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

Optimal. Leaf size=140 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}-\frac{\sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

[Out]

(2*c*Sqrt[f + g*x])/(e^2*g) - ((a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/((e*f -
d*g)*(d + e*x)) + ((c*d*(4*e*f - 3*d*g) - e*(2*b*e*f - b*d*g - a*e*g))*ArcTanh[(
Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*(e*f - d*g)^(3/2))

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Rubi [A]  time = 0.602624, antiderivative size = 140, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.148 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (c d (4 e f-3 d g)-e (-a e g-b d g+2 b e f))}{e^{5/2} (e f-d g)^{3/2}}-\frac{\sqrt{f+g x} \left (a+\frac{d (c d-b e)}{e^2}\right )}{(d+e x) (e f-d g)}+\frac{2 c \sqrt{f+g x}}{e^2 g} \]

Antiderivative was successfully verified.

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

[Out]

(2*c*Sqrt[f + g*x])/(e^2*g) - ((a + (d*(c*d - b*e))/e^2)*Sqrt[f + g*x])/((e*f -
d*g)*(d + e*x)) + ((c*d*(4*e*f - 3*d*g) - e*(2*b*e*f - b*d*g - a*e*g))*ArcTanh[(
Sqrt[e]*Sqrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*(e*f - d*g)^(3/2))

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Rubi in Sympy [A]  time = 128.764, size = 144, normalized size = 1.03 \[ \frac{2 c \sqrt{f + g x}}{e^{2} g} + \frac{g \sqrt{f + g x} \left (a e^{2} - b d e + c d^{2}\right )}{e^{2} \left (d g - e f\right ) \left (d g - e f + e \left (f + g x\right )\right )} + \frac{\left (a e^{2} g + b d e g - 2 b e^{2} f - 3 c d^{2} g + 4 c d e f\right ) \operatorname{atan}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{d g - e f}} \right )}}{e^{\frac{5}{2}} \left (d g - e f\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*sqrt(f + g*x)/(e**2*g) + g*sqrt(f + g*x)*(a*e**2 - b*d*e + c*d**2)/(e**2*(d*
g - e*f)*(d*g - e*f + e*(f + g*x))) + (a*e**2*g + b*d*e*g - 2*b*e**2*f - 3*c*d**
2*g + 4*c*d*e*f)*atan(sqrt(e)*sqrt(f + g*x)/sqrt(d*g - e*f))/(e**(5/2)*(d*g - e*
f)**(3/2))

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Mathematica [A]  time = 0.448304, size = 133, normalized size = 0.95 \[ \frac{\sqrt{f+g x} \left (\frac{e (a e-b d)+c d^2}{(d+e x) (d g-e f)}+\frac{2 c}{g}\right )}{e^2}-\frac{\tanh ^{-1}\left (\frac{\sqrt{e} \sqrt{f+g x}}{\sqrt{e f-d g}}\right ) (e (-a e g-b d g+2 b e f)+c d (3 d g-4 e f))}{e^{5/2} (e f-d g)^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[f + g*x]*((2*c)/g + (c*d^2 + e*(-(b*d) + a*e))/((-(e*f) + d*g)*(d + e*x)))
)/e^2 - ((c*d*(-4*e*f + 3*d*g) + e*(2*b*e*f - b*d*g - a*e*g))*ArcTanh[(Sqrt[e]*S
qrt[f + g*x])/Sqrt[e*f - d*g]])/(e^(5/2)*(e*f - d*g)^(3/2))

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Maple [B]  time = 0.022, size = 371, normalized size = 2.7 \[ 2\,{\frac{c\sqrt{gx+f}}{{e}^{2}g}}+{\frac{ag}{ \left ( dg-ef \right ) \left ( egx+dg \right ) }\sqrt{gx+f}}-{\frac{bdg}{ \left ( dg-ef \right ) e \left ( egx+dg \right ) }\sqrt{gx+f}}+{\frac{c{d}^{2}g}{{e}^{2} \left ( dg-ef \right ) \left ( egx+dg \right ) }\sqrt{gx+f}}+{\frac{ag}{dg-ef}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}+{\frac{bdg}{ \left ( dg-ef \right ) e}\arctan \left ({e\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}} \right ){\frac{1}{\sqrt{ \left ( dg-ef \right ) e}}}}-2\,{\frac{bf}{ \left ( dg-ef \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }-3\,{\frac{c{d}^{2}g}{{e}^{2} \left ( dg-ef \right ) \sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) }+4\,{\frac{cdf}{ \left ( dg-ef \right ) e\sqrt{ \left ( dg-ef \right ) e}}\arctan \left ({\frac{\sqrt{gx+f}e}{\sqrt{ \left ( dg-ef \right ) e}}} \right ) } \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

