3.20.81 \(\int \frac {270+105 x+596 x^2+57 x^3+230 x^4-25 x^5+15 x^7+e^5 (600 x^2+60 x^3+480 x^4-50 x^5)+e^{10} (250 x^4-25 x^5)}{25 x^7} \, dx\)

Optimal. Leaf size=30 \[ \frac {3 x}{5}+\frac {(-5+x) \left (1+e^5+\frac {3+x}{5 x^2}\right )^2}{x^2} \]

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Rubi [B]  time = 0.07, antiderivative size = 78, normalized size of antiderivative = 2.60, number of steps used = 3, number of rules used = 2, integrand size = 77, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {12, 14} \begin {gather*} -\frac {9}{5 x^6}-\frac {21}{25 x^5}-\frac {149+150 e^5}{25 x^4}-\frac {19+20 e^5}{25 x^3}-\frac {23+48 e^5+25 e^{10}}{5 x^2}+\frac {3 x}{5}+\frac {\left (1+e^5\right )^2}{x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(270 + 105*x + 596*x^2 + 57*x^3 + 230*x^4 - 25*x^5 + 15*x^7 + E^5*(600*x^2 + 60*x^3 + 480*x^4 - 50*x^5) +
E^10*(250*x^4 - 25*x^5))/(25*x^7),x]

[Out]

-9/(5*x^6) - 21/(25*x^5) - (149 + 150*E^5)/(25*x^4) - (19 + 20*E^5)/(25*x^3) - (23 + 48*E^5 + 25*E^10)/(5*x^2)
+ (1 + E^5)^2/x + (3*x)/5

Rule 12

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\frac {1}{25} \int \frac {270+105 x+596 x^2+57 x^3+230 x^4-25 x^5+15 x^7+e^5 \left (600 x^2+60 x^3+480 x^4-50 x^5\right )+e^{10} \left (250 x^4-25 x^5\right )}{x^7} \, dx\\ &=\frac {1}{25} \int \left (15+\frac {270}{x^7}+\frac {105}{x^6}+\frac {4 \left (149+150 e^5\right )}{x^5}+\frac {3 \left (19+20 e^5\right )}{x^4}+\frac {10 \left (23+48 e^5+25 e^{10}\right )}{x^3}-\frac {25 \left (1+e^5\right )^2}{x^2}\right ) \, dx\\ &=-\frac {9}{5 x^6}-\frac {21}{25 x^5}-\frac {149+150 e^5}{25 x^4}-\frac {19+20 e^5}{25 x^3}-\frac {23+48 e^5+25 e^{10}}{5 x^2}+\frac {\left (1+e^5\right )^2}{x}+\frac {3 x}{5}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.03, size = 69, normalized size = 2.30 \begin {gather*} \frac {1}{25} \left (-\frac {45}{x^6}-\frac {21}{x^5}+\frac {-149-150 e^5}{x^4}+\frac {-19-20 e^5}{x^3}-\frac {5 \left (23+48 e^5+25 e^{10}\right )}{x^2}+\frac {25 \left (1+e^5\right )^2}{x}+15 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(270 + 105*x + 596*x^2 + 57*x^3 + 230*x^4 - 25*x^5 + 15*x^7 + E^5*(600*x^2 + 60*x^3 + 480*x^4 - 50*x
^5) + E^10*(250*x^4 - 25*x^5))/(25*x^7),x]

[Out]

(-45/x^6 - 21/x^5 + (-149 - 150*E^5)/x^4 + (-19 - 20*E^5)/x^3 - (5*(23 + 48*E^5 + 25*E^10))/x^2 + (25*(1 + E^5
)^2)/x + 15*x)/25

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fricas [B]  time = 0.55, size = 73, normalized size = 2.43 \begin {gather*} \frac {15 \, x^{7} + 25 \, x^{5} - 115 \, x^{4} - 19 \, x^{3} - 149 \, x^{2} + 25 \, {\left (x^{5} - 5 \, x^{4}\right )} e^{10} + 10 \, {\left (5 \, x^{5} - 24 \, x^{4} - 2 \, x^{3} - 15 \, x^{2}\right )} e^{5} - 21 \, x - 45}{25 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*x^5+250*x^4)*exp(5)^2+(-50*x^5+480*x^4+60*x^3+600*x^2)*exp(5)+15*x^7-25*x^5+230*x^4+57*x^
3+596*x^2+105*x+270)/x^7,x, algorithm="fricas")

[Out]

1/25*(15*x^7 + 25*x^5 - 115*x^4 - 19*x^3 - 149*x^2 + 25*(x^5 - 5*x^4)*e^10 + 10*(5*x^5 - 24*x^4 - 2*x^3 - 15*x
^2)*e^5 - 21*x - 45)/x^6

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giac [B]  time = 0.20, size = 76, normalized size = 2.53 \begin {gather*} \frac {3}{5} \, x + \frac {25 \, x^{5} e^{10} + 50 \, x^{5} e^{5} + 25 \, x^{5} - 125 \, x^{4} e^{10} - 240 \, x^{4} e^{5} - 115 \, x^{4} - 20 \, x^{3} e^{5} - 19 \, x^{3} - 150 \, x^{2} e^{5} - 149 \, x^{2} - 21 \, x - 45}{25 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*x^5+250*x^4)*exp(5)^2+(-50*x^5+480*x^4+60*x^3+600*x^2)*exp(5)+15*x^7-25*x^5+230*x^4+57*x^
3+596*x^2+105*x+270)/x^7,x, algorithm="giac")

[Out]

3/5*x + 1/25*(25*x^5*e^10 + 50*x^5*e^5 + 25*x^5 - 125*x^4*e^10 - 240*x^4*e^5 - 115*x^4 - 20*x^3*e^5 - 19*x^3 -
150*x^2*e^5 - 149*x^2 - 21*x - 45)/x^6

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maple [B]  time = 0.07, size = 63, normalized size = 2.10

method	result	size
risch	\(\frac {3 x}{5}+\frac {\left (50 \,{\mathrm e}^{5}+25 \,{\mathrm e}^{10}+25\right ) x^{5}+\left (-125 \,{\mathrm e}^{10}-240 \,{\mathrm e}^{5}-115\right ) x^{4}+\left (-20 \,{\mathrm e}^{5}-19\right ) x^{3}+\left (-150 \,{\mathrm e}^{5}-149\right ) x^{2}-21 x -45}{25 x^{6}}\)	\(63\)
norman	\(\frac {-\frac {9}{5}+\left (-6 \,{\mathrm e}^{5}-\frac {149}{25}\right ) x^{2}+\left (-\frac {4 \,{\mathrm e}^{5}}{5}-\frac {19}{25}\right ) x^{3}+\left (-5 \,{\mathrm e}^{10}-\frac {48 \,{\mathrm e}^{5}}{5}-\frac {23}{5}\right ) x^{4}+\left ({\mathrm e}^{10}+2 \,{\mathrm e}^{5}+1\right ) x^{5}-\frac {21 x}{25}+\frac {3 x^{7}}{5}}{x^{6}}\)	\(65\)
default	\(\frac {3 x}{5}-\frac {21}{25 x^{5}}-\frac {9}{5 x^{6}}-\frac {-50 \,{\mathrm e}^{5}-25 \,{\mathrm e}^{10}-25}{25 x}-\frac {600 \,{\mathrm e}^{5}+596}{100 x^{4}}-\frac {480 \,{\mathrm e}^{5}+250 \,{\mathrm e}^{10}+230}{50 x^{2}}-\frac {60 \,{\mathrm e}^{5}+57}{75 x^{3}}\)	\(67\)
gosper	\(\frac {25 x^{5} {\mathrm e}^{10}+15 x^{7}-125 x^{4} {\mathrm e}^{10}+50 x^{5} {\mathrm e}^{5}-240 x^{4} {\mathrm e}^{5}+25 x^{5}-20 x^{3} {\mathrm e}^{5}-115 x^{4}-150 x^{2} {\mathrm e}^{5}-19 x^{3}-149 x^{2}-21 x -45}{25 x^{6}}\)	\(82\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/25*((-25*x^5+250*x^4)*exp(5)^2+(-50*x^5+480*x^4+60*x^3+600*x^2)*exp(5)+15*x^7-25*x^5+230*x^4+57*x^3+596*
x^2+105*x+270)/x^7,x,method=_RETURNVERBOSE)

