3.870 \(\int \frac{(f+g x)^4}{(d+e x) \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=431 \[ -\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4 \sqrt{a e^2-b d e+c d^2}}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3} \]

[Out]

(g^2*(15*b^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^
2 - 36*d*e*f*g + 11*d^2*g^2))*Sqrt[a + b*x + c*x^2])/(24*c^3*e^3) + (g^3*(24*c*e
*f - 14*c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[a + b*x + c*x^2])/(12*c^2*e^3) + (g^4*(d
+ e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c*e^3) - (g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(
4*b*e*f - b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 -
d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)
))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2)*e^4) + ((
e*f - d*g)^4*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*Sqrt[c*d^2 - b*d*e + a*e^2])

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Rubi [A]  time = 2.30441, antiderivative size = 431, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ -\frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )-6 b c e^2 g^2 (2 a e g-b d g+4 b e f)+5 b^3 e^3 g^3-16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{16 c^{7/2} e^4}+\frac{g^2 \sqrt{a+b x+c x^2} \left (-4 c e g (4 a e g-7 b d g+18 b e f)+15 b^2 e^2 g^2+4 c^2 \left (11 d^2 g^2-36 d e f g+36 e^2 f^2\right )\right )}{24 c^3 e^3}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2} (-5 b e g-14 c d g+24 c e f)}{12 c^2 e^3}+\frac{(e f-d g)^4 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^4 \sqrt{a e^2-b d e+c d^2}}+\frac{g^4 (d+e x)^2 \sqrt{a+b x+c x^2}}{3 c e^3} \]

Antiderivative was successfully verified.

[In]  Int[(f + g*x)^4/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

(g^2*(15*b^2*e^2*g^2 - 4*c*e*g*(18*b*e*f - 7*b*d*g + 4*a*e*g) + 4*c^2*(36*e^2*f^
2 - 36*d*e*f*g + 11*d^2*g^2))*Sqrt[a + b*x + c*x^2])/(24*c^3*e^3) + (g^3*(24*c*e
*f - 14*c*d*g - 5*b*e*g)*(d + e*x)*Sqrt[a + b*x + c*x^2])/(12*c^2*e^3) + (g^4*(d
+ e*x)^2*Sqrt[a + b*x + c*x^2])/(3*c*e^3) - (g*(5*b^3*e^3*g^3 - 6*b*c*e^2*g^2*(
4*b*e*f - b*d*g + 2*a*e*g) - 16*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 -
d^3*g^3) + 8*c^2*e*g*(a*e*g*(4*e*f - d*g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)
))*ArcTanh[(b + 2*c*x)/(2*Sqrt[c]*Sqrt[a + b*x + c*x^2])])/(16*c^(7/2)*e^4) + ((
e*f - d*g)^4*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*Sqrt[c*d^2 - b*d*e + a*e^2])

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Rubi in Sympy [A]  time = 135.431, size = 524, normalized size = 1.22 \[ \frac{3 b g^{3} \left (d g - 4 e f\right ) \sqrt{a + b x + c x^{2}}}{4 c^{2} e^{2}} - \frac{b g^{2} \left (d^{2} g^{2} - 4 d e f g + 6 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{3}{2}} e^{3}} - \frac{b g^{4} \left (- 12 a c + 5 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{16 c^{\frac{7}{2}} e} - \frac{\left (d g - e f\right )^{4} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{4} \sqrt{a e^{2} - b d e + c d^{2}}} + \frac{g^{4} x^{2} \sqrt{a + b x + c x^{2}}}{3 c e} - \frac{g^{3} x \left (d g - 4 e f\right ) \sqrt{a + b x + c x^{2}}}{2 c e^{2}} + \frac{g^{2} \sqrt{a + b x + c x^{2}} \left (d^{2} g^{2} - 4 d e f g + 6 e^{2} f^{2}\right )}{c e^{3}} + \frac{g^{4} \sqrt{a + b x + c x^{2}} \left (- 4 a c + \frac{15 b^{2}}{4} - \frac{5 b c x}{2}\right )}{6 c^{3} e} - \frac{g \left (d g - 2 e f\right ) \left (d^{2} g^{2} - 2 d e f g + 2 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c} e^{4}} - \frac{g^{3} \left (- 4 a c + 3 b^{2}\right ) \left (d g - 4 e f\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}} e^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

3*b*g**3*(d*g - 4*e*f)*sqrt(a + b*x + c*x**2)/(4*c**2*e**2) - b*g**2*(d**2*g**2
- 4*d*e*f*g + 6*e**2*f**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))
/(2*c**(3/2)*e**3) - b*g**4*(-12*a*c + 5*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt
(a + b*x + c*x**2)))/(16*c**(7/2)*e) - (d*g - e*f)**4*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*sqrt
(a*e**2 - b*d*e + c*d**2)) + g**4*x**2*sqrt(a + b*x + c*x**2)/(3*c*e) - g**3*x*(
d*g - 4*e*f)*sqrt(a + b*x + c*x**2)/(2*c*e**2) + g**2*sqrt(a + b*x + c*x**2)*(d*
*2*g**2 - 4*d*e*f*g + 6*e**2*f**2)/(c*e**3) + g**4*sqrt(a + b*x + c*x**2)*(-4*a*
c + 15*b**2/4 - 5*b*c*x/2)/(6*c**3*e) - g*(d*g - 2*e*f)*(d**2*g**2 - 2*d*e*f*g +
2*e**2*f**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(sqrt(c)*e**
4) - g**3*(-4*a*c + 3*b**2)*(d*g - 4*e*f)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a +
b*x + c*x**2)))/(8*c**(5/2)*e**2)

