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

Optimal. Leaf size=172 \[ \frac{\left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{7/2}}-\frac{(2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-8 a b B+5 A b^2\right )}{64 a^3 x^2}+\frac{(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}-\frac{A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4} \]

[Out]

-((5*A*b^2 - 8*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(64*a^3*x^2)
- (A*(a + b*x + c*x^2)^(3/2))/(4*a*x^4) + ((5*A*b - 8*a*B)*(a + b*x + c*x^2)^(3/
2))/(24*a^2*x^3) + ((b^2 - 4*a*c)*(5*A*b^2 - 8*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b
*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(7/2))

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Rubi [A]  time = 0.366619, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ \frac{\left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right ) \tanh ^{-1}\left (\frac{2 a+b x}{2 \sqrt{a} \sqrt{a+b x+c x^2}}\right )}{128 a^{7/2}}-\frac{(2 a+b x) \sqrt{a+b x+c x^2} \left (-4 a A c-8 a b B+5 A b^2\right )}{64 a^3 x^2}+\frac{(5 A b-8 a B) \left (a+b x+c x^2\right )^{3/2}}{24 a^2 x^3}-\frac{A \left (a+b x+c x^2\right )^{3/2}}{4 a x^4} \]

Antiderivative was successfully verified.

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

[Out]

-((5*A*b^2 - 8*a*b*B - 4*a*A*c)*(2*a + b*x)*Sqrt[a + b*x + c*x^2])/(64*a^3*x^2)
- (A*(a + b*x + c*x^2)^(3/2))/(4*a*x^4) + ((5*A*b - 8*a*B)*(a + b*x + c*x^2)^(3/
2))/(24*a^2*x^3) + ((b^2 - 4*a*c)*(5*A*b^2 - 8*a*b*B - 4*a*A*c)*ArcTanh[(2*a + b
*x)/(2*Sqrt[a]*Sqrt[a + b*x + c*x^2])])/(128*a^(7/2))

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Rubi in Sympy [A]  time = 35.7329, size = 165, normalized size = 0.96 \[ - \frac{A \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 a x^{4}} + \frac{\left (5 A b - 8 B a\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 a^{2} x^{3}} - \frac{\left (2 a + b x\right ) \left (- 4 A a c + b \left (5 A b - 8 B a\right )\right ) \sqrt{a + b x + c x^{2}}}{64 a^{3} x^{2}} + \frac{\left (- 4 a c + b^{2}\right ) \left (- 4 A a c + 5 A b^{2} - 8 B a b\right ) \operatorname{atanh}{\left (\frac{2 a + b x}{2 \sqrt{a} \sqrt{a + b x + c x^{2}}} \right )}}{128 a^{\frac{7}{2}}} \]

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**5,x)

[Out]

-A*(a + b*x + c*x**2)**(3/2)/(4*a*x**4) + (5*A*b - 8*B*a)*(a + b*x + c*x**2)**(3
/2)/(24*a**2*x**3) - (2*a + b*x)*(-4*A*a*c + b*(5*A*b - 8*B*a))*sqrt(a + b*x + c
*x**2)/(64*a**3*x**2) + (-4*a*c + b**2)*(-4*A*a*c + 5*A*b**2 - 8*B*a*b)*atanh((2
*a + b*x)/(2*sqrt(a)*sqrt(a + b*x + c*x**2)))/(128*a**(7/2))

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Mathematica [A]  time = 0.477274, size = 193, normalized size = 1.12 \[ \frac{-2 \sqrt{a} \sqrt{a+x (b+c x)} \left (16 a^3 (3 A+4 B x)+8 a^2 x (A (b+3 c x)+2 B x (b+4 c x))-2 a b x^2 (5 A b+26 A c x+12 b B x)+15 A b^3 x^3\right )-3 x^4 \log (x) \left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right )+3 x^4 \left (b^2-4 a c\right ) \left (-4 a A c-8 a b B+5 A b^2\right ) \log \left (2 \sqrt{a} \sqrt{a+x (b+c x)}+2 a+b x\right )}{384 a^{7/2} x^4} \]

Antiderivative was successfully verified.

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

[Out]

(-2*Sqrt[a]*Sqrt[a + x*(b + c*x)]*(15*A*b^3*x^3 + 16*a^3*(3*A + 4*B*x) - 2*a*b*x
^2*(5*A*b + 12*b*B*x + 26*A*c*x) + 8*a^2*x*(A*(b + 3*c*x) + 2*B*x*(b + 4*c*x)))
- 3*(b^2 - 4*a*c)*(5*A*b^2 - 8*a*b*B - 4*a*A*c)*x^4*Log[x] + 3*(b^2 - 4*a*c)*(5*
A*b^2 - 8*a*b*B - 4*a*A*c)*x^4*Log[2*a + b*x + 2*Sqrt[a]*Sqrt[a + x*(b + c*x)]])
/(384*a^(7/2)*x^4)

