3.95.78 \(\int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} (11+3 e^4+3 x)} (24 x+16 x^2+4 x^3)}{9+6 x+x^2} \, dx\)

Optimal. Leaf size=26 \[ \frac {x \left (5-4 x \left (-e^{\frac {11}{3}+e^4+x}+x\right )\right )}{3+x} \]

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Rubi [B]  time = 0.43, antiderivative size = 73, normalized size of antiderivative = 2.81, number of steps used = 13, number of rules used = 8, integrand size = 54, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {27, 6742, 2199, 2194, 2176, 2177, 2178, 1850} \begin {gather*} -4 x^2+4 e^{x+\frac {1}{3} \left (11+3 e^4\right )} x+12 x-12 e^{x+\frac {1}{3} \left (11+3 e^4\right )}+\frac {36 e^{x+\frac {1}{3} \left (11+3 e^4\right )}}{x+3}+\frac {93}{x+3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(15 - 36*x^2 - 8*x^3 + E^((11 + 3*E^4 + 3*x)/3)*(24*x + 16*x^2 + 4*x^3))/(9 + 6*x + x^2),x]

[Out]

-12*E^((11 + 3*E^4)/3 + x) + 12*x + 4*E^((11 + 3*E^4)/3 + x)*x - 4*x^2 + 93/(3 + x) + (36*E^((11 + 3*E^4)/3 +
x))/(3 + x)

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 1850

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^n)^p, x], x] /; FreeQ[
{a, b, n}, x] && PolyQ[Pq, x] && (IGtQ[p, 0] || EqQ[n, 1])

Rule 2176

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.), x_Symbol] :> Simp[((c + d*x)^m
*(b*F^(g*(e + f*x)))^n)/(f*g*n*Log[F]), x] - Dist[(d*m)/(f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*(b*F^(g*(e + f*x
)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && GtQ[m, 0] && IntegerQ[2*m] &&  !$UseGamma === True

Rule 2177

Int[((b_.)*(F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> Simp[((c + d*x)^(m
+ 1)*(b*F^(g*(e + f*x)))^n)/(d*(m + 1)), x] - Dist[(f*g*n*Log[F])/(d*(m + 1)), Int[(c + d*x)^(m + 1)*(b*F^(g*
(e + f*x)))^n, x], x] /; FreeQ[{F, b, c, d, e, f, g, n}, x] && LtQ[m, -1] && IntegerQ[2*m] &&  !$UseGamma ===
True

Rule 2178

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/((c_.) + (d_.)*(x_)), x_Symbol] :> Simp[(F^(g*(e - (c*f)/d))*ExpIntegral
Ei[(f*g*(c + d*x)*Log[F])/d])/d, x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !$UseGamma === True

Rule 2194

Int[((F_)^((c_.)*((a_.) + (b_.)*(x_))))^(n_.), x_Symbol] :> Simp[(F^(c*(a + b*x)))^n/(b*c*n*Log[F]), x] /; Fre
eQ[{F, a, b, c, n}, x]

Rule 2199

Int[(F_)^((c_.)*(v_))*(u_)^(m_.)*(w_), x_Symbol] :> Int[ExpandIntegrand[F^(c*ExpandToSum[v, x]), w*NormalizePo
werOfLinear[u, x]^m, x], x] /; FreeQ[{F, c}, x] && PolynomialQ[w, x] && LinearQ[v, x] && PowerOfLinearQ[u, x]
&& IntegerQ[m] &&  !$UseGamma === True

