∖int∘ ______ a+ c_ x2 + b x _____________

3.779 \(\int \frac{\sqrt{a+\frac{c}{x^2}+\frac{b}{x}}}{d+e x} \, dx\)

Optimal. Leaf size=181 \[ -\frac{\sqrt{a d^2-e (b d-c e)} \tanh ^{-1}\left (\frac{2 a d+\frac{b d-2 c e}{x}-b e}{2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{a d^2-e (b d-c e)}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{e} \]

[Out]

(Sqrt[a]*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/e - (Sqrt[c]*Ar

cTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])])/d - (Sqrt[a*d^2 - e*(b*d

- c*e)]*ArcTanh[(2*a*d - b*e + (b*d - 2*c*e)/x)/(2*Sqrt[a*d^2 - e*(b*d - c*e)]*

Sqrt[a + c/x^2 + b/x])])/(d*e)

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Rubi [A] time = 0.733715, antiderivative size = 181, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292 \[ -\frac{\sqrt{a d^2-e (b d-c e)} \tanh ^{-1}\left (\frac{2 a d+\frac{b d-2 c e}{x}-b e}{2 \sqrt{a+\frac{b}{x}+\frac{c}{x^2}} \sqrt{a d^2-e (b d-c e)}}\right )}{d e}-\frac{\sqrt{c} \tanh ^{-1}\left (\frac{b+\frac{2 c}{x}}{2 \sqrt{c} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{d}+\frac{\sqrt{a} \tanh ^{-1}\left (\frac{2 a+\frac{b}{x}}{2 \sqrt{a} \sqrt{a+\frac{b}{x}+\frac{c}{x^2}}}\right )}{e} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(Sqrt[a]*ArcTanh[(2*a + b/x)/(2*Sqrt[a]*Sqrt[a + c/x^2 + b/x])])/e - (Sqrt[c]*Ar

cTanh[(b + (2*c)/x)/(2*Sqrt[c]*Sqrt[a + c/x^2 + b/x])])/d - (Sqrt[a*d^2 - e*(b*d

- c*e)]*ArcTanh[(2*a*d - b*e + (b*d - 2*c*e)/x)/(2*Sqrt[a*d^2 - e*(b*d - c*e)]*

Sqrt[a + c/x^2 + b/x])])/(d*e)

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Rubi in Sympy [A] time = 32.4875, size = 146, normalized size = 0.81 \[ \frac{\sqrt{a} \operatorname{atanh}{\left (\frac{2 a + \frac{b}{x}}{2 \sqrt{a} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{e} - \frac{\sqrt{c} \operatorname{atanh}{\left (\frac{b + \frac{2 c}{x}}{2 \sqrt{c} \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}}} \right )}}{d} - \frac{\sqrt{a d^{2} - b d e + c e^{2}} \operatorname{atanh}{\left (\frac{2 a d - b e + \frac{b d - 2 c e}{x}}{2 \sqrt{a + \frac{b}{x} + \frac{c}{x^{2}}} \sqrt{a d^{2} - b d e + c e^{2}}} \right )}}{d e} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

sqrt(a)*atanh((2*a + b/x)/(2*sqrt(a)*sqrt(a + b/x + c/x**2)))/e - sqrt(c)*atanh(

(b + 2*c/x)/(2*sqrt(c)*sqrt(a + b/x + c/x**2)))/d - sqrt(a*d**2 - b*d*e + c*e**2

)*atanh((2*a*d - b*e + (b*d - 2*c*e)/x)/(2*sqrt(a + b/x + c/x**2)*sqrt(a*d**2 -

b*d*e + c*e**2)))/(d*e)

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Mathematica [A] time = 0.480269, size = 219, normalized size = 1.21 \[ \frac{x \sqrt{a+\frac{b x+c}{x^2}} \left (-\log (d+e x) \sqrt{a d^2-b d e+c e^2}+\sqrt{a d^2-b d e+c e^2} \log \left (2 \sqrt{x (a x+b)+c} \sqrt{a d^2-b d e+c e^2}-2 a d x-b d+b e x+2 c e\right )+\sqrt{a} d \log \left (2 \sqrt{a} \sqrt{x (a x+b)+c}+2 a x+b\right )-\sqrt{c} e \log \left (2 \sqrt{c} \sqrt{x (a x+b)+c}+b x+2 c\right )+\sqrt{c} e \log (x)\right )}{d e \sqrt{x (a x+b)+c}} \]

Antiderivative was successfully veriﬁed.

