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

Optimal. Leaf size=83 \[ \frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]

[Out]

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

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Rubi [A]  time = 0.100025, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.04, Rules used = {893} \[ \frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 893

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] &
& NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[p] && ((EqQ[p, 1] && I
ntegersQ[m, n]) || (ILtQ[m, 0] && ILtQ[n, 0]))

Rubi steps

\begin{align*} \int \frac{a+b x+c x^2}{(d+e x) (f+g x)} \, dx &=\int \left (\frac{c}{e g}+\frac{c d^2-b d e+a e^2}{e (e f-d g) (d+e x)}+\frac{c f^2-b f g+a g^2}{g (-e f+d g) (f+g x)}\right ) \, dx\\ &=\frac{c x}{e g}+\frac{\left (c d^2-b d e+a e^2\right ) \log (d+e x)}{e^2 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right ) \log (f+g x)}{g^2 (e f-d g)}\\ \end{align*}

Mathematica [A]  time = 0.0538467, size = 85, normalized size = 1.02 \[ -\frac{\log (d+e x) \left (-a e^2+b d e-c d^2\right )}{e^2 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )}{g^2 (e f-d g)}+\frac{c x}{e g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [A]  time = 0.051, size = 142, normalized size = 1.7 \begin{align*}{\frac{cx}{eg}}-{\frac{\ln \left ( ex+d \right ) a}{dg-ef}}+{\frac{\ln \left ( ex+d \right ) bd}{ \left ( dg-ef \right ) e}}-{\frac{\ln \left ( ex+d \right ) c{d}^{2}}{ \left ( dg-ef \right ){e}^{2}}}+{\frac{\ln \left ( gx+f \right ) a}{dg-ef}}-{\frac{\ln \left ( gx+f \right ) bf}{ \left ( dg-ef \right ) g}}+{\frac{\ln \left ( gx+f \right ) c{f}^{2}}{{g}^{2} \left ( dg-ef \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+b*x+a)/(e*x+d)/(g*x+f),x)

[Out]

c*x/e/g-1/(d*g-e*f)*ln(e*x+d)*a+1/(d*g-e*f)/e*ln(e*x+d)*b*d-1/(d*g-e*f)/e^2*ln(e*x+d)*c*d^2+1/(d*g-e*f)*ln(g*x
+f)*a-1/g/(d*g-e*f)*ln(g*x+f)*b*f+1/g^2/(d*g-e*f)*ln(g*x+f)*c*f^2

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Maxima [A]  time = 1.01263, size = 117, normalized size = 1.41 \begin{align*} \frac{{\left (c d^{2} - b d e + a e^{2}\right )} \log \left (e x + d\right )}{e^{3} f - d e^{2} g} - \frac{{\left (c f^{2} - b f g + a g^{2}\right )} \log \left (g x + f\right )}{e f g^{2} - d g^{3}} + \frac{c x}{e g} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c*d^2 - b*d*e + a*e^2)*log(e*x + d)/(e^3*f - d*e^2*g) - (c*f^2 - b*f*g + a*g^2)*log(g*x + f)/(e*f*g^2 - d*g^3
) + c*x/(e*g)

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Fricas [A]  time = 1.30229, size = 198, normalized size = 2.39 \begin{align*} \frac{{\left (c d^{2} - b d e + a e^{2}\right )} g^{2} \log \left (e x + d\right ) +{\left (c e^{2} f g - c d e g^{2}\right )} x -{\left (c e^{2} f^{2} - b e^{2} f g + a e^{2} g^{2}\right )} \log \left (g x + f\right )}{e^{3} f g^{2} - d e^{2} g^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)/(g*x+f),x, algorithm="fricas")

[Out]

((c*d^2 - b*d*e + a*e^2)*g^2*log(e*x + d) + (c*e^2*f*g - c*d*e*g^2)*x - (c*e^2*f^2 - b*e^2*f*g + a*e^2*g^2)*lo
g(g*x + f))/(e^3*f*g^2 - d*e^2*g^3)

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Sympy [B]  time = 11.337, size = 420, normalized size = 5.06 \begin{align*} \frac{c x}{e g} + \frac{\left (a g^{2} - b f g + c f^{2}\right ) \log{\left (x + \frac{a d e g^{2} + a e^{2} f g - 2 b d e f g + c d^{2} f g + c d e f^{2} - \frac{d^{2} e g \left (a g^{2} - b f g + c f^{2}\right )}{d g - e f} + \frac{2 d e^{2} f \left (a g^{2} - b f g + c f^{2}\right )}{d g - e f} - \frac{e^{3} f^{2} \left (a g^{2} - b f g + c f^{2}\right )}{g \left (d g - e f\right )}}{2 a e^{2} g^{2} - b d e g^{2} - b e^{2} f g + c d^{2} g^{2} + c e^{2} f^{2}} \right )}}{g^{2} \left (d g - e f\right )} - \frac{\left (a e^{2} - b d e + c d^{2}\right ) \log{\left (x + \frac{a d e g^{2} + a e^{2} f g - 2 b d e f g + c d^{2} f g + c d e f^{2} + \frac{d^{2} g^{3} \left (a e^{2} - b d e + c d^{2}\right )}{e \left (d g - e f\right )} - \frac{2 d f g^{2} \left (a e^{2} - b d e + c d^{2}\right )}{d g - e f} + \frac{e f^{2} g \left (a e^{2} - b d e + c d^{2}\right )}{d g - e f}}{2 a e^{2} g^{2} - b d e g^{2} - b e^{2} f g + c d^{2} g^{2} + c e^{2} f^{2}} \right )}}{e^{2} \left (d g - e f\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)/(e*x+d)/(g*x+f),x)

[Out]

c*x/(e*g) + (a*g**2 - b*f*g + c*f**2)*log(x + (a*d*e*g**2 + a*e**2*f*g - 2*b*d*e*f*g + c*d**2*f*g + c*d*e*f**2
- d**2*e*g*(a*g**2 - b*f*g + c*f**2)/(d*g - e*f) + 2*d*e**2*f*(a*g**2 - b*f*g + c*f**2)/(d*g - e*f) - e**3*f*
*2*(a*g**2 - b*f*g + c*f**2)/(g*(d*g - e*f)))/(2*a*e**2*g**2 - b*d*e*g**2 - b*e**2*f*g + c*d**2*g**2 + c*e**2*
f**2))/(g**2*(d*g - e*f)) - (a*e**2 - b*d*e + c*d**2)*log(x + (a*d*e*g**2 + a*e**2*f*g - 2*b*d*e*f*g + c*d**2*
f*g + c*d*e*f**2 + d**2*g**3*(a*e**2 - b*d*e + c*d**2)/(e*(d*g - e*f)) - 2*d*f*g**2*(a*e**2 - b*d*e + c*d**2)/
(d*g - e*f) + e*f**2*g*(a*e**2 - b*d*e + c*d**2)/(d*g - e*f))/(2*a*e**2*g**2 - b*d*e*g**2 - b*e**2*f*g + c*d**
2*g**2 + c*e**2*f**2))/(e**2*(d*g - e*f))

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Giac [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)/(e*x+d)/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError