3.2367 \(\int \frac{A+B x}{(d+e x) \left (a+b x+c x^2\right )} \, dx\)

Optimal. Leaf size=146 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]

[Out]

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

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Rubi [A]  time = 0.425262, antiderivative size = 146, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{\sqrt{b^2-4 a c} \left (a e^2-b d e+c d^2\right )}+\frac{(B d-A e) \log \left (a+b x+c x^2\right )}{2 \left (a e^2-b d e+c d^2\right )}-\frac{(B d-A e) \log (d+e x)}{a e^2-b d e+c d^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 67.443, size = 138, normalized size = 0.95 \[ \frac{\left (A e - B d\right ) \log{\left (d + e x \right )}}{a e^{2} - b d e + c d^{2}} - \frac{\left (A e - B d\right ) \log{\left (a + b x + c x^{2} \right )}}{2 \left (a e^{2} - b d e + c d^{2}\right )} + \frac{\left (A b e - 2 A c d - 2 B a e + B b d\right ) \operatorname{atanh}{\left (\frac{b + 2 c x}{\sqrt{- 4 a c + b^{2}}} \right )}}{\sqrt{- 4 a c + b^{2}} \left (a e^{2} - b d e + c d^{2}\right )} \]

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]

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

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Mathematica [A]  time = 0.268483, size = 125, normalized size = 0.86 \[ \frac{\sqrt{4 a c-b^2} (B d-A e) (2 \log (d+e x)-\log (a+x (b+c x)))+2 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) (-2 a B e+A b e-2 A c d+b B d)}{2 \sqrt{4 a c-b^2} \left (e (b d-a e)-c d^2\right )} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [B]  time = 0.008, size = 343, normalized size = 2.4 \[{\frac{\ln \left ( ex+d \right ) Ae}{{e}^{2}a-bde+c{d}^{2}}}-{\frac{\ln \left ( ex+d \right ) Bd}{{e}^{2}a-bde+c{d}^{2}}}-{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Ae}{2\,{e}^{2}a-2\,bde+2\,c{d}^{2}}}+{\frac{\ln \left ( c{x}^{2}+bx+a \right ) Bd}{2\,{e}^{2}a-2\,bde+2\,c{d}^{2}}}-{\frac{Abe}{{e}^{2}a-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}}+2\,{\frac{Acd}{ \left ({e}^{2}a-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }+2\,{\frac{aBe}{ \left ({e}^{2}a-bde+c{d}^{2} \right ) \sqrt{4\,ac-{b}^{2}}}\arctan \left ({\frac{2\,cx+b}{\sqrt{4\,ac-{b}^{2}}}} \right ) }-{\frac{Bbd}{{e}^{2}a-bde+c{d}^{2}}\arctan \left ({(2\,cx+b){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \right ){\frac{1}{\sqrt{4\,ac-{b}^{2}}}}} \]

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]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 4.71915, size = 1, normalized size = 0.01 \[ \left [\frac{{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\right )} e\right )} \log \left (\frac{b^{3} - 4 \, a b c + 2 \,{\left (b^{2} c - 4 \, a c^{2}\right )} x +{\left (2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c\right )} \sqrt{b^{2} - 4 \, a c}}{c x^{2} + b x + a}\right ) + \sqrt{b^{2} - 4 \, a c}{\left ({\left (B d - A e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (B d - A e\right )} \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{b^{2} - 4 \, a c}}, -\frac{2 \,{\left ({\left (B b - 2 \, A c\right )} d -{\left (2 \, B a - A b\right )} e\right )} \arctan \left (-\frac{\sqrt{-b^{2} + 4 \, a c}{\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - \sqrt{-b^{2} + 4 \, a c}{\left ({\left (B d - A e\right )} \log \left (c x^{2} + b x + a\right ) - 2 \,{\left (B d - A e\right )} \log \left (e x + d\right )\right )}}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

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]

Timed out

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GIAC/XCAS [A]  time = 0.335558, size = 209, normalized size = 1.43 \[ \frac{{\left (B d - A e\right )}{\rm ln}\left (c x^{2} + b x + a\right )}{2 \,{\left (c d^{2} - b d e + a e^{2}\right )}} - \frac{{\left (B d e - A e^{2}\right )}{\rm ln}\left ({\left | x e + d \right |}\right )}{c d^{2} e - b d e^{2} + a e^{3}} - \frac{{\left (B b d - 2 \, A c d - 2 \, B a e + A b e\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (c d^{2} - b d e + a e^{2}\right )} \sqrt{-b^{2} + 4 \, a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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