3.91.49 \(\int \frac {100 x^4+e^{x^2} (200 x+60 x^2-200 x^3-40 x^4)}{e^{2 x^2}+10 e^{x^2} x^2+25 x^4} \, dx\)

Optimal. Leaf size=23 \[ \frac {4 x^2 (5+x)}{\frac {e^{x^2}}{5}+x^2} \]

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Rubi [F]  time = 1.27, 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 {100 x^4+e^{x^2} \left (200 x+60 x^2-200 x^3-40 x^4\right )}{e^{2 x^2}+10 e^{x^2} x^2+25 x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(100*x^4 + E^x^2*(200*x + 60*x^2 - 200*x^3 - 40*x^4))/(E^(2*x^2) + 10*E^x^2*x^2 + 25*x^4),x]

[Out]

-200*Defer[Int][x^4/(E^x^2 + 5*x^2)^2, x] + 200*Defer[Int][x^6/(E^x^2 + 5*x^2)^2, x] + 60*Defer[Int][x^2/(E^x^
2 + 5*x^2), x] - 40*Defer[Int][x^4/(E^x^2 + 5*x^2), x] - 500*Defer[Subst][Defer[Int][x/(E^x + 5*x)^2, x], x, x
^2] + 500*Defer[Subst][Defer[Int][x^2/(E^x + 5*x)^2, x], x, x^2] + 100*Defer[Subst][Defer[Int][(E^x + 5*x)^(-1
), x], x, x^2] - 100*Defer[Subst][Defer[Int][x/(E^x + 5*x), x], x, x^2]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {100 x^4+e^{x^2} \left (200 x+60 x^2-200 x^3-40 x^4\right )}{\left (e^{x^2}+5 x^2\right )^2} \, dx\\ &=\int \left (\frac {200 x^3 \left (-5-x+5 x^2+x^3\right )}{\left (e^{x^2}+5 x^2\right )^2}-\frac {20 x \left (-10-3 x+10 x^2+2 x^3\right )}{e^{x^2}+5 x^2}\right ) \, dx\\ &=-\left (20 \int \frac {x \left (-10-3 x+10 x^2+2 x^3\right )}{e^{x^2}+5 x^2} \, dx\right )+200 \int \frac {x^3 \left (-5-x+5 x^2+x^3\right )}{\left (e^{x^2}+5 x^2\right )^2} \, dx\\ &=-\left (20 \int \left (-\frac {10 x}{e^{x^2}+5 x^2}-\frac {3 x^2}{e^{x^2}+5 x^2}+\frac {10 x^3}{e^{x^2}+5 x^2}+\frac {2 x^4}{e^{x^2}+5 x^2}\right ) \, dx\right )+200 \int \left (-\frac {5 x^3}{\left (e^{x^2}+5 x^2\right )^2}-\frac {x^4}{\left (e^{x^2}+5 x^2\right )^2}+\frac {5 x^5}{\left (e^{x^2}+5 x^2\right )^2}+\frac {x^6}{\left (e^{x^2}+5 x^2\right )^2}\right ) \, dx\\ &=-\left (40 \int \frac {x^4}{e^{x^2}+5 x^2} \, dx\right )+60 \int \frac {x^2}{e^{x^2}+5 x^2} \, dx-200 \int \frac {x^4}{\left (e^{x^2}+5 x^2\right )^2} \, dx+200 \int \frac {x^6}{\left (e^{x^2}+5 x^2\right )^2} \, dx+200 \int \frac {x}{e^{x^2}+5 x^2} \, dx-200 \int \frac {x^3}{e^{x^2}+5 x^2} \, dx-1000 \int \frac {x^3}{\left (e^{x^2}+5 x^2\right )^2} \, dx+1000 \int \frac {x^5}{\left (e^{x^2}+5 x^2\right )^2} \, dx\\ &=-\left (40 \int \frac {x^4}{e^{x^2}+5 x^2} \, dx\right )+60 \int \frac {x^2}{e^{x^2}+5 x^2} \, dx+100 \operatorname {Subst}\left (\int \frac {1}{e^x+5 x} \, dx,x,x^2\right )-100 \operatorname {Subst}\left (\int \frac {x}{e^x+5 x} \, dx,x,x^2\right )-200 \int \frac {x^4}{\left (e^{x^2}+5 x^2\right )^2} \, dx+200 \int \frac {x^6}{\left (e^{x^2}+5 x^2\right )^2} \, dx-500 \operatorname {Subst}\left (\int \frac {x}{\left (e^x+5 x\right )^2} \, dx,x,x^2\right )+500 \operatorname {Subst}\left (\int \frac {x^2}{\left (e^x+5 x\right )^2} \, dx,x,x^2\right )\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.24, size = 21, normalized size = 0.91 \begin {gather*} \frac {20 x^2 (5+x)}{e^{x^2}+5 x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(100*x^4 + E^x^2*(200*x + 60*x^2 - 200*x^3 - 40*x^4))/(E^(2*x^2) + 10*E^x^2*x^2 + 25*x^4),x]

