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

Optimal. Leaf size=229 \[ -\frac {\left (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^4 (-e)+b^3 c d\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {\left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac {x \left (2 a b c e-a c^2 d+b^3 (-e)+b^2 c d\right )}{c^4}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 c} \]

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Rubi [A]  time = 0.42, antiderivative size = 229, 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 (-a^2 c^2 e+3 a b^2 c e-2 a b c^2 d+b^3 c d+b^4 (-e)\right ) \log \left (a+b x+c x^2\right )}{2 c^5}-\frac {\left (-5 a^2 b c^2 e+2 a^2 c^3 d-4 a b^2 c^2 d+5 a b^3 c e+b^4 c d+b^5 (-e)\right ) \tanh ^{-1}\left (\frac {b+2 c x}{\sqrt {b^2-4 a c}}\right )}{c^5 \sqrt {b^2-4 a c}}-\frac {x^2 \left (a c e+b^2 (-e)+b c d\right )}{2 c^3}+\frac {x \left (2 a b c e-a c^2 d+b^2 c d+b^3 (-e)\right )}{c^4}+\frac {x^3 (c d-b e)}{3 c^2}+\frac {e x^4}{4 c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.14, size = 222, normalized size = 0.97 \begin {gather*} \frac {6 \left (a^2 c^2 e-3 a b^2 c e+2 a b c^2 d+b^4 e-b^3 c d\right ) \log (a+x (b+c x))+\frac {12 \left (-5 a^2 b c^2 e+2 a^2 c^3 d+5 a b^3 c e-4 a b^2 c^2 d+b^5 (-e)+b^4 c d\right ) \tan ^{-1}\left (\frac {b+2 c x}{\sqrt {4 a c-b^2}}\right )}{\sqrt {4 a c-b^2}}-6 c^2 x^2 \left (a c e+b^2 (-e)+b c d\right )-12 c x \left (-2 a b c e+a c^2 d+b^3 e-b^2 c d\right )+4 c^3 x^3 (c d-b e)+3 c^4 e x^4}{12 c^5} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(-12*c*(-(b^2*c*d) + a*c^2*d + b^3*e - 2*a*b*c*e)*x - 6*c^2*(b*c*d - b^2*e + a*c*e)*x^2 + 4*c^3*(c*d - b*e)*x^
3 + 3*c^4*e*x^4 + (12*(b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ArcTan[(b
+ 2*c*x)/Sqrt[-b^2 + 4*a*c]])/Sqrt[-b^2 + 4*a*c] + 6*(-(b^3*c*d) + 2*a*b*c^2*d + b^4*e - 3*a*b^2*c*e + a^2*c^2
*e)*Log[a + x*(b + c*x)])/(12*c^5)

