3.16.24 \(\int \frac {(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx\) [1524]

Optimal. Leaf size=299 \[ \frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac {e^3 (8 c d-b e) x^3}{3 c}+\frac {e^4 x^4}{2}-\frac {\sqrt {b^2-4 a c} e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^4} \]

[Out]

e*(8*c^3*d^3-b^3*e^3+b*c*e^2*(3*a*e+4*b*d)-2*c^2*d*e*(4*a*e+3*b*d))*x/c^3+1/2*e^2*(12*c^2*d^2+b^2*e^2-2*c*e*(a
*e+2*b*d))*x^2/c^2+1/3*e^3*(-b*e+8*c*d)*x^3/c+1/2*e^4*x^4+1/2*(2*c^4*d^4+b^4*e^4-4*b^2*c*e^3*(a*e+b*d)-4*c^3*d
^2*e*(3*a*e+b*d)+2*c^2*e^2*(a^2*e^2+6*a*b*d*e+3*b^2*d^2))*ln(c*x^2+b*x+a)/c^4-e*(-b*e+2*c*d)*(2*c^2*d^2+b^2*e^
2-2*c*e*(a*e+b*d))*arctanh((2*c*x+b)/(-4*a*c+b^2)^(1/2))*(-4*a*c+b^2)^(1/2)/c^4

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Rubi [A]
time = 0.26, antiderivative size = 299, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {814, 648, 632, 212, 642} \begin {gather*} \frac {\left (2 c^2 e^2 \left (a^2 e^2+6 a b d e+3 b^2 d^2\right )-4 b^2 c e^3 (a e+b d)-4 c^3 d^2 e (3 a e+b d)+b^4 e^4+2 c^4 d^4\right ) \log \left (a+b x+c x^2\right )}{2 c^4}+\frac {e x \left (-2 c^2 d e (4 a e+3 b d)+b c e^2 (3 a e+4 b d)-b^3 e^3+8 c^3 d^3\right )}{c^3}+\frac {e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )}{2 c^2}-\frac {e \sqrt {b^2-4 a c} (2 c d-b e) \left (-2 c e (a e+b d)+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^4}+\frac {e^3 x^3 (8 c d-b e)}{3 c}+\frac {e^4 x^4}{2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2),x]

[Out]

(e*(8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*x)/c^3 + (e^2*(12*c^2*d^2 + b^2
*e^2 - 2*c*e*(2*b*d + a*e))*x^2)/(2*c^2) + (e^3*(8*c*d - b*e)*x^3)/(3*c) + (e^4*x^4)/2 - (Sqrt[b^2 - 4*a*c]*e*
(2*c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/c^4 + ((2*c^4*
d^4 + b^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e
^2))*Log[a + b*x + c*x^2])/(2*c^4)

Rule 212

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

Rule 632

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 642

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 648

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 814

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]

Rubi steps

\begin {align*} \int \frac {(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx &=\int \left (\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right )}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x}{c^2}+\frac {e^3 (8 c d-b e) x^2}{c}+2 e^4 x^3+\frac {-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{c^3 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac {e^3 (8 c d-b e) x^3}{3 c}+\frac {e^4 x^4}{2}+\frac {\int \frac {-4 a b^2 c d e^3+a b^3 e^4-8 a c^2 d e \left (c d^2-a e^2\right )+b c \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) x}{a+b x+c x^2} \, dx}{c^3}\\ &=\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac {e^3 (8 c d-b e) x^3}{3 c}+\frac {e^4 x^4}{2}+\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 c^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 c^4}\\ &=\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac {e^3 (8 c d-b e) x^3}{3 c}+\frac {e^4 x^4}{2}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^4}-\frac {\left (\left (b^2-4 a c\right ) e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right )\right ) \text {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{c^4}\\ &=\frac {e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x}{c^3}+\frac {e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2}{2 c^2}+\frac {e^3 (8 c d-b e) x^3}{3 c}+\frac {e^4 x^4}{2}-\frac {\sqrt {b^2-4 a c} e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^4}+\frac {\left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log \left (a+b x+c x^2\right )}{2 c^4}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 297, normalized size = 0.99 \begin {gather*} \frac {6 c e \left (8 c^3 d^3-b^3 e^3+b c e^2 (4 b d+3 a e)-2 c^2 d e (3 b d+4 a e)\right ) x+3 c^2 e^2 \left (12 c^2 d^2+b^2 e^2-2 c e (2 b d+a e)\right ) x^2+2 c^3 e^3 (8 c d-b e) x^3+3 c^4 e^4 x^4-6 \sqrt {-b^2+4 a c} e (2 c d-b e) \left (2 c^2 d^2+b^2 e^2-2 c e (b d+a e)\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {-b^2+4 a c}}\right )+3 \left (2 c^4 d^4+b^4 e^4-4 b^2 c e^3 (b d+a e)-4 c^3 d^2 e (b d+3 a e)+2 c^2 e^2 \left (3 b^2 d^2+6 a b d e+a^2 e^2\right )\right ) \log (a+x (b+c x))}{6 c^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((b + 2*c*x)*(d + e*x)^4)/(a + b*x + c*x^2),x]