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Mathematica [A]  time = 1.61875, size = 402, normalized size = 0.93 \[ \frac{\frac{3 g \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-8 c^2 e g \left (a e g (4 e f-d g)+b \left (d^2 g^2-4 d e f g+6 e^2 f^2\right )\right )+6 b c e^2 g^2 (2 a e g-b d g+4 b e f)-5 b^3 e^3 g^3+16 c^3 \left (-d^3 g^3+4 d^2 e f g^2-6 d e^2 f^2 g+4 e^3 f^3\right )\right )}{c^{7/2}}+\frac{2 e g^2 \sqrt{a+x (b+c x)} \left (-2 c e g (8 a e g+b (-9 d g+36 e f+5 e g x))+15 b^2 e^2 g^2+4 c^2 \left (6 d^2 g^2-3 d e g (8 f+g x)+2 e^2 \left (18 f^2+6 f g x+g^2 x^2\right )\right )\right )}{c^3}+\frac{48 (e f-d g)^4 \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}-\frac{48 (e f-d g)^4 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}}{48 e^4} \]

Antiderivative was successfully verified.

[In]  Integrate[(f + g*x)^4/((d + e*x)*Sqrt[a + b*x + c*x^2]),x]

[Out]

((2*e*g^2*Sqrt[a + x*(b + c*x)]*(15*b^2*e^2*g^2 - 2*c*e*g*(8*a*e*g + b*(36*e*f -
9*d*g + 5*e*g*x)) + 4*c^2*(6*d^2*g^2 - 3*d*e*g*(8*f + g*x) + 2*e^2*(18*f^2 + 6*
f*g*x + g^2*x^2))))/c^3 + (48*(e*f - d*g)^4*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d)
+ a*e)] + (3*g*(-5*b^3*e^3*g^3 + 6*b*c*e^2*g^2*(4*b*e*f - b*d*g + 2*a*e*g) + 16
*c^3*(4*e^3*f^3 - 6*d*e^2*f^2*g + 4*d^2*e*f*g^2 - d^3*g^3) - 8*c^2*e*g*(a*e*g*(4
*e*f - d*g) + b*(6*e^2*f^2 - 4*d*e*f*g + d^2*g^2)))*Log[b + 2*c*x + 2*Sqrt[c]*Sq
rt[a + x*(b + c*x)]])/c^(7/2) - (48*(e*f - d*g)^4*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)]])/Sqrt[c*d^2 + e
*(-(b*d) + a*e)])/(48*e^4)

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Maple [B]  time = 0.035, size = 1597, normalized size = 3.7 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((g*x+f)^4/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

-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))*f^4+6*g^2/e/c*(c*x^2+b*x+a)^(1/2)*f^2+1/3
*g^4/e*x^2/c*(c*x^2+b*x+a)^(1/2)+5/8*g^4/e*b^2/c^3*(c*x^2+b*x+a)^(1/2)-5/16*g^4/
e*b^3/c^(7/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))-2/3*g^4/e/c^2*a*(c*x^2
+b*x+a)^(1/2)-g^4/e^4*d^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)+4*
g/e*f^3*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-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))*d^4*g^4+g^4/e^3/c*(c*x^2+b*x+a)^(1/2)*d^2+2*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))*d*f-2*g^3/e*a/c^(3/2)*ln((1/2*b+c*
x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f-4*g^3/e^2/c*(c*x^2+b*x+a)^(1/2)*d*f-1/2*g^4/e^
3*b/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d^2-3*g^2/e*b/c^(3/2)*ln
((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*f^2-1/2*g^4/e^2*x/c*(c*x^2+b*x+a)^(1/2
)*d+2*g^3/e*x/c*(c*x^2+b*x+a)^(1/2)*f+3/4*g^4/e^2*b/c^2*(c*x^2+b*x+a)^(1/2)*d-3*
g^3/e*b/c^2*(c*x^2+b*x+a)^(1/2)*f-3/8*g^4/e^2*b^2/c^(5/2)*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))*d+3/2*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))*f-6/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))*d^2*f^2*g^2+4/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))*d*f^3*g+4/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))*d^3*f*g^
3-5/12*g^4/e*b/c^2*x*(c*x^2+b*x+a)^(1/2)+3/4*g^4/e*b/c^(5/2)*a*ln((1/2*b+c*x)/c^
(1/2)+(c*x^2+b*x+a)^(1/2))-6*g^2/e^2*d*f^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^
(1/2))/c^(1/2)+4*g^3/e^3*d^2*f*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/
2)+1/2*g^4/e^2*a/c^(3/2)*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))*d

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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((g*x + f)^4/(sqrt(c*x^2 + b*x + a)*(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((g*x + f)^4/(sqrt(c*x^2 + b*x + a)*(e*x + d)),x, algorithm="fricas")

[Out]

Timed out

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (f + g x\right )^{4}}{\left (d + e x\right ) \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((g*x+f)**4/(e*x+d)/(c*x**2+b*x+a)**(1/2),x)

[Out]

Integral((f + g*x)**4/((d + e*x)*sqrt(a + b*x + c*x**2)), 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((g*x + f)^4/(sqrt(c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError