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

Optimal. Leaf size=204 \[ -\frac {-a b e-a c d+b^2 d}{a^3 x}+\frac {b d-a e}{2 a^2 x^2}+\frac {\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {\log (x) \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right )}{a^4}-\frac {\left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c}}-\frac {d}{3 a x^3} \]

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Rubi [A]  time = 0.29, antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.238, Rules used = {800, 634, 618, 206, 628} \begin {gather*} -\frac {\left (3 a^2 b c e+2 a^2 c^2 d-4 a b^2 c d-a b^3 e+b^4 d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c}}+\frac {\left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {-a b e-a c d+b^2 d}{a^3 x}-\frac {\log (x) \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right )}{a^4}+\frac {b d-a e}{2 a^2 x^2}-\frac {d}{3 a x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-d/(3*a*x^3) + (b*d - a*e)/(2*a^2*x^2) - (b^2*d - a*c*d - a*b*e)/(a^3*x) - ((b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d
- a*b^3*e + 3*a^2*b*c*e)*ArcTanh[(b + 2*c*x)/Sqrt[b^2 - 4*a*c]])/(a^4*Sqrt[b^2 - 4*a*c]) - ((b^3*d - 2*a*b*c*
d - a*b^2*e + a^2*c*e)*Log[x])/a^4 + ((b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*Log[a + b*x + c*x^2])/(2*a^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 {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx &=\int \left (\frac {d}{a x^4}+\frac {-b d+a e}{a^2 x^3}+\frac {b^2 d-a c d-a b e}{a^3 x^2}+\frac {-b^3 d+2 a b c d+a b^2 e-a^2 c e}{a^4 x}+\frac {b^4 d-3 a b^2 c d+a^2 c^2 d-a b^3 e+2 a^2 b c e+c \left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) x}{a^4 \left (a+b x+c x^2\right )}\right ) \, dx\\ &=-\frac {d}{3 a x^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {b^2 d-a c d-a b e}{a^3 x}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)}{a^4}+\frac {\int \frac {b^4 d-3 a b^2 c d+a^2 c^2 d-a b^3 e+2 a^2 b c e+c \left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) x}{a+b x+c x^2} \, dx}{a^4}\\ &=-\frac {d}{3 a x^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {b^2 d-a c d-a b e}{a^3 x}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)}{a^4}+\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \int \frac {b+2 c x}{a+b x+c x^2} \, dx}{2 a^4}+\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \int \frac {1}{a+b x+c x^2} \, dx}{2 a^4}\\ &=-\frac {d}{3 a x^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {b^2 d-a c d-a b e}{a^3 x}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)}{a^4}+\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x+c x^2\right )}{2 a^4}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \operatorname {Subst}\left (\int \frac {1}{b^2-4 a c-x^2} \, dx,x,b+2 c x\right )}{a^4}\\ &=-\frac {d}{3 a x^3}+\frac {b d-a e}{2 a^2 x^2}-\frac {b^2 d-a c d-a b e}{a^3 x}-\frac {\left (b^4 d-4 a b^2 c d+2 a^2 c^2 d-a b^3 e+3 a^2 b c e\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{a^4 \sqrt {b^2-4 a c}}-\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log (x)}{a^4}+\frac {\left (b^3 d-2 a b c d-a b^2 e+a^2 c e\right ) \log \left (a+b x+c x^2\right )}{2 a^4}\\ \end {align*}

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Mathematica [A]  time = 0.14, size = 196, normalized size = 0.96 \begin {gather*} \frac {-\frac {2 a^3 d}{x^3}-6 \log (x) \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right )+3 \left (a^2 c e-a b^2 e-2 a b c d+b^3 d\right ) \log (a+x (b+c x))+\frac {6 \left (3 a^2 b c e+2 a^2 c^2 d-a b^3 e-4 a b^2 c d+b^4 d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}+\frac {3 a^2 (b d-a e)}{x^2}+\frac {6 a \left (a b e+a c d+b^2 (-d)\right )}{x}}{6 a^4} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {d+e x}{x^4 \left (a+b x+c x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 1.54, size = 687, normalized size = 3.37 \begin {gather*} \left [\frac {3 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e\right )} x^{3} \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 ) + 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e\right )} x^{3} \log \left (c x^{2} + b x + a\right ) - 6 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e\right )} x^{3} \log \relax (x) - 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e\right )} x^{2} - 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d + 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}}, -\frac {6 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} - 4 \, a b^{2} c + 2 \, a^{2} c^{2}\right )} d - {\left (a b^{3} - 3 \, a^{2} b c\right )} e\right )} x^{3} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) - 3 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e\right )} x^{3} \log \left (c x^{2} + b x + a\right ) + 6 \, {\left ({\left (b^{5} - 6 \, a b^{3} c + 8 \, a^{2} b c^{2}\right )} d - {\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} e\right )} x^{3} \log \relax (x) + 6 \, {\left ({\left (a b^{4} - 5 \, a^{2} b^{2} c + 4 \, a^{3} c^{2}\right )} d - {\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} e\right )} x^{2} + 2 \, {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} d - 3 \, {\left ({\left (a^{2} b^{3} - 4 \, a^{3} b c\right )} d - {\left (a^{3} b^{2} - 4 \, a^{4} c\right )} e\right )} x}{6 \, {\left (a^{4} b^{2} - 4 \, a^{5} c\right )} x^{3}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^4/(c*x^2+b*x+a),x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.18, size = 214, normalized size = 1.05 \begin {gather*} \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, a^{4}} - \frac {{\left (b^{3} d - 2 \, a b c d - a b^{2} e + a^{2} c e\right )} \log \left ({\left | x \right |}\right )}{a^{4}} + \frac {{\left (b^{4} d - 4 \, a b^{2} c d + 2 \, a^{2} c^{2} d - a b^{3} e + 3 \, a^{2} b c e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} a^{4}} - \frac {2 \, a^{3} d + 6 \, {\left (a b^{2} d - a^{2} c d - a^{2} b e\right )} x^{2} - 3 \, {\left (a^{2} b d - a^{3} e\right )} x}{6 \, a^{4} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)/x^4/(c*x^2+b*x+a),x, algorithm="giac")

[Out]

1/2*(b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)*log(c*x^2 + b*x + a)/a^4 - (b^3*d - 2*a*b*c*d - a*b^2*e + a^2*c*e)
*log(abs(x))/a^4 + (b^4*d - 4*a*b^2*c*d + 2*a^2*c^2*d - a*b^3*e + 3*a^2*b*c*e)*arctan((2*c*x + b)/sqrt(-b^2 +
4*a*c))/(sqrt(-b^2 + 4*a*c)*a^4) - 1/6*(2*a^3*d + 6*(a*b^2*d - a^2*c*d - a^2*b*e)*x^2 - 3*(a^2*b*d - a^3*e)*x)
/(a^4*x^3)

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maple [A]  time = 0.07, size = 381, normalized size = 1.87 \begin {gather*} \frac {3 b c e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}+\frac {2 c^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{2}}-\frac {b^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{3}}-\frac {4 b^{2} c d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{3}}+\frac {b^{4} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, a^{4}}-\frac {c e \ln \relax (x )}{a^{2}}+\frac {c e \ln \left (c \,x^{2}+b x +a \right )}{2 a^{2}}+\frac {b^{2} e \ln \relax (x )}{a^{3}}-\frac {b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 a^{3}}+\frac {2 b c d \ln \relax (x )}{a^{3}}-\frac {b c d \ln \left (c \,x^{2}+b x +a \right )}{a^{3}}-\frac {b^{3} d \ln \relax (x )}{a^{4}}+\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{2 a^{4}}+\frac {b e}{a^{2} x}+\frac {c d}{a^{2} x}-\frac {b^{2} d}{a^{3} x}-\frac {e}{2 a \,x^{2}}+\frac {b d}{2 a^{2} x^{2}}-\frac {d}{3 a \,x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)/x^4/(c*x^2+b*x+a),x)

[Out]

