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

Optimal. Leaf size=131 \[ \frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e \sqrt{a e^2-b d e+c d^2}}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} e} \]

[Out]

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

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Rubi [A]  time = 0.235787, antiderivative size = 131, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{(e f-d g) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e \sqrt{a e^2-b d e+c d^2}}+\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{\sqrt{c} e} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 36.4225, size = 117, normalized size = 0.89 \[ \frac{\left (d g - e f\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e \sqrt{a e^{2} - b d e + c d^{2}}} + \frac{g \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.319102, size = 167, normalized size = 1.27 \[ \frac{\sqrt{c} (d g-e f) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{a e^2-b d e+c d^2}+2 a e-b d+b e x-2 c d x\right )+g \sqrt{a e^2-b d e+c d^2} \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )+\sqrt{c} (e f-d g) \log (d+e x)}{\sqrt{c} e \sqrt{e (a e-b d)+c d^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 349, normalized size = 2.7 \[{\frac{g}{e}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){\frac{1}{\sqrt{c}}}}+{\frac{dg}{{e}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{f}{e}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{ \left ( x+{\frac{d}{e}} \right ) ^{2}c+{\frac{be-2\,cd}{e} \left ( x+{\frac{d}{e}} \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ( x+{\frac{d}{e}} \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 28.5568, size = 1, normalized size = 0.01 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(sqrt(c*d^2 - b*d*e + a*e^2)*g*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a)
- (8*c^2*x^2 + 8*b*c*x + b^2 + 4*a*c)*sqrt(c)) - (e*f - d*g)*sqrt(c)*log(((8*a*
b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e
^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e
+ a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d
^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2
*x^2 + 2*d*e*x + d^2)))/(sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c)*e), -1/2*(2*(e*f -
d*g)*sqrt(c)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*
e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a))) - sqrt(-c*d^2 + b*d*e - a
*e^2)*g*log(-4*(2*c^2*x + b*c)*sqrt(c*x^2 + b*x + a) - (8*c^2*x^2 + 8*b*c*x + b^
2 + 4*a*c)*sqrt(c)))/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(c)*e), 1/2*(2*sqrt(c*d^2
- b*d*e + a*e^2)*g*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2 + b*x + a)*c)) -
(e*f - d*g)*sqrt(-c)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 - (8*c^2*d
^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*b^2 + 4*
a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*a^2*e^3
- (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*c)*d*e^2
)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/(sqrt(c*d^2 - b*d*e + a*
e^2)*sqrt(-c)*e), -((e*f - d*g)*sqrt(-c)*arctan(-1/2*sqrt(-c*d^2 + b*d*e - a*e^2
)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2 + b*x + a)
)) - sqrt(-c*d^2 + b*d*e - a*e^2)*g*arctan(1/2*(2*c*x + b)*sqrt(-c)/(sqrt(c*x^2
+ b*x + a)*c)))/(sqrt(-c*d^2 + b*d*e - a*e^2)*sqrt(-c)*e)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((f + g*x)/((d + e*x)*sqrt(a + b*x + c*x**2)), 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((g*x + f)/(sqrt(c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError