3.2365 \(\int \frac{(A+B x) (d+e x)}{a+b x+c x^2} \, dx\)

Optimal. Leaf size=108 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]

[Out]

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

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Rubi [A]  time = 0.243663, antiderivative size = 108, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.217 \[ -\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{c^2 \sqrt{b^2-4 a c}}+\frac{\log \left (a+b x+c x^2\right ) (A c e-b B e+B c d)}{2 c^2}+\frac{B e x}{c} \]

Antiderivative was successfully verified.

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

[Out]

(B*e*x)/c - ((b^2*B*e - b*c*(B*d + A*e) + 2*c*(A*c*d - a*B*e))*ArcTanh[(b + 2*c*
x)/Sqrt[b^2 - 4*a*c]])/(c^2*Sqrt[b^2 - 4*a*c]) + ((B*c*d - b*B*e + A*c*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. \[ \frac{e \int B\, dx}{c} + \frac{\left (A c e - B b e + B c d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 c^{2}} + \frac{\left (b \left (A c e - B b e + B c d\right ) - 2 c \left (A c d - B a e\right )\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{c^{2} \sqrt{- 4 a c + b^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.162981, size = 108, normalized size = 1. \[ \frac{\frac{2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (2 c (A c d-a B e)-b c (A e+B d)+b^2 B e\right )}{\sqrt{4 a c-b^2}}+\log (a+x (b+c x)) (A c e-b B e+B c d)+2 B c e x}{2 c^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.006, size = 261, normalized size = 2.4 \[{\frac{Bex}{c}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,c}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) bBe}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,c}}+2\,{\frac{Ad}{\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-2\,{\frac{aBe}{c\sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{bAe}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+{\frac{{b}^{2}Be}{{c}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}-{\frac{bBd}{c}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*e*x/c+1/2/c*ln(c*x^2+b*x+a)*A*e-1/2/c^2*ln(c*x^2+b*x+a)*b*B*e+1/2/c*ln(c*x^2+b
*x+a)*B*d+2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*A*d-2/c/(4*a*c
-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*B*e-1/c/(4*a*c-b^2)^(1/2)*arct
an((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*A*e+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(
4*a*c-b^2)^(1/2))*b^2*B*e-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*b*B*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((B*x + A)*(e*x + d)/(c*x^2 + b*x + a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.286018, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) +{\left (2 \, B c e x +{\left (B c d -{\left (B b - A c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{b^{2} - 4 \, a c}}{2 \, \sqrt{b^{2} - 4 \, a c} c^{2}}, -\frac{2 \,{\left ({\left (B b c - 2 \, A c^{2}\right )} d -{\left (B b^{2} -{\left (2 \, B a + A b\right )} c\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) -{\left (2 \, B c e x +{\left (B c d -{\left (B b - A c\right )} e\right )} \log \left (c x^{2} + b x + a\right )\right )} \sqrt{-b^{2} + 4 \, a c}}{2 \, \sqrt{-b^{2} + 4 \, a c} c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

[1/2*(((B*b*c - 2*A*c^2)*d - (B*b^2 - (2*B*a + A*b)*c)*e)*log((b^3 - 4*a*b*c + 2
*(b^2*c - 4*a*c^2)*x + (2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c)*sqrt(b^2 - 4*a*c))/(c
*x^2 + b*x + a)) + (2*B*c*e*x + (B*c*d - (B*b - A*c)*e)*log(c*x^2 + b*x + a))*sq
rt(b^2 - 4*a*c))/(sqrt(b^2 - 4*a*c)*c^2), -1/2*(2*((B*b*c - 2*A*c^2)*d - (B*b^2
- (2*B*a + A*b)*c)*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - (2
*B*c*e*x + (B*c*d - (B*b - A*c)*e)*log(c*x^2 + b*x + a))*sqrt(-b^2 + 4*a*c))/(sq
rt(-b^2 + 4*a*c)*c^2)]

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Sympy [A]  time = 12.3051, size = 677, normalized size = 6.27 \[ \frac{B e x}{c} + \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (- \frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} + \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) \log{\left (x + \frac{2 A a c e - A b c d - B a b e + 2 B a c d - 4 a c^{2} \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right ) + b^{2} c \left (\frac{\sqrt{- 4 a c + b^{2}} \left (A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d\right )}{2 c^{2} \left (4 a c - b^{2}\right )} - \frac{- A c e + B b e - B c d}{2 c^{2}}\right )}{A b c e - 2 A c^{2} d + 2 B a c e - B b^{2} e + B b c d} \right )} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.389755, size = 151, normalized size = 1.4 \[ \frac{B x e}{c} + \frac{{\left (B c d - B b e + A c e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \, c^{2}} - \frac{{\left (B b c d - 2 \, A c^{2} d - B b^{2} e + 2 \, B a c e + A b c e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

B*x*e/c + 1/2*(B*c*d - B*b*e + A*c*e)*ln(c*x^2 + b*x + a)/c^2 - (B*b*c*d - 2*A*c
^2*d - B*b^2*e + 2*B*a*c*e + A*b*c*e)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sq
rt(-b^2 + 4*a*c)*c^2)