3.1539 \(\int \frac{(b+2 c x) (d+e x)^5}{(a+b x+c x^2)^3} \, dx\)

Optimal. Leaf size=293 \[ \frac{5 e^3 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{5 e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{5 e^4 x^2 (2 c d-b e)}{2 c \left (b^2-4 a c\right )}-\frac{5 e (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.677226, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 7, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.269, Rules used = {768, 738, 800, 634, 618, 206, 628} \[ \frac{5 e^3 x \left (-c e (3 a e+2 b d)+b^2 e^2+3 c^2 d^2\right )}{c^2 \left (b^2-4 a c\right )}+\frac{5 e \left (2 b^2 c e^3 (3 a e+b d)-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (a e+2 b d)-b^4 e^4+2 c^4 d^4\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{5 e^4 x^2 (2 c d-b e)}{2 c \left (b^2-4 a c\right )}-\frac{5 e (d+e x)^3 (-2 a e+x (2 c d-b e)+b d)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(d + e*x)^5)/(a + b*x + c*x^2)^3,x]

[Out]

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

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 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 800

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Int[Exp
andIntegrand[((d + e*x)^m*(f + 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] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegerQ[m]

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{(b+2 c x) (d+e x)^5}{\left (a+b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}+\frac{1}{2} (5 e) \int \frac{(d+e x)^4}{\left (a+b x+c x^2\right )^2} \, dx\\ &=-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(5 e) \int \frac{(d+e x)^2 \left (2 \left (c d^2-2 b d e+3 a e^2\right )-2 e (2 c d-b e) x\right )}{a+b x+c x^2} \, dx}{2 \left (b^2-4 a c\right )}\\ &=-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(5 e) \int \left (-\frac{2 e^2 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right )}{c^2}-\frac{2 e^3 (2 c d-b e) x}{c}+\frac{2 \left (c^3 d^4+a b^2 e^4-2 c^2 d^2 e (b d-3 a e)-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x\right )}{c^2 \left (a+b x+c x^2\right )}\right ) \, dx}{2 \left (b^2-4 a c\right )}\\ &=\frac{5 e^3 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 (2 c d-b e) x^2}{2 c \left (b^2-4 a c\right )}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}-\frac{(5 e) \int \frac{c^3 d^4+a b^2 e^4-2 c^2 d^2 e (b d-3 a e)-a c e^3 (2 b d+3 a e)-\left (b^2-4 a c\right ) e^3 (2 c d-b e) x}{a+b x+c x^2} \, dx}{c^2 \left (b^2-4 a c\right )}\\ &=\frac{5 e^3 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 (2 c d-b e) x^2}{2 c \left (b^2-4 a c\right )}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{\left (5 e^4 (2 c d-b e)\right ) \int \frac{b+2 c x}{a+b x+c x^2} \, dx}{2 c^3}-\frac{\left (5 e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right )\right ) \int \frac{1}{a+b x+c x^2} \, dx}{2 c^3 \left (b^2-4 a c\right )}\\ &=\frac{5 e^3 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 (2 c d-b e) x^2}{2 c \left (b^2-4 a c\right )}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}+\frac{\left (5 e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (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^3 \left (b^2-4 a c\right )}\\ &=\frac{5 e^3 \left (3 c^2 d^2+b^2 e^2-c e (2 b d+3 a e)\right ) x}{c^2 \left (b^2-4 a c\right )}+\frac{5 e^4 (2 c d-b e) x^2}{2 c \left (b^2-4 a c\right )}-\frac{(d+e x)^5}{2 \left (a+b x+c x^2\right )^2}-\frac{5 e (d+e x)^3 (b d-2 a e+(2 c d-b e) x)}{2 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )}+\frac{5 e \left (2 c^4 d^4-b^4 e^4-4 c^3 d^2 e (b d-3 a e)-6 a c^2 e^3 (2 b d+a e)+2 b^2 c e^3 (b d+3 a e)\right ) \tanh ^{-1}\left (\frac{b+2 c x}{\sqrt{b^2-4 a c}}\right )}{c^3 \left (b^2-4 a c\right )^{3/2}}+\frac{5 e^4 (2 c d-b e) \log \left (a+b x+c x^2\right )}{2 c^3}\\ \end{align*}