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {15-36 x^2-8 x^3+e^{\frac {1}{3} \left (11+3 e^4+3 x\right )} \left (24 x+16 x^2+4 x^3\right )}{(3+x)^2} \, dx\\ &=\int \left (\frac {4 e^{\frac {11}{3}+e^4+x} x \left (6+4 x+x^2\right )}{(3+x)^2}+\frac {15-36 x^2-8 x^3}{(3+x)^2}\right ) \, dx\\ &=4 \int \frac {e^{\frac {11}{3}+e^4+x} x \left (6+4 x+x^2\right )}{(3+x)^2} \, dx+\int \frac {15-36 x^2-8 x^3}{(3+x)^2} \, dx\\ &=4 \int \left (-2 e^{\frac {1}{3} \left (11+3 e^4\right )+x}+e^{\frac {1}{3} \left (11+3 e^4\right )+x} x-\frac {9 e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{(3+x)^2}+\frac {9 e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{3+x}\right ) \, dx+\int \left (12-8 x-\frac {93}{(3+x)^2}\right ) \, dx\\ &=12 x-4 x^2+\frac {93}{3+x}+4 \int e^{\frac {1}{3} \left (11+3 e^4\right )+x} x \, dx-8 \int e^{\frac {1}{3} \left (11+3 e^4\right )+x} \, dx-36 \int \frac {e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{(3+x)^2} \, dx+36 \int \frac {e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{3+x} \, dx\\ &=-8 e^{\frac {1}{3} \left (11+3 e^4\right )+x}+12 x+4 e^{\frac {1}{3} \left (11+3 e^4\right )+x} x-4 x^2+\frac {93}{3+x}+\frac {36 e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{3+x}+36 e^{\frac {2}{3}+e^4} \text {Ei}(3+x)-4 \int e^{\frac {1}{3} \left (11+3 e^4\right )+x} \, dx-36 \int \frac {e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{3+x} \, dx\\ &=-12 e^{\frac {1}{3} \left (11+3 e^4\right )+x}+12 x+4 e^{\frac {1}{3} \left (11+3 e^4\right )+x} x-4 x^2+\frac {93}{3+x}+\frac {36 e^{\frac {1}{3} \left (11+3 e^4\right )+x}}{3+x}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.17, size = 31, normalized size = 1.19 \begin {gather*} \frac {93+36 x+4 e^{\frac {11}{3}+e^4+x} x^2-4 x^3}{3+x} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(15 - 36*x^2 - 8*x^3 + E^((11 + 3*E^4 + 3*x)/3)*(24*x + 16*x^2 + 4*x^3))/(9 + 6*x + x^2),x]

[Out]

(93 + 36*x + 4*E^(11/3 + E^4 + x)*x^2 - 4*x^3)/(3 + x)

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fricas [A]  time = 0.64, size = 28, normalized size = 1.08 \begin {gather*} -\frac {4 \, x^{3} - 4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )} - 36 \, x - 93}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x, algorithm="fricas")

[Out]

-(4*x^3 - 4*x^2*e^(x + e^4 + 11/3) - 36*x - 93)/(x + 3)

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giac [A]  time = 0.12, size = 28, normalized size = 1.08 \begin {gather*} -\frac {4 \, x^{3} - 4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )} - 36 \, x - 93}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x, algorithm="giac")

[Out]

-(4*x^3 - 4*x^2*e^(x + e^4 + 11/3) - 36*x - 93)/(x + 3)

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maple [A]  time = 2.22, size = 25, normalized size = 0.96

method	result	size
norman	\(\frac {-4 x^{3}+4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} x^{2}-15}{3+x}\)	\(25\)
risch	\(-4 x^{2}+12 x +\frac {93}{3+x}+\frac {4 x^{2} {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{3+x}\)	\(33\)
derivativedivides	\(-\frac {3071}{9 \left (-3 x -9\right )}+\frac {-156 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2}+312 \,{\mathrm e}^{8}+416 \,{\mathrm e}^{4}+\frac {416}{3}}{-3 x -9}+\frac {\left (36 \,{\mathrm e}^{4}+24\right ) \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2}+12 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{3}-72 \,{\mathrm e}^{12}-144 \,{\mathrm e}^{8}-96 \,{\mathrm e}^{4}-\frac {64}{3}}{-3 x -9}-\frac {204 \,{\mathrm e}^{8}}{-3 x -9}-\frac {48 \,{\mathrm e}^{12}}{-3 x -9}-\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9 \left (-3 x -9\right )}+\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{27}-\frac {208 \,{\mathrm e}^{4}}{-3 x -9}+\frac {-72 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2} {\mathrm e}^{4}+144 \,{\mathrm e}^{12}+192 \,{\mathrm e}^{8}+64 \,{\mathrm e}^{4}}{-3 x -9}+\frac {68 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-3 x -9}-612 \left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )-28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-\frac {28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{4}+4\right )}{3 \left (-3 x -9\right )}+28 \left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )+\frac {4 \left (9 \,{\mathrm e}^{4}+3 x +12\right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{3}+\frac {4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (27 \,{\mathrm e}^{12}+54 \,{\mathrm e}^{8}+36 \,{\mathrm e}^{4}+8\right )}{9 \left (-3 x -9\right )}-\frac {4 \left (3 \,{\mathrm e}^{12}+15 \,{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+\frac {44}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{3}-612 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )-252 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )-36 \,{\mathrm e}^{12} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )+504 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-27 x -81}-\left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )\right )+108 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-27 x -81}-\left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )\right )-108 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9}+\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{4}+4\right )}{-81 x -243}-\frac {\left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )\)	\(629\)
default	\(-\frac {3071}{9 \left (-3 x -9\right )}+\frac {-156 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2}+312 \,{\mathrm e}^{8}+416 \,{\mathrm e}^{4}+\frac {416}{3}}{-3 x -9}+\frac {\left (36 \,{\mathrm e}^{4}+24\right ) \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2}+12 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{3}-72 \,{\mathrm e}^{12}-144 \,{\mathrm e}^{8}-96 \,{\mathrm e}^{4}-\frac {64}{3}}{-3 x -9}-\frac {204 \,{\mathrm e}^{8}}{-3 x -9}-\frac {48 \,{\mathrm e}^{12}}{-3 x -9}-\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9 \left (-3 x -9\right )}+\frac {1892 \,{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{27}-\frac {208 \,{\mathrm e}^{4}}{-3 x -9}+\frac {-72 \left ({\mathrm e}^{4}+x +\frac {11}{3}\right )^{2} {\mathrm e}^{4}+144 \,{\mathrm e}^{12}+192 \,{\mathrm e}^{8}+64 \,{\mathrm e}^{4}}{-3 x -9}+\frac {68 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-3 x -9}-612 \left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )-28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}-\frac {28 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{4}+4\right )}{3 \left (-3 x -9\right )}+28 \left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )+\frac {4 \left (9 \,{\mathrm e}^{4}+3 x +12\right ) {\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{3}+\frac {4 \,{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (27 \,{\mathrm e}^{12}+54 \,{\mathrm e}^{8}+36 \,{\mathrm e}^{4}+8\right )}{9 \left (-3 x -9\right )}-\frac {4 \left (3 \,{\mathrm e}^{12}+15 \,{\mathrm e}^{8}+16 \,{\mathrm e}^{4}+\frac {44}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{3}-612 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )-252 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )-36 \,{\mathrm e}^{12} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{-9 x -27}-\frac {{\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )+504 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-27 x -81}-\left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )\right )+108 \,{\mathrm e}^{8} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (3 \,{\mathrm e}^{4}+2\right )}{-27 x -81}-\left (\frac {{\mathrm e}^{4}}{9}+\frac {5}{27}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )\right )-108 \,{\mathrm e}^{4} \left (\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}}}{9}+\frac {{\mathrm e}^{{\mathrm e}^{4}+x +\frac {11}{3}} \left (9 \,{\mathrm e}^{8}+12 \,{\mathrm e}^{4}+4\right )}{-81 x -243}-\frac {\left ({\mathrm e}^{8}+\frac {10 \,{\mathrm e}^{4}}{3}+\frac {16}{9}\right ) {\mathrm e}^{{\mathrm e}^{4}+\frac {2}{3}} \expIntegralEi \left (1, -3-x \right )}{9}\right )\)	\(629\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x,method=_RETURNVERBOSE)

[Out]

(-4*x^3+4*exp(exp(4)+x+11/3)*x^2-15)/(3+x)

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maxima [A]  time = 0.49, size = 32, normalized size = 1.23 \begin {gather*} -4 \, x^{2} + \frac {4 \, x^{2} e^{\left (x + e^{4} + \frac {11}{3}\right )}}{x + 3} + 12 \, x + \frac {93}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x^3+16*x^2+24*x)*exp(exp(4)+x+11/3)-8*x^3-36*x^2+15)/(x^2+6*x+9),x, algorithm="maxima")

[Out]

-4*x^2 + 4*x^2*e^(x + e^4 + 11/3)/(x + 3) + 12*x + 93/(x + 3)

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mupad [B]  time = 5.43, size = 23, normalized size = 0.88 \begin {gather*} \frac {x\,\left (4\,x\,{\mathrm {e}}^{x+{\mathrm {e}}^4+\frac {11}{3}}-4\,x^2+5\right )}{x+3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x + exp(4) + 11/3)*(24*x + 16*x^2 + 4*x^3) - 36*x^2 - 8*x^3 + 15)/(6*x + x^2 + 9),x)

[Out]

(x*(4*x*exp(x + exp(4) + 11/3) - 4*x^2 + 5))/(x + 3)

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sympy [A]  time = 0.13, size = 31, normalized size = 1.19 \begin {gather*} - 4 x^{2} + \frac {4 x^{2} e^{x + \frac {11}{3} + e^{4}}}{x + 3} + 12 x + \frac {93}{x + 3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((4*x**3+16*x**2+24*x)*exp(exp(4)+x+11/3)-8*x**3-36*x**2+15)/(x**2+6*x+9),x)

[Out]

-4*x**2 + 4*x**2*exp(x + 11/3 + exp(4))/(x + 3) + 12*x + 93/(x + 3)

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