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

Optimal. Leaf size=184 \[ \frac{x \left (-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{e^3 g^3}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^4 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )^2}{g^4 (e f-d g)}-\frac{c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac{c^2 x^3}{3 e g} \]

[Out]

((b^2*e^2*g^2 - 2*c*e*g*(b*e*f + b*d*g - a*e*g) + c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*x)/(e^3*g^3) - (c*(c*e*f
+ c*d*g - 2*b*e*g)*x^2)/(2*e^2*g^2) + (c^2*x^3)/(3*e*g) + ((c*d^2 - b*d*e + a*e^2)^2*Log[d + e*x])/(e^4*(e*f -
d*g)) - ((c*f^2 - b*f*g + a*g^2)^2*Log[f + g*x])/(g^4*(e*f - d*g))

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Rubi [A]  time = 0.312182, antiderivative size = 184, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.037, Rules used = {893} \[ \frac{x \left (-2 c e g (-a e g+b d g+b e f)+b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{e^3 g^3}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^2}{e^4 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )^2}{g^4 (e f-d g)}-\frac{c x^2 (-2 b e g+c d g+c e f)}{2 e^2 g^2}+\frac{c^2 x^3}{3 e g} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b^2*e^2*g^2 - 2*c*e*g*(b*e*f + b*d*g - a*e*g) + c^2*(e^2*f^2 + d*e*f*g + d^2*g^2))*x)/(e^3*g^3) - (c*(c*e*f
+ c*d*g - 2*b*e*g)*x^2)/(2*e^2*g^2) + (c^2*x^3)/(3*e*g) + ((c*d^2 - b*d*e + a*e^2)^2*Log[d + e*x])/(e^4*(e*f -
d*g)) - ((c*f^2 - b*f*g + a*g^2)^2*Log[f + g*x])/(g^4*(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{\left (a+b x+c x^2\right )^2}{(d+e x) (f+g x)} \, dx &=\int \left (\frac{b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )}{e^3 g^3}-\frac{c (c e f+c d g-2 b e g) x}{e^2 g^2}+\frac{c^2 x^2}{e g}+\frac{\left (c d^2-b d e+a e^2\right )^2}{e^3 (e f-d g) (d+e x)}+\frac{\left (c f^2-b f g+a g^2\right )^2}{g^3 (-e f+d g) (f+g x)}\right ) \, dx\\ &=\frac{\left (b^2 e^2 g^2-2 c e g (b e f+b d g-a e g)+c^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right ) x}{e^3 g^3}-\frac{c (c e f+c d g-2 b e g) x^2}{2 e^2 g^2}+\frac{c^2 x^3}{3 e g}+\frac{\left (c d^2-b d e+a e^2\right )^2 \log (d+e x)}{e^4 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^2 \log (f+g x)}{g^4 (e f-d g)}\\ \end{align*}

Mathematica [A]  time = 0.160047, size = 177, normalized size = 0.96 \[ -\frac{e g x (d g-e f) \left (6 c e g (2 a e g+b (-2 d g-2 e f+e g x))+6 b^2 e^2 g^2+c^2 \left (6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left (6 f^2-3 f g x+2 g^2 x^2\right )\right )\right )-6 g^4 \log (d+e x) \left (e (a e-b d)+c d^2\right )^2+6 e^4 \log (f+g x) \left (g (a g-b f)+c f^2\right )^2}{6 e^4 g^4 (e f-d g)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(e*g*(-(e*f) + d*g)*x*(6*b^2*e^2*g^2 + 6*c*e*g*(2*a*e*g + b*(-2*e*f - 2*d*g + e*g*x)) + c^2*(6*d^2*g^2 - 3*d*
e*g*(-2*f + g*x) + e^2*(6*f^2 - 3*f*g*x + 2*g^2*x^2))) - 6*(c*d^2 + e*(-(b*d) + a*e))^2*g^4*Log[d + e*x] + 6*e
^4*(c*f^2 + g*(-(b*f) + a*g))^2*Log[f + g*x])/(6*e^4*g^4*(e*f - d*g))