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

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.45, size = 730, normalized size = 3.19 \begin {gather*} \left [\frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} e x^{4} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} x^{3} - 6 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e\right )} x^{2} - 6 \, \sqrt {b^{2} - 4 \, a c} {\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \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 ) + 12 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e\right )} x - 6 \, {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{12 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}, \frac {3 \, {\left (b^{2} c^{4} - 4 \, a c^{5}\right )} e x^{4} + 4 \, {\left ({\left (b^{2} c^{4} - 4 \, a c^{5}\right )} d - {\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} e\right )} x^{3} - 6 \, {\left ({\left (b^{3} c^{3} - 4 \, a b c^{4}\right )} d - {\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} e\right )} x^{2} - 12 \, \sqrt {-b^{2} + 4 \, a c} {\left ({\left (b^{4} c - 4 \, a b^{2} c^{2} + 2 \, a^{2} c^{3}\right )} d - {\left (b^{5} - 5 \, a b^{3} c + 5 \, a^{2} b c^{2}\right )} e\right )} \arctan \left (-\frac {\sqrt {-b^{2} + 4 \, a c} {\left (2 \, c x + b\right )}}{b^{2} - 4 \, a c}\right ) + 12 \, {\left ({\left (b^{4} c^{2} - 5 \, a b^{2} c^{3} + 4 \, a^{2} c^{4}\right )} d - {\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} e\right )} x - 6 \, {\left ({\left (b^{5} c - 6 \, a b^{3} c^{2} + 8 \, a^{2} b c^{3}\right )} d - {\left (b^{6} - 7 \, a b^{4} c + 13 \, a^{2} b^{2} c^{2} - 4 \, a^{3} c^{3}\right )} e\right )} \log \left (c x^{2} + b x + a\right )}{12 \, {\left (b^{2} c^{5} - 4 \, a c^{6}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/12*(3*(b^2*c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c^4)*e)*x^3 - 6*((b^3*c^3 - 4
*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e)*x^2 - 6*sqrt(b^2 - 4*a*c)*((b^4*c - 4*a*b^2*c^2 + 2*a^2*c
^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*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)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e)*x
- 6*((b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*log(c*x^2 + b*
x + a))/(b^2*c^5 - 4*a*c^6), 1/12*(3*(b^2*c^4 - 4*a*c^5)*e*x^4 + 4*((b^2*c^4 - 4*a*c^5)*d - (b^3*c^3 - 4*a*b*c
^4)*e)*x^3 - 6*((b^3*c^3 - 4*a*b*c^4)*d - (b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*e)*x^2 - 12*sqrt(-b^2 + 4*a*c)*(
(b^4*c - 4*a*b^2*c^2 + 2*a^2*c^3)*d - (b^5 - 5*a*b^3*c + 5*a^2*b*c^2)*e)*arctan(-sqrt(-b^2 + 4*a*c)*(2*c*x + b
)/(b^2 - 4*a*c)) + 12*((b^4*c^2 - 5*a*b^2*c^3 + 4*a^2*c^4)*d - (b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*e)*x - 6*((
b^5*c - 6*a*b^3*c^2 + 8*a^2*b*c^3)*d - (b^6 - 7*a*b^4*c + 13*a^2*b^2*c^2 - 4*a^3*c^3)*e)*log(c*x^2 + b*x + a))
/(b^2*c^5 - 4*a*c^6)]

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giac [A]  time = 0.18, size = 247, normalized size = 1.08 \begin {gather*} \frac {3 \, c^{3} x^{4} e + 4 \, c^{3} d x^{3} - 4 \, b c^{2} x^{3} e - 6 \, b c^{2} d x^{2} + 6 \, b^{2} c x^{2} e - 6 \, a c^{2} x^{2} e + 12 \, b^{2} c d x - 12 \, a c^{2} d x - 12 \, b^{3} x e + 24 \, a b c x e}{12 \, c^{4}} - \frac {{\left (b^{3} c d - 2 \, a b c^{2} d - b^{4} e + 3 \, a b^{2} c e - a^{2} c^{2} e\right )} \log \left (c x^{2} + b x + a\right )}{2 \, c^{5}} + \frac {{\left (b^{4} c d - 4 \, a b^{2} c^{2} d + 2 \, a^{2} c^{3} d - b^{5} e + 5 \, a b^{3} c e - 5 \, a^{2} b c^{2} e\right )} \arctan \left (\frac {2 \, c x + b}{\sqrt {-b^{2} + 4 \, a c}}\right )}{\sqrt {-b^{2} + 4 \, a c} c^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/12*(3*c^3*x^4*e + 4*c^3*d*x^3 - 4*b*c^2*x^3*e - 6*b*c^2*d*x^2 + 6*b^2*c*x^2*e - 6*a*c^2*x^2*e + 12*b^2*c*d*x
- 12*a*c^2*d*x - 12*b^3*x*e + 24*a*b*c*x*e)/c^4 - 1/2*(b^3*c*d - 2*a*b*c^2*d - b^4*e + 3*a*b^2*c*e - a^2*c^2*
e)*log(c*x^2 + b*x + a)/c^5 + (b^4*c*d - 4*a*b^2*c^2*d + 2*a^2*c^3*d - b^5*e + 5*a*b^3*c*e - 5*a^2*b*c^2*e)*ar
ctan((2*c*x + b)/sqrt(-b^2 + 4*a*c))/(sqrt(-b^2 + 4*a*c)*c^5)

