3.816 \(\int \frac{(a+b x+c x^2)^3}{(d+e x) (f+g x)} \, dx\)

Optimal. Leaf size=531 \[ -\frac{x \left (-3 c e^2 g^2 \left (a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )+b^2 e^3 g^3 (-3 a e g+b d g+b e f)-3 c^2 e g \left (a e g \left (d^2 g^2+d e f g+e^2 f^2\right )-b \left (d^2 e f g^2+d^3 g^3+d e^2 f^2 g+e^3 f^3\right )\right )+c^3 \left (-\left (d^2 e^2 f^2 g^2+d^3 e f g^3+d^4 g^4+d e^3 f^3 g+e^4 f^4\right )\right )\right )}{e^5 g^5}+\frac{c x^3 \left (-3 c e g (-a e g+b d g+b e f)+3 b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{3 e^3 g^3}+\frac{x^2 \left (-3 c^2 e g \left (a e g (d g+e f)-b \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-3 b c e^2 g^2 (-2 a e g+b d g+b e f)+b^3 e^3 g^3+c^3 \left (-\left (d^2 e f g^2+d^3 g^3+d e^2 f^2 g+e^3 f^3\right )\right )\right )}{2 e^4 g^4}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^6 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )^3}{g^6 (e f-d g)}-\frac{c^2 x^4 (-3 b e g+c d g+c e f)}{4 e^2 g^2}+\frac{c^3 x^5}{5 e g} \]

[Out]

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

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Rubi [A]  time = 0.988291, antiderivative size = 531, 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} \[ -\frac{x \left (-3 c e^2 g^2 \left (a^2 e^2 g^2-2 a b e g (d g+e f)+b^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )+b^2 e^3 g^3 (-3 a e g+b d g+b e f)-3 c^2 e g \left (a e g \left (d^2 g^2+d e f g+e^2 f^2\right )-b \left (d^2 e f g^2+d^3 g^3+d e^2 f^2 g+e^3 f^3\right )\right )+c^3 \left (-\left (d^2 e^2 f^2 g^2+d^3 e f g^3+d^4 g^4+d e^3 f^3 g+e^4 f^4\right )\right )\right )}{e^5 g^5}+\frac{c x^3 \left (-3 c e g (-a e g+b d g+b e f)+3 b^2 e^2 g^2+c^2 \left (d^2 g^2+d e f g+e^2 f^2\right )\right )}{3 e^3 g^3}+\frac{x^2 \left (-3 c^2 e g \left (a e g (d g+e f)-b \left (d^2 g^2+d e f g+e^2 f^2\right )\right )-3 b c e^2 g^2 (-2 a e g+b d g+b e f)+b^3 e^3 g^3+c^3 \left (-\left (d^2 e f g^2+d^3 g^3+d e^2 f^2 g+e^3 f^3\right )\right )\right )}{2 e^4 g^4}+\frac{\log (d+e x) \left (a e^2-b d e+c d^2\right )^3}{e^6 (e f-d g)}-\frac{\log (f+g x) \left (a g^2-b f g+c f^2\right )^3}{g^6 (e f-d g)}-\frac{c^2 x^4 (-3 b e g+c d g+c e f)}{4 e^2 g^2}+\frac{c^3 x^5}{5 e g} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x + c*x^2)^3/((d + e*x)*(f + g*x)),x]

[Out]