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

[Out]

(x*Sqrt[a + (c + b*x)/x^2]*(Sqrt[c]*e*Log[x] - Sqrt[a*d^2 - b*d*e + c*e^2]*Log[d

+ e*x] + Sqrt[a]*d*Log[b + 2*a*x + 2*Sqrt[a]*Sqrt[c + x*(b + a*x)]] - Sqrt[c]*e

*Log[2*c + b*x + 2*Sqrt[c]*Sqrt[c + x*(b + a*x)]] + Sqrt[a*d^2 - b*d*e + c*e^2]*

Log[-(b*d) + 2*c*e - 2*a*d*x + b*e*x + 2*Sqrt[a*d^2 - b*d*e + c*e^2]*Sqrt[c + x*

(b + a*x)]]))/(d*e*Sqrt[c + x*(b + a*x)])

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Maple [B] time = 0.046, size = 385, normalized size = 2.1 \[ -{\frac{x}{{e}^{2}d}\sqrt{{\frac{a{x}^{2}+bx+c}{{x}^{2}}}} \left ( \sqrt{c}\ln \left ({\frac{1}{x} \left ( 2\,c+bx+2\,\sqrt{c}\sqrt{a{x}^{2}+bx+c} \right ) } \right ){e}^{2}\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}-\ln \left ({\frac{1}{2} \left ( 2\,\sqrt{a{x}^{2}+bx+c}\sqrt{a}+2\,ax+b \right ){\frac{1}{\sqrt{a}}}} \right ) \sqrt{a}de\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}-{d}^{2}\ln \left ({\frac{1}{ex+d} \left ( 2\,\sqrt{a{x}^{2}+bx+c}\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}e-2\,adx+bex-bd+2\,ce \right ) } \right ) a+\ln \left ({\frac{1}{ex+d} \left ( 2\,\sqrt{a{x}^{2}+bx+c}\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}e-2\,adx+bex-bd+2\,ce \right ) } \right ) bde-\ln \left ({\frac{1}{ex+d} \left ( 2\,\sqrt{a{x}^{2}+bx+c}\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}e-2\,adx+bex-bd+2\,ce \right ) } \right ) c{e}^{2} \right ){\frac{1}{\sqrt{a{x}^{2}+bx+c}}}{\frac{1}{\sqrt{{\frac{a{d}^{2}-bde+{e}^{2}c}{{e}^{2}}}}}}} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

-((a*x^2+b*x+c)/x^2)^(1/2)*x*(c^(1/2)*ln((2*c+b*x+2*c^(1/2)*(a*x^2+b*x+c)^(1/2))

/x)*e^2*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)-ln(1/2*(2*(a*x^2+b*x+c)^(1/2)*a^(1/2)+2*

a*x+b)/a^(1/2))*a^(1/2)*d*e*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)-d^2*ln((2*(a*x^2+b*x

+c)^(1/2)*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*e-2*a*d*x+b*e*x-b*d+2*c*e)/(e*x+d))*a+

ln((2*(a*x^2+b*x+c)^(1/2)*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*e-2*a*d*x+b*e*x-b*d+2*

c*e)/(e*x+d))*b*d*e-ln((2*(a*x^2+b*x+c)^(1/2)*((a*d^2-b*d*e+c*e^2)/e^2)^(1/2)*e-

2*a*d*x+b*e*x-b*d+2*c*e)/(e*x+d))*c*e^2)/(a*x^2+b*x+c)^(1/2)/d/e^2/((a*d^2-b*d*e

+c*e^2)/e^2)^(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(sqrt(a + b/x + c/x^2)/(e*x + d),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A] time = 74.2548, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(

a)*sqrt((a*x^2 + b*x + c)/x^2)) + sqrt(c)*e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 +

8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2) + sqrt(a*d^2

- b*d*e + c*e^2)*log((8*b*c*d*e - 8*c^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*a^2*d^2 -

8*a*b*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*a*b*d^2 + 4*b*c*e^2 - (3*b^2 + 4*a*c)*

d*e)*x + 4*sqrt(a*d^2 - b*d*e + c*e^2)*((2*a*d - b*e)*x^2 + (b*d - 2*c*e)*x)*sqr

t((a*x^2 + b*x + c)/x^2))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), 1/2*(2*sqrt(-a)*d*a

rctan(1/2*(2*a*x + b)/(sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) + sqrt(c)*e*log(

-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 +

b*x + c)/x^2))/x^2) + sqrt(a*d^2 - b*d*e + c*e^2)*log((8*b*c*d*e - 8*c^2*e^2 - (

b^2 + 4*a*c)*d^2 - (8*a^2*d^2 - 8*a*b*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*a*b*d^

