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

Optimal. Leaf size=116 \[ -\frac{2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}+\frac{2 \sqrt{f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]

[Out]

(2*(b*e*g - c*(e*f + d*g))*Sqrt[f + g*x])/(e^2*g^2) + (2*c*(f + g*x)^(3/2))/(3*e
*g^2) - (2*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*
g]])/(e^(5/2)*Sqrt[e*f - d*g])

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Rubi [A]  time = 0.337769, antiderivative size = 116, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ -\frac{2 \left (a e^2-b d e+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}}+\frac{2 \sqrt{f+g x} (b e g-c (d g+e f))}{e^2 g^2}+\frac{2 c (f+g x)^{3/2}}{3 e g^2} \]

Antiderivative was successfully verified.

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

[Out]

(2*(b*e*g - c*(e*f + d*g))*Sqrt[f + g*x])/(e^2*g^2) + (2*c*(f + g*x)^(3/2))/(3*e
*g^2) - (2*(c*d^2 - b*d*e + a*e^2)*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*
g]])/(e^(5/2)*Sqrt[e*f - d*g])

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Rubi in Sympy [A]  time = 80.0663, size = 107, normalized size = 0.92 \[ \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 e g^{2}} + \frac{2 \sqrt{f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g^{2}} + \frac{2 \left (a e^{2} - b d e + c d^{2}\right ) \operatorname{atan}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{d g - e f}} \right )}}{e^{\frac{5}{2}} \sqrt{d g - e f}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*(f + g*x)**(3/2)/(3*e*g**2) + 2*sqrt(f + g*x)*(b*e*g - c*d*g - c*e*f)/(e**2*
g**2) + 2*(a*e**2 - b*d*e + c*d**2)*atan(sqrt(e)*sqrt(f + g*x)/sqrt(d*g - e*f))/
(e**(5/2)*sqrt(d*g - e*f))

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Mathematica [A]  time = 0.158345, size = 104, normalized size = 0.9 \[ \frac{2 \sqrt{f+g x} (3 b e g+c (-3 d g-2 e f+e g x))}{3 e^2 g^2}-\frac{2 \left (e (a e-b d)+c d^2\right ) \tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right )}{e^{5/2} \sqrt{e f-d g}} \]

Antiderivative was successfully verified.

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

[Out]

(2*Sqrt[f + g*x]*(3*b*e*g + c*(-2*e*f - 3*d*g + e*g*x)))/(3*e^2*g^2) - (2*(c*d^2
+ e*(-(b*d) + a*e))*ArcTanh[(Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*
Sqrt[e*f - d*g])

