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

Optimal. Leaf size=164 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (4 e g (2 a e g-b (d g+e f))+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right )}{4 e^{5/2} g^{5/2}}-\frac{\sqrt{d+e x} \sqrt{f+g x} (-4 b e g+5 c d g+3 c e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g} \]

[Out]

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

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Rubi [A]  time = 0.376869, antiderivative size = 164, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.138 \[ \frac{\tanh ^{-1}\left (\frac{\sqrt{g} \sqrt{d+e x}}{\sqrt{e} \sqrt{f+g x}}\right ) \left (4 e g (2 a e g-b (d g+e f))+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right )}{4 e^{5/2} g^{5/2}}-\frac{\sqrt{d+e x} \sqrt{f+g x} (-4 b e g+5 c d g+3 c e f)}{4 e^2 g^2}+\frac{c (d+e x)^{3/2} \sqrt{f+g x}}{2 e^2 g} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 38.9625, size = 199, normalized size = 1.21 \[ \frac{b \sqrt{d + e x} \sqrt{f + g x}}{e g} + \frac{c x \sqrt{d + e x} \sqrt{f + g x}}{2 e g} - \frac{3 c \sqrt{d + e x} \sqrt{f + g x} \left (d g + e f\right )}{4 e^{2} g^{2}} - \frac{c \left (4 d e f g - 3 \left (d g + e f\right )^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{4 e^{\frac{5}{2}} g^{\frac{5}{2}}} - \frac{2 \left (- a e g + \frac{b \left (d g + e f\right )}{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{e} \sqrt{f + g x}}{\sqrt{g} \sqrt{d + e x}} \right )}}{e^{\frac{3}{2}} g^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.173714, size = 154, normalized size = 0.94 \[ \frac{\log \left (2 \sqrt{e} \sqrt{g} \sqrt{d+e x} \sqrt{f+g x}+d g+e f+2 e g x\right ) \left (4 e g (2 a e g-b (d g+e f))+c \left (3 d^2 g^2+2 d e f g+3 e^2 f^2\right )\right )}{8 e^{5/2} g^{5/2}}+\frac{\sqrt{d+e x} \sqrt{f+g x} (4 b e g+c (-3 d g-3 e f+2 e g x))}{4 e^2 g^2} \]

Antiderivative was successfully verified.

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

[Out]

(Sqrt[d + e*x]*Sqrt[f + g*x]*(4*b*e*g + c*(-3*e*f - 3*d*g + 2*e*g*x)))/(4*e^2*g^
2) + ((c*(3*e^2*f^2 + 2*d*e*f*g + 3*d^2*g^2) + 4*e*g*(2*a*e*g - b*(e*f + d*g)))*
Log[e*f + d*g + 2*e*g*x + 2*Sqrt[e]*Sqrt[g]*Sqrt[d + e*x]*Sqrt[f + g*x]])/(8*e^(
5/2)*g^(5/2))