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

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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((e*x+d)/x^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 = 2.75, size = 1063, normalized size = 5.21 \begin {gather*} \frac {\ln \left (2\,a^2\,b^4\,e+6\,a^4\,c^2\,e-2\,a\,b^5\,d-2\,b^6\,d\,x+2\,a\,b^5\,e\,x+2\,a\,b^4\,d\,\sqrt {b^2-4\,a\,c}+2\,b^5\,d\,x\,\sqrt {b^2-4\,a\,c}+11\,a^2\,b^3\,c\,d-13\,a^3\,b\,c^2\,d-9\,a^3\,b^2\,c\,e+2\,a^3\,c^3\,d\,x-2\,a^2\,b^3\,e\,\sqrt {b^2-4\,a\,c}+a^3\,c^2\,d\,\sqrt {b^2-4\,a\,c}-2\,a\,b^4\,e\,x\,\sqrt {b^2-4\,a\,c}-10\,a^2\,b^3\,c\,e\,x+9\,a^3\,b\,c^2\,e\,x-5\,a^2\,b^2\,c\,d\,\sqrt {b^2-4\,a\,c}-3\,a^3\,c^2\,e\,x\,\sqrt {b^2-4\,a\,c}-17\,a^2\,b^2\,c^2\,d\,x+12\,a\,b^4\,c\,d\,x+3\,a^3\,b\,c\,e\,\sqrt {b^2-4\,a\,c}-8\,a\,b^3\,c\,d\,x\,\sqrt {b^2-4\,a\,c}+7\,a^2\,b\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}+6\,a^2\,b^2\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^4\,d\,\sqrt {b^2-4\,a\,c}-b^5\,d+4\,a^3\,c^2\,e+a\,b^4\,e+6\,a\,b^3\,c\,d-a\,b^3\,e\,\sqrt {b^2-4\,a\,c}-8\,a^2\,b\,c^2\,d-5\,a^2\,b^2\,c\,e+2\,a^2\,c^2\,d\,\sqrt {b^2-4\,a\,c}-4\,a\,b^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a^2\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^5\,c-a^4\,b^2\right )}-\frac {\frac {d}{3\,a}+\frac {x\,\left (a\,e-b\,d\right )}{2\,a^2}-\frac {x^2\,\left (-d\,b^2+a\,e\,b+a\,c\,d\right )}{a^3}}{x^3}-\frac {\ln \left (2\,a^2\,b^4\,e+6\,a^4\,c^2\,e-2\,a\,b^5\,d-2\,b^6\,d\,x+2\,a\,b^5\,e\,x-2\,a\,b^4\,d\,\sqrt {b^2-4\,a\,c}-2\,b^5\,d\,x\,\sqrt {b^2-4\,a\,c}+11\,a^2\,b^3\,c\,d-13\,a^3\,b\,c^2\,d-9\,a^3\,b^2\,c\,e+2\,a^3\,c^3\,d\,x+2\,a^2\,b^3\,e\,\sqrt {b^2-4\,a\,c}-a^3\,c^2\,d\,\sqrt {b^2-4\,a\,c}+2\,a\,b^4\,e\,x\,\sqrt {b^2-4\,a\,c}-10\,a^2\,b^3\,c\,e\,x+9\,a^3\,b\,c^2\,e\,x+5\,a^2\,b^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a^3\,c^2\,e\,x\,\sqrt {b^2-4\,a\,c}-17\,a^2\,b^2\,c^2\,d\,x+12\,a\,b^4\,c\,d\,x-3\,a^3\,b\,c\,e\,\sqrt {b^2-4\,a\,c}+8\,a\,b^3\,c\,d\,x\,\sqrt {b^2-4\,a\,c}-7\,a^2\,b\,c^2\,d\,x\,\sqrt {b^2-4\,a\,c}-6\,a^2\,b^2\,c\,e\,x\,\sqrt {b^2-4\,a\,c}\right )\,\left (b^5\,d+b^4\,d\,\sqrt {b^2-4\,a\,c}-4\,a^3\,c^2\,e-a\,b^4\,e-6\,a\,b^3\,c\,d-a\,b^3\,e\,\sqrt {b^2-4\,a\,c}+8\,a^2\,b\,c^2\,d+5\,a^2\,b^2\,c\,e+2\,a^2\,c^2\,d\,\sqrt {b^2-4\,a\,c}-4\,a\,b^2\,c\,d\,\sqrt {b^2-4\,a\,c}+3\,a^2\,b\,c\,e\,\sqrt {b^2-4\,a\,c}\right )}{2\,\left (4\,a^5\,c-a^4\,b^2\right )}-\frac {\ln \relax (x)\,\left (b^3\,d-a\,\left (e\,b^2+2\,c\,d\,b\right )+a^2\,c\,e\right )}{a^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)/(x^4*(a + b*x + c*x^2)),x)