2 + 4*b*c*e^2 - (3*b^2 + 4*a*c)*d*e)*x + 4*sqrt(a*d^2 - b*d*e + c*e^2)*((2*a*d -

b*e)*x^2 + (b*d - 2*c*e)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(e^2*x^2 + 2*d*e*x + d

^2)))/(d*e), 1/2*(sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2

+ b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) + sqrt(c)*e*log(-(8*b*c*x + (b^2 + 4

*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*x^2 + b*x + c)/x^2))/x^2)

+ 2*sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(b*d - 2*c*e + (2*a*d - b*e)*x)/(sq

rt(-a*d^2 + b*d*e - c*e^2)*x*sqrt((a*x^2 + b*x + c)/x^2))))/(d*e), 1/2*(2*sqrt(-

a)*d*arctan(1/2*(2*a*x + b)/(sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) + sqrt(c)*

e*log(-(8*b*c*x + (b^2 + 4*a*c)*x^2 + 8*c^2 - 4*(b*x^2 + 2*c*x)*sqrt(c)*sqrt((a*

x^2 + b*x + c)/x^2))/x^2) + 2*sqrt(-a*d^2 + b*d*e - c*e^2)*arctan(-1/2*(b*d - 2*

c*e + (2*a*d - b*e)*x)/(sqrt(-a*d^2 + b*d*e - c*e^2)*x*sqrt((a*x^2 + b*x + c)/x^

2))))/(d*e), -1/2*(2*sqrt(-c)*e*arctan(1/2*(b*x + 2*c)/(sqrt(-c)*x*sqrt((a*x^2 +

b*x + c)/x^2))) - sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2 - 4*a*c - 4*(2*a*x^2

+ b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - sqrt(a*d^2 - b*d*e + c*e^2)*log((

8*b*c*d*e - 8*c^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*a^2*d^2 - 8*a*b*d*e + (b^2 + 4*a*

c)*e^2)*x^2 - 2*(4*a*b*d^2 + 4*b*c*e^2 - (3*b^2 + 4*a*c)*d*e)*x + 4*sqrt(a*d^2 -

b*d*e + c*e^2)*((2*a*d - b*e)*x^2 + (b*d - 2*c*e)*x)*sqrt((a*x^2 + b*x + c)/x^2

))/(e^2*x^2 + 2*d*e*x + d^2)))/(d*e), 1/2*(2*sqrt(-a)*d*arctan(1/2*(2*a*x + b)/(

sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) - 2*sqrt(-c)*e*arctan(1/2*(b*x + 2*c)/(

sqrt(-c)*x*sqrt((a*x^2 + b*x + c)/x^2))) + sqrt(a*d^2 - b*d*e + c*e^2)*log((8*b*

c*d*e - 8*c^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*a^2*d^2 - 8*a*b*d*e + (b^2 + 4*a*c)*e

^2)*x^2 - 2*(4*a*b*d^2 + 4*b*c*e^2 - (3*b^2 + 4*a*c)*d*e)*x + 4*sqrt(a*d^2 - b*d

*e + c*e^2)*((2*a*d - b*e)*x^2 + (b*d - 2*c*e)*x)*sqrt((a*x^2 + b*x + c)/x^2))/(

e^2*x^2 + 2*d*e*x + d^2)))/(d*e), -1/2*(2*sqrt(-c)*e*arctan(1/2*(b*x + 2*c)/(sqr

t(-c)*x*sqrt((a*x^2 + b*x + c)/x^2))) - sqrt(a)*d*log(-8*a^2*x^2 - 8*a*b*x - b^2

- 4*a*c - 4*(2*a*x^2 + b*x)*sqrt(a)*sqrt((a*x^2 + b*x + c)/x^2)) - 2*sqrt(-a*d^

2 + b*d*e - c*e^2)*arctan(-1/2*(b*d - 2*c*e + (2*a*d - b*e)*x)/(sqrt(-a*d^2 + b*

d*e - c*e^2)*x*sqrt((a*x^2 + b*x + c)/x^2))))/(d*e), (sqrt(-a)*d*arctan(1/2*(2*a

*x + b)/(sqrt(-a)*x*sqrt((a*x^2 + b*x + c)/x^2))) - sqrt(-c)*e*arctan(1/2*(b*x +

2*c)/(sqrt(-c)*x*sqrt((a*x^2 + b*x + c)/x^2))) + sqrt(-a*d^2 + b*d*e - c*e^2)*a

rctan(-1/2*(b*d - 2*c*e + (2*a*d - b*e)*x)/(sqrt(-a*d^2 + b*d*e - c*e^2)*x*sqrt(

(a*x^2 + b*x + c)/x^2))))/(d*e)]

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

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(a + b/x + c/x**2)/(d + e*x), x)

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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Veriﬁcation of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError

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