3.855 \(\int \frac{(f+g x)^2 \sqrt{a+b x+c x^2}}{d+e x} \, dx\)
Optimal. Leaf size=325 \[ -\frac{\sqrt{a+b x+c x^2} \left (b^2 e^2 g^2-2 c e g x (-b e g-2 c d g+4 c e f)-2 b c e g (2 e f-d g)-8 c^2 (e f-d g)^2\right )}{8 c^2 e^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (g \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-4 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )}{16 c^{5/2} e^4}+\frac{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2}}{3 c e} \]
[Out]
-((b^2*e^2*g^2 - 8*c^2*(e*f - d*g)^2 - 2*b*c*e*g*(2*e*f - d*g) - 2*c*e*g*(4*c*e*
f - 2*c*d*g - b*e*g)*x)*Sqrt[a + b*x + c*x^2])/(8*c^2*e^3) + (g^2*(a + b*x + c*x
^2)^(3/2))/(3*c*e) + (((8*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - a*e))*g*(4*c*e*f - 2*
c*d*g - b*e*g) - 4*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(5/2)*e^4) + (Sqrt[c*d^2 - b*d*e + a*e
^2]*(e*f - d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e
+ a*e^2]*Sqrt[a + b*x + c*x^2])])/e^4
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Rubi [A] time = 1.47167, antiderivative size = 325, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207
\[ -\frac{\sqrt{a+b x+c x^2} \left (b^2 e^2 g^2-2 c e g x (-b e g-2 c d g+4 c e f)-2 b c e g (2 e f-d g)-8 c^2 (e f-d g)^2\right )}{8 c^2 e^3}+\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (g \left (-4 c e (b d-a e)-b^2 e^2+8 c^2 d^2\right ) (-b e g-2 c d g+4 c e f)-4 c e (2 c d-b e) \left (2 c e f^2-b d g^2\right )\right )}{16 c^{5/2} e^4}+\frac{(e f-d g)^2 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2}}{3 c e} \]
Antiderivative was successfully verified.
[In] Int[((f + g*x)^2*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
[Out]
-((b^2*e^2*g^2 - 8*c^2*(e*f - d*g)^2 - 2*b*c*e*g*(2*e*f - d*g) - 2*c*e*g*(4*c*e*
f - 2*c*d*g - b*e*g)*x)*Sqrt[a + b*x + c*x^2])/(8*c^2*e^3) + (g^2*(a + b*x + c*x
^2)^(3/2))/(3*c*e) + (((8*c^2*d^2 - b^2*e^2 - 4*c*e*(b*d - a*e))*g*(4*c*e*f - 2*
c*d*g - b*e*g) - 4*c*e*(2*c*d - b*e)*(2*c*e*f^2 - b*d*g^2))*ArcTanh[(b + 2*c*x)/
(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(5/2)*e^4) + (Sqrt[c*d^2 - b*d*e + a*e
^2]*(e*f - d*g)^2*ArcTanh[(b*d - 2*a*e + (2*c*d - b*e)*x)/(2*Sqrt[c*d^2 - b*d*e
+ a*e^2]*Sqrt[a + b*x + c*x^2])])/e^4
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Rubi in Sympy [A] time = 113.029, size = 366, normalized size = 1.13 \[ - \frac{b g^{2} \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}}}{8 c^{2} e} + \frac{b g^{2} \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}} e} + \frac{\left (d g - e f\right )^{2} \sqrt{a + b x + c x^{2}}}{e^{3}} - \frac{\left (d g - e f\right )^{2} \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{4}} + \frac{g^{2} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 c e} - \frac{g \left (b + 2 c x\right ) \left (d g - 2 e f\right ) \sqrt{a + b x + c x^{2}}}{4 c e^{2}} + \frac{\left (b e - 2 c d\right ) \left (d g - e f\right )^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c} e^{4}} + \frac{g \left (- 4 a c + b^{2}\right ) \left (d g - 2 e f\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((g*x+f)**2*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)
[Out]
-b*g**2*(b + 2*c*x)*sqrt(a + b*x + c*x**2)/(8*c**2*e) + b*g**2*(-4*a*c + b**2)*a
tanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(16*c**(5/2)*e) + (d*g - e*
f)**2*sqrt(a + b*x + c*x**2)/e**3 - (d*g - e*f)**2*sqrt(a*e**2 - b*d*e + c*d**2)
*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**4 + g**2*(a + b*x + c*x**2)**(3/2)/(3*c*e) - g*(b + 2*c*x)*(
d*g - 2*e*f)*sqrt(a + b*x + c*x**2)/(4*c*e**2) + (b*e - 2*c*d)*(d*g - e*f)**2*at
anh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*sqrt(c)*e**4) + g*(-4*a*c
+ b**2)*(d*g - 2*e*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(8*
c**(3/2)*e**2)
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Mathematica [A] time = 0.795448, size = 356, normalized size = 1.1 \[ \frac{-\frac{3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-8 c^2 e \left (a e g (2 e f-d g)+b (e f-d g)^2\right )+2 b c e^2 g (2 a e g-b d g+2 b e f)-b^3 e^3 g^2+16 c^3 d (e f-d g)^2\right )}{c^{5/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left (2 c e g (4 a e g+b (-3 d g+6 e f+e g x))-3 b^2 e^2 g^2+4 c^2 \left (6 d^2 g^2-3 d e g (4 f+g x)+2 e^2 \left (3 f^2+3 f g x+g^2 x^2\right )\right )\right )}{c^2}+48 (e f-d g)^2 \log (d+e x) \sqrt{e (a e-b d)+c d^2}-48 (e f-d g)^2 \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{48 e^4} \]
Antiderivative was successfully verified.
