3.854 \(\int \frac{(f+g x)^3 \sqrt{a+b x+c x^2}}{d+e x} \, dx\)

Optimal. Leaf size=532 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac{b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{128 c^{7/2} e^5}+\frac{\sqrt{a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{64 c^3 e^4}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac{(e f-d g)^3 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^5}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2} \]

[Out]

((5*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(6*b*e*f - 2*b*d*g + a*e*
g) + 16*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(5*b^2*e^2*g^2 - 4
*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x
)*Sqrt[a + b*x + c*x^2])/(64*c^3*e^4) + (g^2*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(a
+ b*x + c*x^2)^(3/2))/(24*c^2*e^2) + (g^3*(d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*
c*e^2) - ((4*c*e*(2*c*d - b*e)*(16*c^2*e^2*f^3 + 5*b^2*d*e*g^3 - 4*c*d*g^2*(6*b*
e*f - 2*b*d*g + a*e*g)) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*g*(5*b
^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g
+ d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7/
2)*e^5) + (Sqrt[c*d^2 - b*d*e + a*e^2]*(e*f - d*g)^3*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^5

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Rubi [A]  time = 3.34148, antiderivative size = 532, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.207 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (4 c e (2 c d-b e) \left (-4 c d g^2 (a e g-2 b d g+6 b e f)+5 b^2 d e g^3+16 c^2 e^2 f^3\right )-2 g \left (-2 c e (b d-a e)-\frac{b^2 e^2}{2}+4 c^2 d^2\right ) \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )\right )}{128 c^{7/2} e^5}+\frac{\sqrt{a+b x+c x^2} \left (2 c e g x \left (-4 c e g (a e g-2 b d g+6 b e f)+5 b^2 e^2 g^2+16 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )-4 b c e^2 g^2 (a e g-2 b d g+6 b e f)+5 b^3 e^3 g^3+16 b c^2 e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+64 c^3 (e f-d g)^3\right )}{64 c^3 e^4}+\frac{g^2 \left (a+b x+c x^2\right )^{3/2} (-5 b e g-14 c d g+24 c e f)}{24 c^2 e^2}+\frac{(e f-d g)^3 \sqrt{a e^2-b d e+c d^2} \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^5}+\frac{g^3 (d+e x) \left (a+b x+c x^2\right )^{3/2}}{4 c e^2} \]

Antiderivative was successfully verified.

[In]  Int[((f + g*x)^3*Sqrt[a + b*x + c*x^2])/(d + e*x),x]

[Out]

((5*b^3*e^3*g^3 + 64*c^3*(e*f - d*g)^3 - 4*b*c*e^2*g^2*(6*b*e*f - 2*b*d*g + a*e*
g) + 16*b*c^2*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2) + 2*c*e*g*(5*b^2*e^2*g^2 - 4
*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))*x
)*Sqrt[a + b*x + c*x^2])/(64*c^3*e^4) + (g^2*(24*c*e*f - 14*c*d*g - 5*b*e*g)*(a
+ b*x + c*x^2)^(3/2))/(24*c^2*e^2) + (g^3*(d + e*x)*(a + b*x + c*x^2)^(3/2))/(4*
c*e^2) - ((4*c*e*(2*c*d - b*e)*(16*c^2*e^2*f^3 + 5*b^2*d*e*g^3 - 4*c*d*g^2*(6*b*
e*f - 2*b*d*g + a*e*g)) - 2*(4*c^2*d^2 - (b^2*e^2)/2 - 2*c*e*(b*d - a*e))*g*(5*b
^2*e^2*g^2 - 4*c*e*g*(6*b*e*f - 2*b*d*g + a*e*g) + 16*c^2*(3*e^2*f^2 - 3*d*e*f*g
+ d^2*g^2)))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(128*c^(7/
2)*e^5) + (Sqrt[c*d^2 - b*d*e + a*e^2]*(e*f - d*g)^3*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^5

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Rubi in Sympy [A]  time = 153.488, size = 578, normalized size = 1.09 \[ - \frac{5 b g^{3} \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{24 c^{2} e} + \frac{b g^{2} \left (b + 2 c x\right ) \left (d g - 3 e f\right ) \sqrt{a + b x + c x^{2}}}{8 c^{2} e^{2}} - \frac{b g^{2} \left (- 4 a c + b^{2}\right ) \left (d g - 3 e f\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{5}{2}} e^{2}} - \frac{\left (d g - e f\right )^{3} \sqrt{a + b x + c x^{2}}}{e^{4}} + \frac{\left (d g - e f\right )^{3} \sqrt{a e^{2} - b d e + c d^{2}} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{5}} + \frac{g^{3} x \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{4 c e} - \frac{g^{2} \left (d g - 3 e f\right ) \left (a + b x + c x^{2}\right )^{\frac{3}{2}}}{3 c e^{2}} + \frac{g \left (b + 2 c x\right ) \sqrt{a + b x + c x^{2}} \left (d^{2} g^{2} - 3 d e f g + 3 e^{2} f^{2}\right )}{4 c e^{3}} + \frac{g^{3} \left (b + 2 c x\right ) \left (- 4 a c + 5 b^{2}\right ) \sqrt{a + b x + c x^{2}}}{64 c^{3} e} - \frac{\left (b e - 2 c d\right ) \left (d g - e f\right )^{3} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 \sqrt{c} e^{5}} - \frac{g \left (- 4 a c + b^{2}\right ) \left (d^{2} g^{2} - 3 d e f g + 3 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{3}{2}} e^{3}} - \frac{g^{3} \left (- 4 a c + b^{2}\right ) \left (- 4 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{128 c^{\frac{7}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**3*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)

[Out]

-5*b*g**3*(a + b*x + c*x**2)**(3/2)/(24*c**2*e) + b*g**2*(b + 2*c*x)*(d*g - 3*e*
f)*sqrt(a + b*x + c*x**2)/(8*c**2*e**2) - b*g**2*(-4*a*c + b**2)*(d*g - 3*e*f)*a
tanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(16*c**(5/2)*e**2) - (d*g -
e*f)**3*sqrt(a + b*x + c*x**2)/e**4 + (d*g - e*f)**3*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**5 + g**3*x*(a + b*x + c*x**2)**(3/2)/(4*c*e) - g**2*(d*g
- 3*e*f)*(a + b*x + c*x**2)**(3/2)/(3*c*e**2) + g*(b + 2*c*x)*sqrt(a + b*x + c*x
**2)*(d**2*g**2 - 3*d*e*f*g + 3*e**2*f**2)/(4*c*e**3) + g**3*(b + 2*c*x)*(-4*a*c
+ 5*b**2)*sqrt(a + b*x + c*x**2)/(64*c**3*e) - (b*e - 2*c*d)*(d*g - e*f)**3*ata
nh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*sqrt(c)*e**5) - g*(-4*a*c
+ b**2)*(d**2*g**2 - 3*d*e*f*g + 3*e**2*f**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(
a + b*x + c*x**2)))/(8*c**(3/2)*e**3) - g**3*(-4*a*c + b**2)*(-4*a*c + 5*b**2)*a
tanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(128*c**(7/2)*e)

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Mathematica [A]  time = 1.92324, size = 553, normalized size = 1.04 \[ \frac{\frac{3 \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-16 c^2 e^2 g \left (a^2 e^2 g^2+2 a b e g (3 e f-d g)+b^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )+8 b^2 c e^3 g^2 (3 a e g-b d g+3 b e f)+64 c^3 e \left (a e g \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )+b (e f-d g)^3\right )-5 b^4 e^4 g^3+128 c^4 d (d g-e f)^3\right )}{c^{7/2}}+\frac{2 e \sqrt{a+x (b+c x)} \left (8 c^2 e g \left (a e g (3 e (8 f+g x)-8 d g)+b \left (6 d^2 g^2-2 d e g (9 f+g x)+e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )-2 b c e^2 g^2 (26 a e g+b (-12 d g+36 e f+5 e g x))+15 b^3 e^3 g^3+16 c^3 \left (-12 d^3 g^3+6 d^2 e g^2 (6 f+g x)-2 d e^2 g \left (18 f^2+9 f g x+2 g^2 x^2\right )+3 e^3 \left (4 f^3+6 f^2 g x+4 f g^2 x^2+g^3 x^3\right )\right )\right )}{c^3}+384 (e f-d g)^3 \log (d+e x) \sqrt{e (a e-b d)+c d^2}+384 (d g-e f)^3 \sqrt{e (a e-b d)+c d^2} \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{384 e^5} \]

Antiderivative was successfully verified.

[In]  Integrate[((f + g*x)^3*Sqrt[a + b*x + c*x^2])/(d + e*x),x]

[Out]

((2*e*Sqrt[a + x*(b + c*x)]*(15*b^3*e^3*g^3 - 2*b*c*e^2*g^2*(26*a*e*g + b*(36*e*
f - 12*d*g + 5*e*g*x)) + 16*c^3*(-12*d^3*g^3 + 6*d^2*e*g^2*(6*f + g*x) - 2*d*e^2
*g*(18*f^2 + 9*f*g*x + 2*g^2*x^2) + 3*e^3*(4*f^3 + 6*f^2*g*x + 4*f*g^2*x^2 + g^3
*x^3)) + 8*c^2*e*g*(a*e*g*(-8*d*g + 3*e*(8*f + g*x)) + b*(6*d^2*g^2 - 2*d*e*g*(9
*f + g*x) + e^2*(18*f^2 + 6*f*g*x + g^2*x^2)))))/c^3 + 384*Sqrt[c*d^2 + e*(-(b*d
) + a*e)]*(e*f - d*g)^3*Log[d + e*x] + (3*(-5*b^4*e^4*g^3 + 128*c^4*d*(-(e*f) +
d*g)^3 + 8*b^2*c*e^3*g^2*(3*b*e*f - b*d*g + 3*a*e*g) - 16*c^2*e^2*g*(a^2*e^2*g^2
+ 2*a*b*e*g*(3*e*f - d*g) + b^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)) + 64*c^3*e*(
b*(e*f - d*g)^3 + a*e*g*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2)))*Log[b + 2*c*x + 2*Sq
rt[c]*Sqrt[a + x*(b + c*x)]])/c^(7/2) + 384*Sqrt[c*d^2 + e*(-(b*d) + a*e)]*(-(e*
f) + d*g)^3*Log[-(b*d) + 2*a*e - 2*c*d*x + b*e*x + 2*Sqrt[c*d^2 + e*(-(b*d) + a*
e)]*Sqrt[a + x*(b + c*x)]])/(384*e^5)

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Maple [B]  time = 0.048, size = 3941, normalized size = 7.4 \[ \text{output too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^3*(c*x^2+b*x+a)^(1/2)/(e*x+d),x)

[Out]

-1/e^4*((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^3*g^3
+3/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*f*g^2-3/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^2*g-3/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*f*g^2+3/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^2*g+1/4*g^3/e^2*b/c^(3/2)*ln((1/2*b
+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a*d-3/4*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*f+3/2/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*f*g
^2-3/2/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^2*g-3/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*f*g^2+3/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^2*g-3
/4*g^2/e^2*d*f/c*(c*x^2+b*x+a)^(1/2)*b-3/2*g^2/e^2*d*f/c^(1/2)*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))*a+3/8*g^2/e^2*d*f/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x
^2+b*x+a)^(1/2))*b^2+1/4*g^3/e^2*b/c*(c*x^2+b*x+a)^(1/2)*x*d-3/4*g^2/e*b/c*(c*x^
2+b*x+a)^(1/2)*x*f+g^2/e*(c*x^2+b*x+a)^(3/2)/c*f-5/128*g^3/e*b^4/c^(7/2)*ln((1/2
*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+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^3+1/e^5*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^4*g^3-1/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^3-1/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^3-1/8*g^3/e/c^(
3/2)*a^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))+3/2*g/e*f^2*(c*x^2+b*x+a)^(
1/2)*x+1/2*g^3/e^3*d^2*(c*x^2+b*x+a)^(1/2)*x-1/3*g^3/e^2*(c*x^2+b*x+a)^(3/2)/c*d
+3/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*f*g
^2-3/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^2
*g+1/4*g^3/e*x*(c*x^2+b*x+a)^(3/2)/c-5/24*g^3/e*b/c^2*(c*x^2+b*x+a)^(3/2)+5/64*g
^3/e*b^3/c^3*(c*x^2+b*x+a)^(1/2)-1/8*g^3/e/c*a*(c*x^2+b*x+a)^(1/2)*x+3/16*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))*f-1/16*g^3/e^2*b^3/c^(5/
2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d+1/8*g^3/e^2*b^2/c^2*(c*x^2+b*x+
a)^(1/2)*d-3/8*g^2/e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*f+3/16*g^3/e*b^2/c^(5/2)*ln((1/
2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+5/32*g^3/e*b^2/c^2*(c*x^2+b*x+a)^(1/2)*x
+1/4*g^3/e^3*d^2/c*(c*x^2+b*x+a)^(1/2)*b+1/2*g^3/e^3*d^2/c^(1/2)*ln((1/2*b+c*x)/
c^(1/2)+(c*x^2+b*x+a)^(1/2))*a-1/8*g^3/e^3*d^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c
*x^2+b*x+a)^(1/2))*b^2+3/4*g/e*f^2/c*(c*x^2+b*x+a)^(1/2)*b+3/2*g/e*f^2/c^(1/2)*l
n((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*a+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^3-3/8*g/e*f^2/c^(3/2)*ln((1/2*b+c*x)/c^(1/2
)+(c*x^2+b*x+a)^(1/2))*b^2-1/16*g^3/e/c^2*a*(c*x^2+b*x+a)^(1/2)*b-3/2*g^2/e^2*d*
f*(c*x^2+b*x+a)^(1/2)*x-1/2/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)*b*d^3*g^3-3/
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*f*g^2+3/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^2*g+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))*a*d^3*g^3-1/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))*b*d^4*g^3+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^3+1/e^6/((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^5*g^3-1/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^3

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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)^3/(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)^3/(e*x + d),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (f + g x\right )^{3} \sqrt{a + b x + c x^{2}}}{d + e x}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**3*(c*x**2+b*x+a)**(1/2)/(e*x+d),x)

[Out]

Integral((f + g*x)**3*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)^3/(e*x + d),x, algorithm="giac")

[Out]

Exception raised: TypeError