3.2373 \(\int \frac{(d+e x)^2}{\left (a+b x+c x^2\right )^{3/2}} \, dx\)

Optimal. Leaf size=129 \[ \frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.212833, antiderivative size = 129, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182 \[ \frac{2 e \sqrt{a+b x+c x^2} (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac{2 (d+e x) (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \sqrt{a+b x+c x^2}}+\frac{e^2 \tanh ^{-1}\left (\frac{b+2 c x}{2 \sqrt{c} \sqrt{a+b x+c x^2}}\right )}{c^{3/2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 28.6245, size = 119, normalized size = 0.92 \[ \frac{2 \left (d + e x\right ) \left (2 a e - b d + x \left (b e - 2 c d\right )\right )}{\left (- 4 a c + b^{2}\right ) \sqrt{a + b x + c x^{2}}} - \frac{2 e \left (b e - 2 c d\right ) \sqrt{a + b x + c x^{2}}}{c \left (- 4 a c + b^{2}\right )} + \frac{e^{2} \operatorname{atanh}{\left (\frac{b + 2 c x}{2 \sqrt{c} \sqrt{a + b x + c x^{2}}} \right )}}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.312026, size = 125, normalized size = 0.97 \[ \frac{\frac{2 \sqrt{c} \left (a b e^2-2 a c e (2 d+e x)+b^2 e^2 x+b c d (d-2 e x)+2 c^2 d^2 x\right )}{\sqrt{a+x (b+c x)}}-e^2 \left (b^2-4 a c\right ) \log \left (2 \sqrt{c} \sqrt{a+x (b+c x)}+b+2 c x\right )}{c^{3/2} \left (4 a c-b^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.011, size = 264, normalized size = 2.1 \[ 2\,{\frac{{d}^{2} \left ( 2\,cx+b \right ) }{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-{\frac{{e}^{2}x}{c}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{e}^{2}b}{2\,{c}^{2}}{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{2}{e}^{2}x}{c \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{\frac{{b}^{3}{e}^{2}}{2\,{c}^{2} \left ( 4\,ac-{b}^{2} \right ) }{\frac{1}{\sqrt{c{x}^{2}+bx+a}}}}+{{e}^{2}\ln \left ({1 \left ({\frac{b}{2}}+cx \right ){\frac{1}{\sqrt{c}}}}+\sqrt{c{x}^{2}+bx+a} \right ){c}^{-{\frac{3}{2}}}}-2\,{\frac{de}{c\sqrt{c{x}^{2}+bx+a}}}-4\,{\frac{bdex}{ \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}}-2\,{\frac{{b}^{2}de}{c \left ( 4\,ac-{b}^{2} \right ) \sqrt{c{x}^{2}+bx+a}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.322419, size = 1, normalized size = 0.01 \[ \left [-\frac{4 \,{\left (b c d^{2} - 4 \, a c d e + a b e^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} +{\left (b^{3} - 4 \, a b c\right )} e^{2} x +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \log \left (-4 \,{\left (2 \, c^{2} x + b c\right )} \sqrt{c x^{2} + b x + a} -{\left (8 \, c^{2} x^{2} + 8 \, b c x + b^{2} + 4 \, a c\right )} \sqrt{c}\right )}{2 \,{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{c}}, -\frac{2 \,{\left (b c d^{2} - 4 \, a c d e + a b e^{2} +{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} x\right )} \sqrt{c x^{2} + b x + a} \sqrt{-c} -{\left ({\left (b^{2} c - 4 \, a c^{2}\right )} e^{2} x^{2} +{\left (b^{3} - 4 \, a b c\right )} e^{2} x +{\left (a b^{2} - 4 \, a^{2} c\right )} e^{2}\right )} \arctan \left (\frac{{\left (2 \, c x + b\right )} \sqrt{-c}}{2 \, \sqrt{c x^{2} + b x + a} c}\right )}{{\left (a b^{2} c - 4 \, a^{2} c^{2} +{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} x^{2} +{\left (b^{3} c - 4 \, a b c^{2}\right )} x\right )} \sqrt{-c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.255856, size = 178, normalized size = 1.38 \[ -\frac{2 \,{\left (\frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )} x}{b^{2} c - 4 \, a c^{2}} + \frac{b c d^{2} - 4 \, a c d e + a b e^{2}}{b^{2} c - 4 \, a c^{2}}\right )}}{\sqrt{c x^{2} + b x + a}} - \frac{e^{2}{\rm ln}\left ({\left | -2 \,{\left (\sqrt{c} x - \sqrt{c x^{2} + b x + a}\right )} \sqrt{c} - b \right |}\right )}{c^{\frac{3}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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