3.2368 \(\int \frac{1}{(d+e x)^2 \sqrt{a+b x+c x^2}} \, dx\)

Optimal. Leaf size=134 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e \sqrt{a+b x+c x^2}}{(d+e x) \left (a e^2-b d e+c d^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.216836, antiderivative size = 134, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ \frac{(2 c d-b e) \tanh ^{-1}\left (\frac{-2 a e+x (2 c d-b e)+b d}{2 \sqrt{a+b x+c x^2} \sqrt{a e^2-b d e+c d^2}}\right )}{2 \left (a e^2-b d e+c d^2\right )^{3/2}}-\frac{e \sqrt{a+b x+c x^2}}{(d+e x) \left (a e^2-b d e+c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 31.5576, size = 116, normalized size = 0.87 \[ - \frac{e \sqrt{a + b x + c x^{2}}}{\left (d + e x\right ) \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (\frac{b e}{2} - c d\right ) \operatorname{atanh}{\left (\frac{2 a e - b d + x \left (b e - 2 c d\right )}{2 \sqrt{a + b x + c x^{2}} \sqrt{a e^{2} - b d e + c d^{2}}} \right )}}{\left (a e^{2} - b d e + c d^{2}\right )^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.239534, size = 159, normalized size = 1.19 \[ \frac{-2 e \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}-(d+e x) (2 c d-b e) \log \left (2 \sqrt{a+x (b+c x)} \sqrt{e (a e-b d)+c d^2}+2 a e-b d+b e x-2 c d x\right )+(d+e x) (2 c d-b e) \log (d+e x)}{2 (d+e x) \left (e (a e-b d)+c d^2\right )^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.017, size = 432, normalized size = 3.2 \[ -{\frac{1}{a{e}^{2}-bde+c{d}^{2}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \left ({\frac{d}{e}}+x \right ) ^{-1}}+{\frac{b}{2\,a{e}^{2}-2\,bde+2\,c{d}^{2}}\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}}-{\frac{cd}{e \left ( a{e}^{2}-bde+c{d}^{2} \right ) }\ln \left ({1 \left ( 2\,{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+2\,\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}\sqrt{c \left ({\frac{d}{e}}+x \right ) ^{2}+{\frac{be-2\,cd}{e} \left ({\frac{d}{e}}+x \right ) }+{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}} \right ) \left ({\frac{d}{e}}+x \right ) ^{-1}} \right ){\frac{1}{\sqrt{{\frac{a{e}^{2}-bde+c{d}^{2}}{{e}^{2}}}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/(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)+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-1/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))*c*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(1/(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 = 0.466834, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \, \sqrt{c d^{2} - b d e + a e^{2}} \sqrt{c x^{2} + b x + a} e +{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \log \left (\frac{{\left (8 \, a b d e - 8 \, a^{2} e^{2} -{\left (b^{2} + 4 \, a c\right )} d^{2} -{\left (8 \, c^{2} d^{2} - 8 \, b c d e +{\left (b^{2} + 4 \, a c\right )} e^{2}\right )} x^{2} - 2 \,{\left (4 \, b c d^{2} + 4 \, a b e^{2} -{\left (3 \, b^{2} + 4 \, a c\right )} d e\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}} + 4 \,{\left (b c d^{3} + 3 \, a b d e^{2} - 2 \, a^{2} e^{3} -{\left (b^{2} + 2 \, a c\right )} d^{2} e +{\left (2 \, c^{2} d^{3} - 3 \, b c d^{2} e - a b e^{3} +{\left (b^{2} + 2 \, a c\right )} d e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right )}{4 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{c d^{2} - b d e + a e^{2}}}, -\frac{2 \, \sqrt{-c d^{2} + b d e - a e^{2}} \sqrt{c x^{2} + b x + a} e +{\left (2 \, c d^{2} - b d e +{\left (2 \, c d e - b e^{2}\right )} x\right )} \arctan \left (-\frac{\sqrt{-c d^{2} + b d e - a e^{2}}{\left (b d - 2 \, a e +{\left (2 \, c d - b e\right )} x\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{c x^{2} + b x + a}}\right )}{2 \,{\left (c d^{3} - b d^{2} e + a d e^{2} +{\left (c d^{2} e - b d e^{2} + a e^{3}\right )} x\right )} \sqrt{-c d^{2} + b d e - a e^{2}}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/((d + e*x)**2*sqrt(a + b*x + c*x**2)), x)

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GIAC/XCAS [A]  time = 11.3923, size = 859, normalized size = 6.41 \[ \frac{{\left (2 \, c^{\frac{3}{2}} d^{2} - 2 \, b \sqrt{c} d e + 2 \, \sqrt{c d^{2} - b d e + a e^{2}} c d{\rm ln}\left ({\left | 2 \, c^{\frac{3}{2}} d^{2} - 2 \, b \sqrt{c} d e - 2 \, \sqrt{c d^{2} - b d e + a e^{2}} c d + 2 \, a \sqrt{c} e^{2} + \sqrt{c d^{2} - b d e + a e^{2}} b e \right |}\right ) - \sqrt{c d^{2} - b d e + a e^{2}} b e{\rm ln}\left ({\left | 2 \, c^{\frac{3}{2}} d^{2} - 2 \, b \sqrt{c} d e - 2 \, \sqrt{c d^{2} - b d e + a e^{2}} c d + 2 \, a \sqrt{c} e^{2} + \sqrt{c d^{2} - b d e + a e^{2}} b e \right |}\right ) + 2 \, a \sqrt{c} e^{2}\right )}{\rm sign}\left (\frac{1}{x e + d}\right )}{2 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2} + 2 \, a c d^{2} e^{2} - 2 \, a b d e^{3} + a^{2} e^{4}\right )}} - \frac{\sqrt{c d^{2} - b d e + a e^{2}}{\left (2 \, c d e - b e^{2}\right )}{\rm ln}\left ({\left | 2 \,{\left (c d^{2} - b d e + a e^{2}\right )}{\left (\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}} + \frac{\sqrt{c d^{2} e^{2} - b d e^{3} + a e^{4}} e^{\left (-1\right )}}{x e + d}\right )} - \sqrt{c d^{2} - b d e + a e^{2}}{\left (2 \, c d - b e\right )} \right |}\right )}{2 \,{\left (c^{2} d^{4} e - 2 \, b c d^{3} e^{2} + b^{2} d^{2} e^{3} + 2 \, a c d^{2} e^{3} - 2 \, a b d e^{4} + a^{2} e^{5}\right )}{\rm sign}\left (\frac{1}{x e + d}\right )} - \frac{\sqrt{c - \frac{2 \, c d}{x e + d} + \frac{c d^{2}}{{\left (x e + d\right )}^{2}} + \frac{b e}{x e + d} - \frac{b d e}{{\left (x e + d\right )}^{2}} + \frac{a e^{2}}{{\left (x e + d\right )}^{2}}}}{c d^{2}{\rm sign}\left (\frac{1}{x e + d}\right ) - b d e{\rm sign}\left (\frac{1}{x e + d}\right ) + a e^{2}{\rm sign}\left (\frac{1}{x e + d}\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/(sqrt(c*x^2 + b*x + a)*(e*x + d)^2),x, algorithm="giac")

[Out]

1/2*(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e + 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d*ln(abs(
2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqrt(c
)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) - sqrt(c*d^2 - b*d*e + a*e^2)*b*e*ln(a
bs(2*c^(3/2)*d^2 - 2*b*sqrt(c)*d*e - 2*sqrt(c*d^2 - b*d*e + a*e^2)*c*d + 2*a*sqr
t(c)*e^2 + sqrt(c*d^2 - b*d*e + a*e^2)*b*e)) + 2*a*sqrt(c)*e^2)*sign(1/(x*e + d)
)/(c^2*d^4 - 2*b*c*d^3*e + b^2*d^2*e^2 + 2*a*c*d^2*e^2 - 2*a*b*d*e^3 + a^2*e^4)
- 1/2*sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d*e - b*e^2)*ln(abs(2*(c*d^2 - b*d*e + a*
e^2)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e
+ d)^2 + a*e^2/(x*e + d)^2) + sqrt(c*d^2*e^2 - b*d*e^3 + a*e^4)*e^(-1)/(x*e + d)
) - sqrt(c*d^2 - b*d*e + a*e^2)*(2*c*d - b*e)))/((c^2*d^4*e - 2*b*c*d^3*e^2 + b^
2*d^2*e^3 + 2*a*c*d^2*e^3 - 2*a*b*d*e^4 + a^2*e^5)*sign(1/(x*e + d))) - sqrt(c -
2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2 + a*e^2
/(x*e + d)^2)/(c*d^2*sign(1/(x*e + d)) - b*d*e*sign(1/(x*e + d)) + a*e^2*sign(1/
(x*e + d)))