3.64.40 \(\int \frac {6-18 x+18 x^2-6 x^3+e^4 (-15 x^2+5 x^3)}{-5+15 x-15 x^2+5 x^3} \, dx\)

Optimal. Leaf size=21 \[ 4-\frac {6 x}{5}+\frac {e^4 x^3}{(1-x)^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 36, normalized size of antiderivative = 1.71, number of steps used = 2, number of rules used = 1, integrand size = 48, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.021, Rules used = {2074} \begin {gather*} -\frac {1}{5} \left (6-5 e^4\right ) x-\frac {3 e^4}{1-x}+\frac {e^4}{(1-x)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(6 - 18*x + 18*x^2 - 6*x^3 + E^4*(-15*x^2 + 5*x^3))/(-5 + 15*x - 15*x^2 + 5*x^3),x]

[Out]

E^4/(1 - x)^2 - (3*E^4)/(1 - x) - ((6 - 5*E^4)*x)/5

Rule 2074

Int[(P_)^(p_)*(Q_)^(q_.), x_Symbol] :> With[{PP = Factor[P]}, Int[ExpandIntegrand[PP^p*Q^q, x], x] /;  !SumQ[N
onfreeFactors[PP, x]]] /; FreeQ[q, x] && PolyQ[P, x] && PolyQ[Q, x] && IntegerQ[p] && NeQ[P, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (\frac {1}{5} \left (-6+5 e^4\right )-\frac {2 e^4}{(-1+x)^3}-\frac {3 e^4}{(-1+x)^2}\right ) \, dx\\ &=\frac {e^4}{(1-x)^2}-\frac {3 e^4}{1-x}-\frac {1}{5} \left (6-5 e^4\right ) x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.02, size = 32, normalized size = 1.52 \begin {gather*} \frac {1}{5} \left (6-6 x+\frac {5 e^4 \left (-3+6 x-3 x^2+x^3\right )}{(-1+x)^2}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(6 - 18*x + 18*x^2 - 6*x^3 + E^4*(-15*x^2 + 5*x^3))/(-5 + 15*x - 15*x^2 + 5*x^3),x]

[Out]

(6 - 6*x + (5*E^4*(-3 + 6*x - 3*x^2 + x^3))/(-1 + x)^2)/5

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fricas [B]  time = 0.56, size = 43, normalized size = 2.05 \begin {gather*} -\frac {6 \, x^{3} - 12 \, x^{2} - 5 \, {\left (x^{3} - 2 \, x^{2} + 4 \, x - 2\right )} e^{4} + 6 \, x}{5 \, {\left (x^{2} - 2 \, x + 1\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3-15*x^2)*exp(1)^4-6*x^3+18*x^2-18*x+6)/(5*x^3-15*x^2+15*x-5),x, algorithm="fricas")

[Out]

-1/5*(6*x^3 - 12*x^2 - 5*(x^3 - 2*x^2 + 4*x - 2)*e^4 + 6*x)/(x^2 - 2*x + 1)

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giac [A]  time = 0.12, size = 24, normalized size = 1.14 \begin {gather*} x e^{4} - \frac {6}{5} \, x + \frac {3 \, x e^{4} - 2 \, e^{4}}{{\left (x - 1\right )}^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3-15*x^2)*exp(1)^4-6*x^3+18*x^2-18*x+6)/(5*x^3-15*x^2+15*x-5),x, algorithm="giac")

[Out]

x*e^4 - 6/5*x + (3*x*e^4 - 2*e^4)/(x - 1)^2

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maple [A]  time = 0.07, size = 22, normalized size = 1.05

method	result	size
norman	\(\frac {\frac {18 x}{5}+\left ({\mathrm e}^{4}-\frac {6}{5}\right ) x^{3}-\frac {12}{5}}{\left (x -1\right )^{2}}\)	\(22\)
default	\(x \,{\mathrm e}^{4}-\frac {6 x}{5}+\frac {3 \,{\mathrm e}^{4}}{x -1}+\frac {{\mathrm e}^{4}}{\left (x -1\right )^{2}}\)	\(26\)
risch	\(x \,{\mathrm e}^{4}-\frac {6 x}{5}+\frac {3 x \,{\mathrm e}^{4}-2 \,{\mathrm e}^{4}}{x^{2}-2 x +1}\)	\(30\)
gosper	\(\frac {5 x^{3} {\mathrm e}^{4}-6 x^{3}+18 x -12}{5 x^{2}-10 x +5}\)	\(32\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((5*x^3-15*x^2)*exp(1)^4-6*x^3+18*x^2-18*x+6)/(5*x^3-15*x^2+15*x-5),x,method=_RETURNVERBOSE)

[Out]

(18/5*x+(exp(1)^4-6/5)*x^3-12/5)/(x-1)^2

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maxima [A]  time = 0.36, size = 31, normalized size = 1.48 \begin {gather*} \frac {1}{5} \, x {\left (5 \, e^{4} - 6\right )} + \frac {3 \, x e^{4} - 2 \, e^{4}}{x^{2} - 2 \, x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x^3-15*x^2)*exp(1)^4-6*x^3+18*x^2-18*x+6)/(5*x^3-15*x^2+15*x-5),x, algorithm="maxima")

[Out]

1/5*x*(5*e^4 - 6) + (3*x*e^4 - 2*e^4)/(x^2 - 2*x + 1)

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mupad [B]  time = 0.08, size = 24, normalized size = 1.14 \begin {gather*} x\,\left ({\mathrm {e}}^4-\frac {6}{5}\right )-\frac {2\,{\mathrm {e}}^4-3\,x\,{\mathrm {e}}^4}{{\left (x-1\right )}^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(18*x + exp(4)*(15*x^2 - 5*x^3) - 18*x^2 + 6*x^3 - 6)/(15*x - 15*x^2 + 5*x^3 - 5),x)

[Out]

x*(exp(4) - 6/5) - (2*exp(4) - 3*x*exp(4))/(x - 1)^2

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sympy [A]  time = 0.15, size = 29, normalized size = 1.38 \begin {gather*} - x \left (\frac {6}{5} - e^{4}\right ) - \frac {- 3 x e^{4} + 2 e^{4}}{x^{2} - 2 x + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((5*x**3-15*x**2)*exp(1)**4-6*x**3+18*x**2-18*x+6)/(5*x**3-15*x**2+15*x-5),x)

[Out]

-x*(6/5 - exp(4)) - (-3*x*exp(4) + 2*exp(4))/(x**2 - 2*x + 1)

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