[Out]

(log(2*a^2*b^4*e + 6*a^4*c^2*e - 2*a*b^5*d - 2*b^6*d*x + 2*a*b^5*e*x + 2*a*b^4*d*(b^2 - 4*a*c)^(1/2) + 2*b^5*d
*x*(b^2 - 4*a*c)^(1/2) + 11*a^2*b^3*c*d - 13*a^3*b*c^2*d - 9*a^3*b^2*c*e + 2*a^3*c^3*d*x - 2*a^2*b^3*e*(b^2 -
4*a*c)^(1/2) + a^3*c^2*d*(b^2 - 4*a*c)^(1/2) - 2*a*b^4*e*x*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^3*c*e*x + 9*a^3*b*c^
2*e*x - 5*a^2*b^2*c*d*(b^2 - 4*a*c)^(1/2) - 3*a^3*c^2*e*x*(b^2 - 4*a*c)^(1/2) - 17*a^2*b^2*c^2*d*x + 12*a*b^4*
c*d*x + 3*a^3*b*c*e*(b^2 - 4*a*c)^(1/2) - 8*a*b^3*c*d*x*(b^2 - 4*a*c)^(1/2) + 7*a^2*b*c^2*d*x*(b^2 - 4*a*c)^(1
/2) + 6*a^2*b^2*c*e*x*(b^2 - 4*a*c)^(1/2))*(b^4*d*(b^2 - 4*a*c)^(1/2) - b^5*d + 4*a^3*c^2*e + a*b^4*e + 6*a*b^
3*c*d - a*b^3*e*(b^2 - 4*a*c)^(1/2) - 8*a^2*b*c^2*d - 5*a^2*b^2*c*e + 2*a^2*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^
2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*c*e*(b^2 - 4*a*c)^(1/2)))/(2*(4*a^5*c - a^4*b^2)) - (d/(3*a) + (x*(a*e - b
*d))/(2*a^2) - (x^2*(a*b*e - b^2*d + a*c*d))/a^3)/x^3 - (log(2*a^2*b^4*e + 6*a^4*c^2*e - 2*a*b^5*d - 2*b^6*d*x
+ 2*a*b^5*e*x - 2*a*b^4*d*(b^2 - 4*a*c)^(1/2) - 2*b^5*d*x*(b^2 - 4*a*c)^(1/2) + 11*a^2*b^3*c*d - 13*a^3*b*c^2
*d - 9*a^3*b^2*c*e + 2*a^3*c^3*d*x + 2*a^2*b^3*e*(b^2 - 4*a*c)^(1/2) - a^3*c^2*d*(b^2 - 4*a*c)^(1/2) + 2*a*b^4
*e*x*(b^2 - 4*a*c)^(1/2) - 10*a^2*b^3*c*e*x + 9*a^3*b*c^2*e*x + 5*a^2*b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^3*c^2*
e*x*(b^2 - 4*a*c)^(1/2) - 17*a^2*b^2*c^2*d*x + 12*a*b^4*c*d*x - 3*a^3*b*c*e*(b^2 - 4*a*c)^(1/2) + 8*a*b^3*c*d*
x*(b^2 - 4*a*c)^(1/2) - 7*a^2*b*c^2*d*x*(b^2 - 4*a*c)^(1/2) - 6*a^2*b^2*c*e*x*(b^2 - 4*a*c)^(1/2))*(b^5*d + b^
4*d*(b^2 - 4*a*c)^(1/2) - 4*a^3*c^2*e - a*b^4*e - 6*a*b^3*c*d - a*b^3*e*(b^2 - 4*a*c)^(1/2) + 8*a^2*b*c^2*d +
5*a^2*b^2*c*e + 2*a^2*c^2*d*(b^2 - 4*a*c)^(1/2) - 4*a*b^2*c*d*(b^2 - 4*a*c)^(1/2) + 3*a^2*b*c*e*(b^2 - 4*a*c)^
(1/2)))/(2*(4*a^5*c - a^4*b^2)) - (log(x)*(b^3*d - a*(b^2*e + 2*b*c*d) + a^2*c*e))/a^4

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sympy [F(-1)]  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((e*x+d)/x**4/(c*x**2+b*x+a),x)

[Out]

Timed out

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