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Maple [B]  time = 0.056, size = 444, normalized size = 2.4 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,eg}}+{\frac{bc{x}^{2}}{eg}}-{\frac{{c}^{2}{x}^{2}d}{2\,{e}^{2}g}}-{\frac{{c}^{2}{x}^{2}f}{2\,e{g}^{2}}}+2\,{\frac{acx}{eg}}+{\frac{{b}^{2}x}{eg}}-2\,{\frac{bcdx}{{e}^{2}g}}-2\,{\frac{bcfx}{e{g}^{2}}}+{\frac{{c}^{2}{d}^{2}x}{{e}^{3}g}}+{\frac{{c}^{2}dfx}{{e}^{2}{g}^{2}}}+{\frac{{c}^{2}{f}^{2}x}{e{g}^{3}}}-{\frac{\ln \left ( ex+d \right ){a}^{2}}{dg-ef}}+2\,{\frac{\ln \left ( ex+d \right ) abd}{ \left ( dg-ef \right ) e}}-2\,{\frac{\ln \left ( ex+d \right ) ac{d}^{2}}{{e}^{2} \left ( dg-ef \right ) }}-{\frac{\ln \left ( ex+d \right ){b}^{2}{d}^{2}}{{e}^{2} \left ( dg-ef \right ) }}+2\,{\frac{\ln \left ( ex+d \right ){d}^{3}bc}{{e}^{3} \left ( dg-ef \right ) }}-{\frac{\ln \left ( ex+d \right ){c}^{2}{d}^{4}}{{e}^{4} \left ( dg-ef \right ) }}+{\frac{\ln \left ( gx+f \right ){a}^{2}}{dg-ef}}-2\,{\frac{\ln \left ( gx+f \right ) abf}{ \left ( dg-ef \right ) g}}+2\,{\frac{\ln \left ( gx+f \right ) ac{f}^{2}}{{g}^{2} \left ( dg-ef \right ) }}+{\frac{\ln \left ( gx+f \right ){b}^{2}{f}^{2}}{{g}^{2} \left ( dg-ef \right ) }}-2\,{\frac{\ln \left ( gx+f \right ) bc{f}^{3}}{{g}^{3} \left ( dg-ef \right ) }}+{\frac{\ln \left ( gx+f \right ){c}^{2}{f}^{4}}{{g}^{4} \left ( dg-ef \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.973157, size = 344, normalized size = 1.87 \begin{align*} \frac{{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} \log \left (e x + d\right )}{e^{5} f - d e^{4} g} - \frac{{\left (c^{2} f^{4} - 2 \, b c f^{3} g - 2 \, a b f g^{3} + a^{2} g^{4} +{\left (b^{2} + 2 \, a c\right )} f^{2} g^{2}\right )} \log \left (g x + f\right )}{e f g^{4} - d g^{5}} + \frac{2 \, c^{2} e^{2} g^{2} x^{3} - 3 \,{\left (c^{2} e^{2} f g +{\left (c^{2} d e - 2 \, b c e^{2}\right )} g^{2}\right )} x^{2} + 6 \,{\left (c^{2} e^{2} f^{2} +{\left (c^{2} d e - 2 \, b c e^{2}\right )} f g +{\left (c^{2} d^{2} - 2 \, b c d e +{\left (b^{2} + 2 \, a c\right )} e^{2}\right )} g^{2}\right )} x}{6 \, e^{3} g^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*log(e*x + d)/(e^5*f - d*e^4*g) - (c^2*
f^4 - 2*b*c*f^3*g - 2*a*b*f*g^3 + a^2*g^4 + (b^2 + 2*a*c)*f^2*g^2)*log(g*x + f)/(e*f*g^4 - d*g^5) + 1/6*(2*c^2
*e^2*g^2*x^3 - 3*(c^2*e^2*f*g + (c^2*d*e - 2*b*c*e^2)*g^2)*x^2 + 6*(c^2*e^2*f^2 + (c^2*d*e - 2*b*c*e^2)*f*g +
(c^2*d^2 - 2*b*c*d*e + (b^2 + 2*a*c)*e^2)*g^2)*x)/(e^3*g^3)

