{x^{5} \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x+A)/x**5/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((A + B*x)/(x**5*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.285149, size = 1, normalized size = 0. \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((B*x + A)/(sqrt(c*x^2 + b*x + a)*x^5),x, algorithm="giac")

[Out]

Done