[Out]

(20*x^2*(5 + x))/(E^x^2 + 5*x^2)

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fricas [A]  time = 0.62, size = 23, normalized size = 1.00 \begin {gather*} \frac {20 \, {\left (x^{3} + 5 \, x^{2}\right )}}{5 \, x^{2} + e^{\left (x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^4-200*x^3+60*x^2+200*x)*exp(x^2)+100*x^4)/(exp(x^2)^2+10*x^2*exp(x^2)+25*x^4),x, algorithm="
fricas")

[Out]

20*(x^3 + 5*x^2)/(5*x^2 + e^(x^2))

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giac [A]  time = 0.20, size = 23, normalized size = 1.00 \begin {gather*} \frac {20 \, {\left (x^{3} + 5 \, x^{2}\right )}}{5 \, x^{2} + e^{\left (x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^4-200*x^3+60*x^2+200*x)*exp(x^2)+100*x^4)/(exp(x^2)^2+10*x^2*exp(x^2)+25*x^4),x, algorithm="
giac")

[Out]

20*(x^3 + 5*x^2)/(5*x^2 + e^(x^2))

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maple [A]  time = 0.05, size = 21, normalized size = 0.91

method	result	size
risch	\(\frac {20 x^{2} \left (5+x \right )}{5 x^{2}+{\mathrm e}^{x^{2}}}\)	\(21\)
norman	\(\frac {20 x^{3}+100 x^{2}}{5 x^{2}+{\mathrm e}^{x^{2}}}\)	\(25\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-40*x^4-200*x^3+60*x^2+200*x)*exp(x^2)+100*x^4)/(exp(x^2)^2+10*x^2*exp(x^2)+25*x^4),x,method=_RETURNVERB
OSE)

[Out]

20*x^2*(5+x)/(5*x^2+exp(x^2))

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maxima [A]  time = 0.39, size = 23, normalized size = 1.00 \begin {gather*} \frac {20 \, {\left (x^{3} + 5 \, x^{2}\right )}}{5 \, x^{2} + e^{\left (x^{2}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x^4-200*x^3+60*x^2+200*x)*exp(x^2)+100*x^4)/(exp(x^2)^2+10*x^2*exp(x^2)+25*x^4),x, algorithm="
maxima")

[Out]

20*(x^3 + 5*x^2)/(5*x^2 + e^(x^2))

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mupad [B]  time = 7.98, size = 20, normalized size = 0.87 \begin {gather*} \frac {20\,x^2\,\left (x+5\right )}{{\mathrm {e}}^{x^2}+5\,x^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(x^2)*(200*x + 60*x^2 - 200*x^3 - 40*x^4) + 100*x^4)/(exp(2*x^2) + 10*x^2*exp(x^2) + 25*x^4),x)

[Out]

(20*x^2*(x + 5))/(exp(x^2) + 5*x^2)

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sympy [A]  time = 0.12, size = 19, normalized size = 0.83 \begin {gather*} \frac {20 x^{3} + 100 x^{2}}{5 x^{2} + e^{x^{2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-40*x**4-200*x**3+60*x**2+200*x)*exp(x**2)+100*x**4)/(exp(x**2)**2+10*x**2*exp(x**2)+25*x**4),x)

[Out]

(20*x**3 + 100*x**2)/(5*x**2 + exp(x**2))

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