3.66.26 \(\int \frac {10 x+x^2+e^x (-76-17 x-x^2)}{475+190 x+24 x^2+x^3+e^x (95+19 x+x^2)} \, dx\)

Optimal. Leaf size=24 \[ -4-\log \left (5+e^x+x\right )+\log \left (4 \left (-5-x+(10+x)^2\right )\right ) \]

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Rubi [F]  time = 0.44, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {10 x+x^2+e^x \left (-76-17 x-x^2\right )}{475+190 x+24 x^2+x^3+e^x \left (95+19 x+x^2\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(10*x + x^2 + E^x*(-76 - 17*x - x^2))/(475 + 190*x + 24*x^2 + x^3 + E^x*(95 + 19*x + x^2)),x]

[Out]

-x + Log[95 + 19*x + x^2] + 4*Defer[Int][(5 + E^x + x)^(-1), x] + Defer[Int][x/(5 + E^x + x), x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {10 x+x^2+e^x \left (-76-17 x-x^2\right )}{\left (5+e^x+x\right ) \left (95+19 x+x^2\right )} \, dx\\ &=\int \left (\frac {4+x}{5+e^x+x}+\frac {-76-17 x-x^2}{95+19 x+x^2}\right ) \, dx\\ &=\int \frac {4+x}{5+e^x+x} \, dx+\int \frac {-76-17 x-x^2}{95+19 x+x^2} \, dx\\ &=\int \left (\frac {4}{5+e^x+x}+\frac {x}{5+e^x+x}\right ) \, dx+\int \left (-1+\frac {19+2 x}{95+19 x+x^2}\right ) \, dx\\ &=-x+4 \int \frac {1}{5+e^x+x} \, dx+\int \frac {x}{5+e^x+x} \, dx+\int \frac {19+2 x}{95+19 x+x^2} \, dx\\ &=-x+\log \left (95+19 x+x^2\right )+4 \int \frac {1}{5+e^x+x} \, dx+\int \frac {x}{5+e^x+x} \, dx\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.09, size = 19, normalized size = 0.79 \begin {gather*} -\log \left (5+e^x+x\right )+\log \left (95+19 x+x^2\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*x + x^2 + E^x*(-76 - 17*x - x^2))/(475 + 190*x + 24*x^2 + x^3 + E^x*(95 + 19*x + x^2)),x]

[Out]

-Log[5 + E^x + x] + Log[95 + 19*x + x^2]

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fricas [A]  time = 0.57, size = 18, normalized size = 0.75 \begin {gather*} \log \left (x^{2} + 19 \, x + 95\right ) - \log \left (x + e^{x} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-17*x-76)*exp(x)+x^2+10*x)/((x^2+19*x+95)*exp(x)+x^3+24*x^2+190*x+475),x, algorithm="fricas")

[Out]

log(x^2 + 19*x + 95) - log(x + e^x + 5)

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giac [A]  time = 0.16, size = 22, normalized size = 0.92 \begin {gather*} \log \left (x^{2} + 19 \, x + 95\right ) - \log \left (-x - e^{x} - 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-17*x-76)*exp(x)+x^2+10*x)/((x^2+19*x+95)*exp(x)+x^3+24*x^2+190*x+475),x, algorithm="giac")

[Out]

log(x^2 + 19*x + 95) - log(-x - e^x - 5)

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maple [A]  time = 0.06, size = 19, normalized size = 0.79

method	result	size
norman	\(-\ln \left ({\mathrm e}^{x}+5+x \right )+\ln \left (x^{2}+19 x +95\right )\)	\(19\)
risch	\(-\ln \left ({\mathrm e}^{x}+5+x \right )+\ln \left (x^{2}+19 x +95\right )\)	\(19\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-x^2-17*x-76)*exp(x)+x^2+10*x)/((x^2+19*x+95)*exp(x)+x^3+24*x^2+190*x+475),x,method=_RETURNVERBOSE)

[Out]

-ln(exp(x)+5+x)+ln(x^2+19*x+95)

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maxima [A]  time = 0.43, size = 18, normalized size = 0.75 \begin {gather*} \log \left (x^{2} + 19 \, x + 95\right ) - \log \left (x + e^{x} + 5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x^2-17*x-76)*exp(x)+x^2+10*x)/((x^2+19*x+95)*exp(x)+x^3+24*x^2+190*x+475),x, algorithm="maxima")

[Out]

log(x^2 + 19*x + 95) - log(x + e^x + 5)

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mupad [B]  time = 0.52, size = 18, normalized size = 0.75 \begin {gather*} \ln \left (x^2+19\,x+95\right )-\ln \left (x+{\mathrm {e}}^x+5\right ) \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x - exp(x)*(17*x + x^2 + 76) + x^2)/(190*x + exp(x)*(19*x + x^2 + 95) + 24*x^2 + x^3 + 475),x)

[Out]

log(19*x + x^2 + 95) - log(x + exp(x) + 5)

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sympy [A]  time = 0.15, size = 17, normalized size = 0.71 \begin {gather*} - \log {\left (x + e^{x} + 5 \right )} + \log {\left (x^{2} + 19 x + 95 \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-x**2-17*x-76)*exp(x)+x**2+10*x)/((x**2+19*x+95)*exp(x)+x**3+24*x**2+190*x+475),x)

[Out]

-log(x + exp(x) + 5) + log(x**2 + 19*x + 95)

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