[Out]

(6*c*e*(8*c^3*d^3 - b^3*e^3 + b*c*e^2*(4*b*d + 3*a*e) - 2*c^2*d*e*(3*b*d + 4*a*e))*x + 3*c^2*e^2*(12*c^2*d^2 +
b^2*e^2 - 2*c*e*(2*b*d + a*e))*x^2 + 2*c^3*e^3*(8*c*d - b*e)*x^3 + 3*c^4*e^4*x^4 - 6*Sqrt[-b^2 + 4*a*c]*e*(2*
c*d - b*e)*(2*c^2*d^2 + b^2*e^2 - 2*c*e*(b*d + a*e))*ArcTan[(b + 2*c*x)/Sqrt[-b^2 + 4*a*c]] + 3*(2*c^4*d^4 + b
^4*e^4 - 4*b^2*c*e^3*(b*d + a*e) - 4*c^3*d^2*e*(b*d + 3*a*e) + 2*c^2*e^2*(3*b^2*d^2 + 6*a*b*d*e + a^2*e^2))*Lo
g[a + x*(b + c*x)])/(6*c^4)

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Maple [A]
time = 1.04, size = 474, normalized size = 1.59

method	result	size
default	\(\frac {e \left (\frac {1}{2} c^{3} x^{4} e^{3}-\frac {1}{3} b \,c^{2} e^{3} x^{3}+\frac {8}{3} c^{3} d \,e^{2} x^{3}-a \,c^{2} e^{3} x^{2}+\frac {1}{2} b^{2} c \,e^{3} x^{2}-2 b \,c^{2} d \,e^{2} x^{2}+6 c^{3} d^{2} e \,x^{2}+3 a b c \,e^{3} x -8 d \,e^{2} c^{2} a x -b^{3} e^{3} x +4 b^{2} d \,e^{2} c x -6 b \,c^{2} d^{2} e x +8 c^{3} d^{3} x \right )}{c^{3}}+\frac {\frac {\left (2 e^{4} a^{2} c^{2}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} c^{3} a +b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) \ln \left (c \,x^{2}+b x +a \right )}{2 c}+\frac {2 \left (-3 c \,e^{4} a^{2} b +8 a^{2} c^{2} d \,e^{3}+a \,b^{3} e^{4}-4 a \,b^{2} c d \,e^{3}+6 a b \,c^{2} d^{2} e^{2}-8 a \,c^{3} d^{3} e +d^{4} b \,c^{3}-\frac {\left (2 e^{4} a^{2} c^{2}-4 a \,b^{2} c \,e^{4}+12 a b \,c^{2} d \,e^{3}-12 d^{2} e^{2} c^{3} a +b^{4} e^{4}-4 b^{3} c d \,e^{3}+6 b^{2} c^{2} d^{2} e^{2}-4 b \,c^{3} d^{3} e +2 c^{4} d^{4}\right ) b}{2 c}\right ) \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}}{c^{3}}\)	\(474\)
risch	\(\text {Expression too large to display}\)	\(5278\)

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),x,method=_RETURNVERBOSE)

