3.41.6 $\int \frac{-4-40x-100x^2+x^5+10x^6+25x^7+e^{\frac{4x^2}{1+5x}}(8x^6+20x^7)}{x^5+10x^6+25x^7}dx$

Optimal. Leaf size=24 $$-\frac{e^9}{2}+e^{\frac{4x}{5+\frac{1}{x}}}+\frac{1}{x^4}+x$$

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Rubi [A]  time = 0.74, antiderivative size = 19, normalized size of antiderivative = 0.79, number of steps used = 15, number of rules used = 6, integrand size = 66, $\frac{\text{number of rules}}{\text{integrand size}}$ = 0.091, Rules used = {1594, 27, 6742, 44, 43, 6706} $$\begin{array}{c}\frac{1}{x^4}+e^{\frac{4x^2}{5x+1}}+x\end{array}$$

Antiderivative was successfully verified.

[In]

Int[(-4 - 40*x - 100*x^2 + x^5 + 10*x^6 + 25*x^7 + E^((4*x^2)/(1 + 5*x))*(8*x^6 + 20*x^7))/(x^5 + 10*x^6 + 25*
x^7),x]

[Out]

E^((4*x^2)/(1 + 5*x)) + x^(-4) + x

Rule 27

Int[(u_.)*((a_) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[u*Cancel[(b/2 + c*x)^(2*p)/c^p], x] /; Fr
eeQ[{a, b, c}, x] && EqQ[b^2 - 4*a*c, 0] && IntegerQ[p]

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 44

Int[((a_) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d*
x)^n, x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rule 1594

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.) + (c_.)*(x_)^(r_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^
(q - p) + c*x^(r - p))^n, x] /; FreeQ[{a, b, c, p, q, r}, x] && IntegerQ[n] && PosQ[q - p] && PosQ[r - p]

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]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

$$\begin{array}{c}\begin{array}{rl}\text{integral}& =\int \frac{-4-40x-100x^2+x^5+10x^6+25x^7+e^{\frac{4x^2}{1+5x}}(8x^6+20x^7)}{x^5(1+10x+25x^2)}dx\\ & =\int \frac{-4-40x-100x^2+x^5+10x^6+25x^7+e^{\frac{4x^2}{1+5x}}(8x^6+20x^7)}{x^5(1+5x{)}^2}dx\\ & =\int (\frac{1}{(1+5x{)}^2}-\frac{4}{x^5(1+5x{)}^2}-\frac{40}{x^4(1+5x{)}^2}-\frac{100}{x^3(1+5x{)}^2}+\frac{10x}{(1+5x{)}^2}+\frac{25x^2}{(1+5x{)}^2}+\frac{4e^{\frac{4x^2}{1+5x}}x(2+5x)}{(1+5x{)}^2})dx\\ & =-\frac{1}{5(1+5x)}-4\int \frac{1}{x^5(1+5x{)}^2}dx+4\int \frac{e^{\frac{4x^2}{1+5x}}x(2+5x)}{(1+5x{)}^2}dx+10\int \frac{x}{(1+5x{)}^2}dx+25\int \frac{x^2}{(1+5x{)}^2}dx-40\int \frac{1}{x^4(1+5x{)}^2}dx-100\int \frac{1}{x^3(1+5x{)}^2}dx\\ & =e^{\frac{4x^2}{1+5x}}-\frac{1}{5(1+5x)}-4\int (\frac{1}{x^5}-\frac{10}{x^4}+\frac{75}{x^3}-\frac{500}{x^2}+\frac{3125}{x}-\frac{3125}{(1+5x{)}^2}-\frac{15625}{1+5x})dx+10\int (-\frac{1}{5(1+5x{)}^2}+\frac{1}{5(1+5x)})dx+25\int (\frac{1}{25}+\frac{1}{25(1+5x{)}^2}-\frac{2}{25(1+5x)})dx-40\int (\frac{1}{x^4}-\frac{10}{x^3}+\frac{75}{x^2}-\frac{500}{x}+\frac{625}{(1+5x{)}^2}+\frac{2500}{1+5x})dx-100\int (\frac{1}{x^3}-\frac{10}{x^2}+\frac{75}{x}-\frac{125}{(1+5x{)}^2}-\frac{375}{1+5x})dx\\ & =e^{\frac{4x^2}{1+5x}}+\frac{1}{x^4}+x\end{array}\end{array}$$

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Mathematica [A]  time = 0.22, size = 19, normalized size = 0.79 $$\begin{array}{c}{e}^{\frac{4x^2}{1+5x}}+\frac{1}{x^4}+x\end{array}$$

Antiderivative was successfully verified.

