3.1579 \(\int \frac{b+2 c x}{(d+e x)^3 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=225 \[ \frac{\sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{3 e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.63905, antiderivative size = 225, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{\sqrt{a+b x+c x^2} \left (-4 c e (2 a e+b d)+3 b^2 e^2+4 c^2 d^2\right )}{4 (d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{3 e \left (b^2-4 a c\right ) (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{8 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{2 (d+e x)^2 \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 121.967, size = 206, normalized size = 0.92 \[ - \frac{3 e \left (- 4 a c + b^{2}\right ) \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{8 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{\sqrt{a + b x + c x^{2}} \left (- 2 a c e^{2} + \frac{3 b^{2} e^{2}}{4} - b c d e + c^{2} d^{2}\right )}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}} - \frac{\left (\frac{b e}{2} - c d\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right )^{2} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-3*e*(-4*a*c + b**2)*(b*e - 2*c*d)*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)))/(8*(a*e**2 - b*d*e + c*d**2)*
*(5/2)) + sqrt(a + b*x + c*x**2)*(-2*a*c*e**2 + 3*b**2*e**2/4 - b*c*d*e + c**2*d
**2)/((d + e*x)*(a*e**2 - b*d*e + c*d**2)**2) - (b*e/2 - c*d)*sqrt(a + b*x + c*x
**2)/((d + e*x)**2*(a*e**2 - b*d*e + c*d**2))

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Mathematica [A]  time = 0.760509, size = 240, normalized size = 1.07 \[ \frac{-3 e \left (b^2-4 a c\right ) (d+e x)^2 (b e-2 c d) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+3 e \left (b^2-4 a c\right ) (d+e x)^2 (b e-2 c d) \log (d+e x)+2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2} \left (-2 c e (2 a e (d+2 e x)+b d (5 d+2 e x))+b e^2 (-2 a e+5 b d+3 b e x)+4 c^2 d^2 (2 d+e x)\right )}{8 (d+e x)^2 \left (e (a e-b d)+c d^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.019, size = 1588, normalized size = 7.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/2/e/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b
*d*e+c*d^2)/e^2)^(1/2)*b+1/e^2/(a*e^2-b*d*e+c*d^2)/(d/e+x)^2*(c*(d/e+x)^2+(b*e-2
*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c*d+3/4*e/(a*e^2-b*d*e+c*d^2)^2/(
d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*b^2-3/(
a*e^2-b*d*e+c*d^2)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d
^2)/e^2)^(1/2)*b*c*d+3/e/(a*e^2-b*d*e+c*d^2)^2/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/
e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*c^2*d^2-3/8*e/(a*e^2-b*d*e+c*d^2)^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*(d/e+x)
+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d
*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b^3+9/4/(a*e^2-b*d*e+c*d^2)^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*(d/e+x)+2*((a*e^2-b*d*e
+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(
1/2))/(d/e+x))*b^2*c*d-9/2/e/(a*e^2-b*d*e+c*d^2)^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^
2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e
+x))*b*c^2*d^2+3/e^2/(a*e^2-b*d*e+c*d^2)^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)
*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*c^3
*d^3+3/2/e*c/(a*e^2-b*d*e+c*d^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x
)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*b-3/e^2*c^2/(
a*e^2-b*d*e+c*d^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*(d/e+x)+2*((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*(c*(d/e+x)^2+(b*e-2*c*d
)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(1/2))/(d/e+x))*d-2*c/e/(a*e^2-b*d*e+c*d^2)
/(d/e+x)*(c*(d/e+x)^2+(b*e-2*c*d)/e*(d/e+x)+(a*e^2-b*d*e+c*d^2)/e^2)^(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((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.798048, size = 1, normalized size = 0. \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="fricas")

[Out]

[1/16*(4*(8*c^2*d^3 - 10*b*c*d^2*e - 2*a*b*e^3 + (5*b^2 - 4*a*c)*d*e^2 + (4*c^2*
d^2*e - 4*b*c*d*e^2 + (3*b^2 - 8*a*c)*e^3)*x)*sqrt(c*d^2 - b*d*e + a*e^2)*sqrt(c
*x^2 + b*x + a) + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*a*b*c)*d^2*e^2 + (2*(b
^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^2 + 2*(2*(b^2*c - 4*a*c^2)*d^2*e^
2 - (b^3 - 4*a*b*c)*d*e^3)*x)*log(((8*a*b*d*e - 8*a^2*e^2 - (b^2 + 4*a*c)*d^2 -
(8*c^2*d^2 - 8*b*c*d*e + (b^2 + 4*a*c)*e^2)*x^2 - 2*(4*b*c*d^2 + 4*a*b*e^2 - (3*
b^2 + 4*a*c)*d*e)*x)*sqrt(c*d^2 - b*d*e + a*e^2) + 4*(b*c*d^3 + 3*a*b*d*e^2 - 2*
a^2*e^3 - (b^2 + 2*a*c)*d^2*e + (2*c^2*d^3 - 3*b*c*d^2*e - a*b*e^3 + (b^2 + 2*a*
c)*d*e^2)*x)*sqrt(c*x^2 + b*x + a))/(e^2*x^2 + 2*d*e*x + d^2)))/((c^2*d^6 - 2*b*
c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2
*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a*c)*d^2*e^4)*x^2 + 2*(c^2*d^5*e
- 2*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b^2 + 2*a*c)*d^3*e^3)*x)*sqrt(c*
d^2 - b*d*e + a*e^2)), 1/8*(2*(8*c^2*d^3 - 10*b*c*d^2*e - 2*a*b*e^3 + (5*b^2 - 4
*a*c)*d*e^2 + (4*c^2*d^2*e - 4*b*c*d*e^2 + (3*b^2 - 8*a*c)*e^3)*x)*sqrt(-c*d^2 +
b*d*e - a*e^2)*sqrt(c*x^2 + b*x + a) + 3*(2*(b^2*c - 4*a*c^2)*d^3*e - (b^3 - 4*
a*b*c)*d^2*e^2 + (2*(b^2*c - 4*a*c^2)*d*e^3 - (b^3 - 4*a*b*c)*e^4)*x^2 + 2*(2*(b
^2*c - 4*a*c^2)*d^2*e^2 - (b^3 - 4*a*b*c)*d*e^3)*x)*arctan(-1/2*sqrt(-c*d^2 + b*
d*e - a*e^2)*(b*d - 2*a*e + (2*c*d - b*e)*x)/((c*d^2 - b*d*e + a*e^2)*sqrt(c*x^2
+ b*x + a))))/((c^2*d^6 - 2*b*c*d^5*e - 2*a*b*d^3*e^3 + a^2*d^2*e^4 + (b^2 + 2*
a*c)*d^4*e^2 + (c^2*d^4*e^2 - 2*b*c*d^3*e^3 - 2*a*b*d*e^5 + a^2*e^6 + (b^2 + 2*a
*c)*d^2*e^4)*x^2 + 2*(c^2*d^5*e - 2*b*c*d^4*e^2 - 2*a*b*d^2*e^4 + a^2*d*e^5 + (b
^2 + 2*a*c)*d^3*e^3)*x)*sqrt(-c*d^2 + b*d*e - a*e^2))]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral((b + 2*c*x)/((d + e*x)**3*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 0.608652, size = 4, normalized size = 0.02 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x + b)/(sqrt(c*x^2 + b*x + a)*(e*x + d)^3),x, algorithm="giac")

[Out]

sage0*x