[Out]

e/c^3*(1/2*c^3*x^4*e^3-1/3*b*c^2*e^3*x^3+8/3*c^3*d*e^2*x^3-a*c^2*e^3*x^2+1/2*b^2*c*e^3*x^2-2*b*c^2*d*e^2*x^2+6
*c^3*d^2*e*x^2+3*a*b*c*e^3*x-8*d*e^2*c^2*a*x-b^3*e^3*x+4*b^2*d*e^2*c*x-6*b*c^2*d^2*e*x+8*c^3*d^3*x)+1/c^3*(1/2
*(2*a^2*c^2*e^4-4*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-4*b^3*c*d*e^3+6*b^2*c^2*d^2*e^2-4*b*c^
3*d^3*e+2*c^4*d^4)/c*ln(c*x^2+b*x+a)+2*(-3*c*e^4*a^2*b+8*a^2*c^2*d*e^3+a*b^3*e^4-4*a*b^2*c*d*e^3+6*a*b*c^2*d^2
*e^2-8*a*c^3*d^3*e+d^4*b*c^3-1/2*(2*a^2*c^2*e^4-4*a*b^2*c*e^4+12*a*b*c^2*d*e^3-12*a*c^3*d^2*e^2+b^4*e^4-4*b^3*
c*d*e^3+6*b^2*c^2*d^2*e^2-4*b*c^3*d^3*e+2*c^4*d^4)*b/c)/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2)))

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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),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 deta

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Fricas [A]
time = 3.07, size = 683, normalized size = 2.28 \begin {gather*} \left [\frac {48 \, c^{4} d^{3} x e + 3 \, {\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c - a c^{2}\right )} d e^{3} - {\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \sqrt {b^{2} - 4 \, a c} \log \left (\frac {2 \, c^{2} x^{2} + 2 \, b c x + b^{2} - 2 \, a c - \sqrt {b^{2} - 4 \, a c} {\left (2 \, c x + b\right )}}{c x^{2} + b x + a}\right ) + {\left (3 \, c^{4} x^{4} - 2 \, b c^{3} x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} x\right )} e^{4} + 4 \, {\left (4 \, c^{4} d x^{3} - 3 \, b c^{3} d x^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d x\right )} e^{3} + 36 \, {\left (c^{4} d^{2} x^{2} - b c^{3} d^{2} x\right )} e^{2} + 3 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{4}}, \frac {48 \, c^{4} d^{3} x e - 6 \, {\left (4 \, c^{3} d^{3} e - 6 \, b c^{2} d^{2} e^{2} + 4 \, {\left (b^{2} c - a c^{2}\right )} d e^{3} - {\left (b^{3} - 2 \, a b c\right )} e^{4}\right )} \sqrt {-b^{2} + 4 \, a c} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + {\left (3 \, c^{4} x^{4} - 2 \, b c^{3} x^{3} + 3 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} x^{2} - 6 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} x\right )} e^{4} + 4 \, {\left (4 \, c^{4} d x^{3} - 3 \, b c^{3} d x^{2} + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d x\right )} e^{3} + 36 \, {\left (c^{4} d^{2} x^{2} - b c^{3} d^{2} x\right )} e^{2} + 3 \, {\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d^{2} e^{2} - 4 \, {\left (b^{3} c - 3 \, a b c^{2}\right )} d e^{3} + {\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{6 \, c^{4}}\right ] \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),x, algorithm="fricas")

[Out]

[1/6*(48*c^4*d^3*x*e + 3*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - a*c^2)*d*e^3 - (b^3 - 2*a*b*c)*e^4)*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)) + (3*c
^4*x^4 - 2*b*c^3*x^3 + 3*(b^2*c^2 - 2*a*c^3)*x^2 - 6*(b^3*c - 3*a*b*c^2)*x)*e^4 + 4*(4*c^4*d*x^3 - 3*b*c^3*d*x
^2 + 6*(b^2*c^2 - 2*a*c^3)*d*x)*e^3 + 36*(c^4*d^2*x^2 - b*c^3*d^2*x)*e^2 + 3*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b
^2*c^2 - 2*a*c^3)*d^2*e^2 - 4*(b^3*c - 3*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^4)*log(c*x^2 + b*x +
a))/c^4, 1/6*(48*c^4*d^3*x*e - 6*(4*c^3*d^3*e - 6*b*c^2*d^2*e^2 + 4*(b^2*c - a*c^2)*d*e^3 - (b^3 - 2*a*b*c)*e
^4)*sqrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + (3*c^4*x^4 - 2*b*c^3*x^3 + 3*(b
^2*c^2 - 2*a*c^3)*x^2 - 6*(b^3*c - 3*a*b*c^2)*x)*e^4 + 4*(4*c^4*d*x^3 - 3*b*c^3*d*x^2 + 6*(b^2*c^2 - 2*a*c^3)*
d*x)*e^3 + 36*(c^4*d^2*x^2 - b*c^3*d^2*x)*e^2 + 3*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*(b^2*c^2 - 2*a*c^3)*d^2*e^2 -
4*(b^3*c - 3*a*b*c^2)*d*e^3 + (b^4 - 4*a*b^2*c + 2*a^2*c^2)*e^4)*log(c*x^2 + b*x + a))/c^4]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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),x)