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Fricas [A]  time = 2.99518, size = 639, normalized size = 3.47 \begin{align*} \frac{6 \,{\left (c^{2} d^{4} - 2 \, b c d^{3} e - 2 \, a b d e^{3} + a^{2} e^{4} +{\left (b^{2} + 2 \, a c\right )} d^{2} e^{2}\right )} g^{4} \log \left (e x + d\right ) + 2 \,{\left (c^{2} e^{4} f g^{3} - c^{2} d e^{3} g^{4}\right )} x^{3} - 3 \,{\left (c^{2} e^{4} f^{2} g^{2} - 2 \, b c e^{4} f g^{3} -{\left (c^{2} d^{2} e^{2} - 2 \, b c d e^{3}\right )} g^{4}\right )} x^{2} + 6 \,{\left (c^{2} e^{4} f^{3} g - 2 \, b c e^{4} f^{2} g^{2} +{\left (b^{2} + 2 \, a c\right )} e^{4} f g^{3} -{\left (c^{2} d^{3} e - 2 \, b c d^{2} e^{2} +{\left (b^{2} + 2 \, a c\right )} d e^{3}\right )} g^{4}\right )} x - 6 \,{\left (c^{2} e^{4} f^{4} - 2 \, b c e^{4} f^{3} g - 2 \, a b e^{4} f g^{3} + a^{2} e^{4} g^{4} +{\left (b^{2} + 2 \, a c\right )} e^{4} f^{2} g^{2}\right )} \log \left (g x + f\right )}{6 \,{\left (e^{5} f g^{4} - d e^{4} g^{5}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*(6*(c^2*d^4 - 2*b*c*d^3*e - 2*a*b*d*e^3 + a^2*e^4 + (b^2 + 2*a*c)*d^2*e^2)*g^4*log(e*x + d) + 2*(c^2*e^4*f
*g^3 - c^2*d*e^3*g^4)*x^3 - 3*(c^2*e^4*f^2*g^2 - 2*b*c*e^4*f*g^3 - (c^2*d^2*e^2 - 2*b*c*d*e^3)*g^4)*x^2 + 6*(c
^2*e^4*f^3*g - 2*b*c*e^4*f^2*g^2 + (b^2 + 2*a*c)*e^4*f*g^3 - (c^2*d^3*e - 2*b*c*d^2*e^2 + (b^2 + 2*a*c)*d*e^3)
*g^4)*x - 6*(c^2*e^4*f^4 - 2*b*c*e^4*f^3*g - 2*a*b*e^4*f*g^3 + a^2*e^4*g^4 + (b^2 + 2*a*c)*e^4*f^2*g^2)*log(g*
x + f))/(e^5*f*g^4 - d*e^4*g^5)

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Sympy [B]  time = 98.6109, size = 989, normalized size = 5.38 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

c**2*x**3/(3*e*g) + (a*g**2 - b*f*g + c*f**2)**2*log(x + (a**2*d*e**3*g**4 + a**2*e**4*f*g**3 - 4*a*b*d*e**3*f
*g**3 + 2*a*c*d**2*e**2*f*g**3 + 2*a*c*d*e**3*f**2*g**2 + b**2*d**2*e**2*f*g**3 + b**2*d*e**3*f**2*g**2 - 2*b*
c*d**3*e*f*g**3 - 2*b*c*d*e**3*f**3*g + c**2*d**4*f*g**3 + c**2*d*e**3*f**4 - d**2*e**3*g*(a*g**2 - b*f*g + c*
f**2)**2/(d*g - e*f) + 2*d*e**4*f*(a*g**2 - b*f*g + c*f**2)**2/(d*g - e*f) - e**5*f**2*(a*g**2 - b*f*g + c*f**
2)**2/(g*(d*g - e*f)))/(2*a**2*e**4*g**4 - 2*a*b*d*e**3*g**4 - 2*a*b*e**4*f*g**3 + 2*a*c*d**2*e**2*g**4 + 2*a*
c*e**4*f**2*g**2 + b**2*d**2*e**2*g**4 + b**2*e**4*f**2*g**2 - 2*b*c*d**3*e*g**4 - 2*b*c*e**4*f**3*g + c**2*d*
*4*g**4 + c**2*e**4*f**4))/(g**4*(d*g - e*f)) + x**2*(2*b*c*e*g - c**2*d*g - c**2*e*f)/(2*e**2*g**2) + x*(2*a*
c*e**2*g**2 + b**2*e**2*g**2 - 2*b*c*d*e*g**2 - 2*b*c*e**2*f*g + c**2*d**2*g**2 + c**2*d*e*f*g + c**2*e**2*f**
2)/(e**3*g**3) - (a*e**2 - b*d*e + c*d**2)**2*log(x + (a**2*d*e**3*g**4 + a**2*e**4*f*g**3 - 4*a*b*d*e**3*f*g*
*3 + 2*a*c*d**2*e**2*f*g**3 + 2*a*c*d*e**3*f**2*g**2 + b**2*d**2*e**2*f*g**3 + b**2*d*e**3*f**2*g**2 - 2*b*c*d
**3*e*f*g**3 - 2*b*c*d*e**3*f**3*g + c**2*d**4*f*g**3 + c**2*d*e**3*f**4 + d**2*g**5*(a*e**2 - b*d*e + c*d**2)
**2/(e*(d*g - e*f)) - 2*d*f*g**4*(a*e**2 - b*d*e + c*d**2)**2/(d*g - e*f) + e*f**2*g**3*(a*e**2 - b*d*e + c*d*
*2)**2/(d*g - e*f))/(2*a**2*e**4*g**4 - 2*a*b*d*e**3*g**4 - 2*a*b*e**4*f*g**3 + 2*a*c*d**2*e**2*g**4 + 2*a*c*e
**4*f**2*g**2 + b**2*d**2*e**2*g**4 + b**2*e**4*f**2*g**2 - 2*b*c*d**3*e*g**4 - 2*b*c*e**4*f**3*g + c**2*d**4*
g**4 + c**2*e**4*f**4))/(e**4*(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)^2/(e*x+d)/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError