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

Optimal. Leaf size=270 \[ \frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac{3 g^2 \sqrt{a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac{(e f-d g)^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3 \sqrt{a e^2-b d e+c d^2}}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2} \]

[Out]

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

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Rubi [A]  time = 1.239, antiderivative size = 270, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.172 \[ \frac{g \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{8 c^{5/2} e^3}+\frac{3 g^2 \sqrt{a+b x+c x^2} (-b e g-2 c d g+4 c e f)}{4 c^2 e^2}+\frac{(e f-d g)^3 \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{e^3 \sqrt{a e^2-b d e+c d^2}}+\frac{g^3 (d+e x) \sqrt{a+b x+c x^2}}{2 c e^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 94.1935, size = 332, normalized size = 1.23 \[ - \frac{3 b g^{3} \sqrt{a + b x + c x^{2}}}{4 c^{2} e} + \frac{b g^{2} \left (d g - 3 e f\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{2 c^{\frac{3}{2}} e^{2}} + \frac{\left (d g - e f\right )^{3} \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{e^{3} \sqrt{a e^{2} - b d e + c d^{2}}} + \frac{g^{3} x \sqrt{a + b x + c x^{2}}}{2 c e} - \frac{g^{2} \left (d g - 3 e f\right ) \sqrt{a + b x + c x^{2}}}{c e^{2}} + \frac{g \left (d^{2} g^{2} - 3 d e f g + 3 e^{2} f^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{\sqrt{c} e^{3}} + \frac{g^{3} \left (- 4 a c + 3 b^{2}\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{8 c^{\frac{5}{2}} e} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*b*g**3*sqrt(a + b*x + c*x**2)/(4*c**2*e) + b*g**2*(d*g - 3*e*f)*atanh((b + 2*
c*x)/(2*sqrt(c)*sqrt(a + b*x + c*x**2)))/(2*c**(3/2)*e**2) + (d*g - e*f)**3*atan
h((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**3*sqrt(a*e**2 - b*d*e + c*d**2)) + g**3*x*sqrt(a + b*x + c*x**2)
/(2*c*e) - g**2*(d*g - 3*e*f)*sqrt(a + b*x + c*x**2)/(c*e**2) + g*(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)))/(s
qrt(c)*e**3) + g**3*(-4*a*c + 3*b**2)*atanh((b + 2*c*x)/(2*sqrt(c)*sqrt(a + b*x
+ c*x**2)))/(8*c**(5/2)*e)

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Mathematica [A]  time = 0.633402, size = 269, normalized size = 1. \[ \frac{\frac{g \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right ) \left (-4 c e g (a e g-b d g+3 b e f)+3 b^2 e^2 g^2+8 c^2 \left (d^2 g^2-3 d e f g+3 e^2 f^2\right )\right )}{c^{5/2}}+\frac{2 e g^2 \sqrt{a+x (b+c x)} (2 c (-2 d g+6 e f+e g x)-3 b e g)}{c^2}+\frac{8 (e f-d g)^3 \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}+\frac{8 (d g-e f)^3 \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}}{8 e^3} \]

Antiderivative was successfully verified.

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

[Out]

((2*e*g^2*Sqrt[a + x*(b + c*x)]*(-3*b*e*g + 2*c*(6*e*f - 2*d*g + e*g*x)))/c^2 +
(8*(e*f - d*g)^3*Log[d + e*x])/Sqrt[c*d^2 + e*(-(b*d) + a*e)] + (g*(3*b^2*e^2*g^
2 - 4*c*e*g*(3*b*e*f - b*d*g + a*e*g) + 8*c^2*(3*e^2*f^2 - 3*d*e*f*g + d^2*g^2))
*Log[b + 2*c*x + 2*Sqrt[c]*Sqrt[a + x*(b + c*x)]])/c^(5/2) + (8*(-(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)]])/Sqrt[c*d^2 + e*(-(b*d) + a*e)])/(8*e^3)

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Maple [B]  time = 0.022, size = 1007, 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)^3/(e*x+d)/(c*x^2+b*x+a)^(1/2),x)

[Out]

g^3/e^3*d^2*ln((1/2*b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-g^3/e^2/c*(c*x^2
+b*x+a)^(1/2)*d+3*g^2/e/c*(c*x^2+b*x+a)^(1/2)*f+1/2*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-3/2*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+1/2*g^3/e*x/c*(c*x^2+b*x+a)^(1/2)-3/4*g^3/e*b/c^2*(c*x^2+b
*x+a)^(1/2)+3/8*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))-1/
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))+3*g/e*f^2*ln((1/2*
b+c*x)/c^(1/2)+(c*x^2+b*x+a)^(1/2))/c^(1/2)-3*g^2/e^2*d*f*ln((1/2*b+c*x)/c^(1/2)
+(c*x^2+b*x+a)^(1/2))/c^(1/2)+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))*d^3*g^3-3/
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))*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))*d*f^2*g-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^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((g*x + f)^3/(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)^3/(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. \[ \int \frac{\left (f + g x\right )^{3}}{\left (d + e x\right ) \sqrt{a + b x + c x^{2}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((f + g*x)**3/((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)^3/(sqrt(c*x^2 + b*x + a)*(e*x + d)),x, algorithm="giac")

[Out]

Exception raised: TypeError