3.94.46 \(\int \frac {1}{15} (-4+42 x+75 x^2+e^4 (-4+50 x)) \, dx\)

Optimal. Leaf size=18 \[ \frac {1}{3} x \left (1+e^4+x\right ) \left (-\frac {4}{5}+5 x\right ) \]

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Rubi [A]  time = 0.01, antiderivative size = 34, normalized size of antiderivative = 1.89, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {12} \begin {gather*} \frac {5 x^3}{3}+\frac {7 x^2}{5}-\frac {4 x}{15}+\frac {1}{375} e^4 (2-25 x)^2 \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-4 + 42*x + 75*x^2 + E^4*(-4 + 50*x))/15,x]

[Out]

(E^4*(2 - 25*x)^2)/375 - (4*x)/15 + (7*x^2)/5 + (5*x^3)/3

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{15} \int \left (-4+42 x+75 x^2+e^4 (-4+50 x)\right ) \, dx\\ &=\frac {1}{375} e^4 (2-25 x)^2-\frac {4 x}{15}+\frac {7 x^2}{5}+\frac {5 x^3}{3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.00, size = 32, normalized size = 1.78 \begin {gather*} \frac {1}{15} \left (-4 x-4 e^4 x+21 x^2+25 e^4 x^2+25 x^3\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-4 + 42*x + 75*x^2 + E^4*(-4 + 50*x))/15,x]

[Out]

(-4*x - 4*E^4*x + 21*x^2 + 25*E^4*x^2 + 25*x^3)/15

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fricas [B]  time = 0.85, size = 27, normalized size = 1.50 \begin {gather*} \frac {5}{3} \, x^{3} + \frac {7}{5} \, x^{2} + \frac {1}{15} \, {\left (25 \, x^{2} - 4 \, x\right )} e^{4} - \frac {4}{15} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(50*x-4)*exp(4)+5*x^2+14/5*x-4/15,x, algorithm="fricas")

[Out]

5/3*x^3 + 7/5*x^2 + 1/15*(25*x^2 - 4*x)*e^4 - 4/15*x

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giac [B]  time = 0.22, size = 27, normalized size = 1.50 \begin {gather*} \frac {5}{3} \, x^{3} + \frac {7}{5} \, x^{2} + \frac {1}{15} \, {\left (25 \, x^{2} - 4 \, x\right )} e^{4} - \frac {4}{15} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(50*x-4)*exp(4)+5*x^2+14/5*x-4/15,x, algorithm="giac")

[Out]

5/3*x^3 + 7/5*x^2 + 1/15*(25*x^2 - 4*x)*e^4 - 4/15*x

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maple [A]  time = 0.02, size = 23, normalized size = 1.28

method	result	size
gosper	\(\frac {x \left (25 x \,{\mathrm e}^{4}+25 x^{2}-4 \,{\mathrm e}^{4}+21 x -4\right )}{15}\)	\(23\)
norman	\(\left (-\frac {4 \,{\mathrm e}^{4}}{15}-\frac {4}{15}\right ) x +\left (\frac {5 \,{\mathrm e}^{4}}{3}+\frac {7}{5}\right ) x^{2}+\frac {5 x^{3}}{3}\)	\(25\)
risch	\(\frac {5 x^{2} {\mathrm e}^{4}}{3}-\frac {4 x \,{\mathrm e}^{4}}{15}+\frac {5 x^{3}}{3}+\frac {7 x^{2}}{5}-\frac {4 x}{15}\)	\(27\)
default	\(\frac {{\mathrm e}^{4} \left (25 x^{2}-4 x \right )}{15}+\frac {5 x^{3}}{3}+\frac {7 x^{2}}{5}-\frac {4 x}{15}\)	\(28\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/15*(50*x-4)*exp(4)+5*x^2+14/5*x-4/15,x,method=_RETURNVERBOSE)

[Out]

1/15*x*(25*x*exp(4)+25*x^2-4*exp(4)+21*x-4)

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maxima [B]  time = 0.37, size = 27, normalized size = 1.50 \begin {gather*} \frac {5}{3} \, x^{3} + \frac {7}{5} \, x^{2} + \frac {1}{15} \, {\left (25 \, x^{2} - 4 \, x\right )} e^{4} - \frac {4}{15} \, x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(50*x-4)*exp(4)+5*x^2+14/5*x-4/15,x, algorithm="maxima")

[Out]

5/3*x^3 + 7/5*x^2 + 1/15*(25*x^2 - 4*x)*e^4 - 4/15*x

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mupad [B]  time = 0.04, size = 13, normalized size = 0.72 \begin {gather*} \frac {x\,\left (25\,x-4\right )\,\left (x+{\mathrm {e}}^4+1\right )}{15} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((14*x)/5 + 5*x^2 + (exp(4)*(50*x - 4))/15 - 4/15,x)

[Out]

(x*(25*x - 4)*(x + exp(4) + 1))/15

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sympy [A]  time = 0.06, size = 32, normalized size = 1.78 \begin {gather*} \frac {5 x^{3}}{3} + x^{2} \left (\frac {7}{5} + \frac {5 e^{4}}{3}\right ) + x \left (- \frac {4 e^{4}}{15} - \frac {4}{15}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/15*(50*x-4)*exp(4)+5*x**2+14/5*x-4/15,x)

[Out]

5*x**3/3 + x**2*(7/5 + 5*exp(4)/3) + x*(-4*exp(4)/15 - 4/15)

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