3.1012 \(\int \frac{A+B x}{\sqrt{x} \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=180 \[ \frac{\sqrt{2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

[Out]

(Sqrt[2]*(B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(
B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi [A]  time = 0.554577, antiderivative size = 180, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.13 \[ \frac{\sqrt{2} \left (B-\frac{b B-2 A c}{\sqrt{b^2-4 a c}}\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{c} \sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\sqrt{2} \left (\frac{b B-2 A c}{\sqrt{b^2-4 a c}}+B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{\sqrt{b^2-4 a c}+b}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[2]*(B - (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/
Sqrt[b - Sqrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b - Sqrt[b^2 - 4*a*c]]) + (Sqrt[2]*(
B + (b*B - 2*A*c)/Sqrt[b^2 - 4*a*c])*ArcTan[(Sqrt[2]*Sqrt[c]*Sqrt[x])/Sqrt[b + S
qrt[b^2 - 4*a*c]]])/(Sqrt[c]*Sqrt[b + Sqrt[b^2 - 4*a*c]])

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Rubi in Sympy [A]  time = 53.8751, size = 189, normalized size = 1.05 \[ - \frac{\sqrt{2} \left (2 A c - B b - B \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b + \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{b + \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} + \frac{\sqrt{2} \left (2 A c - B b + B \sqrt{- 4 a c + b^{2}}\right ) \operatorname{atan}{\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b - \sqrt{- 4 a c + b^{2}}}} \right )}}{\sqrt{c} \sqrt{b - \sqrt{- 4 a c + b^{2}}} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-sqrt(2)*(2*A*c - B*b - B*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*sqrt(x)/sqrt
(b + sqrt(-4*a*c + b**2)))/(sqrt(c)*sqrt(b + sqrt(-4*a*c + b**2))*sqrt(-4*a*c +
b**2)) + sqrt(2)*(2*A*c - B*b + B*sqrt(-4*a*c + b**2))*atan(sqrt(2)*sqrt(c)*sqrt
(x)/sqrt(b - sqrt(-4*a*c + b**2)))/(sqrt(c)*sqrt(b - sqrt(-4*a*c + b**2))*sqrt(-
4*a*c + b**2))

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Mathematica [A]  time = 0.18222, size = 181, normalized size = 1.01 \[ \frac{\sqrt{2} \left (\frac{\left (B \sqrt{b^2-4 a c}+2 A c-b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{b-\sqrt{b^2-4 a c}}}\right )}{\sqrt{b-\sqrt{b^2-4 a c}}}+\frac{\left (B \sqrt{b^2-4 a c}-2 A c+b B\right ) \tan ^{-1}\left (\frac{\sqrt{2} \sqrt{c} \sqrt{x}}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{\sqrt{b^2-4 a c}+b}}\right )}{\sqrt{c} \sqrt{b^2-4 a c}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.031, size = 337, normalized size = 1.9 \[ -2\,{\frac{c\sqrt{2}A}{\sqrt{-4\,ac+{b}^{2}}\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}{\it Artanh} \left ({\frac{c\sqrt{x}\sqrt{2}}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }-{\sqrt{2}B{\it Artanh} \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\sqrt{2}Bb{\it Artanh} \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( -b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}-2\,{\frac{c\sqrt{2}A}{\sqrt{-4\,ac+{b}^{2}}\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}\arctan \left ({\frac{c\sqrt{x}\sqrt{2}}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}} \right ) }+{\sqrt{2}B\arctan \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}}+{\sqrt{2}Bb\arctan \left ({c\sqrt{2}\sqrt{x}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \right ){\frac{1}{\sqrt{-4\,ac+{b}^{2}}}}{\frac{1}{\sqrt{ \left ( b+\sqrt{-4\,ac+{b}^{2}} \right ) c}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1
/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*A-2^(1/2)/((-b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B+1/(-4*
a*c+b^2)^(1/2)*2^(1/2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctanh(c*x^(1/2)*2^(1/
2)/((-b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b-2*c/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1
/2))*A+2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4
*a*c+b^2)^(1/2))*c)^(1/2))*B+1/(-4*a*c+b^2)^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2)
)*c)^(1/2)*arctan(c*x^(1/2)*2^(1/2)/((b+(-4*a*c+b^2)^(1/2))*c)^(1/2))*B*b

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*A*sqrt(x)/a - integrate((A*c*x^(3/2) - (B*a - A*b)*sqrt(x))/(a*c*x^2 + a*b*x +
a^2), x)

