3.22.14 \(\int \frac {10 e^4 x^5+e^{4+\frac {-5625-150 x^3+4 x^5-x^6}{x^4}} (112500+750 x^3+20 x^5-10 x^6)}{x^5} \, dx\)

Optimal. Leaf size=26 \[ 5 e^4 \left (e^{4 x-\left (\frac {75}{x^2}+x\right )^2}+2 x\right ) \]

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Rubi [A]  time = 0.33, antiderivative size = 31, normalized size of antiderivative = 1.19, number of steps used = 3, number of rules used = 2, integrand size = 56, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.036, Rules used = {14, 6706} \begin {gather*} 5 e^{-\frac {5625}{x^4}-x^2+4 x-\frac {150}{x}+4}+10 e^4 x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(10*E^4*x^5 + E^(4 + (-5625 - 150*x^3 + 4*x^5 - x^6)/x^4)*(112500 + 750*x^3 + 20*x^5 - 10*x^6))/x^5,x]

[Out]

5*E^(4 - 5625/x^4 - 150/x + 4*x - x^2) + 10*E^4*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]]

Rule 6706

Int[(F_)^(v_)*(u_), x_Symbol] :> With[{q = DerivativeDivides[v, u, x]}, Simp[(q*F^v)/Log[F], x] /;  !FalseQ[q]
] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (10 e^4-\frac {10 e^{4-\frac {5625}{x^4}-\frac {150}{x}+4 x-x^2} \left (-11250-75 x^3-2 x^5+x^6\right )}{x^5}\right ) \, dx\\ &=10 e^4 x-10 \int \frac {e^{4-\frac {5625}{x^4}-\frac {150}{x}+4 x-x^2} \left (-11250-75 x^3-2 x^5+x^6\right )}{x^5} \, dx\\ &=5 e^{4-\frac {5625}{x^4}-\frac {150}{x}+4 x-x^2}+10 e^4 x\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.21, size = 31, normalized size = 1.19 \begin {gather*} e^4 \left (5 e^{-\frac {5625}{x^4}-\frac {150}{x}+4 x-x^2}+10 x\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(10*E^4*x^5 + E^(4 + (-5625 - 150*x^3 + 4*x^5 - x^6)/x^4)*(112500 + 750*x^3 + 20*x^5 - 10*x^6))/x^5,
x]

[Out]

E^4*(5*E^(-5625/x^4 - 150/x + 4*x - x^2) + 10*x)

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fricas [A]  time = 0.75, size = 34, normalized size = 1.31 \begin {gather*} 10 \, x e^{4} + 5 \, e^{\left (-\frac {x^{6} - 4 \, x^{5} - 4 \, x^{4} + 150 \, x^{3} + 5625}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^6+20*x^5+750*x^3+112500)*exp(4)*exp((-x^6+4*x^5-150*x^3-5625)/x^4)+10*x^5*exp(4))/x^5,x, alg
orithm="fricas")

[Out]

10*x*e^4 + 5*e^(-(x^6 - 4*x^5 - 4*x^4 + 150*x^3 + 5625)/x^4)

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giac [A]  time = 0.35, size = 34, normalized size = 1.31 \begin {gather*} 10 \, x e^{4} + 5 \, e^{\left (-\frac {x^{6} - 4 \, x^{5} - 4 \, x^{4} + 150 \, x^{3} + 5625}{x^{4}}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^6+20*x^5+750*x^3+112500)*exp(4)*exp((-x^6+4*x^5-150*x^3-5625)/x^4)+10*x^5*exp(4))/x^5,x, alg
orithm="giac")

[Out]

10*x*e^4 + 5*e^(-(x^6 - 4*x^5 - 4*x^4 + 150*x^3 + 5625)/x^4)

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maple [A]  time = 1.62, size = 35, normalized size = 1.35

method	result	size
risch	\(10 x \,{\mathrm e}^{4}+5 \,{\mathrm e}^{-\frac {x^{6}-4 x^{5}-4 x^{4}+150 x^{3}+5625}{x^{4}}}\)	\(35\)
norman	\(\frac {10 x^{5} {\mathrm e}^{4}+5 x^{4} {\mathrm e}^{4} {\mathrm e}^{\frac {-x^{6}+4 x^{5}-150 x^{3}-5625}{x^{4}}}}{x^{4}}\)	\(42\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-10*x^6+20*x^5+750*x^3+112500)*exp(4)*exp((-x^6+4*x^5-150*x^3-5625)/x^4)+10*x^5*exp(4))/x^5,x,method=_RE
TURNVERBOSE)

[Out]

10*x*exp(4)+5*exp(-(x^6-4*x^5-4*x^4+150*x^3+5625)/x^4)

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maxima [A]  time = 0.81, size = 29, normalized size = 1.12 \begin {gather*} 10 \, x e^{4} + 5 \, e^{\left (-x^{2} + 4 \, x - \frac {150}{x} - \frac {5625}{x^{4}} + 4\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x^6+20*x^5+750*x^3+112500)*exp(4)*exp((-x^6+4*x^5-150*x^3-5625)/x^4)+10*x^5*exp(4))/x^5,x, alg
orithm="maxima")

[Out]

10*x*e^4 + 5*e^(-x^2 + 4*x - 150/x - 5625/x^4 + 4)

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mupad [B]  time = 1.23, size = 32, normalized size = 1.23 \begin {gather*} 10\,x\,{\mathrm {e}}^4+5\,{\mathrm {e}}^{4\,x}\,{\mathrm {e}}^4\,{\mathrm {e}}^{-x^2}\,{\mathrm {e}}^{-\frac {150}{x}}\,{\mathrm {e}}^{-\frac {5625}{x^4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((10*x^5*exp(4) + exp(-(150*x^3 - 4*x^5 + x^6 + 5625)/x^4)*exp(4)*(750*x^3 + 20*x^5 - 10*x^6 + 112500))/x^5
,x)

[Out]

10*x*exp(4) + 5*exp(4*x)*exp(4)*exp(-x^2)*exp(-150/x)*exp(-5625/x^4)

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sympy [A]  time = 0.27, size = 31, normalized size = 1.19 \begin {gather*} 10 x e^{4} + 5 e^{4} e^{\frac {- x^{6} + 4 x^{5} - 150 x^{3} - 5625}{x^{4}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-10*x**6+20*x**5+750*x**3+112500)*exp(4)*exp((-x**6+4*x**5-150*x**3-5625)/x**4)+10*x**5*exp(4))/x*
*5,x)

[Out]

10*x*exp(4) + 5*exp(4)*exp((-x**6 + 4*x**5 - 150*x**3 - 5625)/x**4)

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