[Out]

Timed out

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Giac [A]
time = 1.55, size = 401, normalized size = 1.34 \begin {gather*} \frac {{\left (2 \, c^{4} d^{4} - 4 \, b c^{3} d^{3} e + 6 \, b^{2} c^{2} d^{2} e^{2} - 12 \, a c^{3} d^{2} e^{2} - 4 \, b^{3} c d e^{3} + 12 \, a b c^{2} d e^{3} + b^{4} e^{4} - 4 \, a b^{2} c e^{4} + 2 \, a^{2} c^{2} e^{4}\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{4}} + \frac {{\left (4 \, b^{2} c^{3} d^{3} e - 16 \, a c^{4} d^{3} e - 6 \, b^{3} c^{2} d^{2} e^{2} + 24 \, a b c^{3} d^{2} e^{2} + 4 \, b^{4} c d e^{3} - 20 \, a b^{2} c^{2} d e^{3} + 16 \, a^{2} c^{3} d e^{3} - b^{5} e^{4} + 6 \, a b^{3} c e^{4} - 8 \, a^{2} b c^{2} e^{4}\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{4}} + \frac {3 \, c^{4} x^{4} e^{4} + 16 \, c^{4} d x^{3} e^{3} + 36 \, c^{4} d^{2} x^{2} e^{2} + 48 \, c^{4} d^{3} x e - 2 \, b c^{3} x^{3} e^{4} - 12 \, b c^{3} d x^{2} e^{3} - 36 \, b c^{3} d^{2} x e^{2} + 3 \, b^{2} c^{2} x^{2} e^{4} - 6 \, a c^{3} x^{2} e^{4} + 24 \, b^{2} c^{2} d x e^{3} - 48 \, a c^{3} d x e^{3} - 6 \, b^{3} c x e^{4} + 18 \, a b c^{2} x e^{4}}{6 \, c^{4}} \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),x, algorithm="giac")

[Out]

1/2*(2*c^4*d^4 - 4*b*c^3*d^3*e + 6*b^2*c^2*d^2*e^2 - 12*a*c^3*d^2*e^2 - 4*b^3*c*d*e^3 + 12*a*b*c^2*d*e^3 + b^4
*e^4 - 4*a*b^2*c*e^4 + 2*a^2*c^2*e^4)*log(c*x^2 + b*x + a)/c^4 + (4*b^2*c^3*d^3*e - 16*a*c^4*d^3*e - 6*b^3*c^2
*d^2*e^2 + 24*a*b*c^3*d^2*e^2 + 4*b^4*c*d*e^3 - 20*a*b^2*c^2*d*e^3 + 16*a^2*c^3*d*e^3 - b^5*e^4 + 6*a*b^3*c*e^
4 - 8*a^2*b*c^2*e^4)*arctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^4) + 1/6*(3*c^4*x^4*e^4 + 16
*c^4*d*x^3*e^3 + 36*c^4*d^2*x^2*e^2 + 48*c^4*d^3*x*e - 2*b*c^3*x^3*e^4 - 12*b*c^3*d*x^2*e^3 - 36*b*c^3*d^2*x*e
^2 + 3*b^2*c^2*x^2*e^4 - 6*a*c^3*x^2*e^4 + 24*b^2*c^2*d*x*e^3 - 48*a*c^3*d*x*e^3 - 6*b^3*c*x*e^4 + 18*a*b*c^2*
x*e^4)/c^4