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Maple [A]  time = 0.014, size = 189, normalized size = 1.6 \[{\frac{2\,c}{3\,e{g}^{2}} \left ( gx+f \right ) ^{{\frac{3}{2}}}}+2\,{\frac{b\sqrt{gx+f}}{eg}}-2\,{\frac{cd\sqrt{gx+f}}{g{e}^{2}}}-2\,{\frac{cf\sqrt{gx+f}}{e{g}^{2}}}+2\,{\frac{a}{\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-2\,{\frac{bd}{e\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+2\,{\frac{c{d}^{2}}{{e}^{2}\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2/3*c*(g*x+f)^(3/2)/e/g^2+2/g/e*b*(g*x+f)^(1/2)-2/g*c/e^2*d*(g*x+f)^(1/2)-2/g^2*
c/e*f*(g*x+f)^(1/2)+2/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(
1/2))*a-2/e/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*b*d+
2/e^2/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*c*d^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((c*x^2 + b*x + a)/((e*x + d)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.284794, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (c d^{2} - b d e + a e^{2}\right )} g^{2} \log \left (\frac{\sqrt{e^{2} f - d e g}{\left (e g x + 2 \, e f - d g\right )} - 2 \,{\left (e^{2} f - d e g\right )} \sqrt{g x + f}}{e x + d}\right ) + 2 \,{\left (c e g x - 2 \, c e f - 3 \,{\left (c d - b e\right )} g\right )} \sqrt{e^{2} f - d e g} \sqrt{g x + f}}{3 \, \sqrt{e^{2} f - d e g} e^{2} g^{2}}, -\frac{2 \,{\left (3 \,{\left (c d^{2} - b d e + a e^{2}\right )} g^{2} \arctan \left (-\frac{e f - d g}{\sqrt{-e^{2} f + d e g} \sqrt{g x + f}}\right ) -{\left (c e g x - 2 \, c e f - 3 \,{\left (c d - b e\right )} g\right )} \sqrt{-e^{2} f + d e g} \sqrt{g x + f}\right )}}{3 \, \sqrt{-e^{2} f + d e g} e^{2} g^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [A]  time = 30.853, size = 257, normalized size = 2.22 \[ \frac{2 c \left (f + g x\right )^{\frac{3}{2}}}{3 e g^{2}} - \frac{2 \left (a e^{2} - b d e + c d^{2}\right ) \left (\begin{cases} \frac{\operatorname{atan}{\left (\frac{1}{\sqrt{\frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{\frac{e}{d g - e f}} \left (d g - e f\right )} & \text{for}\: \frac{e}{d g - e f} > 0 \\- \frac{\operatorname{acoth}{\left (\frac{1}{\sqrt{- \frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{- \frac{e}{d g - e f}} \left (d g - e f\right )} & \text{for}\: \frac{1}{f + g x} > - \frac{e}{d g - e f} \wedge \frac{e}{d g - e f} < 0 \\- \frac{\operatorname{atanh}{\left (\frac{1}{\sqrt{- \frac{e}{d g - e f}} \sqrt{f + g x}} \right )}}{\sqrt{- \frac{e}{d g - e f}} \left (d g - e f\right )} & \text{for}\: \frac{e}{d g - e f} < 0 \wedge \frac{1}{f + g x} < - \frac{e}{d g - e f} \end{cases}\right )}{e^{2}} + \frac{2 \sqrt{f + g x} \left (b e g - c d g - c e f\right )}{e^{2} g^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*(f + g*x)**(3/2)/(3*e*g**2) - 2*(a*e**2 - b*d*e + c*d**2)*Piecewise((atan(1/
(sqrt(e/(d*g - e*f))*sqrt(f + g*x)))/(sqrt(e/(d*g - e*f))*(d*g - e*f)), e/(d*g -
e*f) > 0), (-acoth(1/(sqrt(-e/(d*g - e*f))*sqrt(f + g*x)))/(sqrt(-e/(d*g - e*f)
)*(d*g - e*f)), (e/(d*g - e*f) < 0) & (1/(f + g*x) > -e/(d*g - e*f))), (-atanh(1
/(sqrt(-e/(d*g - e*f))*sqrt(f + g*x)))/(sqrt(-e/(d*g - e*f))*(d*g - e*f)), (e/(d
*g - e*f) < 0) & (1/(f + g*x) < -e/(d*g - e*f))))/e**2 + 2*sqrt(f + g*x)*(b*e*g
- c*d*g - c*e*f)/(e**2*g**2)

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GIAC/XCAS [A]  time = 0.265168, size = 173, normalized size = 1.49 \[ \frac{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right ) e^{\left (-2\right )}}{\sqrt{d g e - f e^{2}}} - \frac{2 \,{\left (3 \, \sqrt{g x + f} c d g^{5} e -{\left (g x + f\right )}^{\frac{3}{2}} c g^{4} e^{2} + 3 \, \sqrt{g x + f} c f g^{4} e^{2} - 3 \, \sqrt{g x + f} b g^{5} e^{2}\right )} e^{\left (-3\right )}}{3 \, g^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*(c*d^2 - b*d*e + a*e^2)*arctan(sqrt(g*x + f)*e/sqrt(d*g*e - f*e^2))*e^(-2)/sqr
t(d*g*e - f*e^2) - 2/3*(3*sqrt(g*x + f)*c*d*g^5*e - (g*x + f)^(3/2)*c*g^4*e^2 +
3*sqrt(g*x + f)*c*f*g^4*e^2 - 3*sqrt(g*x + f)*b*g^5*e^2)*e^(-3)/g^6