3.1578 \(\int \frac{b+2 c x}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=141 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.291157, antiderivative size = 141, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.107 \[ \frac{\sqrt{a+b x+c x^2} (2 c d-b e)}{(d+e x) \left (a e^2-b d e+c d^2\right )}-\frac{e \left (b^2-4 a c\right ) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 35.7617, size = 126, normalized size = 0.89 \[ \frac{e \left (- 4 a c + b^{2}\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} - \frac{\left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} \]

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)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.506476, size = 176, normalized size = 1.25 \[ \frac{\frac{e \left (b^2-4 a c\right ) \log (d+e x)}{\sqrt{e (a e-b d)+c d^2}}+\frac{e \left (4 a c-b^2\right ) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )}{\sqrt{e (a e-b d)+c d^2}}+\frac{2 \sqrt{a+x (b+c x)} (b e-2 c d)}{d+e x}}{2 \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.015, size = 860, normalized size = 6.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-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+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+1/2/(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))*b^2-2/e/(a*e^2-b*d*e+c*d^2)/((a*e^2-b*d*e+c*d^2)/e^2)^(1/2)*l
n((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+2/e^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^2*d^2-2*c/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*(
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))

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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)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.546653, size = 1, normalized size = 0.01 \[ \left [\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a}{\left (2 \, c d - b e\right )} -{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} - 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, \frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a}{\left (2 \, c d - b e\right )} +{\left ({\left (b^{2} - 4 \, a c\right )} e^{2} x +{\left (b^{2} - 4 \, a c\right )} d e\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{2 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

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)^2),x, algorithm="giac")

[Out]

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