3.5.95 \(\int \frac {-15 x^3+e x (-20+3 x^2)}{e x^7} \, dx\)

Optimal. Leaf size=16 \[ \frac {-1+\frac {5}{e}+\frac {4}{x^2}}{x^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 18, normalized size of antiderivative = 1.12, number of steps used = 3, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {12, 14} \begin {gather*} \frac {4}{x^5}+\frac {5-e}{e x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(-15*x^3 + E*x*(-20 + 3*x^2))/(E*x^7),x]

[Out]

4/x^5 + (5 - E)/(E*x^3)

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 {\int \frac {-15 x^3+e x \left (-20+3 x^2\right )}{x^7} \, dx}{e}\\ &=\frac {\int \left (-\frac {20 e}{x^6}+\frac {3 (-5+e)}{x^4}\right ) \, dx}{e}\\ &=\frac {4}{x^5}+\frac {5-e}{e x^3}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.01, size = 18, normalized size = 1.12 \begin {gather*} \frac {4}{x^5}+\frac {5-e}{e x^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(-15*x^3 + E*x*(-20 + 3*x^2))/(E*x^7),x]

[Out]

4/x^5 + (5 - E)/(E*x^3)

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fricas [A]  time = 0.76, size = 21, normalized size = 1.31 \begin {gather*} \frac {{\left (5 \, x^{2} - {\left (x^{2} - 4\right )} e\right )} e^{\left (-1\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-20)*exp(log(x)+1)-15*x^3)/x^6/exp(log(x)+1),x, algorithm="fricas")

[Out]

(5*x^2 - (x^2 - 4)*e)*e^(-1)/x^5

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giac [A]  time = 0.30, size = 23, normalized size = 1.44 \begin {gather*} -\frac {{\left (x^{2} e - 5 \, x^{2} - 4 \, e\right )} e^{\left (-1\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-20)*exp(log(x)+1)-15*x^3)/x^6/exp(log(x)+1),x, algorithm="giac")

[Out]

-(x^2*e - 5*x^2 - 4*e)*e^(-1)/x^5

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maple [A]  time = 0.08, size = 19, normalized size = 1.19

method	result	size
default	\(\frac {4}{x^{5}}-\frac {1}{x^{3}}+\frac {5 \,{\mathrm e}^{-1}}{x^{3}}\)	\(19\)
norman	\(\frac {4 x -\left ({\mathrm e}-5\right ) {\mathrm e}^{-1} x^{3}}{x^{6}}\)	\(22\)
risch	\(\frac {{\mathrm e}^{-1} \left (\left (5-{\mathrm e}\right ) x^{2}+4 \,{\mathrm e}\right )}{x^{5}}\)	\(22\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((3*x^2-20)*exp(ln(x)+1)-15*x^3)/x^6/exp(ln(x)+1),x,method=_RETURNVERBOSE)

[Out]

4/x^5-1/x^3+5*exp(-1)/x^3

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maxima [A]  time = 0.41, size = 20, normalized size = 1.25 \begin {gather*} -\frac {{\left (x^{2} {\left (e - 5\right )} - 4 \, e\right )} e^{\left (-1\right )}}{x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x^2-20)*exp(log(x)+1)-15*x^3)/x^6/exp(log(x)+1),x, algorithm="maxima")

[Out]

-(x^2*(e - 5) - 4*e)*e^(-1)/x^5

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mupad [B]  time = 0.07, size = 16, normalized size = 1.00 \begin {gather*} \frac {\left (5\,{\mathrm {e}}^{-1}-1\right )\,x^2+4}{x^5} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(- log(x) - 1)*(exp(log(x) + 1)*(3*x^2 - 20) - 15*x^3))/x^6,x)

[Out]

(x^2*(5*exp(-1) - 1) + 4)/x^5

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sympy [A]  time = 0.22, size = 20, normalized size = 1.25 \begin {gather*} - \frac {x^{2} \left (-5 + e\right ) - 4 e}{e x^{5}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((3*x**2-20)*exp(ln(x)+1)-15*x**3)/x**6/exp(ln(x)+1),x)

[Out]

-(x**2*(-5 + E) - 4*E)*exp(-1)/x**5

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