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

Optimal. Leaf size=591 \[ \frac{\log \left (a+b x+c x^2\right ) \left (c^2 e \left (a^2 e^2 h+2 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )-b^2 c e^2 (3 a e h+3 b d h+b e g)-c^3 \left (a e \left (3 d^2 h+3 d e g+e^2 f\right )+b d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^4 e^3 h+c^4 d^2 (d g+3 e f)\right )}{2 c^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b c^2 e \left (5 a^2 e^2 h+4 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )+c^3 \left (2 a^2 e^2 (3 d h+e g)+3 a b e \left (3 d^2 h+3 d e g+e^2 f\right )+b^2 d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^3 c e^2 (5 a e h+3 b d h+b e g)-c^4 d \left (2 a \left (d^2 h+3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )+b^5 \left (-e^3\right ) h+2 c^5 d^3 f\right )}{c^5 \sqrt{b^2-4 a c}}+\frac{e x^2 \left (-c e (a e h+3 b d h+b e g)+b^2 e^2 h+c^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )}{2 c^3}-\frac{x \left (c^2 e \left (a e (3 d h+e g)+b \left (3 d^2 h+3 d e g+e^2 f\right )\right )-b c e^2 (2 a e h+3 b d h+b e g)+b^3 e^3 h+c^3 (-d) \left (d^2 h+3 d e g+3 e^2 f\right )\right )}{c^4}+\frac{e^2 x^3 (-b e h+3 c d h+c e g)}{3 c^2}+\frac{e^3 h x^4}{4 c} \]

[Out]

-(((b^3*e^3*h - c^3*d*(3*e^2*f + 3*d*e*g + d^2*h) - b*c*e^2*(b*e*g + 3*b*d*h + 2*a*e*h) + c^2*e*(a*e*(e*g + 3*
d*h) + b*(e^2*f + 3*d*e*g + 3*d^2*h)))*x)/c^4) + (e*(b^2*e^2*h + c^2*(e^2*f + 3*d*e*g + 3*d^2*h) - c*e*(b*e*g
+ 3*b*d*h + a*e*h))*x^2)/(2*c^3) + (e^2*(c*e*g + 3*c*d*h - b*e*h)*x^3)/(3*c^2) + (e^3*h*x^4)/(4*c) - ((2*c^5*d
^3*f - b^5*e^3*h + b^3*c*e^2*(b*e*g + 3*b*d*h + 5*a*e*h) - c^4*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f + 3*d*e*g +
d^2*h)) - b*c^2*e*(5*a^2*e^2*h + 4*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) + c^3*(2*a^2*e^2*(e
*g + 3*d*h) + b^2*d*(3*e^2*f + 3*d*e*g + d^2*h) + 3*a*b*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*ArcTanh[(b + 2*c*x)/Sq
rt[b^2 - 4*a*c]])/(c^5*Sqrt[b^2 - 4*a*c]) + ((c^4*d^2*(3*e*f + d*g) + b^4*e^3*h - b^2*c*e^2*(b*e*g + 3*b*d*h +
3*a*e*h) + c^2*e*(a^2*e^2*h + 2*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) - c^3*(b*d*(3*e^2*f +
3*d*e*g + d^2*h) + a*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*Log[a + b*x + c*x^2])/(2*c^5)

________________________________________________________________________________________

Rubi [A]  time = 1.42811, antiderivative size = 591, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167, Rules used = {1628, 634, 618, 206, 628} \[ \frac{\log \left (a+b x+c x^2\right ) \left (c^2 e \left (a^2 e^2 h+2 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )-b^2 c e^2 (3 a e h+3 b d h+b e g)-c^3 \left (a e \left (3 d^2 h+3 d e g+e^2 f\right )+b d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^4 e^3 h+c^4 d^2 (d g+3 e f)\right )}{2 c^5}-\frac{\tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right ) \left (-b c^2 e \left (5 a^2 e^2 h+4 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )+c^3 \left (2 a^2 e^2 (3 d h+e g)+3 a b e \left (3 d^2 h+3 d e g+e^2 f\right )+b^2 d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^3 c e^2 (5 a e h+3 b d h+b e g)-c^4 d \left (2 a \left (d^2 h+3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )+b^5 \left (-e^3\right ) h+2 c^5 d^3 f\right )}{c^5 \sqrt{b^2-4 a c}}+\frac{e x^2 \left (-c e (a e h+3 b d h+b e g)+b^2 e^2 h+c^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )}{2 c^3}-\frac{x \left (c^2 e \left (a e (3 d h+e g)+b \left (3 d^2 h+3 d e g+e^2 f\right )\right )-b c e^2 (2 a e h+3 b d h+b e g)+b^3 e^3 h+c^3 (-d) \left (d^2 h+3 d e g+3 e^2 f\right )\right )}{c^4}+\frac{e^2 x^3 (-b e h+3 c d h+c e g)}{3 c^2}+\frac{e^3 h x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((b^3*e^3*h - c^3*d*(3*e^2*f + 3*d*e*g + d^2*h) - b*c*e^2*(b*e*g + 3*b*d*h + 2*a*e*h) + c^2*e*(a*e*(e*g + 3*
d*h) + b*(e^2*f + 3*d*e*g + 3*d^2*h)))*x)/c^4) + (e*(b^2*e^2*h + c^2*(e^2*f + 3*d*e*g + 3*d^2*h) - c*e*(b*e*g
+ 3*b*d*h + a*e*h))*x^2)/(2*c^3) + (e^2*(c*e*g + 3*c*d*h - b*e*h)*x^3)/(3*c^2) + (e^3*h*x^4)/(4*c) - ((2*c^5*d
^3*f - b^5*e^3*h + b^3*c*e^2*(b*e*g + 3*b*d*h + 5*a*e*h) - c^4*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f + 3*d*e*g +
d^2*h)) - b*c^2*e*(5*a^2*e^2*h + 4*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) + c^3*(2*a^2*e^2*(e
*g + 3*d*h) + b^2*d*(3*e^2*f + 3*d*e*g + d^2*h) + 3*a*b*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*ArcTanh[(b + 2*c*x)/Sq
rt[b^2 - 4*a*c]])/(c^5*Sqrt[b^2 - 4*a*c]) + ((c^4*d^2*(3*e*f + d*g) + b^4*e^3*h - b^2*c*e^2*(b*e*g + 3*b*d*h +
3*a*e*h) + c^2*e*(a^2*e^2*h + 2*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) - c^3*(b*d*(3*e^2*f +
3*d*e*g + d^2*h) + a*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*Log[a + b*x + c*x^2])/(2*c^5)

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 634

Int[((d_.) + (e_.)*(x_))/((a_) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Dist[(2*c*d - b*e)/(2*c), Int[1/(a +
b*x + c*x^2), x], x] + Dist[e/(2*c), Int[(b + 2*c*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e}, x] &
& NeQ[2*c*d - b*e, 0] && NeQ[b^2 - 4*a*c, 0] &&  !NiceSqrtQ[b^2 - 4*a*c]

Rule 618

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(-1), x_Symbol] :> Dist[-2, Subst[Int[1/Simp[b^2 - 4*a*c - x^2, x], x]
, x, b + 2*c*x], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 628

Int[((d_) + (e_.)*(x_))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(d*Log[RemoveContent[a + b*x +
c*x^2, x]])/b, x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[2*c*d - b*e, 0]

Rubi steps

\begin{align*} \int \frac{(d+e x)^3 \left (f+g x+h x^2\right )}{a+b x+c x^2} \, dx &=\int \left (-\frac{b^3 e^3 h-c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-b c e^2 (b e g+3 b d h+2 a e h)+c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )}{c^4}+\frac{e \left (b^2 e^2 h+c^2 \left (e^2 f+3 d e g+3 d^2 h\right )-c e (b e g+3 b d h+a e h)\right ) x}{c^3}+\frac{e^2 (c e g+3 c d h-b e h) x^2}{c^2}+\frac{e^3 h x^3}{c}+\frac{c^4 d^3 f+a b^3 e^3 h-a c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-a b c e^2 (b e g+3 b d h+2 a e h)+a c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )+\left (c^4 d^2 (3 e f+d g)+b^4 e^3 h-b^2 c e^2 (b e g+3 b d h+3 a e h)+c^2 e \left (a^2 e^2 h+2 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )-c^3 \left (b d \left (3 e^2 f+3 d e g+d^2 h\right )+a e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{c^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac{\left (b^3 e^3 h-c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-b c e^2 (b e g+3 b d h+2 a e h)+c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{c^4}+\frac{e \left (b^2 e^2 h+c^2 \left (e^2 f+3 d e g+3 d^2 h\right )-c e (b e g+3 b d h+a e h)\right ) x^2}{2 c^3}+\frac{e^2 (c e g+3 c d h-b e h) x^3}{3 c^2}+\frac{e^3 h x^4}{4 c}+\frac{\int \frac{c^4 d^3 f+a b^3 e^3 h-a c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-a b c e^2 (b e g+3 b d h+2 a e h)+a c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )+\left (c^4 d^2 (3 e f+d g)+b^4 e^3 h-b^2 c e^2 (b e g+3 b d h+3 a e h)+c^2 e \left (a^2 e^2 h+2 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )-c^3 \left (b d \left (3 e^2 f+3 d e g+d^2 h\right )+a e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{a+b x+c x^2} \, dx}{c^4}\\ &=-\frac{\left (b^3 e^3 h-c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-b c e^2 (b e g+3 b d h+2 a e h)+c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{c^4}+\frac{e \left (b^2 e^2 h+c^2 \left (e^2 f+3 d e g+3 d^2 h\right )-c e (b e g+3 b d h+a e h)\right ) x^2}{2 c^3}+\frac{e^2 (c e g+3 c d h-b e h) x^3}{3 c^2}+\frac{e^3 h x^4}{4 c}+\frac{\left (c^4 d^2 (3 e f+d g)+b^4 e^3 h-b^2 c e^2 (b e g+3 b d h+3 a e h)+c^2 e \left (a^2 e^2 h+2 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )-c^3 \left (b d \left (3 e^2 f+3 d e g+d^2 h\right )+a e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^5}+\frac{\left (2 c^5 d^3 f-b^5 e^3 h+b^3 c e^2 (b e g+3 b d h+5 a e h)-c^4 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f+3 d e g+d^2 h\right )\right )-b c^2 e \left (5 a^2 e^2 h+4 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )+c^3 \left (2 a^2 e^2 (e g+3 d h)+b^2 d \left (3 e^2 f+3 d e g+d^2 h\right )+3 a b e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^5}\\ &=-\frac{\left (b^3 e^3 h-c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-b c e^2 (b e g+3 b d h+2 a e h)+c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{c^4}+\frac{e \left (b^2 e^2 h+c^2 \left (e^2 f+3 d e g+3 d^2 h\right )-c e (b e g+3 b d h+a e h)\right ) x^2}{2 c^3}+\frac{e^2 (c e g+3 c d h-b e h) x^3}{3 c^2}+\frac{e^3 h x^4}{4 c}+\frac{\left (c^4 d^2 (3 e f+d g)+b^4 e^3 h-b^2 c e^2 (b e g+3 b d h+3 a e h)+c^2 e \left (a^2 e^2 h+2 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )-c^3 \left (b d \left (3 e^2 f+3 d e g+d^2 h\right )+a e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac{\left (2 c^5 d^3 f-b^5 e^3 h+b^3 c e^2 (b e g+3 b d h+5 a e h)-c^4 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f+3 d e g+d^2 h\right )\right )-b c^2 e \left (5 a^2 e^2 h+4 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )+c^3 \left (2 a^2 e^2 (e g+3 d h)+b^2 d \left (3 e^2 f+3 d e g+d^2 h\right )+3 a b e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \operatorname{Subst}\left (\int \frac{1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^5}\\ &=-\frac{\left (b^3 e^3 h-c^3 d \left (3 e^2 f+3 d e g+d^2 h\right )-b c e^2 (b e g+3 b d h+2 a e h)+c^2 e \left (a e (e g+3 d h)+b \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) x}{c^4}+\frac{e \left (b^2 e^2 h+c^2 \left (e^2 f+3 d e g+3 d^2 h\right )-c e (b e g+3 b d h+a e h)\right ) x^2}{2 c^3}+\frac{e^2 (c e g+3 c d h-b e h) x^3}{3 c^2}+\frac{e^3 h x^4}{4 c}-\frac{\left (2 c^5 d^3 f-b^5 e^3 h+b^3 c e^2 (b e g+3 b d h+5 a e h)-c^4 d \left (b d (3 e f+d g)+2 a \left (3 e^2 f+3 d e g+d^2 h\right )\right )-b c^2 e \left (5 a^2 e^2 h+4 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )+c^3 \left (2 a^2 e^2 (e g+3 d h)+b^2 d \left (3 e^2 f+3 d e g+d^2 h\right )+3 a b e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^5 \sqrt{b^2-4 a c}}+\frac{\left (c^4 d^2 (3 e f+d g)+b^4 e^3 h-b^2 c e^2 (b e g+3 b d h+3 a e h)+c^2 e \left (a^2 e^2 h+2 a b e (e g+3 d h)+b^2 \left (e^2 f+3 d e g+3 d^2 h\right )\right )-c^3 \left (b d \left (3 e^2 f+3 d e g+d^2 h\right )+a e \left (e^2 f+3 d e g+3 d^2 h\right )\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^5}\\ \end{align*}

Mathematica [A]  time = 0.630873, size = 585, normalized size = 0.99 \[ \frac{6 \log (a+x (b+c x)) \left (c^2 e \left (a^2 e^2 h+2 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )-b^2 c e^2 (3 a e h+3 b d h+b e g)-c^3 \left (a e \left (3 d^2 h+3 d e g+e^2 f\right )+b d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^4 e^3 h+c^4 d^2 (d g+3 e f)\right )+\frac{12 \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right ) \left (-b c^2 e \left (5 a^2 e^2 h+4 a b e (3 d h+e g)+b^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )+c^3 \left (2 a^2 e^2 (3 d h+e g)+3 a b e \left (3 d^2 h+3 d e g+e^2 f\right )+b^2 d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+b^3 c e^2 (5 a e h+3 b d h+b e g)-c^4 d \left (2 a \left (d^2 h+3 d e g+3 e^2 f\right )+b d (d g+3 e f)\right )+b^5 \left (-e^3\right ) h+2 c^5 d^3 f\right )}{\sqrt{4 a c-b^2}}+6 c^2 e x^2 \left (-c e (a e h+3 b d h+b e g)+b^2 e^2 h+c^2 \left (3 d^2 h+3 d e g+e^2 f\right )\right )+12 c x \left (-c^2 e \left (a e (3 d h+e g)+b \left (3 d^2 h+3 d e g+e^2 f\right )\right )+b c e^2 (2 a e h+3 b d h+b e g)-b^3 e^3 h+c^3 d \left (d^2 h+3 d e g+3 e^2 f\right )\right )+4 c^3 e^2 x^3 (-b e h+3 c d h+c e g)+3 c^4 e^3 h x^4}{12 c^5} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(12*c*(-(b^3*e^3*h) + c^3*d*(3*e^2*f + 3*d*e*g + d^2*h) + b*c*e^2*(b*e*g + 3*b*d*h + 2*a*e*h) - c^2*e*(a*e*(e*
g + 3*d*h) + b*(e^2*f + 3*d*e*g + 3*d^2*h)))*x + 6*c^2*e*(b^2*e^2*h + c^2*(e^2*f + 3*d*e*g + 3*d^2*h) - c*e*(b
*e*g + 3*b*d*h + a*e*h))*x^2 + 4*c^3*e^2*(c*e*g + 3*c*d*h - b*e*h)*x^3 + 3*c^4*e^3*h*x^4 + (12*(2*c^5*d^3*f -
b^5*e^3*h + b^3*c*e^2*(b*e*g + 3*b*d*h + 5*a*e*h) - c^4*d*(b*d*(3*e*f + d*g) + 2*a*(3*e^2*f + 3*d*e*g + d^2*h)
) - b*c^2*e*(5*a^2*e^2*h + 4*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) + c^3*(2*a^2*e^2*(e*g + 3*
d*h) + b^2*d*(3*e^2*f + 3*d*e*g + d^2*h) + 3*a*b*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*ArcTan[(b + 2*c*x)/Sqrt[-b^2
+ 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(c^4*d^2*(3*e*f + d*g) + b^4*e^3*h - b^2*c*e^2*(b*e*g + 3*b*d*h + 3*a*e*h) +
c^2*e*(a^2*e^2*h + 2*a*b*e*(e*g + 3*d*h) + b^2*(e^2*f + 3*d*e*g + 3*d^2*h)) - c^3*(b*d*(3*e^2*f + 3*d*e*g + d
^2*h) + a*e*(e^2*f + 3*d*e*g + 3*d^2*h)))*Log[a + x*(b + c*x)])/(12*c^5)

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Maple [B]  time = 0.181, size = 1738, normalized size = 2.9 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3/c^2*x^3*b*e^3*h+1/c*x^3*d*e^2*h-1/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e^3*h-1/c
^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*e^3*f-3/2/c^2*ln(c*x^2+b*x+a)*b*d*e^2*f-2/c/(4*a*
c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^3*h+3/2/c^3*ln(c*x^2+b*x+a)*b^2*d^2*e*h+3/2/c^3*ln(c*x^2+
b*x+a)*b^2*d*e^2*g-3/2/c^2*ln(c*x^2+b*x+a)*b*d^2*e*g+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2
))*a^2*e^3*g-3/2/c^4*ln(c*x^2+b*x+a)*b^3*d*e^2*h-3/2/c^2*ln(c*x^2+b*x+a)*a*d^2*e*h-3/2/c^2*ln(c*x^2+b*x+a)*a*d
*e^2*g-3/c^2*b*d*e^2*g*x+3/c^3*b^2*d*e^2*h*x-3/c^2*b*d^2*e*h*x+3/c*d*e^2*f*x+1/c^3*b^2*e^3*g*x-1/c^2*b*e^3*f*x
+3/c*d^2*e*g*x-1/c^4*b^3*e^3*h*x+1/2/c^3*x^2*b^2*e^3*h-1/2/c^2*x^2*b*e^3*g+3/2/c*x^2*d^2*e*h+3/2/c*x^2*d*e^2*g
-1/c^2*a*e^3*g*x-1/2/c^2*x^2*a*e^3*h-1/2/c^2*ln(c*x^2+b*x+a)*a*e^3*f+1/2/c^5*ln(c*x^2+b*x+a)*b^4*e^3*h+1/2/c^3
*ln(c*x^2+b*x+a)*b^2*e^3*f-1/2/c^2*ln(c*x^2+b*x+a)*b*d^3*h-1/2/c^4*ln(c*x^2+b*x+a)*b^3*e^3*g+1/2/c^3*ln(c*x^2+
b*x+a)*a^2*e^3*h+3/2/c*ln(c*x^2+b*x+a)*d^2*e*f+1/3/c*x^3*e^3*g+1/2/c*x^2*e^3*f+1/c*d^3*h*x+2/(4*a*c-b^2)^(1/2)
*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^3*f+1/2/c*ln(c*x^2+b*x+a)*d^3*g-12/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*a*b^2*d*e^2*h+9/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d^2*e*h+9/
c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d*e^2*g-3/2/c^2*x^2*b*d*e^2*h+2/c^3*a*b*e^3*h*x-
3/c^2*a*d*e^2*h*x+1/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d^3*h-3/2/c^4*ln(c*x^2+b*x+a
)*a*b^2*e^3*h+1/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*e^3*g+1/c^3*ln(c*x^2+b*x+a)*a*b*
e^3*g-1/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^3*g+3/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*b^4*d*e^2*h+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*e^3*f-6/c/(4
*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^2*e*g-6/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^
2)^(1/2))*a*d*e^2*f-3/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^2*e*f-3/c^3/(4*a*c-b^2)^(1/2
)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*d^2*e*h-3/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*
b^3*d*e^2*g-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*e^3*g+3/c^3*ln(c*x^2+b*x+a)*a*b*
d*e^2*h-5/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*b*e^3*h+6/c^2/(4*a*c-b^2)^(1/2)*arctan
((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*d*e^2*h+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d^2*
e*g+3/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^2*d*e^2*f+5/c^4/(4*a*c-b^2)^(1/2)*arctan((2*
c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^3*h+1/4*e^3*h*x^4/c

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 9.13653, size = 4321, normalized size = 7.31 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*(b^2*c^4 - 4*a*c^5)*e^3*h*x^4 + 4*((b^2*c^4 - 4*a*c^5)*e^3*g + (3*(b^2*c^4 - 4*a*c^5)*d*e^2 - (b^3*c^
3 - 4*a*b*c^4)*e^3)*h)*x^3 + 6*((b^2*c^4 - 4*a*c^5)*e^3*f + (3*(b^2*c^4 - 4*a*c^5)*d*e^2 - (b^3*c^3 - 4*a*b*c^
4)*e^3)*g + (3*(b^2*c^4 - 4*a*c^5)*d^2*e - 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)
*e^3)*h)*x^2 - 6*sqrt(b^2 - 4*a*c)*((2*c^5*d^3 - 3*b*c^4*d^2*e + 3*(b^2*c^3 - 2*a*c^4)*d*e^2 - (b^3*c^2 - 3*a*
b*c^3)*e^3)*f - (b*c^4*d^3 - 3*(b^2*c^3 - 2*a*c^4)*d^2*e + 3*(b^3*c^2 - 3*a*b*c^3)*d*e^2 - (b^4*c - 4*a*b^2*c^
2 + 2*a^2*c^3)*e^3)*g + ((b^2*c^3 - 2*a*c^4)*d^3 - 3*(b^3*c^2 - 3*a*b*c^3)*d^2*e + 3*(b^4*c - 4*a*b^2*c^2 + 2*
a^2*c^3)*d*e^2 - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e^3)*h)*log((2*c^2*x^2 + 2*b*c*x + b^2 - 2*a*c + sqrt(b^2 - 4
*a*c)*(2*c*x + b))/(c*x^2 + b*x + a)) + 12*((3*(b^2*c^4 - 4*a*c^5)*d*e^2 - (b^3*c^3 - 4*a*b*c^4)*e^3)*f + (3*(
b^2*c^4 - 4*a*c^5)*d^2*e - 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e^3)*g + ((b^2*
c^4 - 4*a*c^5)*d^3 - 3*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 3*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (b^5*c - 6*
a*b^3*c^2 + 8*a^2*b*c^3)*e^3)*h)*x + 6*((3*(b^2*c^4 - 4*a*c^5)*d^2*e - 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^
2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e^3)*f + ((b^2*c^4 - 4*a*c^5)*d^3 - 3*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 3*(b^4*c^2 -
5*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e^3)*g - ((b^3*c^3 - 4*a*b*c^4)*d^3 - 3*(
b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e + 3*(b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d*e^2 - (b^6 - 7*a*b^4*c + 13
*a^2*b^2*c^2 - 4*a^3*c^3)*e^3)*h)*log(c*x^2 + b*x + a))/(b^2*c^5 - 4*a*c^6), 1/12*(3*(b^2*c^4 - 4*a*c^5)*e^3*h
*x^4 + 4*((b^2*c^4 - 4*a*c^5)*e^3*g + (3*(b^2*c^4 - 4*a*c^5)*d*e^2 - (b^3*c^3 - 4*a*b*c^4)*e^3)*h)*x^3 + 6*((b
^2*c^4 - 4*a*c^5)*e^3*f + (3*(b^2*c^4 - 4*a*c^5)*d*e^2 - (b^3*c^3 - 4*a*b*c^4)*e^3)*g + (3*(b^2*c^4 - 4*a*c^5)
*d^2*e - 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e^3)*h)*x^2 - 12*sqrt(-b^2 + 4*a*
c)*((2*c^5*d^3 - 3*b*c^4*d^2*e + 3*(b^2*c^3 - 2*a*c^4)*d*e^2 - (b^3*c^2 - 3*a*b*c^3)*e^3)*f - (b*c^4*d^3 - 3*(
b^2*c^3 - 2*a*c^4)*d^2*e + 3*(b^3*c^2 - 3*a*b*c^3)*d*e^2 - (b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*e^3)*g + ((b^2*c^
3 - 2*a*c^4)*d^3 - 3*(b^3*c^2 - 3*a*b*c^3)*d^2*e + 3*(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d*e^2 - (b^5 - 5*a*b^3*
c + 5*a^2*b*c^2)*e^3)*h)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 12*((3*(b^2*c^4 - 4*a*c^5)*d*
e^2 - (b^3*c^3 - 4*a*b*c^4)*e^3)*f + (3*(b^2*c^4 - 4*a*c^5)*d^2*e - 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^2 -
5*a*b^2*c^3 + 4*a^2*c^4)*e^3)*g + ((b^2*c^4 - 4*a*c^5)*d^3 - 3*(b^3*c^3 - 4*a*b*c^4)*d^2*e + 3*(b^4*c^2 - 5*a
*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e^3)*h)*x + 6*((3*(b^2*c^4 - 4*a*c^5)*d^2*e
- 3*(b^3*c^3 - 4*a*b*c^4)*d*e^2 + (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e^3)*f + ((b^2*c^4 - 4*a*c^5)*d^3 - 3*(b
^3*c^3 - 4*a*b*c^4)*d^2*e + 3*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d*e^2 - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*
e^3)*g - ((b^3*c^3 - 4*a*b*c^4)*d^3 - 3*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^2*e + 3*(b^5*c - 6*a*b^3*c^2 + 8
*a^2*b*c^3)*d*e^2 - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e^3)*h)*log(c*x^2 + b*x + a))/(b^2*c^5 - 4*
a*c^6)]

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Sympy [B]  time = 83.9893, size = 4962, normalized size = 8.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e**3*h - 6*a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h +
12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f
+ 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g +
3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*
b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*e*f - 2*c**5*d**3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e
**3*h - 3*a*b**2*c*e**3*h + 6*a*b*c**2*d*e**2*h + 2*a*b*c**2*e**3*g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g -
a*c**3*e**3*f + b**4*e**3*h - 3*b**3*c*d*e**2*h - b**3*c*e**3*g + 3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*g
+ b**2*c**2*e**3*f - b*c**3*d**3*h - 3*b*c**3*d**2*e*g - 3*b*c**3*d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/(2
*c**5))*log(x + (2*a**3*c**2*e**3*h - 4*a**2*b**2*c*e**3*h + 9*a**2*b*c**2*d*e**2*h + 3*a**2*b*c**2*e**3*g - 6
*a**2*c**3*d**2*e*h - 6*a**2*c**3*d*e**2*g - 2*a**2*c**3*e**3*f + a*b**4*e**3*h - 3*a*b**3*c*d*e**2*h - a*b**3
*c*e**3*g + 3*a*b**2*c**2*d**2*e*h + 3*a*b**2*c**2*d*e**2*g + a*b**2*c**2*e**3*f - a*b*c**3*d**3*h - 3*a*b*c**
3*d**2*e*g - 3*a*b*c**3*d*e**2*f - 4*a*c**5*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e**3*h - 6*a**2*c**3*d*e**2*h
- 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*a*b*c**3*d**2*e
*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d*e**2*f + b**5*
e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3*c**2*e**3*f -
b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*e*f - 2*c**5*d*
*3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e**3*h - 3*a*b**2*c*e**3*h + 6*a*b*c**2*d*e**2*h + 2*a*b*c**2*e**3*
g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g - a*c**3*e**3*f + b**4*e**3*h - 3*b**3*c*d*e**2*h - b**3*c*e**3*g +
3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*g + b**2*c**2*e**3*f - b*c**3*d**3*h - 3*b*c**3*d**2*e*g - 3*b*c**3*
d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/(2*c**5)) + 2*a*c**4*d**3*g + 6*a*c**4*d**2*e*f + b**2*c**4*(-sqrt(-
4*a*c + b**2)*(5*a**2*b*c**2*e**3*h - 6*a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*b**
2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*a*c
**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b**3*
c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*c**
3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*e*f - 2*c**5*d**3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e**3*h -
3*a*b**2*c*e**3*h + 6*a*b*c**2*d*e**2*h + 2*a*b*c**2*e**3*g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g - a*c**3*e
**3*f + b**4*e**3*h - 3*b**3*c*d*e**2*h - b**3*c*e**3*g + 3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*g + b**2*c
**2*e**3*f - b*c**3*d**3*h - 3*b*c**3*d**2*e*g - 3*b*c**3*d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/(2*c**5))
- b*c**4*d**3*f)/(5*a**2*b*c**2*e**3*h - 6*a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*
b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*
a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b*
*3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*
c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*e*f - 2*c**5*d**3*f)) + (sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e**3
*h - 6*a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e
**3*g - 9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g +
6*a*c**4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**
2*g + b**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*
c**4*d**2*e*f - 2*c**5*d**3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e**3*h - 3*a*b**2*c*e**3*h + 6*a*b*c**2*d*
e**2*h + 2*a*b*c**2*e**3*g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g - a*c**3*e**3*f + b**4*e**3*h - 3*b**3*c*d*
e**2*h - b**3*c*e**3*g + 3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*g + b**2*c**2*e**3*f - b*c**3*d**3*h - 3*b*
c**3*d**2*e*g - 3*b*c**3*d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/(2*c**5))*log(x + (2*a**3*c**2*e**3*h - 4*a
**2*b**2*c*e**3*h + 9*a**2*b*c**2*d*e**2*h + 3*a**2*b*c**2*e**3*g - 6*a**2*c**3*d**2*e*h - 6*a**2*c**3*d*e**2*
g - 2*a**2*c**3*e**3*f + a*b**4*e**3*h - 3*a*b**3*c*d*e**2*h - a*b**3*c*e**3*g + 3*a*b**2*c**2*d**2*e*h + 3*a*
b**2*c**2*d*e**2*g + a*b**2*c**2*e**3*f - a*b*c**3*d**3*h - 3*a*b*c**3*d**2*e*g - 3*a*b*c**3*d*e**2*f - 4*a*c*
*5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e**3*h - 6*a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h
+ 12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3
*f + 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g
+ 3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g -
3*b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*e*f - 2*c**5*d**3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2
*e**3*h - 3*a*b**2*c*e**3*h + 6*a*b*c**2*d*e**2*h + 2*a*b*c**2*e**3*g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g
- a*c**3*e**3*f + b**4*e**3*h - 3*b**3*c*d*e**2*h - b**3*c*e**3*g + 3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*
g + b**2*c**2*e**3*f - b*c**3*d**3*h - 3*b*c**3*d**2*e*g - 3*b*c**3*d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/
(2*c**5)) + 2*a*c**4*d**3*g + 6*a*c**4*d**2*e*f + b**2*c**4*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e**3*h - 6*a**
2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g - 9*
a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**4*d
*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b**3
*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d**2*
e*f - 2*c**5*d**3*f)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e**3*h - 3*a*b**2*c*e**3*h + 6*a*b*c**2*d*e**2*h + 2
*a*b*c**2*e**3*g - 3*a*c**3*d**2*e*h - 3*a*c**3*d*e**2*g - a*c**3*e**3*f + b**4*e**3*h - 3*b**3*c*d*e**2*h - b
**3*c*e**3*g + 3*b**2*c**2*d**2*e*h + 3*b**2*c**2*d*e**2*g + b**2*c**2*e**3*f - b*c**3*d**3*h - 3*b*c**3*d**2*
e*g - 3*b*c**3*d*e**2*f + c**4*d**3*g + 3*c**4*d**2*e*f)/(2*c**5)) - b*c**4*d**3*f)/(5*a**2*b*c**2*e**3*h - 6*
a**2*c**3*d*e**2*h - 2*a**2*c**3*e**3*g - 5*a*b**3*c*e**3*h + 12*a*b**2*c**2*d*e**2*h + 4*a*b**2*c**2*e**3*g -
9*a*b*c**3*d**2*e*h - 9*a*b*c**3*d*e**2*g - 3*a*b*c**3*e**3*f + 2*a*c**4*d**3*h + 6*a*c**4*d**2*e*g + 6*a*c**
4*d*e**2*f + b**5*e**3*h - 3*b**4*c*d*e**2*h - b**4*c*e**3*g + 3*b**3*c**2*d**2*e*h + 3*b**3*c**2*d*e**2*g + b
**3*c**2*e**3*f - b**2*c**3*d**3*h - 3*b**2*c**3*d**2*e*g - 3*b**2*c**3*d*e**2*f + b*c**4*d**3*g + 3*b*c**4*d*
*2*e*f - 2*c**5*d**3*f)) + e**3*h*x**4/(4*c) - x**3*(b*e**3*h - 3*c*d*e**2*h - c*e**3*g)/(3*c**2) - x**2*(a*c*
e**3*h - b**2*e**3*h + 3*b*c*d*e**2*h + b*c*e**3*g - 3*c**2*d**2*e*h - 3*c**2*d*e**2*g - c**2*e**3*f)/(2*c**3)
+ x*(2*a*b*c*e**3*h - 3*a*c**2*d*e**2*h - a*c**2*e**3*g - b**3*e**3*h + 3*b**2*c*d*e**2*h + b**2*c*e**3*g - 3
*b*c**2*d**2*e*h - 3*b*c**2*d*e**2*g - b*c**2*e**3*f + c**3*d**3*h + 3*c**3*d**2*e*g + 3*c**3*d*e**2*f)/c**4