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Mupad [B]
time = 2.09, size = 726, normalized size = 2.43 \begin {gather*} x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{c}-\frac {2\,b\,e^4}{c}\right )}{c}+\frac {2\,a\,e^4}{c}-\frac {4\,d\,e^2\,\left (b\,e+3\,c\,d\right )}{c}\right )}{c}-\frac {a\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{c}-\frac {2\,b\,e^4}{c}\right )}{c}+\frac {2\,d^2\,e\,\left (3\,b\,e+4\,c\,d\right )}{c}\right )+x^3\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{3\,c}-\frac {2\,b\,e^4}{3\,c}\right )-x^2\,\left (\frac {b\,\left (\frac {b\,e^4+8\,c\,d\,e^3}{c}-\frac {2\,b\,e^4}{c}\right )}{2\,c}+\frac {a\,e^4}{c}-\frac {2\,d\,e^2\,\left (b\,e+3\,c\,d\right )}{c}\right )+\frac {e^4\,x^4}{2}+\frac {\ln \left (b\,\sqrt {b^2-4\,a\,c}-4\,a\,c+b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,e^4+2\,c^4\,d^4+b^3\,e^4\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c^2\,e^4-12\,a\,c^3\,d^2\,e^2+6\,b^2\,c^2\,d^2\,e^2-4\,a\,b^2\,c\,e^4-4\,b\,c^3\,d^3\,e-4\,b^3\,c\,d\,e^3-4\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}+12\,a\,b\,c^2\,d\,e^3+4\,a\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}-4\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}+6\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}-2\,a\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^4}+\frac {\ln \left (4\,a\,c+b\,\sqrt {b^2-4\,a\,c}-b^2+2\,c\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,e^4+2\,c^4\,d^4-b^3\,e^4\,\sqrt {b^2-4\,a\,c}+2\,a^2\,c^2\,e^4-12\,a\,c^3\,d^2\,e^2+6\,b^2\,c^2\,d^2\,e^2-4\,a\,b^2\,c\,e^4-4\,b\,c^3\,d^3\,e-4\,b^3\,c\,d\,e^3+4\,c^3\,d^3\,e\,\sqrt {b^2-4\,a\,c}+12\,a\,b\,c^2\,d\,e^3-4\,a\,c^2\,d\,e^3\,\sqrt {b^2-4\,a\,c}+4\,b^2\,c\,d\,e^3\,\sqrt {b^2-4\,a\,c}-6\,b\,c^2\,d^2\,e^2\,\sqrt {b^2-4\,a\,c}+2\,a\,b\,c\,e^4\,\sqrt {b^2-4\,a\,c}\right )}{2\,c^4} \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),x)

[Out]

x*((b*((b*((b*e^4 + 8*c*d*e^3)/c - (2*b*e^4)/c))/c + (2*a*e^4)/c - (4*d*e^2*(b*e + 3*c*d))/c))/c - (a*((b*e^4
+ 8*c*d*e^3)/c - (2*b*e^4)/c))/c + (2*d^2*e*(3*b*e + 4*c*d))/c) + x^3*((b*e^4 + 8*c*d*e^3)/(3*c) - (2*b*e^4)/(
3*c)) - x^2*((b*((b*e^4 + 8*c*d*e^3)/c - (2*b*e^4)/c))/(2*c) + (a*e^4)/c - (2*d*e^2*(b*e + 3*c*d))/c) + (e^4*x
^4)/2 + (log(b*(b^2 - 4*a*c)^(1/2) - 4*a*c + b^2 + 2*c*x*(b^2 - 4*a*c)^(1/2))*(b^4*e^4 + 2*c^4*d^4 + b^3*e^4*(
b^2 - 4*a*c)^(1/2) + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*
b^3*c*d*e^3 - 4*c^3*d^3*e*(b^2 - 4*a*c)^(1/2) + 12*a*b*c^2*d*e^3 + 4*a*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) - 4*b^2*c
*d*e^3*(b^2 - 4*a*c)^(1/2) + 6*b*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) - 2*a*b*c*e^4*(b^2 - 4*a*c)^(1/2)))/(2*c^4) +
(log(4*a*c + b*(b^2 - 4*a*c)^(1/2) - b^2 + 2*c*x*(b^2 - 4*a*c)^(1/2))*(b^4*e^4 + 2*c^4*d^4 - b^3*e^4*(b^2 - 4
*a*c)^(1/2) + 2*a^2*c^2*e^4 - 12*a*c^3*d^2*e^2 + 6*b^2*c^2*d^2*e^2 - 4*a*b^2*c*e^4 - 4*b*c^3*d^3*e - 4*b^3*c*d
*e^3 + 4*c^3*d^3*e*(b^2 - 4*a*c)^(1/2) + 12*a*b*c^2*d*e^3 - 4*a*c^2*d*e^3*(b^2 - 4*a*c)^(1/2) + 4*b^2*c*d*e^3*
(b^2 - 4*a*c)^(1/2) - 6*b*c^2*d^2*e^2*(b^2 - 4*a*c)^(1/2) + 2*a*b*c*e^4*(b^2 - 4*a*c)^(1/2)))/(2*c^4)

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