2*c*(g*x+f)^(1/2)/e^2/g+g/(d*g-e*f)*(g*x+f)^(1/2)/(e*g*x+d*g)*a-g/e/(d*g-e*f)*(g
*x+f)^(1/2)/(e*g*x+d*g)*b*d+g/e^2/(d*g-e*f)*(g*x+f)^(1/2)/(e*g*x+d*g)*c*d^2+g/(d
*g-e*f)/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*a+g/e/(d
*g-e*f)/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*b*d-2/(d
*g-e*f)/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*b*f-3*g/
e^2/(d*g-e*f)/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2))*c*
d^2+4/e/(d*g-e*f)/((d*g-e*f)*e)^(1/2)*arctan((g*x+f)^(1/2)*e/((d*g-e*f)*e)^(1/2)
)*c*d*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((c*x^2 + b*x + a)/((e*x + d)^2*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.29409, size = 1, normalized size = 0.01 \[ \left [\frac{2 \,{\left (2 \, c d e f -{\left (3 \, c d^{2} - b d e + a e^{2}\right )} g + 2 \,{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{e^{2} f - d e g} \sqrt{g x + f} -{\left (2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} f g -{\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} +{\left (2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f g -{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \log \left (\frac{\sqrt{e^{2} f - d e g}{\left (e g x + 2 \, e f - d g\right )} - 2 \,{\left (e^{2} f - d e g\right )} \sqrt{g x + f}}{e x + d}\right )}{2 \,{\left (d e^{3} f g - d^{2} e^{2} g^{2} +{\left (e^{4} f g - d e^{3} g^{2}\right )} x\right )} \sqrt{e^{2} f - d e g}}, \frac{{\left (2 \, c d e f -{\left (3 \, c d^{2} - b d e + a e^{2}\right )} g + 2 \,{\left (c e^{2} f - c d e g\right )} x\right )} \sqrt{-e^{2} f + d e g} \sqrt{g x + f} +{\left (2 \,{\left (2 \, c d^{2} e - b d e^{2}\right )} f g -{\left (3 \, c d^{3} - b d^{2} e - a d e^{2}\right )} g^{2} +{\left (2 \,{\left (2 \, c d e^{2} - b e^{3}\right )} f g -{\left (3 \, c d^{2} e - b d e^{2} - a e^{3}\right )} g^{2}\right )} x\right )} \arctan \left (-\frac{e f - d g}{\sqrt{-e^{2} f + d e g} \sqrt{g x + f}}\right )}{{\left (d e^{3} f g - d^{2} e^{2} g^{2} +{\left (e^{4} f g - d e^{3} g^{2}\right )} x\right )} \sqrt{-e^{2} f + d e g}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-2)]  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)**2/(g*x+f)**(1/2),x)

[Out]

Exception raised: ValueError

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GIAC/XCAS [A]  time = 0.266678, size = 236, normalized size = 1.69 \[ \frac{2 \, \sqrt{g x + f} c e^{\left (-2\right )}}{g} - \frac{{\left (3 \, c d^{2} g - 4 \, c d f e - b d g e + 2 \, b f e^{2} - a g e^{2}\right )} \arctan \left (\frac{\sqrt{g x + f} e}{\sqrt{d g e - f e^{2}}}\right )}{{\left (d g e^{2} - f e^{3}\right )} \sqrt{d g e - f e^{2}}} + \frac{\sqrt{g x + f} c d^{2} g - \sqrt{g x + f} b d g e + \sqrt{g x + f} a g e^{2}}{{\left (d g e^{2} - f e^{3}\right )}{\left (d g +{\left (g x + f\right )} e - f e\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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