3.14.32 \(\int \frac {(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, dx\)

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

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Rubi [A]  time = 0.40, 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 = {800, 634, 618, 206, 628} \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^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 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 \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 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 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]

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 ) \operatorname {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 = 0.22, size = 297, normalized size = 0.99 \begin {gather*} \frac {3 \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 (a+x (b+c x))+6 c 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 )+3 c^2 e^2 x^2 \left (-2 c e (a e+2 b d)+b^2 e^2+12 c^2 d^2\right )-6 e \sqrt {4 a c-b^2} (2 c d-b e) \left (-2 c e (a e+b d)+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 )+2 c^3 e^3 x^3 (8 c d-b e)+3 c^4 e^4 x^4}{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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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(b+2 c x) (d+e x)^4}{a+b x+c x^2} \, 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),x]

[Out]

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

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fricas [A]  time = 0.44, size = 709, normalized size = 2.37 \begin {gather*} \left [\frac {3 \, c^{4} e^{4} x^{4} + 2 \, {\left (8 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (12 \, c^{4} d^{2} e^{2} - 4 \, b c^{3} d e^{3} + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{4}\right )} x^{2} + 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 ) + 6 \, {\left (8 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 4 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} x + 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 {3 \, c^{4} e^{4} x^{4} + 2 \, {\left (8 \, c^{4} d e^{3} - b c^{3} e^{4}\right )} x^{3} + 3 \, {\left (12 \, c^{4} d^{2} e^{2} - 4 \, b c^{3} d e^{3} + {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} e^{4}\right )} x^{2} - 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 ) + 6 \, {\left (8 \, c^{4} d^{3} e - 6 \, b c^{3} d^{2} e^{2} + 4 \, {\left (b^{2} c^{2} - 2 \, a c^{3}\right )} d e^{3} - {\left (b^{3} c - 3 \, a b c^{2}\right )} e^{4}\right )} x + 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*(3*c^4*e^4*x^4 + 2*(8*c^4*d*e^3 - b*c^3*e^4)*x^3 + 3*(12*c^4*d^2*e^2 - 4*b*c^3*d*e^3 + (b^2*c^2 - 2*a*c^3
)*e^4)*x^2 + 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)) + 6*(8*c^4*d^3*e
- 6*b*c^3*d^2*e^2 + 4*(b^2*c^2 - 2*a*c^3)*d*e^3 - (b^3*c - 3*a*b*c^2)*e^4)*x + 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*(3*c^4*e^4*x^4 + 2*(8*c^4*d*e^3 - b*c^3*e^4)*x^3 + 3*(12*c^4*d^2*e^2 - 4*b*c^3*d*e^3 + (b^2
*c^2 - 2*a*c^3)*e^4)*x^2 - 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)*s
qrt(-b^2 + 4*a*c)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b)/(b^2 - 4*a*c)) + 6*(8*c^4*d^3*e - 6*b*c^3*d^2*e^2 + 4
*(b^2*c^2 - 2*a*c^3)*d*e^3 - (b^3*c - 3*a*b*c^2)*e^4)*x + 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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giac [A]  time = 0.16, 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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maple [B]  time = 0.05, size = 781, normalized size = 2.61 \begin {gather*} \frac {e^{4} x^{4}}{2}-\frac {b \,e^{4} x^{3}}{3 c}+\frac {8 d \,e^{3} x^{3}}{3}-\frac {8 a^{2} b \,e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {16 a^{2} d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {6 a \,b^{3} e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {20 a \,b^{2} d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {24 a b \,d^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}-\frac {a \,e^{4} x^{2}}{c}-\frac {16 a \,d^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}}-\frac {b^{5} e^{4} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}+\frac {4 b^{4} d \,e^{3} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {6 b^{3} d^{2} e^{2} \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {4 b^{2} d^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c}+\frac {b^{2} e^{4} x^{2}}{2 c^{2}}-\frac {2 b d \,e^{3} x^{2}}{c}+6 d^{2} e^{2} x^{2}+\frac {a^{2} e^{4} \ln \left (c \,x^{2}+b x +a \right )}{c^{2}}-\frac {2 a \,b^{2} e^{4} \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}+\frac {6 a b d \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{c^{2}}+\frac {3 a b \,e^{4} x}{c^{2}}-\frac {6 a \,d^{2} e^{2} \ln \left (c \,x^{2}+b x +a \right )}{c}-\frac {8 a d \,e^{3} x}{c}+\frac {b^{4} e^{4} \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}-\frac {2 b^{3} d \,e^{3} \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}-\frac {b^{3} e^{4} x}{c^{3}}+\frac {3 b^{2} d^{2} e^{2} \ln \left (c \,x^{2}+b x +a \right )}{c^{2}}+\frac {4 b^{2} d \,e^{3} x}{c^{2}}-\frac {2 b \,d^{3} e \ln \left (c \,x^{2}+b x +a \right )}{c}-\frac {6 b \,d^{2} e^{2} x}{c}+d^{4} \ln \left (c \,x^{2}+b x +a \right )+8 d^{3} e x \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),x)

