3.1586 \(\int \frac{b+2 c x}{(d+e x) \left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=166 \[ -\frac{2 \left (\left (b^2-4 a c\right ) (c d-b e)-c e x \left (b^2-4 a c\right )\right )}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \]

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

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Rubi [A]  time = 0.342678, antiderivative size = 166, 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 \[ -\frac{2 (-b e+c d-c e x)}{\sqrt{a+b x+c x^2} \left (a e^2-b d e+c d^2\right )}-\frac{e (2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{\left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

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Rubi in Sympy [A]  time = 57.467, size = 128, normalized size = 0.77 \[ - \frac{e \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} + \frac{2 \left (b e - c d + c e x\right )}{\sqrt{a + b x + c x^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-e*(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)))/(a*e**2 - b*d*e + c*d**2)**(3/2) + 2*(b*e - c*d
+ c*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 = 0.891757, size = 167, normalized size = 1.01 \[ \frac{2 (b e-c d+c e x)}{\sqrt{a+x (b+c x)} \left (e (a e-b d)+c d^2\right )}+\frac{e (b e-2 c d) \log (d+e x)}{\left (e (a e-b d)+c d^2\right )^{3/2}}+\frac{e (2 c d-b e) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.013, size = 1084, normalized size = 6.5 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.665293, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}}{\left (c e x - c d + b e\right )} \sqrt{c x^{2} + b x + a} -{\left (2 \, a c d e - a b e^{2} +{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} +{\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} - 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{2 \,{\left (a c d^{2} - a b d e + a^{2} e^{2} +{\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} +{\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, \frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}}{\left (c e x - c d + b e\right )} \sqrt{c x^{2} + b x + a} +{\left (2 \, a c d e - a b e^{2} +{\left (2 \, c^{2} d e - b c e^{2}\right )} x^{2} +{\left (2 \, b c d e - b^{2} e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{{\left (a c d^{2} - a b d e + a^{2} e^{2} +{\left (c^{2} d^{2} - b c d e + a c e^{2}\right )} x^{2} +{\left (b c d^{2} - b^{2} d e + a b e^{2}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

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)),x, algorithm="fricas")

[Out]

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.287484, size = 676, normalized size = 4.07 \[ \frac{2 \,{\left (\frac{{\left (b^{2} c^{2} d^{2} e - 4 \, a c^{3} d^{2} e - b^{3} c d e^{2} + 4 \, a b c^{2} d e^{2} + a b^{2} c e^{3} - 4 \, a^{2} c^{2} e^{3}\right )} x}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}} - \frac{b^{2} c^{2} d^{3} - 4 \, a c^{3} d^{3} - 2 \, b^{3} c d^{2} e + 8 \, a b c^{2} d^{2} e + b^{4} d e^{2} - 3 \, a b^{2} c d e^{2} - 4 \, a^{2} c^{2} d e^{2} - a b^{3} e^{3} + 4 \, a^{2} b c e^{3}}{b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} - 2 \, b^{3} c d^{3} e + 8 \, a b c^{2} d^{3} e + b^{4} d^{2} e^{2} - 2 \, a b^{2} c d^{2} e^{2} - 8 \, a^{2} c^{2} d^{2} e^{2} - 2 \, a b^{3} d e^{3} + 8 \, a^{2} b c d e^{3} + a^{2} b^{2} e^{4} - 4 \, a^{3} c e^{4}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{2 \,{\left (2 \, c d e - b e^{2}\right )} \arctan \left (-\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e - a e^{2}}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-c d^{2} + b d e - a e^{2}}} \]

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)),x, algorithm="giac")

[Out]

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