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Giac [A]  time = 1.28457, size = 1041, normalized size = 1.76 \begin{align*} \frac{3 \, c^{3} h x^{4} e^{3} + 12 \, c^{3} d h x^{3} e^{2} + 18 \, c^{3} d^{2} h x^{2} e + 12 \, c^{3} d^{3} h x + 4 \, c^{3} g x^{3} e^{3} - 4 \, b c^{2} h x^{3} e^{3} + 18 \, c^{3} d g x^{2} e^{2} - 18 \, b c^{2} d h x^{2} e^{2} + 36 \, c^{3} d^{2} g x e - 36 \, b c^{2} d^{2} h x e + 6 \, c^{3} f x^{2} e^{3} - 6 \, b c^{2} g x^{2} e^{3} + 6 \, b^{2} c h x^{2} e^{3} - 6 \, a c^{2} h x^{2} e^{3} + 36 \, c^{3} d f x e^{2} - 36 \, b c^{2} d g x e^{2} + 36 \, b^{2} c d h x e^{2} - 36 \, a c^{2} d h x e^{2} - 12 \, b c^{2} f x e^{3} + 12 \, b^{2} c g x e^{3} - 12 \, a c^{2} g x e^{3} - 12 \, b^{3} h x e^{3} + 24 \, a b c h x e^{3}}{12 \, c^{4}} + \frac{{\left (c^{4} d^{3} g - b c^{3} d^{3} h + 3 \, c^{4} d^{2} f e - 3 \, b c^{3} d^{2} g e + 3 \, b^{2} c^{2} d^{2} h e - 3 \, a c^{3} d^{2} h e - 3 \, b c^{3} d f e^{2} + 3 \, b^{2} c^{2} d g e^{2} - 3 \, a c^{3} d g e^{2} - 3 \, b^{3} c d h e^{2} + 6 \, a b c^{2} d h e^{2} + b^{2} c^{2} f e^{3} - a c^{3} f e^{3} - b^{3} c g e^{3} + 2 \, a b c^{2} g e^{3} + b^{4} h e^{3} - 3 \, a b^{2} c h e^{3} + a^{2} c^{2} h e^{3}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac{{\left (2 \, c^{5} d^{3} f - b c^{4} d^{3} g + b^{2} c^{3} d^{3} h - 2 \, a c^{4} d^{3} h - 3 \, b c^{4} d^{2} f e + 3 \, b^{2} c^{3} d^{2} g e - 6 \, a c^{4} d^{2} g e - 3 \, b^{3} c^{2} d^{2} h e + 9 \, a b c^{3} d^{2} h e + 3 \, b^{2} c^{3} d f e^{2} - 6 \, a c^{4} d f e^{2} - 3 \, b^{3} c^{2} d g e^{2} + 9 \, a b c^{3} d g e^{2} + 3 \, b^{4} c d h e^{2} - 12 \, a b^{2} c^{2} d h e^{2} + 6 \, a^{2} c^{3} d h e^{2} - b^{3} c^{2} f e^{3} + 3 \, a b c^{3} f e^{3} + b^{4} c g e^{3} - 4 \, a b^{2} c^{2} g e^{3} + 2 \, a^{2} c^{3} g e^{3} - b^{5} h e^{3} + 5 \, a b^{3} c h e^{3} - 5 \, a^{2} b c^{2} h e^{3}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{\sqrt{-b^{2} + 4 \, a c} c^{5}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^3*h*x^4*e^3 + 12*c^3*d*h*x^3*e^2 + 18*c^3*d^2*h*x^2*e + 12*c^3*d^3*h*x + 4*c^3*g*x^3*e^3 - 4*b*c^2*h
*x^3*e^3 + 18*c^3*d*g*x^2*e^2 - 18*b*c^2*d*h*x^2*e^2 + 36*c^3*d^2*g*x*e - 36*b*c^2*d^2*h*x*e + 6*c^3*f*x^2*e^3
- 6*b*c^2*g*x^2*e^3 + 6*b^2*c*h*x^2*e^3 - 6*a*c^2*h*x^2*e^3 + 36*c^3*d*f*x*e^2 - 36*b*c^2*d*g*x*e^2 + 36*b^2*
c*d*h*x*e^2 - 36*a*c^2*d*h*x*e^2 - 12*b*c^2*f*x*e^3 + 12*b^2*c*g*x*e^3 - 12*a*c^2*g*x*e^3 - 12*b^3*h*x*e^3 + 2
4*a*b*c*h*x*e^3)/c^4 + 1/2*(c^4*d^3*g - b*c^3*d^3*h + 3*c^4*d^2*f*e - 3*b*c^3*d^2*g*e + 3*b^2*c^2*d^2*h*e - 3*
a*c^3*d^2*h*e - 3*b*c^3*d*f*e^2 + 3*b^2*c^2*d*g*e^2 - 3*a*c^3*d*g*e^2 - 3*b^3*c*d*h*e^2 + 6*a*b*c^2*d*h*e^2 +
b^2*c^2*f*e^3 - a*c^3*f*e^3 - b^3*c*g*e^3 + 2*a*b*c^2*g*e^3 + b^4*h*e^3 - 3*a*b^2*c*h*e^3 + a^2*c^2*h*e^3)*log
(c*x^2 + b*x + a)/c^5 + (2*c^5*d^3*f - b*c^4*d^3*g + b^2*c^3*d^3*h - 2*a*c^4*d^3*h - 3*b*c^4*d^2*f*e + 3*b^2*c
^3*d^2*g*e - 6*a*c^4*d^2*g*e - 3*b^3*c^2*d^2*h*e + 9*a*b*c^3*d^2*h*e + 3*b^2*c^3*d*f*e^2 - 6*a*c^4*d*f*e^2 - 3
*b^3*c^2*d*g*e^2 + 9*a*b*c^3*d*g*e^2 + 3*b^4*c*d*h*e^2 - 12*a*b^2*c^2*d*h*e^2 + 6*a^2*c^3*d*h*e^2 - b^3*c^2*f*
e^3 + 3*a*b*c^3*f*e^3 + b^4*c*g*e^3 - 4*a*b^2*c^2*g*e^3 + 2*a^2*c^3*g*e^3 - b^5*h*e^3 + 5*a*b^3*c*h*e^3 - 5*a^
2*b*c^2*h*e^3)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)