3.1526 \(\int \frac{(b+2 c x) (d+e x)^2}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=114 \[ \frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^2}-\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2}+e x \left (4 d-\frac{b e}{c}\right )+e^2 x^2 \]

[Out]

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

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Rubi [A]  time = 0.289835, antiderivative size = 114, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192 \[ \frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log \left (a+b x+c x^2\right )}{2 c^2}-\frac{e \sqrt{b^2-4 a c} (2 c d-b e) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^2}+e x \left (4 d-\frac{b e}{c}\right )+e^2 x^2 \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ 2 e^{2} \int x\, dx - \frac{\left (b e - 4 c d\right ) \int e\, dx}{c} + \frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2}} + \frac{\left (- 2 a c e^{2} + b^{2} e^{2} - 2 b c d e + 2 c^{2} d^{2}\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} \]

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),x)

[Out]

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

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Mathematica [A]  time = 0.172012, size = 111, normalized size = 0.97 \[ \frac{\left (-2 c e (a e+b d)+b^2 e^2+2 c^2 d^2\right ) \log (a+x (b+c x))+2 e \sqrt{4 a c-b^2} (b e-2 c d) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )+2 c e x (-b e+4 c d+c e x)}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 264, normalized size = 2.3 \[{e}^{2}{x}^{2}-{\frac{b{e}^{2}x}{c}}+4\,dex-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) a{e}^{2}}{c}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ){b}^{2}{e}^{2}}{2\,{c}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bde}{c}}+\ln \left ( c{x}^{2}+bx+a \right ){d}^{2}+4\,{\frac{ab{e}^{2}}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-8\,{\frac{aed}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{{b}^{3}{e}^{2}}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{{b}^{2}de}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) } \]

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),x)

[Out]

e^2*x^2-1/c*e^2*b*x+4*d*e*x-1/c*ln(c*x^2+b*x+a)*a*e^2+1/2/c^2*ln(c*x^2+b*x+a)*b^
2*e^2-1/c*ln(c*x^2+b*x+a)*b*d*e+ln(c*x^2+b*x+a)*d^2+4/c/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^2-8/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*
c-b^2)^(1/2))*a*d*e-1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
b^3*e^2+2/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.277298, size = 1, normalized size = 0.01 \[ \left [\frac{2 \, c^{2} e^{2} x^{2} -{\left (2 \, c d e - b e^{2}\right )} \sqrt{b^{2} - 4 \, a c} \log \left (\frac{2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c + \sqrt{b^{2} - 4 \, a c}{\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + 2 \,{\left (4 \, c^{2} d e - b c e^{2}\right )} x +{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}}, \frac{2 \, c^{2} e^{2} x^{2} - 2 \,{\left (2 \, c d e - b e^{2}\right )} \sqrt{-b^{2} + 4 \, a c} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right ) + 2 \,{\left (4 \, c^{2} d e - b c e^{2}\right )} x +{\left (2 \, c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} - 2 \, a c\right )} e^{2}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{2}}\right ] \]

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),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 7.10918, size = 337, normalized size = 2.96 \[ e^{2} x^{2} + \left (- \frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c^{2}} - \frac{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}}{2 c^{2}}\right ) \log{\left (x + \frac{a e^{2} - c d^{2} + c \left (- \frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c^{2}} - \frac{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}}{2 c^{2}}\right )}{b e^{2} - 2 c d e} \right )} + \left (\frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c^{2}} - \frac{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}}{2 c^{2}}\right ) \log{\left (x + \frac{a e^{2} - c d^{2} + c \left (\frac{e \sqrt{- 4 a c + b^{2}} \left (b e - 2 c d\right )}{2 c^{2}} - \frac{2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}}{2 c^{2}}\right )}{b e^{2} - 2 c d e} \right )} - \frac{x \left (b e^{2} - 4 c d e\right )}{c} \]

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),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.273772, size = 194, normalized size = 1.7 \[ \frac{{\left (2 \, c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, a c e^{2}\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} + \frac{{\left (2 \, b^{2} c d e - 8 \, a c^{2} d e - b^{3} e^{2} + 4 \, a b c e^{2}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} + \frac{c^{2} x^{2} e^{2} + 4 \, c^{2} d x e - b c x e^{2}}{c^{2}} \]

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),x, algorithm="giac")

[Out]

1/2*(2*c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*a*c*e^2)*ln(c*x^2 + b*x + a)/c^2 + (2*b
^2*c*d*e - 8*a*c^2*d*e - b^3*e^2 + 4*a*b*c*e^2)*arctan((2*c*x + b)/sqrt(-b^2 + 4
*a*c))/(sqrt(-b^2 + 4*a*c)*c^2) + (c^2*x^2*e^2 + 4*c^2*d*x*e - b*c*x*e^2)/c^2