[Out]

3*e^4/c^2*a*b*x-e^4/c^3*b^3*x-1/3*e^4/c*x^3*b-e^4/c*x^2*a+1/2*e^4/c^2*x^2*b^2+1/c^2*ln(c*x^2+b*x+a)*a^2*e^4+1/
2/c^4*ln(c*x^2+b*x+a)*b^4*e^4+4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*d*e^3-6/c^2/(4*a
*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^3*d^2*e^2-8/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-
b^2)^(1/2))*a^2*b*e^4+16/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*d*e^3+6/c^3/(4*a*c-b^2)^(
1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e^4+4/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b
^2*d^3*e+6/c^2*ln(c*x^2+b*x+a)*a*b*d*e^3-20/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^2*d*
e^3+24/c/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b*d^2*e^2-2*e^3/c*x^2*b*d+8*e*d^3*x+8/3*e^3*x
^3*d+6*e^2*x^2*d^2+ln(c*x^2+b*x+a)*d^4+3/c^2*ln(c*x^2+b*x+a)*b^2*d^2*e^2-2/c*ln(c*x^2+b*x+a)*b*d^3*e-16/(4*a*c
-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*d^3*e-8*e^3/c*a*d*x+4*e^3/c^2*b^2*d*x-6/c*ln(c*x^2+b*x+a)*a*
d^2*e^2-2/c^3*ln(c*x^2+b*x+a)*b^3*d*e^3-6*e^2/c*b*d^2*x-1/c^4/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(
1/2))*b^5*e^4-2/c^3*ln(c*x^2+b*x+a)*a*b^2*e^4+1/2*e^4*x^4

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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 details)Is 4*a*c-b^2 positive or negative?

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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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sympy [B]  time = 12.74, size = 1056, normalized size = 3.53 \begin {gather*} \frac {e^{4} x^{4}}{2} + x^{3} \left (- \frac {b e^{4}}{3 c} + \frac {8 d e^{3}}{3}\right ) + x^{2} \left (- \frac {a e^{4}}{c} + \frac {b^{2} e^{4}}{2 c^{2}} - \frac {2 b d e^{3}}{c} + 6 d^{2} e^{2}\right ) + x \left (\frac {3 a b e^{4}}{c^{2}} - \frac {8 a d e^{3}}{c} - \frac {b^{3} e^{4}}{c^{3}} + \frac {4 b^{2} d e^{3}}{c^{2}} - \frac {6 b d^{2} e^{2}}{c} + 8 d^{3} e\right ) + \left (- \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {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}}{2 c^{4}}\right ) \log {\left (x + \frac {a^{2} c e^{4} - a b^{2} e^{4} + 4 a b c d e^{3} - 6 a c^{2} d^{2} e^{2} + c^{3} d^{4} - c^{3} \left (- \frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {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}}{2 c^{4}}\right )}{2 a b c e^{4} - 4 a c^{2} d e^{3} - b^{3} e^{4} + 4 b^{2} c d e^{3} - 6 b c^{2} d^{2} e^{2} + 4 c^{3} d^{3} e} \right )} + \left (\frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {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}}{2 c^{4}}\right ) \log {\left (x + \frac {a^{2} c e^{4} - a b^{2} e^{4} + 4 a b c d e^{3} - 6 a c^{2} d^{2} e^{2} + c^{3} d^{4} - c^{3} \left (\frac {e \sqrt {- 4 a c + b^{2}} \left (b e - 2 c d\right ) \left (2 a c e^{2} - b^{2} e^{2} + 2 b c d e - 2 c^{2} d^{2}\right )}{2 c^{4}} + \frac {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}}{2 c^{4}}\right )}{2 a b c e^{4} - 4 a c^{2} d e^{3} - b^{3} e^{4} + 4 b^{2} c d e^{3} - 6 b c^{2} d^{2} e^{2} + 4 c^{3} d^{3} e} \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)

[Out]

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

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