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Maple [B]  time = 0., size = 425, normalized size = 2.6 \[{\frac{1}{8\,{g}^{2}{e}^{2}} \left ( 8\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) a{e}^{2}{g}^{2}-4\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) bde{g}^{2}-4\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) b{e}^{2}fg+3\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) c{d}^{2}{g}^{2}+2\,c\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) dfeg+3\,\ln \left ( 1/2\,{\frac{2\,egx+2\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}+dg+ef}{\sqrt{eg}}} \right ) c{e}^{2}{f}^{2}+4\,c\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }xeg\sqrt{eg}+8\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}beg-6\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}cdg-6\,\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }\sqrt{eg}cef \right ) \sqrt{ex+d}\sqrt{gx+f}{\frac{1}{\sqrt{ \left ( ex+d \right ) \left ( gx+f \right ) }}}{\frac{1}{\sqrt{eg}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/8*(8*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2
))*a*e^2*g^2-4*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e
*g)^(1/2))*b*d*e*g^2-4*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)+d*g
+e*f)/(e*g)^(1/2))*b*e^2*f*g+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(
1/2)+d*g+e*f)/(e*g)^(1/2))*c*d^2*g^2+2*c*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f))^(1/
2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*d*f*e*g+3*ln(1/2*(2*e*g*x+2*((e*x+d)*(g*x+f
))^(1/2)*(e*g)^(1/2)+d*g+e*f)/(e*g)^(1/2))*c*e^2*f^2+4*c*((e*x+d)*(g*x+f))^(1/2)
*x*e*g*(e*g)^(1/2)+8*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)*b*e*g-6*((e*x+d)*(g*x+f
))^(1/2)*(e*g)^(1/2)*c*d*g-6*((e*x+d)*(g*x+f))^(1/2)*(e*g)^(1/2)*c*e*f)*(e*x+d)^
(1/2)*(g*x+f)^(1/2)/(e*g)^(1/2)/g^2/e^2/((e*x+d)*(g*x+f))^(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((c*x^2 + b*x + a)/(sqrt(e*x + d)*sqrt(g*x + f)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.49738, size = 1, normalized size = 0.01 \[ \left [\frac{4 \,{\left (2 \, c e g x - 3 \, c e f -{\left (3 \, c d - 4 \, b e\right )} g\right )} \sqrt{e g} \sqrt{e x + d} \sqrt{g x + f} +{\left (3 \, c e^{2} f^{2} + 2 \,{\left (c d e - 2 \, b e^{2}\right )} f g +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} g^{2}\right )} \log \left (4 \,{\left (2 \, e^{2} g^{2} x + e^{2} f g + d e g^{2}\right )} \sqrt{e x + d} \sqrt{g x + f} +{\left (8 \, e^{2} g^{2} x^{2} + e^{2} f^{2} + 6 \, d e f g + d^{2} g^{2} + 8 \,{\left (e^{2} f g + d e g^{2}\right )} x\right )} \sqrt{e g}\right )}{16 \, \sqrt{e g} e^{2} g^{2}}, \frac{2 \,{\left (2 \, c e g x - 3 \, c e f -{\left (3 \, c d - 4 \, b e\right )} g\right )} \sqrt{-e g} \sqrt{e x + d} \sqrt{g x + f} +{\left (3 \, c e^{2} f^{2} + 2 \,{\left (c d e - 2 \, b e^{2}\right )} f g +{\left (3 \, c d^{2} - 4 \, b d e + 8 \, a e^{2}\right )} g^{2}\right )} \arctan \left (\frac{{\left (2 \, e g x + e f + d g\right )} \sqrt{-e g}}{2 \, \sqrt{e x + d} \sqrt{g x + f} e g}\right )}{8 \, \sqrt{-e g} e^{2} g^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/16*(4*(2*c*e*g*x - 3*c*e*f - (3*c*d - 4*b*e)*g)*sqrt(e*g)*sqrt(e*x + d)*sqrt(
g*x + f) + (3*c*e^2*f^2 + 2*(c*d*e - 2*b*e^2)*f*g + (3*c*d^2 - 4*b*d*e + 8*a*e^2
)*g^2)*log(4*(2*e^2*g^2*x + e^2*f*g + d*e*g^2)*sqrt(e*x + d)*sqrt(g*x + f) + (8*
e^2*g^2*x^2 + e^2*f^2 + 6*d*e*f*g + d^2*g^2 + 8*(e^2*f*g + d*e*g^2)*x)*sqrt(e*g)
))/(sqrt(e*g)*e^2*g^2), 1/8*(2*(2*c*e*g*x - 3*c*e*f - (3*c*d - 4*b*e)*g)*sqrt(-e
*g)*sqrt(e*x + d)*sqrt(g*x + f) + (3*c*e^2*f^2 + 2*(c*d*e - 2*b*e^2)*f*g + (3*c*
d^2 - 4*b*d*e + 8*a*e^2)*g^2)*arctan(1/2*(2*e*g*x + e*f + d*g)*sqrt(-e*g)/(sqrt(
e*x + d)*sqrt(g*x + f)*e*g)))/(sqrt(-e*g)*e^2*g^2)]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.287325, size = 242, normalized size = 1.48 \[ \frac{1}{4} \, \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \sqrt{x e + d}{\left (\frac{2 \,{\left (x e + d\right )} c e^{\left (-3\right )}}{g} - \frac{{\left (5 \, c d g^{2} e^{5} + 3 \, c f g e^{6} - 4 \, b g^{2} e^{6}\right )} e^{\left (-8\right )}}{g^{3}}\right )} - \frac{{\left (3 \, c d^{2} g^{2} + 2 \, c d f g e - 4 \, b d g^{2} e + 3 \, c f^{2} e^{2} - 4 \, b f g e^{2} + 8 \, a g^{2} e^{2}\right )} e^{\left (-\frac{5}{2}\right )}{\rm ln}\left ({\left | -\sqrt{x e + d} \sqrt{g} e^{\frac{1}{2}} + \sqrt{{\left (x e + d\right )} g e - d g e + f e^{2}} \right |}\right )}{4 \, g^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/4*sqrt((x*e + d)*g*e - d*g*e + f*e^2)*sqrt(x*e + d)*(2*(x*e + d)*c*e^(-3)/g -
(5*c*d*g^2*e^5 + 3*c*f*g*e^6 - 4*b*g^2*e^6)*e^(-8)/g^3) - 1/4*(3*c*d^2*g^2 + 2*c
*d*f*g*e - 4*b*d*g^2*e + 3*c*f^2*e^2 - 4*b*f*g*e^2 + 8*a*g^2*e^2)*e^(-5/2)*ln(ab
s(-sqrt(x*e + d)*sqrt(g)*e^(1/2) + sqrt((x*e + d)*g*e - d*g*e + f*e^2)))/g^(5/2)