3.14.48 \(\int \frac {(b+2 c x) (d+e x)^4}{(a+b x+c x^2)^3} \, dx\)
Optimal. Leaf size=197 \[ \frac {2 e (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {2 e^3 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {2 e (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^4 \log \left (a+b x+c x^2\right )}{c^2}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2} \]
________________________________________________________________________________________
Rubi [A] time = 0.33, antiderivative size = 197, normalized size of antiderivative = 1.00,
number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used =
{768, 738, 773, 634, 618, 206, 628} \begin {gather*} \frac {2 e (2 c d-b e) \left (-2 c e (b d-3 a e)-b^2 e^2+2 c^2 d^2\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {2 e^3 x (2 c d-b e)}{c \left (b^2-4 a c\right )}-\frac {2 e (d+e x)^2 (-2 a e+x (2 c d-b e)+b d)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^4 \log \left (a+b x+c x^2\right )}{c^2}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2} \end {gather*}
Antiderivative was successfully verified.
[In]
Int[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x]
[Out]
(2*e^3*(2*c*d - b*e)*x)/(c*(b^2 - 4*a*c)) - (d + e*x)^4/(2*(a + b*x + c*x^2)^2) - (2*e*(d + e*x)^2*(b*d - 2*a*
e + (2*c*d - b*e)*x))/((b^2 - 4*a*c)*(a + b*x + c*x^2)) + (2*e*(2*c*d - b*e)*(2*c^2*d^2 - b^2*e^2 - 2*c*e*(b*d
- 3*a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(c^2*(b^2 - 4*a*c)^(3/2)) + (e^4*Log[a + b*x + c*x^2])/c^2
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 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 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]
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 738
Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(
d*b - 2*a*e + (2*c*d - b*e)*x)*(a + b*x + c*x^2)^(p + 1))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/((p + 1)*(b^2 -
4*a*c)), Int[(d + e*x)^(m - 2)*Simp[e*(2*a*e*(m - 1) + b*d*(2*p - m + 4)) - 2*c*d^2*(2*p + 3) + e*(b*e - 2*d*
c)*(m + 2*p + 2)*x, x]*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] &
& NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, b, c, d,
e, m, p, x]
Rule 768
Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Sim
p[(g*(d + e*x)^m*(a + b*x + c*x^2)^(p + 1))/(2*c*(p + 1)), x] - Dist[(e*g*m)/(2*c*(p + 1)), Int[(d + e*x)^(m -
1)*(a + b*x + c*x^2)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && EqQ[2*c*f - b*g, 0] && LtQ[p, -1]
&& GtQ[m, 0]
Rule 773
Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/
c, x] + Dist[1/c, Int[(c*d*f - a*e*g + (c*e*f + c*d*g - b*e*g)*x)/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c,
