3.1587 \(\int \frac{b+2 c x}{(d+e x)^2 \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=248 \[ -\frac{e \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}}-\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) (d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}+\frac{3 e^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2} \]

[Out]

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

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Rubi [A]  time = 0.685731, antiderivative size = 248, 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 \[ -\frac{e \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{5/2}}+\frac{3 e^2 \sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )^2}-\frac{2 (-b e+c d-c e x)}{(d+e x) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 135.425, size = 209, normalized size = 0.84 \[ - \frac{3 e^{2} \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )^{2}} + \frac{e \left (- 4 a c e^{2} + 3 b^{2} e^{2} - 8 b c d e + 8 c^{2} d^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{5}{2}}} + \frac{2 \left (b e - c d + c e x\right )}{\left (d + e x\right ) \sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 1.25503, size = 286, normalized size = 1.15 \[ \frac{1}{2} \left (\frac{e \log (d+e x) \left (4 c e (a e+2 b d)-3 b^2 e^2-8 c^2 d^2\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{e \left (-4 c e (a e+2 b d)+3 b^2 e^2+8 c^2 d^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{5/2}}+\frac{2 c e \left (2 a e (2 d+e x)+b \left (4 d^2+4 d e x-3 e^2 x^2\right )\right )-2 b e^2 (a e+2 b d+3 b e x)-4 c^2 d \left (d^2-d e x-3 e^2 x^2\right )}{(d+e x) \sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )^2}\right ) \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.022, size = 2293, normalized size = 9.3 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-2*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))+2*c/(a*e^2-b*d*e+c*d^
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)+6/(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^2*d^2-6*c/(a*e^2-b*d*e+c*d^2)/(
4*a*c-b^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^2
+3/2*e^2/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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^4+3/2*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))*b^2+2/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)*c*d+6*e/(a*e^2-b*d*e+c*d^2)^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*c*d+18/(a*e^2-b*d*e+c*d
^2)^2/(4*a*c-b^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^2*c^2*d^2-12*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*b+36/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*b*c^3*d^2-
24/e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*c^4*d^3-9*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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^3*c*d-6*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*d-12/e/(a*e^2-b*d*e+c*d^2)^2/
(4*a*c-b^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*
c^3*d^3+24/e*c^3/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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)*x*d+12/e*c^2/(a*e^2-b*d*e+c*d^2)/(4*a*c-b^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*d+3*e^2/(a*e
^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*b^3*c-1/(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)*b-6/(a*e^2-b*d*e+c*d^2)^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^2*d^2-3/2*e^2/(a*e^2-b*d*e
+c*d^2)^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^2-
18*e/(a*e^2-b*d*e+c*d^2)^2/(4*a*c-b^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)*x*b^2*c^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((2*c*x + b)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.65738, 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)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^2),x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((2*c*x + b)/((c*x^2 + b*x + a)^(3/2)*(e*x + d)^2), x)