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Fricas [A]  time = 0.519006, size = 2129, normalized size = 11.83 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B
^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c
^2))*log(sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2)*c + (4*(2*
B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c
+ A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a
*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^
3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b + A^3*B*b*c - A^4*c^2)*
sqrt(x)) - 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c + (a*b^2*c - 4*a^2*c
^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*
c - 4*a^2*c^2))*log(-sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^2 + A^3*b^2
)*c + (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B^4*a^2 - 2*
A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (4*A*B*a - A
^2*b)*c + (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^
2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b + A^3*B*b*c
- A^4*c^2)*sqrt(x)) + 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)*c - (a*b^2
*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^
3)))/(a*b^2*c - 4*a^2*c^2))*log(sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 - (4*A*B^2*a^
2 + A^3*b^2)*c - (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3)*c)*sqrt((B
^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-(B^2*a*b - (
4*A*B*a - A^2*b)*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c
^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2 - A*B^3*a*b
+ A^3*B*b*c - A^4*c^2)*sqrt(x)) - 1/2*sqrt(2)*sqrt(-(B^2*a*b - (4*A*B*a - A^2*b)
*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2
- 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2))*log(-sqrt(2)*(A*B^2*a*b^2 + 4*A^3*a*c^2 -
(4*A*B^2*a^2 + A^3*b^2)*c - (4*(2*B*a^3 - A*a^2*b)*c^2 - (2*B*a^2*b^2 - A*a*b^3
)*c)*sqrt((B^4*a^2 - 2*A^2*B^2*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))*sqrt(-
(B^2*a*b - (4*A*B*a - A^2*b)*c - (a*b^2*c - 4*a^2*c^2)*sqrt((B^4*a^2 - 2*A^2*B^2
*a*c + A^4*c^2)/(a^2*b^2*c^2 - 4*a^3*c^3)))/(a*b^2*c - 4*a^2*c^2)) - 4*(B^4*a^2
- A*B^3*a*b + A^3*B*b*c - A^4*c^2)*sqrt(x))

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Sympy [A]  time = 104.647, size = 1130, normalized size = 6.28 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

A*Piecewise((4*I*sqrt(b)*c*sqrt(x)*sqrt(1/c)/(I*b**(5/2)*sqrt(1/c) + 2*I*b**(3/2
)*c*x*sqrt(1/c)) + sqrt(2)*b*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(I*b*
*(5/2)*sqrt(1/c) + 2*I*b**(3/2)*c*x*sqrt(1/c)) - sqrt(2)*b*log(sqrt(2)*I*sqrt(b)
*sqrt(1/c)/2 + sqrt(x))/(I*b**(5/2)*sqrt(1/c) + 2*I*b**(3/2)*c*x*sqrt(1/c)) + 2*
sqrt(2)*c*x*log(-sqrt(2)*I*sqrt(b)*sqrt(1/c)/2 + sqrt(x))/(I*b**(5/2)*sqrt(1/c)
+ 2*I*b**(3/2)*c*x*sqrt(1/c)) - 2*sqrt(2)*c*x*log(sqrt(2)*I*sqrt(b)*sqrt(1/c)/2
+ sqrt(x))/(I*b**(5/2)*sqrt(1/c) + 2*I*b**(3/2)*c*x*sqrt(1/c)), Eq(a, b**2/(4*c)
)), (-2/(b*sqrt(x)) + I*log(-I*sqrt(b)*sqrt(1/c) + sqrt(x))/(b**(3/2)*sqrt(1/c))
- I*log(I*sqrt(b)*sqrt(1/c) + sqrt(x))/(b**(3/2)*sqrt(1/c)), Eq(a, 0)), (-I*log
(-I*sqrt(a)*sqrt(1/b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)) + I*log(I*sqrt(a)*sqrt(1/
b) + sqrt(x))/(sqrt(a)*b*sqrt(1/b)), Eq(c, 0)), (sqrt(2)*b*sqrt(-b/c - sqrt(-4*a
*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a*s
qrt(-4*a*c + b**2)) - sqrt(2)*b*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) +
sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*c + b**2)) - sqrt(
2)*b*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c + sqrt(-
4*a*c + b**2)/c)/2)/(4*a*sqrt(-4*a*c + b**2)) + sqrt(2)*b*sqrt(-b/c + sqrt(-4*a*
c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a*sq
rt(-4*a*c + b**2)) - sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sq
rt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a) + sqrt(2)*sqrt(-b/c - sqrt(-4*
a*c + b**2)/c)*log(sqrt(x) + sqrt(2)*sqrt(-b/c - sqrt(-4*a*c + b**2)/c)/2)/(4*a)
- sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log(sqrt(x) - sqrt(2)*sqrt(-b/c +
sqrt(-4*a*c + b**2)/c)/2)/(4*a) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)*log
(sqrt(x) + sqrt(2)*sqrt(-b/c + sqrt(-4*a*c + b**2)/c)/2)/(4*a), True)) + 2*B*Roo
tSum(_t**4*(256*a**2*c**3 - 128*a*b**2*c**2 + 16*b**4*c) + _t**2*(-16*a*b*c + 4*
b**3) + a, Lambda(_t, _t*log(64*_t**3*a*c**2 - 16*_t**3*b**2*c - 2*_t*b + sqrt(x
))))

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GIAC/XCAS [A]  time = 92.7098, size = 1, normalized size = 0.01 \[ \mathit{Done} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Done