Mathematica [A]  time = 0.916702, size = 480, normalized size = 1.64 \[ \frac{\frac{e \left (b c^2 \left (31 a^2 e^4-10 a c d e^2 (7 d+10 e x)-5 c^2 d^3 (d-4 e x)\right )-2 c^3 \left (a^2 e^3 (40 d+9 e x)-10 a c d^2 e (4 d+5 e x)+5 c^2 d^4 x\right )+2 b^2 c^2 e \left (a e^2 (25 d+17 e x)-5 c d^2 (d+4 e x)\right )+b^3 c e^2 \left (10 c d (d+3 e x)-13 a e^2\right )-b^4 c e^3 (5 d+8 e x)+b^5 e^4\right )}{\left (b^2-4 a c\right ) (a+x (b+c x))}-\frac{c^2 e^3 \left (a^2 e (5 d+e x)+10 a b d (d+e x)+10 b^2 d^2 x\right )-b c e^4 \left (2 a^2 e+a b (5 d+3 e x)+5 b^2 d x\right )+b^3 e^5 (a+b x)-10 c^3 d^2 e^2 (a (d+e x)+b d x)+c^4 d^4 (d+5 e x)}{(a+x (b+c x))^2}-\frac{10 c e \left (-2 b^2 c e^3 (3 a e+b d)+4 c^3 d^2 e (b d-3 a e)+6 a c^2 e^3 (a e+2 b d)+b^4 e^4-2 c^4 d^4\right ) \tan ^{-1}\left (\frac{b+2 c x}{\sqrt{4 a c-b^2}}\right )}{\left (4 a c-b^2\right )^{3/2}}+5 c e^4 (2 c d-b e) \log (a+x (b+c x))+4 c^2 e^5 x}{2 c^4} \]

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(d + e*x)^5)/(a + b*x + c*x^2)^3,x]

[Out]

(4*c^2*e^5*x - (b^3*e^5*(a + b*x) + c^4*d^4*(d + 5*e*x) - 10*c^3*d^2*e^2*(b*d*x + a*(d + e*x)) + c^2*e^3*(10*b
^2*d^2*x + 10*a*b*d*(d + e*x) + a^2*e*(5*d + e*x)) - b*c*e^4*(2*a^2*e + 5*b^2*d*x + a*b*(5*d + 3*e*x)))/(a + x
*(b + c*x))^2 + (e*(b^5*e^4 - b^4*c*e^3*(5*d + 8*e*x) + b^3*c*e^2*(-13*a*e^2 + 10*c*d*(d + 3*e*x)) - 2*c^3*(5*
c^2*d^4*x - 10*a*c*d^2*e*(4*d + 5*e*x) + a^2*e^3*(40*d + 9*e*x)) + b*c^2*(31*a^2*e^4 - 5*c^2*d^3*(d - 4*e*x) -
10*a*c*d*e^2*(7*d + 10*e*x)) + 2*b^2*c^2*e*(-5*c*d^2*(d + 4*e*x) + a*e^2*(25*d + 17*e*x))))/((b^2 - 4*a*c)*(a
+ x*(b + c*x))) - (10*c*e*(-2*c^4*d^4 + b^4*e^4 + 4*c^3*d^2*e*(b*d - 3*a*e) + 6*a*c^2*e^3*(2*b*d + a*e) - 2*b
^2*c*e^3*(b*d + 3*a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]])/(-b^2 + 4*a*c)^(3/2) + 5*c*e^4*(2*c*d - b*e)*L
og[a + x*(b + c*x)])/(2*c^4)

________________________________________________________________________________________

Maple [B]  time = 0.017, size = 1921, normalized size = 6.6 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(e*x+d)^5/(c*x^2+b*x+a)^3,x)

[Out]

