3.28.90 \(\int \frac {-3+7 x-91 x^2-101 x^3+x^4+4 x^5}{3 x+3 x^2} \, dx\)

Optimal. Leaf size=31 \[ 2 x-\left (-16+\frac {1}{3} (-1+x) x\right ) \left (1-x^2\right )+\log \left (5+\frac {5}{x}\right ) \]

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Rubi [A]  time = 0.05, antiderivative size = 35, normalized size of antiderivative = 1.13, number of steps used = 3, number of rules used = 2, integrand size = 35, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.057, Rules used = {1593, 1620} \begin {gather*} \frac {x^4}{3}-\frac {x^3}{3}-\frac {49 x^2}{3}+\frac {7 x}{3}-\log (x)+\log (x+1) \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-3 + 7*x - 91*x^2 - 101*x^3 + x^4 + 4*x^5)/(3*x + 3*x^2),x]

[Out]

(7*x)/3 - (49*x^2)/3 - x^3/3 + x^4/3 - Log[x] + Log[1 + x]

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 1620

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[Px*(a + b*x)
^m*(c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && PolyQ[Px, x] && (IntegersQ[m, n] || IGtQ[m, -2]) &&
GtQ[Expon[Px, x], 2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {-3+7 x-91 x^2-101 x^3+x^4+4 x^5}{x (3+3 x)} \, dx\\ &=\int \left (\frac {7}{3}-\frac {1}{x}-\frac {98 x}{3}-x^2+\frac {4 x^3}{3}+\frac {1}{1+x}\right ) \, dx\\ &=\frac {7 x}{3}-\frac {49 x^2}{3}-\frac {x^3}{3}+\frac {x^4}{3}-\log (x)+\log (1+x)\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 35, normalized size = 1.13 \begin {gather*} \frac {7 x}{3}-\frac {49 x^2}{3}-\frac {x^3}{3}+\frac {x^4}{3}-\log (x)+\log (1+x) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-3 + 7*x - 91*x^2 - 101*x^3 + x^4 + 4*x^5)/(3*x + 3*x^2),x]

[Out]

(7*x)/3 - (49*x^2)/3 - x^3/3 + x^4/3 - Log[x] + Log[1 + x]

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fricas [A]  time = 0.86, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, x^{4} - \frac {1}{3} \, x^{3} - \frac {49}{3} \, x^{2} + \frac {7}{3} \, x + \log \left (x + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+x^4-101*x^3-91*x^2+7*x-3)/(3*x^2+3*x),x, algorithm="fricas")

[Out]

1/3*x^4 - 1/3*x^3 - 49/3*x^2 + 7/3*x + log(x + 1) - log(x)

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giac [A]  time = 0.25, size = 29, normalized size = 0.94 \begin {gather*} \frac {1}{3} \, x^{4} - \frac {1}{3} \, x^{3} - \frac {49}{3} \, x^{2} + \frac {7}{3} \, x + \log \left ({\left | x + 1 \right |}\right ) - \log \left ({\left | x \right |}\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+x^4-101*x^3-91*x^2+7*x-3)/(3*x^2+3*x),x, algorithm="giac")

[Out]

1/3*x^4 - 1/3*x^3 - 49/3*x^2 + 7/3*x + log(abs(x + 1)) - log(abs(x))

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maple [A]  time = 0.45, size = 28, normalized size = 0.90

method	result	size
default	\(\frac {x^{4}}{3}-\frac {x^{3}}{3}-\frac {49 x^{2}}{3}+\frac {7 x}{3}-\ln \relax (x )+\ln \left (x +1\right )\)	\(28\)
norman	\(\frac {x^{4}}{3}-\frac {x^{3}}{3}-\frac {49 x^{2}}{3}+\frac {7 x}{3}-\ln \relax (x )+\ln \left (x +1\right )\)	\(28\)
risch	\(\frac {x^{4}}{3}-\frac {x^{3}}{3}-\frac {49 x^{2}}{3}+\frac {7 x}{3}-\ln \relax (x )+\ln \left (x +1\right )\)	\(28\)
meijerg	\(-\ln \relax (x )+\ln \left (x +1\right )-\frac {x \left (-15 x^{3}+20 x^{2}-30 x +60\right )}{45}+\frac {x \left (4 x^{2}-6 x +12\right )}{36}+\frac {101 x \left (-3 x +6\right )}{18}-\frac {91 x}{3}\)	\(52\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((4*x^5+x^4-101*x^3-91*x^2+7*x-3)/(3*x^2+3*x),x,method=_RETURNVERBOSE)

[Out]

1/3*x^4-1/3*x^3-49/3*x^2+7/3*x-ln(x)+ln(x+1)

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maxima [A]  time = 0.54, size = 27, normalized size = 0.87 \begin {gather*} \frac {1}{3} \, x^{4} - \frac {1}{3} \, x^{3} - \frac {49}{3} \, x^{2} + \frac {7}{3} \, x + \log \left (x + 1\right ) - \log \relax (x) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x^5+x^4-101*x^3-91*x^2+7*x-3)/(3*x^2+3*x),x, algorithm="maxima")

[Out]

1/3*x^4 - 1/3*x^3 - 49/3*x^2 + 7/3*x + log(x + 1) - log(x)

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mupad [B]  time = 0.05, size = 30, normalized size = 0.97 \begin {gather*} \frac {7\,x}{3}-\frac {49\,x^2}{3}-\frac {x^3}{3}+\frac {x^4}{3}-\mathrm {atan}\left (x\,2{}\mathrm {i}+1{}\mathrm {i}\right )\,2{}\mathrm {i} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((7*x - 91*x^2 - 101*x^3 + x^4 + 4*x^5 - 3)/(3*x + 3*x^2),x)

[Out]

(7*x)/3 - atan(x*2i + 1i)*2i - (49*x^2)/3 - x^3/3 + x^4/3

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sympy [A]  time = 0.09, size = 29, normalized size = 0.94 \begin {gather*} \frac {x^{4}}{3} - \frac {x^{3}}{3} - \frac {49 x^{2}}{3} + \frac {7 x}{3} - \log {\relax (x )} + \log {\left (x + 1 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((4*x**5+x**4-101*x**3-91*x**2+7*x-3)/(3*x**2+3*x),x)

[Out]

x**4/3 - x**3/3 - 49*x**2/3 + 7*x/3 - log(x) + log(x + 1)

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