-(((b^2*e^3*g^3*(b*e*f + b*d*g - 3*a*e*g) - c^3*(e^4*f^4 + d*e^3*f^3*g + d^2*e^2*f^2*g^2 + d^3*e*f*g^3 + d^4*g
^4) - 3*c*e^2*g^2*(a^2*e^2*g^2 - 2*a*b*e*g*(e*f + d*g) + b^2*(e^2*f^2 + d*e*f*g + d^2*g^2)) - 3*c^2*e*g*(a*e*g
*(e^2*f^2 + d*e*f*g + d^2*g^2) - b*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3)))*x)/(e^5*g^5)) + ((b^3*e^3
*g^3 - 3*b*c*e^2*g^2*(b*e*f + b*d*g - 2*a*e*g) - c^3*(e^3*f^3 + d*e^2*f^2*g + d^2*e*f*g^2 + d^3*g^3) - 3*c^2*e
*g*(a*e*g*(e*f + d*g) - b*(e^2*f^2 + d*e*f*g + d^2*g^2)))*x^2)/(2*e^4*g^4) + (c*(3*b^2*e^2*g^2 - 3*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^3)/(3*e^3*g^3) - (c^2*(c*e*f + c*d*g - 3*b*e*g)*x^4)
/(4*e^2*g^2) + (c^3*x^5)/(5*e*g) + ((c*d^2 - b*d*e + a*e^2)^3*Log[d + e*x])/(e^6*(e*f - d*g)) - ((c*f^2 - b*f*
g + a*g^2)^3*Log[f + g*x])/(g^6*(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 )^3}{(d+e x) (f+g x)} \, dx &=\int \left (\frac{-b^2 e^3 g^3 (b e f+b d g-3 a e g)+c^3 \left (e^4 f^4+d e^3 f^3 g+d^2 e^2 f^2 g^2+d^3 e f g^3+d^4 g^4\right )+3 c e^2 g^2 \left (a^2 e^2 g^2-2 a b e g (e f+d g)+b^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right )+3 c^2 e g \left (a e g \left (e^2 f^2+d e f g+d^2 g^2\right )-b \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )\right )}{e^5 g^5}+\frac{\left (b^3 e^3 g^3-3 b c e^2 g^2 (b e f+b d g-2 a e g)-c^3 \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )-3 c^2 e g \left (a e g (e f+d g)-b \left (e^2 f^2+d e f g+d^2 g^2\right )\right )\right ) x}{e^4 g^4}+\frac{c \left (3 b^2 e^2 g^2-3 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^2}{e^3 g^3}-\frac{c^2 (c e f+c d g-3 b e g) x^3}{e^2 g^2}+\frac{c^3 x^4}{e g}+\frac{\left (c d^2-b d e+a e^2\right )^3}{e^5 (e f-d g) (d+e x)}+\frac{\left (c f^2-b f g+a g^2\right )^3}{g^5 (-e f+d g) (f+g x)}\right ) \, dx\\ &=-\frac{\left (b^2 e^3 g^3 (b e f+b d g-3 a e g)-c^3 \left (e^4 f^4+d e^3 f^3 g+d^2 e^2 f^2 g^2+d^3 e f g^3+d^4 g^4\right )-3 c e^2 g^2 \left (a^2 e^2 g^2-2 a b e g (e f+d g)+b^2 \left (e^2 f^2+d e f g+d^2 g^2\right )\right )-3 c^2 e g \left (a e g \left (e^2 f^2+d e f g+d^2 g^2\right )-b \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )\right )\right ) x}{e^5 g^5}+\frac{\left (b^3 e^3 g^3-3 b c e^2 g^2 (b e f+b d g-2 a e g)-c^3 \left (e^3 f^3+d e^2 f^2 g+d^2 e f g^2+d^3 g^3\right )-3 c^2 e g \left (a e g (e f+d g)-b \left (e^2 f^2+d e f g+d^2 g^2\right )\right )\right ) x^2}{2 e^4 g^4}+\frac{c \left (3 b^2 e^2 g^2-3 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^3}{3 e^3 g^3}-\frac{c^2 (c e f+c d g-3 b e g) x^4}{4 e^2 g^2}+\frac{c^3 x^5}{5 e g}+\frac{\left (c d^2-b d e+a e^2\right )^3 \log (d+e x)}{e^6 (e f-d g)}-\frac{\left (c f^2-b f g+a g^2\right )^3 \log (f+g x)}{g^6 (e f-d g)}\\ \end{align*}

Mathematica [A]  time = 0.465518, size = 476, normalized size = 0.9 \[ -\frac{e g x \left (-30 c e^2 g^2 (e f-d g) \left (6 a^2 e^2 g^2+6 a b e g (-2 d g-2 e f+e g x)+b^2 \left (6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left (6 f^2-3 f g x+2 g^2 x^2\right )\right )\right )-30 b^2 e^3 g^3 (e f-d g) (6 a e g+b (-2 d g-2 e f+e g x))+15 c^2 e g \left (b \left (-4 d^2 e^2 g^4 x^2+6 d^3 e g^4 x-12 d^4 g^4+3 d e^3 g^4 x^3+e^4 f \left (-6 f^2 g x+12 f^3+4 f g^2 x^2-3 g^3 x^3\right )\right )-2 a e g (e f-d g) \left (6 d^2 g^2-3 d e g (g x-2 f)+e^2 \left (6 f^2-3 f g x+2 g^2 x^2\right )\right )\right )+c^3 \left (-15 d^2 e^3 g^5 x^3+20 d^3 e^2 g^5 x^2-30 d^4 e g^5 x+60 d^5 g^5+12 d e^4 g^5 x^4+e^5 f \left (-20 f^2 g^2 x^2+30 f^3 g x-60 f^4+15 f g^3 x^3-12 g^4 x^4\right )\right )\right )-60 g^6 \log (d+e x) \left (e (a e-b d)+c d^2\right )^3+60 e^6 \log (f+g x) \left (g (a g-b f)+c f^2\right )^3}{60 e^6 g^6 (e f-d g)} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x + c*x^2)^3/((d + e*x)*(f + g*x)),x]

[Out]

-(e*g*x*(-30*b^2*e^3*g^3*(e*f - d*g)*(6*a*e*g + b*(-2*e*f - 2*d*g + e*g*x)) + c^3*(60*d^5*g^5 - 30*d^4*e*g^5*x
+ 20*d^3*e^2*g^5*x^2 - 15*d^2*e^3*g^5*x^3 + 12*d*e^4*g^5*x^4 + e^5*f*(-60*f^4 + 30*f^3*g*x - 20*f^2*g^2*x^2 +
15*f*g^3*x^3 - 12*g^4*x^4)) - 30*c*e^2*g^2*(e*f - d*g)*(6*a^2*e^2*g^2 + 6*a*b*e*g*(-2*e*f - 2*d*g + e*g*x) +
b^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))) + 15*c^2*e*g*(-2*a*e*g*(e*f - d*g)
*(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)) + b*(-12*d^4*g^4 + 6*d^3*e*g^4*x - 4*d
^2*e^2*g^4*x^2 + 3*d*e^3*g^4*x^3 + e^4*f*(12*f^3 - 6*f^2*g*x + 4*f*g^2*x^2 - 3*g^3*x^3)))) - 60*(c*d^2 + e*(-(
b*d) + a*e))^3*g^6*Log[d + e*x] + 60*e^6*(c*f^2 + g*(-(b*f) + a*g))^3*Log[f + g*x])/(60*e^6*g^6*(e*f - d*g))

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Maple [B]  time = 0.068, size = 1232, normalized size = 2.3 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.03556, size = 973, normalized size = 1.83 \begin{align*} \frac{{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} \log \left (e x + d\right )}{e^{7} f - d e^{6} g} - \frac{{\left (c^{3} f^{6} - 3 \, b c^{2} f^{5} g - 3 \, a^{2} b f g^{5} + a^{3} g^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} f^{4} g^{2} -{\left (b^{3} + 6 \, a b c\right )} f^{3} g^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} f^{2} g^{4}\right )} \log \left (g x + f\right )}{e f g^{6} - d g^{7}} + \frac{12 \, c^{3} e^{4} g^{4} x^{5} - 15 \,{\left (c^{3} e^{4} f g^{3} +{\left (c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} g^{4}\right )} x^{4} + 20 \,{\left (c^{3} e^{4} f^{2} g^{2} +{\left (c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} f g^{3} +{\left (c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} g^{4}\right )} x^{3} - 30 \,{\left (c^{3} e^{4} f^{3} g +{\left (c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} f^{2} g^{2} +{\left (c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} f g^{3} +{\left (c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} g^{4}\right )} x^{2} + 60 \,{\left (c^{3} e^{4} f^{4} +{\left (c^{3} d e^{3} - 3 \, b c^{2} e^{4}\right )} f^{3} g +{\left (c^{3} d^{2} e^{2} - 3 \, b c^{2} d e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{4}\right )} f^{2} g^{2} +{\left (c^{3} d^{3} e - 3 \, b c^{2} d^{2} e^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{4}\right )} f g^{3} +{\left (c^{3} d^{4} - 3 \, b c^{2} d^{3} e + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{4}\right )} g^{4}\right )} x}{60 \, e^{5} g^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)/(g*x+f),x, algorithm="maxima")

[Out]

(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*e^3 + 3*(
a*b^2 + a^2*c)*d^2*e^4)*log(e*x + d)/(e^7*f - d*e^6*g) - (c^3*f^6 - 3*b*c^2*f^5*g - 3*a^2*b*f*g^5 + a^3*g^6 +
3*(b^2*c + a*c^2)*f^4*g^2 - (b^3 + 6*a*b*c)*f^3*g^3 + 3*(a*b^2 + a^2*c)*f^2*g^4)*log(g*x + f)/(e*f*g^6 - d*g^7
) + 1/60*(12*c^3*e^4*g^4*x^5 - 15*(c^3*e^4*f*g^3 + (c^3*d*e^3 - 3*b*c^2*e^4)*g^4)*x^4 + 20*(c^3*e^4*f^2*g^2 +
(c^3*d*e^3 - 3*b*c^2*e^4)*f*g^3 + (c^3*d^2*e^2 - 3*b*c^2*d*e^3 + 3*(b^2*c + a*c^2)*e^4)*g^4)*x^3 - 30*(c^3*e^4
*f^3*g + (c^3*d*e^3 - 3*b*c^2*e^4)*f^2*g^2 + (c^3*d^2*e^2 - 3*b*c^2*d*e^3 + 3*(b^2*c + a*c^2)*e^4)*f*g^3 + (c^
3*d^3*e - 3*b*c^2*d^2*e^2 + 3*(b^2*c + a*c^2)*d*e^3 - (b^3 + 6*a*b*c)*e^4)*g^4)*x^2 + 60*(c^3*e^4*f^4 + (c^3*d
*e^3 - 3*b*c^2*e^4)*f^3*g + (c^3*d^2*e^2 - 3*b*c^2*d*e^3 + 3*(b^2*c + a*c^2)*e^4)*f^2*g^2 + (c^3*d^3*e - 3*b*c
^2*d^2*e^2 + 3*(b^2*c + a*c^2)*d*e^3 - (b^3 + 6*a*b*c)*e^4)*f*g^3 + (c^3*d^4 - 3*b*c^2*d^3*e + 3*(b^2*c + a*c^
2)*d^2*e^2 - (b^3 + 6*a*b*c)*d*e^3 + 3*(a*b^2 + a^2*c)*e^4)*g^4)*x)/(e^5*g^5)

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Fricas [A]  time = 20.5178, size = 1465, normalized size = 2.76 \begin{align*} \frac{60 \,{\left (c^{3} d^{6} - 3 \, b c^{2} d^{5} e - 3 \, a^{2} b d e^{5} + a^{3} e^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{4} e^{2} -{\left (b^{3} + 6 \, a b c\right )} d^{3} e^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} d^{2} e^{4}\right )} g^{6} \log \left (e x + d\right ) + 12 \,{\left (c^{3} e^{6} f g^{5} - c^{3} d e^{5} g^{6}\right )} x^{5} - 15 \,{\left (c^{3} e^{6} f^{2} g^{4} - 3 \, b c^{2} e^{6} f g^{5} -{\left (c^{3} d^{2} e^{4} - 3 \, b c^{2} d e^{5}\right )} g^{6}\right )} x^{4} + 20 \,{\left (c^{3} e^{6} f^{3} g^{3} - 3 \, b c^{2} e^{6} f^{2} g^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6} f g^{5} -{\left (c^{3} d^{3} e^{3} - 3 \, b c^{2} d^{2} e^{4} + 3 \,{\left (b^{2} c + a c^{2}\right )} d e^{5}\right )} g^{6}\right )} x^{3} - 30 \,{\left (c^{3} e^{6} f^{4} g^{2} - 3 \, b c^{2} e^{6} f^{3} g^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6} f^{2} g^{4} -{\left (b^{3} + 6 \, a b c\right )} e^{6} f g^{5} -{\left (c^{3} d^{4} e^{2} - 3 \, b c^{2} d^{3} e^{3} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{2} e^{4} -{\left (b^{3} + 6 \, a b c\right )} d e^{5}\right )} g^{6}\right )} x^{2} + 60 \,{\left (c^{3} e^{6} f^{5} g - 3 \, b c^{2} e^{6} f^{4} g^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6} f^{3} g^{3} -{\left (b^{3} + 6 \, a b c\right )} e^{6} f^{2} g^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6} f g^{5} -{\left (c^{3} d^{5} e - 3 \, b c^{2} d^{4} e^{2} + 3 \,{\left (b^{2} c + a c^{2}\right )} d^{3} e^{3} -{\left (b^{3} + 6 \, a b c\right )} d^{2} e^{4} + 3 \,{\left (a b^{2} + a^{2} c\right )} d e^{5}\right )} g^{6}\right )} x - 60 \,{\left (c^{3} e^{6} f^{6} - 3 \, b c^{2} e^{6} f^{5} g - 3 \, a^{2} b e^{6} f g^{5} + a^{3} e^{6} g^{6} + 3 \,{\left (b^{2} c + a c^{2}\right )} e^{6} f^{4} g^{2} -{\left (b^{3} + 6 \, a b c\right )} e^{6} f^{3} g^{3} + 3 \,{\left (a b^{2} + a^{2} c\right )} e^{6} f^{2} g^{4}\right )} \log \left (g x + f\right )}{60 \,{\left (e^{7} f g^{6} - d e^{6} g^{7}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+b*x+a)^3/(e*x+d)/(g*x+f),x, algorithm="fricas")

[Out]

1/60*(60*(c^3*d^6 - 3*b*c^2*d^5*e - 3*a^2*b*d*e^5 + a^3*e^6 + 3*(b^2*c + a*c^2)*d^4*e^2 - (b^3 + 6*a*b*c)*d^3*
e^3 + 3*(a*b^2 + a^2*c)*d^2*e^4)*g^6*log(e*x + d) + 12*(c^3*e^6*f*g^5 - c^3*d*e^5*g^6)*x^5 - 15*(c^3*e^6*f^2*g
^4 - 3*b*c^2*e^6*f*g^5 - (c^3*d^2*e^4 - 3*b*c^2*d*e^5)*g^6)*x^4 + 20*(c^3*e^6*f^3*g^3 - 3*b*c^2*e^6*f^2*g^4 +
3*(b^2*c + a*c^2)*e^6*f*g^5 - (c^3*d^3*e^3 - 3*b*c^2*d^2*e^4 + 3*(b^2*c + a*c^2)*d*e^5)*g^6)*x^3 - 30*(c^3*e^6
*f^4*g^2 - 3*b*c^2*e^6*f^3*g^3 + 3*(b^2*c + a*c^2)*e^6*f^2*g^4 - (b^3 + 6*a*b*c)*e^6*f*g^5 - (c^3*d^4*e^2 - 3*
b*c^2*d^3*e^3 + 3*(b^2*c + a*c^2)*d^2*e^4 - (b^3 + 6*a*b*c)*d*e^5)*g^6)*x^2 + 60*(c^3*e^6*f^5*g - 3*b*c^2*e^6*
f^4*g^2 + 3*(b^2*c + a*c^2)*e^6*f^3*g^3 - (b^3 + 6*a*b*c)*e^6*f^2*g^4 + 3*(a*b^2 + a^2*c)*e^6*f*g^5 - (c^3*d^5
*e - 3*b*c^2*d^4*e^2 + 3*(b^2*c + a*c^2)*d^3*e^3 - (b^3 + 6*a*b*c)*d^2*e^4 + 3*(a*b^2 + a^2*c)*d*e^5)*g^6)*x -
60*(c^3*e^6*f^6 - 3*b*c^2*e^6*f^5*g - 3*a^2*b*e^6*f*g^5 + a^3*e^6*g^6 + 3*(b^2*c + a*c^2)*e^6*f^4*g^2 - (b^3
+ 6*a*b*c)*e^6*f^3*g^3 + 3*(a*b^2 + a^2*c)*e^6*f^2*g^4)*log(g*x + f))/(e^7*f*g^6 - d*e^6*g^7)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+b*x+a)**3/(e*x+d)/(g*x+f),x)

[Out]

Timed out

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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)^3/(e*x+d)/(g*x+f),x, algorithm="giac")

[Out]

Exception raised: NotImplementedError