d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0]
Rubi steps
\begin {align*} \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}+(2 e) \int \frac {(d+e x)^3}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {2 e (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 e) \int \frac {(d+e x) \left (2 c d^2-e (3 b d-4 a e)-e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{b^2-4 a c}\\ &=\frac {2 e^3 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {2 e (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac {(2 e) \int \frac {a e^2 (2 c d-b e)+c d \left (2 c d^2-e (3 b d-4 a e)\right )+\left (-c d e (2 c d-b e)+b e^2 (2 c d-b e)+c e \left (2 c d^2-e (3 b d-4 a e)\right )\right ) x}{a+b x+c x^2} \, dx}{c \left (b^2-4 a c\right )}\\ &=\frac {2 e^3 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {2 e (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^4 \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{c^2}-\frac {\left (e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 e^3 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {2 e (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {e^4 \log \left (a+b x+c x^2\right )}{c^2}+\frac {\left (2 e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^2 \left (b^2-4 a c\right )}\\ &=\frac {2 e^3 (2 c d-b e) x}{c \left (b^2-4 a c\right )}-\frac {(d+e x)^4}{2 \left (a+b x+c x^2\right )^2}-\frac {2 e (d+e x)^2 (b d-2 a e+(2 c d-b e) x)}{\left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac {2 e (2 c d-b e) \left (2 c^2 d^2-b^2 e^2-2 c e (b d-3 a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^2 \left (b^2-4 a c\right )^{3/2}}+\frac {e^4 \log \left (a+b x+c x^2\right )}{c^2}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.54, size = 334, normalized size = 1.70 \begin {gather*} \frac {\frac {-c e^3 \left (a^2 e+2 a b (2 d+e x)+4 b^2 d x\right )+b^2 e^4 (a+b x)+2 c^2 d e^2 (3 a d+2 a e x+3 b d x)-c^3 d^3 (d+4 e x)}{(a+x (b+c x))^2}-\frac {e \left (8 c^2 \left (2 a^2 e^3-a c d e (6 d+5 e x)+c^2 d^3 x\right )+2 b^2 c e \left (c d (3 d+8 e x)-5 a e^2\right )+4 b c^2 \left (a e^2 (7 d+5 e x)+c d^2 (d-3 e x)\right )+b^4 e^3-2 b^3 c e^2 (2 d+3 e x)\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}+\frac {4 c e (b e-2 c d) \left (2 c e (b d-3 a e)+b^2 e^2-2 c^2 d^2\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+2 c e^4 \log (a+x (b+c x))}{2 c^3} \end {gather*}
Antiderivative was successfully verified.
[In]
Integrate[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x]
[Out]
((b^2*e^4*(a + b*x) - c^3*d^3*(d + 4*e*x) + 2*c^2*d*e^2*(3*a*d + 3*b*d*x + 2*a*e*x) - c*e^3*(a^2*e + 4*b^2*d*x