50/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^3*a*b*d-21/2/c^2/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^2*a*b^3-60/c*e^4/(4*a*
c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d-25/2/c^2/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^2*b^4*d+30/c
^2*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2+10/c^2*e^4/(4*a*c-b^2)^(3/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*b^3*d+7/2/c^3/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^3*e^5+5*c^2/(c*x^2+b*x+a)^2*e/(4*a*c-b^2
)*x^3*d^4+4/c^2/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^3*b^4-23/2/c^2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*b*e^5+20/c*e^
4/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*d-10/c^2*e^5/(4*a*c-b^2)*ln(c*x^2+b*x+a)*a*b-5/c^2*e^4/(4*a*c-b^2)*ln(c*x^2+b*
x+a)*b^2*d+30/c/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^3*d*e^4+20/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x^3*b^2*d^2-30/(c*x^2
+b*x+a)^2*e^3/(4*a*c-b^2)*x*a^2*d^2+5/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)*x*b^2*d^4+5/2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*
a*b*d^4*e-5/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x^2*b^2*d^3+40/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^2*a^2*d+7/2/c^3/(
c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^2*b^5+7/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^3+1/2/(c*x^2+b*x+a)^2/(4*a*c-b^
2)*b^2*d^5+2*e^5/c^2*x-2*c/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a*d^5-5/c^3*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a
*c-b^2)^(1/2))*b^4+5/2/c^3*e^5/(4*a*c-b^2)*ln(c*x^2+b*x+a)*b^3-30/c*e^5/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*
a*c-b^2)^(1/2))*a^2+10*c*e/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*d^4-20*e^2/(4*a*c-b^2)^(3/2)*
arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b*d^3+60*e^3/(4*a*c-b^2)^(3/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^2-2
0/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*d^3*e^2+9/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^3*a^2-40*c/(c*x^2+b*x+a)^2*e^2/(
4*a*c-b^2)*x^2*a*d^3-10*c/(c*x^2+b*x+a)^2*e^2/(4*a*c-b^2)*x^3*b*d^3-13/2/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^2
*a^2*b-17/c/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x^3*a*b^2-50*c/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x^3*a*d^2-15/c/(c*x
^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^3*b^3*d-15/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x^2*a*b*d^2-30/(c*x^2+b*x+a)^2*e^2/(4
*a*c-b^2)*x*a*b*d^3-25/c^2/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a*b^3*d+70/c/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x*a^
2*b*d+30/c/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x*a*b^2*d^2+25/c/(c*x^2+b*x+a)^2*e^4/(4*a*c-b^2)*x^2*a*b^2*d-26/c^2
/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a^2*b^2-25/2/c^2/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b^2*d*e^4+15/2*c/(c*x^2+b*
x+a)^2*e/(4*a*c-b^2)*x^2*b*d^4+7/c^3/(c*x^2+b*x+a)^2*e^5/(4*a*c-b^2)*x*a*b^4-5*c/(c*x^2+b*x+a)^2*e/(4*a*c-b^2)
*x*a*d^4+15/c/(c*x^2+b*x+a)^2/(4*a*c-b^2)*a^2*b*d^2*e^3+15/c/(c*x^2+b*x+a)^2*e^3/(4*a*c-b^2)*x^2*b^3*d^2

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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((2*c*x+b)*(e*x+d)^5/(c*x^2+b*x+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.76782, size = 7375, normalized size = 25.17 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^5/(c*x^2+b*x+a)^3,x, algorithm="fricas")

[Out]

[1/2*(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^5 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5*x^4 - (b^4
*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^5 - 5*(a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e + 40*(a^2*b^2*c^3 - 4*a^3*c^4)*d^3*e^
2 - 30*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^3 + 5*(5*a^2*b^4*c - 32*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^4 - (7*a^2*b^5
- 51*a^3*b^3*c + 92*a^4*b*c^2)*e^5 - 2*(5*(b^2*c^5 - 4*a*c^6)*d^4*e - 10*(b^3*c^4 - 4*a*b*c^5)*d^3*e^2 + 10*(2
*b^4*c^3 - 13*a*b^2*c^4 + 20*a^2*c^5)*d^2*e^3 - 5*(3*b^5*c^2 - 22*a*b^3*c^3 + 40*a^2*b*c^4)*d*e^4 + (2*b^6*c -
21*a*b^4*c^2 + 77*a^2*b^2*c^3 - 100*a^3*c^4)*e^5)*x^3 - (15*(b^3*c^4 - 4*a*b*c^5)*d^4*e - 10*(b^4*c^3 + 4*a*b
^2*c^4 - 32*a^2*c^5)*d^3*e^2 + 30*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^2*e^3 - 5*(5*b^6*c - 30*a*b^4*c^2 +
24*a^2*b^2*c^3 + 64*a^3*c^4)*d*e^4 + (7*b^7 - 57*a*b^5*c + 135*a^2*b^3*c^2 - 76*a^3*b*c^3)*e^5)*x^2 + 5*(2*a^2
*c^4*d^4*e - 4*a^2*b*c^3*d^3*e^2 + 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2)*d*e^4 - (a^2*b^4 - 6*a^3*b
^2*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*c^3 - 6*a*b*c^4)*d*e^4 - (b
^4*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12*a*b*c^4*d^2*e^3 + 2*(b^
4*c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*e - 4*(
b^3*c^3 + 2*a*b*c^4)*d^3*e^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*b^3*c^2 - 12*a^2*b*c^3)*d*e
^4 - (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*d^4*e - 4*a*b^2*c^3*d^3*e^2 + 12*a
^2*b*c^3*d^2*e^3 + 2*(a*b^4*c - 6*a^2*b^2*c^2)*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e^5)*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*(5*(b^4*
c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^4*e - 30*(a*b^3*c^3 - 4*a^2*b*c^4)*d^3*e^2 + 30*(a*b^4*c^2 - 5*a^2*b^2*c^3 +
4*a^3*c^4)*d^2*e^3 - 5*(5*a*b^5*c - 34*a^2*b^3*c^2 + 56*a^3*b*c^3)*d*e^4 + (7*a*b^6 - 56*a^2*b^4*c + 127*a^3*b
^2*c^2 - 60*a^4*c^3)*e^5)*x + 5*(2*(a^2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 1
6*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)
*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (
2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2
*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*log(c*x^2 + b*x + a))/(a^2*b^4
*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a
^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4 + 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x), 1/
2*(4*(b^4*c^3 - 8*a*b^2*c^4 + 16*a^2*c^5)*e^5*x^5 + 8*(b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5*x^4 - (b^4*c^