[In] Integrate[((f + g*x)^2*Sqrt[a + b*x + c*x^2])/(d + e*x),x]
[Out]
((2*e*Sqrt[a + x*(b + c*x)]*(-3*b^2*e^2*g^2 + 2*c*e*g*(4*a*e*g + b*(6*e*f - 3*d*
g + e*g*x)) + 4*c^2*(6*d^2*g^2 - 3*d*e*g*(4*f + g*x) + 2*e^2*(3*f^2 + 3*f*g*x +
g^2*x^2))))/c^2 + 48*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(e*f - d*g)^2*Log[d + e*x] -
(3*(-(b^3*e^3*g^2) + 16*c^3*d*(e*f - d*g)^2 + 2*b*c*e^2*g*(2*b*e*f - b*d*g + 2*
a*e*g) - 8*c^2*e*(b*(e*f - d*g)^2 + a*e*g*(2*e*f - d*g)))*Log[b + 2*c*x + 2*Sqrt
[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) - 48*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(e*f - d
*g)^2*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*Sq
rt[a + x*(b + c*x)]])/(48*e^4)
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Maple [B] time = 0.022, size = 2602, normalized size = 8. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((g*x+f)^2*(c*x^2+b*x+a)^(1/2)/(e*x+d),x)
[Out]
1/3*g^2*(c*x^2+b*x+a)^(3/2)/c/e+1/e^3*((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)*d^2*g^2+1/e*((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)*f^2-1/e^3/((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))*a*d^2*g^2+1/e^
4/((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))*b*d^3*g^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))*b*d*f^2-1/e^5/((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))*c*d^4*g^2-1/e^3/((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))*c*d^2*f^2-1/2*g^2/e^2*d*(c*x^2+b*x+a)^(1/2)*x+g/e*f*(c*x
^2+b*x+a)^(1/2)*x-2/e^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)*d*f*g+1/2/e*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))/c^(1/2)*b*f^2-1/e^4*ln((1/2*(b
*e-2*c*d)/e+c*(x+d/e))/c^(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))*c^(1/2)*d^3*g^2-1/e^2*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))*c^(1/2)*d*f^
2-1/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))*a*f^2+1/16*g^2/e*b^3/c^(5/2)*ln((1/2*b+c
*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-1/e^2*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))/c^(1/2)*b*d*f
*g+2/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))*a*d*f*g-2/e^3/((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))*b*d^2*f*g+2/e^4/((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))*c*d^3*f*g-1/8*g^2/e
*b^2/c^2*(c*x^2+b*x+a)^(1/2)-1/4*g^2/e*b/c*(c*x^2+b*x+a)^(1/2)*x-1/4*g^2/e*b/c^(
3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/4*g^2/e^2*d/c*(c*x^2+b*x+a)
^(1/2)*b-1/2*g^2/e^2*d/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+1/8
*g^2/e^2*d/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2+1/2*g/e*f/c*(
c*x^2+b*x+a)^(1/2)*b+g/e*f/c^(1/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a
-1/4*g/e*f/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*b^2+1/2/e^3*ln((1
/2*(b*e-2*c*d)/e+c*(x+d/e))/c^(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))/c^(1/2)*b*d^2*g^2+2/e^3*ln((1/2*(b*e-2*c*d)/e+c*(x+d/e))/
c^(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))*c^(1/
2)*d^2*f*g
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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(c*x^2 + b*x + a)*(g*x + f)^2/(e*x + d),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2 + b*x + a)*(g*x + f)^2/(e*x + d),x, algorithm="fricas")
[Out]
Timed out
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (f + g x\right )^{2} \sqrt{a + b x + c x^{2}}}{d + e x}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((g*x+f)**2*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)
[Out]
Integral((f + g*x)**2*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(c*x^2 + b*x + a)*(g*x + f)^2/(e*x + d),x, algorithm="giac")
[Out]
Exception raised: TypeError