[Out]

3/5*x+1/25*((50*exp(5)+25*exp(10)+25)*x^5+(-125*exp(10)-240*exp(5)-115)*x^4+(-20*exp(5)-19)*x^3+(-150*exp(5)-1
49)*x^2-21*x-45)/x^6

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maxima [B]  time = 0.68, size = 64, normalized size = 2.13 \begin {gather*} \frac {3}{5} \, x + \frac {25 \, x^{5} {\left (e^{10} + 2 \, e^{5} + 1\right )} - 5 \, x^{4} {\left (25 \, e^{10} + 48 \, e^{5} + 23\right )} - x^{3} {\left (20 \, e^{5} + 19\right )} - x^{2} {\left (150 \, e^{5} + 149\right )} - 21 \, x - 45}{25 \, x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*x^5+250*x^4)*exp(5)^2+(-50*x^5+480*x^4+60*x^3+600*x^2)*exp(5)+15*x^7-25*x^5+230*x^4+57*x^
3+596*x^2+105*x+270)/x^7,x, algorithm="maxima")

[Out]

3/5*x + 1/25*(25*x^5*(e^10 + 2*e^5 + 1) - 5*x^4*(25*e^10 + 48*e^5 + 23) - x^3*(20*e^5 + 19) - x^2*(150*e^5 + 1
49) - 21*x - 45)/x^6

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mupad [B]  time = 1.18, size = 63, normalized size = 2.10 \begin {gather*} \frac {3\,x}{5}-\frac {\left (-50\,{\mathrm {e}}^5-25\,{\mathrm {e}}^{10}-25\right )\,x^5+\left (240\,{\mathrm {e}}^5+125\,{\mathrm {e}}^{10}+115\right )\,x^4+\left (20\,{\mathrm {e}}^5+19\right )\,x^3+\left (150\,{\mathrm {e}}^5+149\right )\,x^2+21\,x+45}{25\,x^6} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((21*x)/5 + (exp(10)*(250*x^4 - 25*x^5))/25 + (596*x^2)/25 + (57*x^3)/25 + (46*x^4)/5 - x^5 + (3*x^7)/5 +
(exp(5)*(600*x^2 + 60*x^3 + 480*x^4 - 50*x^5))/25 + 54/5)/x^7,x)

[Out]

(3*x)/5 - (21*x - x^5*(50*exp(5) + 25*exp(10) + 25) + x^4*(240*exp(5) + 125*exp(10) + 115) + x^3*(20*exp(5) +
19) + x^2*(150*exp(5) + 149) + 45)/(25*x^6)

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sympy [B]  time = 1.77, size = 70, normalized size = 2.33 \begin {gather*} \frac {3 x}{5} + \frac {x^{5} \left (25 + 50 e^{5} + 25 e^{10}\right ) + x^{4} \left (- 125 e^{10} - 240 e^{5} - 115\right ) + x^{3} \left (- 20 e^{5} - 19\right ) + x^{2} \left (- 150 e^{5} - 149\right ) - 21 x - 45}{25 x^{6}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/25*((-25*x**5+250*x**4)*exp(5)**2+(-50*x**5+480*x**4+60*x**3+600*x**2)*exp(5)+15*x**7-25*x**5+230*
x**4+57*x**3+596*x**2+105*x+270)/x**7,x)

[Out]

3*x/5 + (x**5*(25 + 50*exp(5) + 25*exp(10)) + x**4*(-125*exp(10) - 240*exp(5) - 115) + x**3*(-20*exp(5) - 19)
+ x**2*(-150*exp(5) - 149) - 21*x - 45)/(25*x**6)

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