+ 2*a*b*(2*d + e*x)))/(a + x*(b + c*x))^2 - (e*(b^4*e^3 - 2*b^3*c*e^2*(2*d + 3*e*x) + 8*c^2*(2*a^2*e^3 + c^2*
d^3*x - a*c*d*e*(6*d + 5*e*x)) + 4*b*c^2*(c*d^2*(d - 3*e*x) + a*e^2*(7*d + 5*e*x)) + 2*b^2*c*e*(-5*a*e^2 + c*d
*(3*d + 8*e*x))))/((b^2 - 4*a*c)*(a + x*(b + c*x))) + (4*c*e*(-2*c*d + b*e)*(-2*c^2*d^2 + b^2*e^2 + 2*c*e*(b*d
- 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 2*c*e^4*Log[a + x*(b + c*x)])/(2*c^3
)
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) (d+e x)^4}{\left (a+b x+c x^2\right )^3} \, dx \end {gather*}
Verification is not applicable to the result.
[In]
IntegrateAlgebraic[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x]
[Out]
IntegrateAlgebraic[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3, x]
________________________________________________________________________________________
fricas [B] time = 0.47, size = 2400, normalized size = 12.18
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="fricas")
[Out]
[-1/2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*d^4 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e - 24*(a^2*b^2*c^2 - 4*a^3*
c^3)*d^2*e^2 + 12*(a^2*b^3*c - 4*a^3*b*c^2)*d*e^3 - (5*a^2*b^4 - 32*a^3*b^2*c + 48*a^4*c^2)*e^4 + 2*(4*(b^2*c^
4 - 4*a*c^5)*d^3*e - 6*(b^3*c^3 - 4*a*b*c^4)*d^2*e^2 + 4*(2*b^4*c^2 - 13*a*b^2*c^3 + 20*a^2*c^4)*d*e^3 - (3*b^
5*c - 22*a*b^3*c^2 + 40*a^2*b*c^3)*e^4)*x^3 + (12*(b^3*c^3 - 4*a*b*c^4)*d^3*e - 6*(b^4*c^2 + 4*a*b^2*c^3 - 32*
a^2*c^4)*d^2*e^2 + 12*(b^5*c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^3 - (5*b^6 - 30*a*b^4*c + 24*a^2*b^2*c^2 + 64*a^
3*c^3)*e^4)*x^2 - 2*(4*a^2*c^3*d^3*e - 6*a^2*b*c^2*d^2*e^2 + 12*a^3*c^2*d*e^3 + (a^2*b^3 - 6*a^3*b*c)*e^4 + (4
*c^5*d^3*e - 6*b*c^4*d^2*e^2 + 12*a*c^4*d*e^3 + (b^3*c^2 - 6*a*b*c^3)*e^4)*x^4 + 2*(4*b*c^4*d^3*e - 6*b^2*c^3*
d^2*e^2 + 12*a*b*c^3*d*e^3 + (b^4*c - 6*a*b^2*c^2)*e^4)*x^3 + (4*(b^2*c^3 + 2*a*c^4)*d^3*e - 6*(b^3*c^2 + 2*a*
b*c^3)*d^2*e^2 + 12*(a*b^2*c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 4*a*b^3*c - 12*a^2*b*c^2)*e^4)*x^2 + 2*(4*a*b*c^3*d
^3*e - 6*a*b^2*c^2*d^2*e^2 + 12*a^2*b*c^2*d*e^3 + (a*b^4 - 6*a^2*b^2*c)*e^4)*x)*sqrt(b^2 - 4*a*c)*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)) + 2*(4*(b^4*c^2 - 5*a*b^2*c^3 +
4*a^2*c^4)*d^3*e - 18*(a*b^3*c^2 - 4*a^2*b*c^3)*d^2*e^2 + 12*(a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d*e^3 - (5
*a*b^5 - 34*a^2*b^3*c + 56*a^3*b*c^2)*e^4)*x - 2*((b^4*c^2 - 8*a*b^2*c^3 + 16*a^2*c^4)*e^4*x^4 + 2*(b^5*c - 8*
a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b
*c^2)*e^4*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4)*log(c*x^2 + b*x + a))/(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 1
6*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2
- 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3 + 16*a^3*b*c^4)*x), -1/2*((b^4*c^2 - 8*a*b^2*c
^3 + 16*a^2*c^4)*d^4 + 4*(a*b^3*c^2 - 4*a^2*b*c^3)*d^3*e - 24*(a^2*b^2*c^2 - 4*a^3*c^3)*d^2*e^2 + 12*(a^2*b^3*
c - 4*a^3*b*c^2)*d*e^3 - (5*a^2*b^4 - 32*a^3*b^2*c + 48*a^4*c^2)*e^4 + 2*(4*(b^2*c^4 - 4*a*c^5)*d^3*e - 6*(b^3
*c^3 - 4*a*b*c^4)*d^2*e^2 + 4*(2*b^4*c^2 - 13*a*b^2*c^3 + 20*a^2*c^4)*d*e^3 - (3*b^5*c - 22*a*b^3*c^2 + 40*a^2
*b*c^3)*e^4)*x^3 + (12*(b^3*c^3 - 4*a*b*c^4)*d^3*e - 6*(b^4*c^2 + 4*a*b^2*c^3 - 32*a^2*c^4)*d^2*e^2 + 12*(b^5*
c - 5*a*b^3*c^2 + 4*a^2*b*c^3)*d*e^3 - (5*b^6 - 30*a*b^4*c + 24*a^2*b^2*c^2 + 64*a^3*c^3)*e^4)*x^2 - 4*(4*a^2*
c^3*d^3*e - 6*a^2*b*c^2*d^2*e^2 + 12*a^3*c^2*d*e^3 + (a^2*b^3 - 6*a^3*b*c)*e^4 + (4*c^5*d^3*e - 6*b*c^4*d^2*e^
2 + 12*a*c^4*d*e^3 + (b^3*c^2 - 6*a*b*c^3)*e^4)*x^4 + 2*(4*b*c^4*d^3*e - 6*b^2*c^3*d^2*e^2 + 12*a*b*c^3*d*e^3
+ (b^4*c - 6*a*b^2*c^2)*e^4)*x^3 + (4*(b^2*c^3 + 2*a*c^4)*d^3*e - 6*(b^3*c^2 + 2*a*b*c^3)*d^2*e^2 + 12*(a*b^2*
c^2 + 2*a^2*c^3)*d*e^3 + (b^5 - 4*a*b^3*c - 12*a^2*b*c^2)*e^4)*x^2 + 2*(4*a*b*c^3*d^3*e - 6*a*b^2*c^2*d^2*e^2
+ 12*a^2*b*c^2*d*e^3 + (a*b^4 - 6*a^2*b^2*c)*e^4)*x)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)
/(b^2 - 4*a*c)) + 2*(4*(b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d^3*e - 18*(a*b^3*c^2 - 4*a^2*b*c^3)*d^2*e^2 + 12*(
a*b^4*c - 5*a^2*b^2*c^2 + 4*a^3*c^3)*d*e^3 - (5*a*b^5 - 34*a^2*b^3*c + 56*a^3*b*c^2)*e^4)*x - 2*((b^4*c^2 - 8*
a*b^2*c^3 + 16*a^2*c^4)*e^4*x^4 + 2*(b^5*c - 8*a*b^3*c^2 + 16*a^2*b*c^3)*e^4*x^3 + (b^6 - 6*a*b^4*c + 32*a^3*c
^3)*e^4*x^2 + 2*(a*b^5 - 8*a^2*b^3*c + 16*a^3*b*c^2)*e^4*x + (a^2*b^4 - 8*a^3*b^2*c + 16*a^4*c^2)*e^4)*log(c*x
^2 + b*x + a))/(a^2*b^4*c^2 - 8*a^3*b^2*c^3 + 16*a^4*c^4 + (b^4*c^4 - 8*a*b^2*c^5 + 16*a^2*c^6)*x^4 + 2*(b^5*c
^3 - 8*a*b^3*c^4 + 16*a^2*b*c^5)*x^3 + (b^6*c^2 - 6*a*b^4*c^3 + 32*a^3*c^5)*x^2 + 2*(a*b^5*c^2 - 8*a^2*b^3*c^3
+ 16*a^3*b*c^4)*x)]
________________________________________________________________________________________
giac [B] time = 0.18, size = 439, normalized size = 2.23 \begin {gather*} -\frac {2 \, {\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 12 \, a c^{2} d e^{3} + b^{3} e^{4} - 6 \, a b c e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{2} - 4 \, a c^{3}\right )} \sqrt {-b^{2} + 4 \, a c}} + \frac {e^{4} \log \left (c x^{2} + b x + a\right )}{c^{2}} - \frac {b^{2} c^{2} d^{4} - 4 \, a c^{3} d^{4} + 4 \, a b c^{2} d^{3} e - 24 \, a^{2} c^{2} d^{2} e^{2} + 12 \, a^{2} b c d e^{3} - 5 \, a^{2} b^{2} e^{4} + 12 \, a^{3} c e^{4} + 2 \, {\left (4 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 8 \, b^{2} c^{2} d e^{3} - 20 \, a c^{3} d e^{3} - 3 \, b^{3} c e^{4} + 10 \, a b c^{2} e^{4}\right )} x^{3} + {\left (12 \, b c^{3} d^{3} e - 6 \, b^{2} c^{2} d^{2} e^{2} - 48 \, a c^{3} d^{2} e^{2} + 12 \, b^{3} c d e^{3} - 12 \, a b c^{2} d e^{3} - 5 \, b^{4} e^{4} + 10 \, a b^{2} c e^{4} + 16 \, a^{2} c^{2} e^{4}\right )} x^{2} + 2 \, {\left (4 \, b^{2} c^{2} d^{3} e - 4 \, a c^{3} d^{3} e - 18 \, a b c^{2} d^{2} e^{2} + 12 \, a b^{2} c d e^{3} - 12 \, a^{2} c^{2} d e^{3} - 5 \, a b^{3} e^{4} + 14 \, a^{2} b c e^{4}\right )} x}{2 \, {\left (c x^{2} + b x + a\right )}^{2} {\left (b^{2} - 4 \, a c\right )} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="giac")
[Out]
-2*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 12*a*c^2*d*e^3 + b^3*e^4 - 6*a*b*c*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a
*c))/((b^2*c^2 - 4*a*c^3)*sqrt(-b^2 + 4*a*c)) + e^4*log(c*x^2 + b*x + a)/c^2 - 1/2*(b^2*c^2*d^4 - 4*a*c^3*d^4
+ 4*a*b*c^2*d^3*e - 24*a^2*c^2*d^2*e^2 + 12*a^2*b*c*d*e^3 - 5*a^2*b^2*e^4 + 12*a^3*c*e^4 + 2*(4*c^4*d^3*e - 6*
b*c^3*d^2*e^2 + 8*b^2*c^2*d*e^3 - 20*a*c^3*d*e^3 - 3*b^3*c*e^4 + 10*a*b*c^2*e^4)*x^3 + (12*b*c^3*d^3*e - 6*b^2
*c^2*d^2*e^2 - 48*a*c^3*d^2*e^2 + 12*b^3*c*d*e^3 - 12*a*b*c^2*d*e^3 - 5*b^4*e^4 + 10*a*b^2*c*e^4 + 16*a^2*c^2*
e^4)*x^2 + 2*(4*b^2*c^2*d^3*e - 4*a*c^3*d^3*e - 18*a*b*c^2*d^2*e^2 + 12*a*b^2*c*d*e^3 - 12*a^2*c^2*d*e^3 - 5*a
*b^3*e^4 + 14*a^2*b*c*e^4)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)*c^2)
________________________________________________________________________________________
maple [B] time = 0.06, size = 649, normalized size = 3.29 \begin {gather*} -\frac {12 a b \,e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c}+\frac {24 a d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {2 b^{3} e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}} c^{2}}-\frac {12 b \,d^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {8 c \,d^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\left (4 a c -b^{2}\right )^{\frac {3}{2}}}+\frac {4 a \,e^{4} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c}-\frac {b^{2} e^{4} \ln \left (c \,x^{2}+b x +a \right )}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {\frac {\left (10 a b c \,e^{3}-20 c^{2} a d \,e^{2}-3 b^{3} e^{3}+8 b^{2} c d \,e^{2}-6 b \,c^{2} d^{2} e +4 c^{3} d^{3}\right ) e \,x^{3}}{\left (4 a c -b^{2}\right ) c}+\frac {\left (16 a^{2} c^{2} e^{3}+10 a \,b^{2} c \,e^{3}-12 a b \,c^{2} d \,e^{2}-48 a \,c^{3} d^{2} e -5 b^{4} e^{3}+12 b^{3} c d \,e^{2}-6 b^{2} c^{2} d^{2} e +12 b \,c^{3} d^{3}\right ) e \,x^{2}}{2 \left (4 a c -b^{2}\right ) c^{2}}+\frac {\left (14 a^{2} b c \,e^{3}-12 a^{2} c^{2} d \,e^{2}-5 a \,b^{3} e^{3}+12 a \,b^{2} c d \,e^{2}-18 a b \,c^{2} d^{2} e -4 a \,c^{3} d^{3}+4 b^{2} c^{2} d^{3}\right ) e x}{\left (4 a c -b^{2}\right ) c^{2}}+\frac {12 a^{3} c \,e^{4}-5 a^{2} b^{2} e^{4}+12 a^{2} b c d \,e^{3}-24 a^{2} c^{2} d^{2} e^{2}+4 a b \,c^{2} d^{3} e -4 a \,c^{3} d^{4}+b^{2} c^{2} d^{4}}{2 \left (4 a c -b^{2}\right ) c^{2}}}{\left (c \,x^{2}+b x +a \right )^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^3,x)
[Out]
(1/c*e*(10*a*b*c*e^3-20*a*c^2*d*e^2-3*b^3*e^3+8*b^2*c*d*e^2-6*b*c^2*d^2*e+4*c^3*d^3)/(4*a*c-b^2)*x^3+1/2*e*(16
*a^2*c^2*e^3+10*a*b^2*c*e^3-12*a*b*c^2*d*e^2-48*a*c^3*d^2*e-5*b^4*e^3+12*b^3*c*d*e^2-6*b^2*c^2*d^2*e+12*b*c^3*
d^3)/c^2/(4*a*c-b^2)*x^2+e*(14*a^2*b*c*e^3-12*a^2*c^2*d*e^2-5*a*b^3*e^3+12*a*b^2*c*d*e^2-18*a*b*c^2*d^2*e-4*a*
c^3*d^3+4*b^2*c^2*d^3)/(4*a*c-b^2)/c^2*x+1/2*(12*a^3*c*e^4-5*a^2*b^2*e^4+12*a^2*b*c*d*e^3-24*a^2*c^2*d^2*e^2+4
*a*b*c^2*d^3*e-4*a*c^3*d^4+b^2*c^2*d^4)/c^2/(4*a*c-b^2))/(c*x^2+b*x+a)^2+4*e^4/(4*a*c-b^2)/c*ln(c*x^2+b*x+a)*a
-e^4/(4*a*c-b^2)/c^2*ln(c*x^2+b*x+a)*b^2-12*e^4/(4*a*c-b^2)^(3/2)/c*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b+24
*e^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d-12*e^2/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*
c-b^2)^(1/2))*b*d^2+8*e/(4*a*c-b^2)^(3/2)*c*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^3+2*e^4/(4*a*c-b^2)^(3/2)/c^
2*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3
________________________________________________________________________________________
maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)^4/(c*x^2+b*x+a)^3,x, algorithm="maxima")
[Out]
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(4*a*c-b^2>0)', see `assume?` f
or more details)Is 4*a*c-b^2 positive or negative?
________________________________________________________________________________________
mupad [B] time = 2.92, size = 753, normalized size = 3.82 \begin {gather*} \frac {2\,e\,\mathrm {atan}\left (\frac {c^2\,\left (\frac {2\,e\,x\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c\,{\left (4\,a\,c-b^2\right )}^3}-\frac {e\,\left (b\,e-2\,c\,d\right )\,\left (b^3\,c-4\,a\,b\,c^2\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^3\,{\left (4\,a\,c-b^2\right )}^4}\right )\,{\left (4\,a\,c-b^2\right )}^{5/2}}{b^3\,e^4-6\,b\,c^2\,d^2\,e^2-6\,a\,b\,c\,e^4+4\,c^3\,d^3\,e+12\,a\,c^2\,d\,e^3}\right )\,\left (b\,e-2\,c\,d\right )\,\left (b^2\,e^2+2\,b\,c\,d\,e-2\,c^2\,d^2-6\,a\,c\,e^2\right )}{c^2\,{\left (4\,a\,c-b^2\right )}^{3/2}}-\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (-128\,a^3\,c^3\,e^4+96\,a^2\,b^2\,c^2\,e^4-24\,a\,b^4\,c\,e^4+2\,b^6\,e^4\right )}{2\,\left (64\,a^3\,c^5-48\,a^2\,b^2\,c^4+12\,a\,b^4\,c^3-b^6\,c^2\right )}-\frac {\frac {x\,\left (-14\,a^2\,b\,c\,e^4+12\,a^2\,c^2\,d\,e^3+5\,a\,b^3\,e^4-12\,a\,b^2\,c\,d\,e^3+18\,a\,b\,c^2\,d^2\,e^2+4\,a\,c^3\,d^3\,e-4\,b^2\,c^2\,d^3\,e\right )}{c^2\,\left (4\,a\,c-b^2\right )}-\frac {12\,a^3\,c\,e^4-5\,a^2\,b^2\,e^4+12\,a^2\,b\,c\,d\,e^3-24\,a^2\,c^2\,d^2\,e^2+4\,a\,b\,c^2\,d^3\,e-4\,a\,c^3\,d^4+b^2\,c^2\,d^4}{2\,c^2\,\left (4\,a\,c-b^2\right )}+\frac {x^2\,\left (-16\,a^2\,c^2\,e^4-10\,a\,b^2\,c\,e^4+12\,a\,b\,c^2\,d\,e^3+48\,a\,c^3\,d^2\,e^2+5\,b^4\,e^4-12\,b^3\,c\,d\,e^3+6\,b^2\,c^2\,d^2\,e^2-12\,b\,c^3\,d^3\,e\right )}{2\,c^2\,\left (4\,a\,c-b^2\right )}+\frac {e\,x^3\,\left (3\,b^3\,e^3-8\,b^2\,c\,d\,e^2+6\,b\,c^2\,d^2\,e-10\,a\,b\,c\,e^3-4\,c^3\,d^3+20\,a\,c^2\,d\,e^2\right )}{c\,\left (4\,a\,c-b^2\right )}}{x^2\,\left (b^2+2\,a\,c\right )+a^2+c^2\,x^4+2\,a\,b\,x+2\,b\,c\,x^3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int(((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2)^3,x)
[Out]
(2*e*atan((c^2*((2*e*x*(b*e - 2*c*d)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c*(4*a*c - b^2)^3) - (e*(
b*e - 2*c*d)*(b^3*c - 4*a*b*c^2)*(b^2*e^2 - 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c^3*(4*a*c - b^2)^4))*(4*a*c
- b^2)^(5/2))/(b^3*e^4 + 4*c^3*d^3*e - 6*b*c^2*d^2*e^2 - 6*a*b*c*e^4 + 12*a*c^2*d*e^3))*(b*e - 2*c*d)*(b^2*e^2
- 2*c^2*d^2 - 6*a*c*e^2 + 2*b*c*d*e))/(c^2*(4*a*c - b^2)^(3/2)) - (log(a + b*x + c*x^2)*(2*b^6*e^4 - 128*a^3*
c^3*e^4 + 96*a^2*b^2*c^2*e^4 - 24*a*b^4*c*e^4))/(2*(64*a^3*c^5 - b^6*c^2 + 12*a*b^4*c^3 - 48*a^2*b^2*c^4)) - (
(x*(5*a*b^3*e^4 + 12*a^2*c^2*d*e^3 - 4*b^2*c^2*d^3*e - 14*a^2*b*c*e^4 + 4*a*c^3*d^3*e - 12*a*b^2*c*d*e^3 + 18*
a*b*c^2*d^2*e^2))/(c^2*(4*a*c - b^2)) - (12*a^3*c*e^4 - 4*a*c^3*d^4 - 5*a^2*b^2*e^4 + b^2*c^2*d^4 - 24*a^2*c^2
*d^2*e^2 + 4*a*b*c^2*d^3*e + 12*a^2*b*c*d*e^3)/(2*c^2*(4*a*c - b^2)) + (x^2*(5*b^4*e^4 - 16*a^2*c^2*e^4 + 48*a
*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 10*a*b^2*c*e^4 - 12*b*c^3*d^3*e - 12*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3))/(2*c^
2*(4*a*c - b^2)) + (e*x^3*(3*b^3*e^3 - 4*c^3*d^3 - 10*a*b*c*e^3 + 20*a*c^2*d*e^2 + 6*b*c^2*d^2*e - 8*b^2*c*d*e
^2))/(c*(4*a*c - b^2)))/(x^2*(2*a*c + b^2) + a^2 + c^2*x^4 + 2*a*b*x + 2*b*c*x^3)
________________________________________________________________________________________
sympy [B] time = 133.50, size = 1545, normalized size = 7.84
result too large to display
Verification of antiderivative is not currently implemented for this CAS.
[In]
integrate((2*c*x+b)*(e*x+d)**4/(c*x**2+b*x+a)**3,x)
[Out]
(e**4/c**2 - e*sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2
*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e**4/c**2 - e*sqrt(-(4*a*c
- b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a**2*b**
2*c**2 + 12*a*b**4*c - b**6))) + 16*a**2*c*e**4 + 8*a*b**2*c**2*(e**4/c**2 - e*sqrt(-(4*a*c - b**2)**3)*(b*e -
2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4
*c - b**6))) - 2*a*b**2*e**4 - 12*a*b*c*d*e**3 - b**4*c*(e**4/c**2 - e*sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*
(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**
6))) + 6*b**2*c*d**2*e**2 - 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 2*b**3*e**4 + 12*b*c**2*d**2*
e**2 - 8*c**3*d**3*e)) + (e**4/c**2 + e*sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c
*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6)))*log(x + (-16*a**2*c**3*(e*
*4/c**2 + e*sqrt(-(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(6
4*a**3*c**3 - 48*a**2*b**2*c**2 + 12*a*b**4*c - b**6))) + 16*a**2*c*e**4 + 8*a*b**2*c**2*(e**4/c**2 + e*sqrt(-
(4*a*c - b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a
**2*b**2*c**2 + 12*a*b**4*c - b**6))) - 2*a*b**2*e**4 - 12*a*b*c*d*e**3 - b**4*c*(e**4/c**2 + e*sqrt(-(4*a*c -
b**2)**3)*(b*e - 2*c*d)*(6*a*c*e**2 - b**2*e**2 - 2*b*c*d*e + 2*c**2*d**2)/(c**2*(64*a**3*c**3 - 48*a**2*b**2
*c**2 + 12*a*b**4*c - b**6))) + 6*b**2*c*d**2*e**2 - 4*b*c**2*d**3*e)/(12*a*b*c*e**4 - 24*a*c**2*d*e**3 - 2*b*
*3*e**4 + 12*b*c**2*d**2*e**2 - 8*c**3*d**3*e)) + (12*a**3*c*e**4 - 5*a**2*b**2*e**4 + 12*a**2*b*c*d*e**3 - 24
*a**2*c**2*d**2*e**2 + 4*a*b*c**2*d**3*e - 4*a*c**3*d**4 + b**2*c**2*d**4 + x**3*(20*a*b*c**2*e**4 - 40*a*c**3
*d*e**3 - 6*b**3*c*e**4 + 16*b**2*c**2*d*e**3 - 12*b*c**3*d**2*e**2 + 8*c**4*d**3*e) + x**2*(16*a**2*c**2*e**4
+ 10*a*b**2*c*e**4 - 12*a*b*c**2*d*e**3 - 48*a*c**3*d**2*e**2 - 5*b**4*e**4 + 12*b**3*c*d*e**3 - 6*b**2*c**2*
d**2*e**2 + 12*b*c**3*d**3*e) + x*(28*a**2*b*c*e**4 - 24*a**2*c**2*d*e**3 - 10*a*b**3*e**4 + 24*a*b**2*c*d*e**
3 - 36*a*b*c**2*d**2*e**2 - 8*a*c**3*d**3*e + 8*b**2*c**2*d**3*e))/(8*a**3*c**3 - 2*a**2*b**2*c**2 + x**4*(8*a
*c**5 - 2*b**2*c**4) + x**3*(16*a*b*c**4 - 4*b**3*c**3) + x**2*(16*a**2*c**4 + 4*a*b**2*c**3 - 2*b**4*c**2) +
x*(16*a**2*b*c**3 - 4*a*b**3*c**2))
________________________________________________________________________________________