3 - 8*a*b^2*c^4 + 16*a^2*c^5)*d^5 - 5*(a*b^3*c^3 - 4*a^2*b*c^4)*d^4*e + 40*(a^2*b^2*c^3 - 4*a^3*c^4)*d^3*e^2 -
30*(a^2*b^3*c^2 - 4*a^3*b*c^3)*d^2*e^3 + 5*(5*a^2*b^4*c - 32*a^3*b^2*c^2 + 48*a^4*c^3)*d*e^4 - (7*a^2*b^5 - 5
1*a^3*b^3*c + 92*a^4*b*c^2)*e^5 - 2*(5*(b^2*c^5 - 4*a*c^6)*d^4*e - 10*(b^3*c^4 - 4*a*b*c^5)*d^3*e^2 + 10*(2*b^
4*c^3 - 13*a*b^2*c^4 + 20*a^2*c^5)*d^2*e^3 - 5*(3*b^5*c^2 - 22*a*b^3*c^3 + 40*a^2*b*c^4)*d*e^4 + (2*b^6*c - 21
*a*b^4*c^2 + 77*a^2*b^2*c^3 - 100*a^3*c^4)*e^5)*x^3 - (15*(b^3*c^4 - 4*a*b*c^5)*d^4*e - 10*(b^4*c^3 + 4*a*b^2*
c^4 - 32*a^2*c^5)*d^3*e^2 + 30*(b^5*c^2 - 5*a*b^3*c^3 + 4*a^2*b*c^4)*d^2*e^3 - 5*(5*b^6*c - 30*a*b^4*c^2 + 24*
a^2*b^2*c^3 + 64*a^3*c^4)*d*e^4 + (7*b^7 - 57*a*b^5*c + 135*a^2*b^3*c^2 - 76*a^3*b*c^3)*e^5)*x^2 + 10*(2*a^2*c
^4*d^4*e - 4*a^2*b*c^3*d^3*e^2 + 12*a^3*c^3*d^2*e^3 + 2*(a^2*b^3*c - 6*a^3*b*c^2)*d*e^4 - (a^2*b^4 - 6*a^3*b^2
*c + 6*a^4*c^2)*e^5 + (2*c^6*d^4*e - 4*b*c^5*d^3*e^2 + 12*a*c^5*d^2*e^3 + 2*(b^3*c^3 - 6*a*b*c^4)*d*e^4 - (b^4
*c^2 - 6*a*b^2*c^3 + 6*a^2*c^4)*e^5)*x^4 + 2*(2*b*c^5*d^4*e - 4*b^2*c^4*d^3*e^2 + 12*a*b*c^4*d^2*e^3 + 2*(b^4*
c^2 - 6*a*b^2*c^3)*d*e^4 - (b^5*c - 6*a*b^3*c^2 + 6*a^2*b*c^3)*e^5)*x^3 + (2*(b^2*c^4 + 2*a*c^5)*d^4*e - 4*(b^
3*c^3 + 2*a*b*c^4)*d^3*e^2 + 12*(a*b^2*c^3 + 2*a^2*c^4)*d^2*e^3 + 2*(b^5*c - 4*a*b^3*c^2 - 12*a^2*b*c^3)*d*e^4
- (b^6 - 4*a*b^4*c - 6*a^2*b^2*c^2 + 12*a^3*c^3)*e^5)*x^2 + 2*(2*a*b*c^4*d^4*e - 4*a*b^2*c^3*d^3*e^2 + 12*a^2
*b*c^3*d^2*e^3 + 2*(a*b^4*c - 6*a^2*b^2*c^2)*d*e^4 - (a*b^5 - 6*a^2*b^3*c + 6*a^3*b*c^2)*e^5)*x)*sqrt(-b^2 + 4
*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) - 2*(5*(b^4*c^3 - 5*a*b^2*c^4 + 4*a^2*c^5)*d^4*e -
30*(a*b^3*c^3 - 4*a^2*b*c^4)*d^3*e^2 + 30*(a*b^4*c^2 - 5*a^2*b^2*c^3 + 4*a^3*c^4)*d^2*e^3 - 5*(5*a*b^5*c - 34
*a^2*b^3*c^2 + 56*a^3*b*c^3)*d*e^4 + (7*a*b^6 - 56*a^2*b^4*c + 127*a^3*b^2*c^2 - 60*a^4*c^3)*e^5)*x + 5*(2*(a^
2*b^4*c - 8*a^3*b^2*c^2 + 16*a^4*c^3)*d*e^4 - (a^2*b^5 - 8*a^3*b^3*c + 16*a^4*b*c^2)*e^5 + (2*(b^4*c^3 - 8*a*b
^2*c^4 + 16*a^2*c^5)*d*e^4 - (b^5*c^2 - 8*a*b^3*c^3 + 16*a^2*b*c^4)*e^5)*x^4 + 2*(2*(b^5*c^2 - 8*a*b^3*c^3 + 1
6*a^2*b*c^4)*d*e^4 - (b^6*c - 8*a*b^4*c^2 + 16*a^2*b^2*c^3)*e^5)*x^3 + (2*(b^6*c - 6*a*b^4*c^2 + 32*a^3*c^4)*d
*e^4 - (b^7 - 6*a*b^5*c + 32*a^3*b*c^3)*e^5)*x^2 + 2*(2*(a*b^5*c - 8*a^2*b^3*c^2 + 16*a^3*b*c^3)*d*e^4 - (a*b^
6 - 8*a^2*b^4*c + 16*a^3*b^2*c^2)*e^5)*x)*log(c*x^2 + b*x + a))/(a^2*b^4*c^3 - 8*a^3*b^2*c^4 + 16*a^4*c^5 + (b
^4*c^5 - 8*a*b^2*c^6 + 16*a^2*c^7)*x^4 + 2*(b^5*c^4 - 8*a*b^3*c^5 + 16*a^2*b*c^6)*x^3 + (b^6*c^3 - 6*a*b^4*c^4
+ 32*a^3*c^6)*x^2 + 2*(a*b^5*c^3 - 8*a^2*b^3*c^4 + 16*a^3*b*c^5)*x)]

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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((2*c*x+b)*(e*x+d)**5/(c*x**2+b*x+a)**3,x)

[Out]

Timed out

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Giac [B]  time = 1.19468, size = 859, normalized size = 2.93 \begin{align*} -\frac{5 \,{\left (2 \, c^{4} d^{4} e - 4 \, b c^{3} d^{3} e^{2} + 12 \, a c^{3} d^{2} e^{3} + 2 \, b^{3} c d e^{4} - 12 \, a b c^{2} d e^{4} - b^{4} e^{5} + 6 \, a b^{2} c e^{5} - 6 \, a^{2} c^{2} e^{5}\right )} \arctan \left (\frac{2 \, c x + b}{\sqrt{-b^{2} + 4 \, a c}}\right )}{{\left (b^{2} c^{3} - 4 \, a c^{4}\right )} \sqrt{-b^{2} + 4 \, a c}} + \frac{2 \, x e^{5}}{c^{2}} + \frac{5 \,{\left (2 \, c d e^{4} - b e^{5}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{3}} - \frac{b^{2} c^{3} d^{5} - 4 \, a c^{4} d^{5} + 5 \, a b c^{3} d^{4} e - 40 \, a^{2} c^{3} d^{3} e^{2} + 30 \, a^{2} b c^{2} d^{2} e^{3} - 25 \, a^{2} b^{2} c d e^{4} + 60 \, a^{3} c^{2} d e^{4} + 7 \, a^{2} b^{3} e^{5} - 23 \, a^{3} b c e^{5} + 2 \,{\left (5 \, c^{5} d^{4} e - 10 \, b c^{4} d^{3} e^{2} + 20 \, b^{2} c^{3} d^{2} e^{3} - 50 \, a c^{4} d^{2} e^{3} - 15 \, b^{3} c^{2} d e^{4} + 50 \, a b c^{3} d e^{4} + 4 \, b^{4} c e^{5} - 17 \, a b^{2} c^{2} e^{5} + 9 \, a^{2} c^{3} e^{5}\right )} x^{3} +{\left (15 \, b c^{4} d^{4} e - 10 \, b^{2} c^{3} d^{3} e^{2} - 80 \, a c^{4} d^{3} e^{2} + 30 \, b^{3} c^{2} d^{2} e^{3} - 30 \, a b c^{3} d^{2} e^{3} - 25 \, b^{4} c d e^{4} + 50 \, a b^{2} c^{2} d e^{4} + 80 \, a^{2} c^{3} d e^{4} + 7 \, b^{5} e^{5} - 21 \, a b^{3} c e^{5} - 13 \, a^{2} b c^{2} e^{5}\right )} x^{2} + 2 \,{\left (5 \, b^{2} c^{3} d^{4} e - 5 \, a c^{4} d^{4} e - 30 \, a b c^{3} d^{3} e^{2} + 30 \, a b^{2} c^{2} d^{2} e^{3} - 30 \, a^{2} c^{3} d^{2} e^{3} - 25 \, a b^{3} c d e^{4} + 70 \, a^{2} b c^{2} d e^{4} + 7 \, a b^{4} e^{5} - 26 \, a^{2} b^{2} c e^{5} + 7 \, a^{3} c^{2} e^{5}\right )} x}{2 \,{\left (c x^{2} + b x + a\right )}^{2}{\left (b^{2} - 4 \, a c\right )} c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(e*x+d)^5/(c*x^2+b*x+a)^3,x, algorithm="giac")

[Out]

-5*(2*c^4*d^4*e - 4*b*c^3*d^3*e^2 + 12*a*c^3*d^2*e^3 + 2*b^3*c*d*e^4 - 12*a*b*c^2*d*e^4 - b^4*e^5 + 6*a*b^2*c*
e^5 - 6*a^2*c^2*e^5)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/((b^2*c^3 - 4*a*c^4)*sqrt(-b^2 + 4*a*c)) + 2*x*e^5
/c^2 + 5/2*(2*c*d*e^4 - b*e^5)*log(c*x^2 + b*x + a)/c^3 - 1/2*(b^2*c^3*d^5 - 4*a*c^4*d^5 + 5*a*b*c^3*d^4*e - 4
0*a^2*c^3*d^3*e^2 + 30*a^2*b*c^2*d^2*e^3 - 25*a^2*b^2*c*d*e^4 + 60*a^3*c^2*d*e^4 + 7*a^2*b^3*e^5 - 23*a^3*b*c*
e^5 + 2*(5*c^5*d^4*e - 10*b*c^4*d^3*e^2 + 20*b^2*c^3*d^2*e^3 - 50*a*c^4*d^2*e^3 - 15*b^3*c^2*d*e^4 + 50*a*b*c^
3*d*e^4 + 4*b^4*c*e^5 - 17*a*b^2*c^2*e^5 + 9*a^2*c^3*e^5)*x^3 + (15*b*c^4*d^4*e - 10*b^2*c^3*d^3*e^2 - 80*a*c^
4*d^3*e^2 + 30*b^3*c^2*d^2*e^3 - 30*a*b*c^3*d^2*e^3 - 25*b^4*c*d*e^4 + 50*a*b^2*c^2*d*e^4 + 80*a^2*c^3*d*e^4 +
7*b^5*e^5 - 21*a*b^3*c*e^5 - 13*a^2*b*c^2*e^5)*x^2 + 2*(5*b^2*c^3*d^4*e - 5*a*c^4*d^4*e - 30*a*b*c^3*d^3*e^2
+ 30*a*b^2*c^2*d^2*e^3 - 30*a^2*c^3*d^2*e^3 - 25*a*b^3*c*d*e^4 + 70*a^2*b*c^2*d*e^4 + 7*a*b^4*e^5 - 26*a^2*b^2
*c*e^5 + 7*a^3*c^2*e^5)*x)/((c*x^2 + b*x + a)^2*(b^2 - 4*a*c)*c^3)