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Maple [B]  time = 0.019, size = 569, normalized size = 3.3 \[ -{\frac{A}{4\,a{x}^{4}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,Ab}{24\,{a}^{2}{x}^{3}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{b}^{2}A}{32\,{a}^{3}{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,A{b}^{3}}{64\,{a}^{4}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,A{b}^{4}}{64\,{a}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{5\,A{b}^{4}}{128}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{7}{2}}}}-{\frac{5\,A{b}^{3}cx}{64\,{a}^{4}}\sqrt{c{x}^{2}+bx+a}}+{\frac{7\,A{b}^{2}c}{32\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{3\,A{b}^{2}c}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{Ac}{8\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{Abc}{16\,{a}^{3}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Axb{c}^{2}}{16\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{A{c}^{2}}{8\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{A{c}^{2}}{8}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}}-{\frac{B}{3\,a{x}^{3}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{Bb}{4\,{a}^{2}{x}^{2}} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}-{\frac{{b}^{2}B}{8\,{a}^{3}x} \left ( c{x}^{2}+bx+a \right ) ^{{\frac{3}{2}}}}+{\frac{B{b}^{3}}{8\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{B{b}^{3}}{16}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{5}{2}}}}+{\frac{Bx{b}^{2}c}{8\,{a}^{3}}\sqrt{c{x}^{2}+bx+a}}-{\frac{Bbc}{4\,{a}^{2}}\sqrt{c{x}^{2}+bx+a}}+{\frac{Bbc}{4}\ln \left ({\frac{1}{x} \left ( 2\,a+bx+2\,\sqrt{a}\sqrt{c{x}^{2}+bx+a} \right ) } \right ){a}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*A*(c*x^2+b*x+a)^(3/2)/a/x^4+5/24*A*b/a^2/x^3*(c*x^2+b*x+a)^(3/2)-5/32*A*b^2
/a^3/x^2*(c*x^2+b*x+a)^(3/2)+5/64*A*b^3/a^4/x*(c*x^2+b*x+a)^(3/2)-5/64*A*b^4/a^4
*(c*x^2+b*x+a)^(1/2)+5/128*A*b^4/a^(7/2)*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/
2))/x)-5/64*A*b^3/a^4*c*(c*x^2+b*x+a)^(1/2)*x+7/32*A*b^2/a^3*c*(c*x^2+b*x+a)^(1/
2)-3/16*A*b^2/a^(5/2)*c*ln((2*a+b*x+2*a^(1/2)*(c*x^2+b*x+a)^(1/2))/x)+1/8*A/a^2*
c/x^2*(c*x^2+b*x+a)^(3/2)-1/16*A/a^3*c*b/x*(c*x^2+b*x+a)^(3/2)+1/16*A/a^3*c^2*b*
(c*x^2+b*x+a)^(1/2)*x-1/8*A/a^2*c^2*(c*x^2+b*x+a)^(1/2)+1/8*A/a^(3/2)*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)^(3/2)+1/4*B*b/
a^2/x^2*(c*x^2+b*x+a)^(3/2)-1/8*B*b^2/a^3/x*(c*x^2+b*x+a)^(3/2)+1/8*B*b^3/a^3*(c
*x^2+b*x+a)^(1/2)-1/16*B*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*B*b^2/a^3*c*(c*x^2+b*x+a)^(1/2)*x-1/4*B*b/a^2*c*(c*x^2+b*x+a)^(1/2)+1/4*B
*b/a^(3/2)*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^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.335975, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \,{\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{4} \log \left (-\frac{4 \,{\left (a b x + 2 \, a^{2}\right )} \sqrt{c x^{2} + b x + a} +{\left (8 \, a b x +{\left (b^{2} + 4 \, a c\right )} x^{2} + 8 \, a^{2}\right )} \sqrt{a}}{x^{2}}\right ) + 4 \,{\left (48 \, A a^{3} -{\left (24 \, B a b^{2} - 15 \, A b^{3} - 4 \,{\left (16 \, B a^{2} - 13 \, A a b\right )} c\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2} + 12 \, A a^{2} c\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{a}}{768 \, a^{\frac{7}{2}} x^{4}}, -\frac{3 \,{\left (8 \, B a b^{3} - 5 \, A b^{4} - 16 \, A a^{2} c^{2} - 8 \,{\left (4 \, B a^{2} b - 3 \, A a b^{2}\right )} c\right )} x^{4} \arctan \left (\frac{{\left (b x + 2 \, a\right )} \sqrt{-a}}{2 \, \sqrt{c x^{2} + b x + a} a}\right ) + 2 \,{\left (48 \, A a^{3} -{\left (24 \, B a b^{2} - 15 \, A b^{3} - 4 \,{\left (16 \, B a^{2} - 13 \, A a b\right )} c\right )} x^{3} + 2 \,{\left (8 \, B a^{2} b - 5 \, A a b^{2} + 12 \, A a^{2} c\right )} x^{2} + 8 \,{\left (8 \, B a^{3} + A a^{2} b\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-a}}{384 \, \sqrt{-a} a^{3} x^{4}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[-1/768*(3*(8*B*a*b^3 - 5*A*b^4 - 16*A*a^2*c^2 - 8*(4*B*a^2*b - 3*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 - (24*B*a*b^2 - 15*A*b^3 - 4*(16*B*a^2 - 13*
A*a*b)*c)*x^3 + 2*(8*B*a^2*b - 5*A*a*b^2 + 12*A*a^2*c)*x^2 + 8*(8*B*a^3 + A*a^2*
b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(a))/(a^(7/2)*x^4), -1/384*(3*(8*B*a*b^3 - 5*A*b
^4 - 16*A*a^2*c^2 - 8*(4*B*a^2*b - 3*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 - (24*B*a*b^2 - 15*A*b^3 - 4*(16*B
*a^2 - 13*A*a*b)*c)*x^3 + 2*(8*B*a^2*b - 5*A*a*b^2 + 12*A*a^2*c)*x^2 + 8*(8*B*a^
3 + A*a^2*b)*x)*sqrt(c*x^2 + b*x + a)*sqrt(-a))/(sqrt(-a)*a^3*x^4)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.285617, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done