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maple [B]  time = 0.06, size = 445, normalized size = 1.94 \begin {gather*} \frac {e \,x^{4}}{4 c}-\frac {b e \,x^{3}}{3 c^{2}}+\frac {d \,x^{3}}{3 c}-\frac {5 a^{2} b e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}+\frac {2 a^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{2}}+\frac {5 a \,b^{3} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}-\frac {4 a \,b^{2} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{3}}-\frac {a e \,x^{2}}{2 c^{2}}-\frac {b^{5} e \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{5}}+\frac {b^{4} d \arctan \left (\frac {2 c x +b}{\sqrt {4 a c -b^{2}}}\right )}{\sqrt {4 a c -b^{2}}\, c^{4}}+\frac {b^{2} e \,x^{2}}{2 c^{3}}-\frac {b d \,x^{2}}{2 c^{2}}+\frac {a^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{3}}-\frac {3 a \,b^{2} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}+\frac {a b d \ln \left (c \,x^{2}+b x +a \right )}{c^{3}}+\frac {2 a b e x}{c^{3}}-\frac {a d x}{c^{2}}+\frac {b^{4} e \ln \left (c \,x^{2}+b x +a \right )}{2 c^{5}}-\frac {b^{3} d \ln \left (c \,x^{2}+b x +a \right )}{2 c^{4}}-\frac {b^{3} e x}{c^{4}}+\frac {b^{2} d x}{c^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/4/c*e*x^4-1/3/c^2*x^3*b*e+1/3/c*d*x^3-1/2*a/c^2*e*x^2+1/2/c^3*x^2*b^2*e-1/2/c^2*x^2*b*d+2/c^3*a*b*e*x-a/c^2*
d*x-1/c^4*b^3*e*x+1/c^3*b^2*d*x+1/2/c^3*ln(c*x^2+b*x+a)*e*a^2-3/2/c^4*ln(c*x^2+b*x+a)*a*b^2*e+1/c^3*ln(c*x^2+b
*x+a)*a*b*d+1/2/c^5*ln(c*x^2+b*x+a)*b^4*e-1/2/c^4*ln(c*x^2+b*x+a)*b^3*d-5/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+
b)/(4*a*c-b^2)^(1/2))*a^2*b*e+2/c^2/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a^2*d+5/c^4/(4*a*c-b
^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*a*b^3*e-4/c^3/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/
2))*a*b^2*d-1/c^5/(4*a*c-b^2)^(1/2)*arctan((2*c*x+b)/(4*a*c-b^2)^(1/2))*b^5*e+1/c^4/(4*a*c-b^2)^(1/2)*arctan((
2*c*x+b)/(4*a*c-b^2)^(1/2))*b^4*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(x^4*(e*x+d)/(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 = 0.23, size = 302, normalized size = 1.32 \begin {gather*} x^3\,\left (\frac {d}{3\,c}-\frac {b\,e}{3\,c^2}\right )+x\,\left (\frac {b\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}+\frac {a\,e}{c^2}\right )}{c}-\frac {a\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{c}\right )-x^2\,\left (\frac {b\,\left (\frac {d}{c}-\frac {b\,e}{c^2}\right )}{2\,c}+\frac {a\,e}{2\,c^2}\right )+\frac {\ln \left (c\,x^2+b\,x+a\right )\,\left (4\,e\,a^3\,c^3-13\,e\,a^2\,b^2\,c^2+8\,d\,a^2\,b\,c^3+7\,e\,a\,b^4\,c-6\,d\,a\,b^3\,c^2-e\,b^6+d\,b^5\,c\right )}{2\,\left (4\,a\,c^6-b^2\,c^5\right )}+\frac {e\,x^4}{4\,c}-\frac {\mathrm {atan}\left (\frac {b}{\sqrt {4\,a\,c-b^2}}+\frac {2\,c\,x}{\sqrt {4\,a\,c-b^2}}\right )\,\left (5\,e\,a^2\,b\,c^2-2\,d\,a^2\,c^3-5\,e\,a\,b^3\,c+4\,d\,a\,b^2\,c^2+e\,b^5-d\,b^4\,c\right )}{c^5\,\sqrt {4\,a\,c-b^2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x^3*(d/(3*c) - (b*e)/(3*c^2)) + x*((b*((b*(d/c - (b*e)/c^2))/c + (a*e)/c^2))/c - (a*(d/c - (b*e)/c^2))/c) - x^
2*((b*(d/c - (b*e)/c^2))/(2*c) + (a*e)/(2*c^2)) + (log(a + b*x + c*x^2)*(4*a^3*c^3*e - b^6*e + b^5*c*d - 13*a^
2*b^2*c^2*e + 7*a*b^4*c*e - 6*a*b^3*c^2*d + 8*a^2*b*c^3*d))/(2*(4*a*c^6 - b^2*c^5)) + (e*x^4)/(4*c) - (atan(b/
(4*a*c - b^2)^(1/2) + (2*c*x)/(4*a*c - b^2)^(1/2))*(b^5*e - 2*a^2*c^3*d - b^4*c*d - 5*a*b^3*c*e + 4*a*b^2*c^2*
d + 5*a^2*b*c^2*e))/(c^5*(4*a*c - b^2)^(1/2))

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sympy [B]  time = 3.92, size = 1100, normalized size = 4.80 \begin {gather*} x^{3} \left (- \frac {b e}{3 c^{2}} + \frac {d}{3 c}\right ) + x^{2} \left (- \frac {a e}{2 c^{2}} + \frac {b^{2} e}{2 c^{3}} - \frac {b d}{2 c^{2}}\right ) + x \left (\frac {2 a b e}{c^{3}} - \frac {a d}{c^{2}} - \frac {b^{3} e}{c^{4}} + \frac {b^{2} d}{c^{3}}\right ) + \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} e - 4 a^{2} b^{2} c e + 3 a^{2} b c^{2} d + a b^{4} e - a b^{3} c d - 4 a c^{5} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) + b^{2} c^{4} \left (- \frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right )}{5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d} \right )} + \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) \log {\left (x + \frac {2 a^{3} c^{2} e - 4 a^{2} b^{2} c e + 3 a^{2} b c^{2} d + a b^{4} e - a b^{3} c d - 4 a c^{5} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right ) + b^{2} c^{4} \left (\frac {\sqrt {- 4 a c + b^{2}} \left (5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d\right )}{2 c^{5} \left (4 a c - b^{2}\right )} + \frac {a^{2} c^{2} e - 3 a b^{2} c e + 2 a b c^{2} d + b^{4} e - b^{3} c d}{2 c^{5}}\right )}{5 a^{2} b c^{2} e - 2 a^{2} c^{3} d - 5 a b^{3} c e + 4 a b^{2} c^{2} d + b^{5} e - b^{4} c d} \right )} + \frac {e x^{4}}{4 c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x**3*(-b*e/(3*c**2) + d/(3*c)) + x**2*(-a*e/(2*c**2) + b**2*e/(2*c**3) - b*d/(2*c**2)) + x*(2*a*b*e/c**3 - a*d
/c**2 - b**3*e/c**4 + b**2*d/c**3) + (-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4
*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**
4*e - b**3*c*d)/(2*c**5))*log(x + (2*a**3*c**2*e - 4*a**2*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*e - a*b**3*c*d -
4*a*c**5*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b
**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)) +
b**2*c**4*(-sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e -
b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)))
/(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)) + (sqrt(-4*a*c + b**2
)*(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2
)) + (a**2*c**2*e - 3*a*b**2*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5))*log(x + (2*a**3*c**2*e - 4*a**2
*b**2*c*e + 3*a**2*b*c**2*d + a*b**4*e - a*b**3*c*d - 4*a*c**5*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*
c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2
*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)) + b**2*c**4*(sqrt(-4*a*c + b**2)*(5*a**2*b*c**2*e - 2*a**2*
c**3*d - 5*a*b**3*c*e + 4*a*b**2*c**2*d + b**5*e - b**4*c*d)/(2*c**5*(4*a*c - b**2)) + (a**2*c**2*e - 3*a*b**2
*c*e + 2*a*b*c**2*d + b**4*e - b**3*c*d)/(2*c**5)))/(5*a**2*b*c**2*e - 2*a**2*c**3*d - 5*a*b**3*c*e + 4*a*b**2
*c**2*d + b**5*e - b**4*c*d)) + e*x**4/(4*c)

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