[In]

Integrate[(-4 - 40*x - 100*x^2 + x^5 + 10*x^6 + 25*x^7 + E^((4*x^2)/(1 + 5*x))*(8*x^6 + 20*x^7))/(x^5 + 10*x^6
+ 25*x^7),x]

[Out]

E^((4*x^2)/(1 + 5*x)) + x^(-4) + x

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fricas [A]  time = 0.48, size = 26, normalized size = 1.08 $$\begin{array}{c}\frac{x^5+x^4{e}^{\left(\frac{4x^2}{5x+1}\right)}+1}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^7+8*x^6)*exp(4*x^2/(1+5*x))+25*x^7+10*x^6+x^5-100*x^2-40*x-4)/(25*x^7+10*x^6+x^5),x, algorith
m="fricas")

[Out]

(x^5 + x^4*e^(4*x^2/(5*x + 1)) + 1)/x^4

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giac [A]  time = 0.14, size = 26, normalized size = 1.08 $$\begin{array}{c}\frac{x^5+x^4{e}^{\left(\frac{4x^2}{5x+1}\right)}+1}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^7+8*x^6)*exp(4*x^2/(1+5*x))+25*x^7+10*x^6+x^5-100*x^2-40*x-4)/(25*x^7+10*x^6+x^5),x, algorith
m="giac")

[Out]

(x^5 + x^4*e^(4*x^2/(5*x + 1)) + 1)/x^4

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maple [A]  time = 0.24, size = 19, normalized size = 0.79

method	result	size
risch	$\frac{1}{x^4}+x+{\mathrm{e}}^{\frac{4x^2}{1+5x}}$	$19$
norman	$\frac{1+x^4{\mathrm{e}}^{\frac{4x^2}{1+5x}}+x^5+5x+5x^6+5x^5{\mathrm{e}}^{\frac{4x^2}{1+5x}}}{x^4(1+5x)}$	$60$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((20*x^7+8*x^6)*exp(4*x^2/(1+5*x))+25*x^7+10*x^6+x^5-100*x^2-40*x-4)/(25*x^7+10*x^6+x^5),x,method=_RETURNV
ERBOSE)

[Out]

1/x^4+x+exp(4*x^2/(1+5*x))

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maxima [B]  time = 0.43, size = 101, normalized size = 4.21 $$\begin{array}{c}x-\frac{37500x^4+3750x^3-250x^2+25x-3}{3(5x^5+x^4)}+\frac{40(1500x^3+150x^2-10x+1)}{3(5x^4+x^3)}-\frac{50(150x^2+15x-1)}{5x^3+x^2}+e^{(\frac{4}{5}x+\frac{4}{25(5x+1)}-\frac{4}{25})}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x^7+8*x^6)*exp(4*x^2/(1+5*x))+25*x^7+10*x^6+x^5-100*x^2-40*x-4)/(25*x^7+10*x^6+x^5),x, algorith
m="maxima")

[Out]

x - 1/3*(37500*x^4 + 3750*x^3 - 250*x^2 + 25*x - 3)/(5*x^5 + x^4) + 40/3*(1500*x^3 + 150*x^2 - 10*x + 1)/(5*x^
4 + x^3) - 50*(150*x^2 + 15*x - 1)/(5*x^3 + x^2) + e^(4/5*x + 4/25/(5*x + 1) - 4/25)

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mupad [B]  time = 2.70, size = 18, normalized size = 0.75 $$\begin{array}{c}x+{\mathrm{e}}^{\frac{4x^2}{5x+1}}+\frac{1}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((4*x^2)/(5*x + 1))*(8*x^6 + 20*x^7) - 40*x - 100*x^2 + x^5 + 10*x^6 + 25*x^7 - 4)/(x^5 + 10*x^6 + 25*
x^7),x)

[Out]

x + exp((4*x^2)/(5*x + 1)) + 1/x^4

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sympy [A]  time = 0.18, size = 17, normalized size = 0.71 $$\begin{array}{c}x+e^{\frac{4x^2}{5x+1}}+\frac{1}{x^4}\end{array}$$

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((20*x**7+8*x**6)*exp(4*x**2/(1+5*x))+25*x**7+10*x**6+x**5-100*x**2-40*x-4)/(25*x**7+10*x**6+x**5),x
)

[Out]

x + exp(4*x**2/(5*x + 1)) + x**(-4)

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