3.84.22 \(\int \frac {262144+x^5-2 x^6+e^{12 x+10 x^4} (458752-1376256 x-4587520 x^4)+e^{18 x+15 x^4} (229376-1032192 x-3440640 x^4)+e^{6 x+5 x^4} (524288-786432 x-2621440 x^4)+e^{24 x+20 x^4} (71680-430080 x-1433600 x^4)+e^{30 x+25 x^4} (14336-107520 x-358400 x^4)+e^{36 x+30 x^4} (1792-16128 x-53760 x^4)+e^{42 x+35 x^4} (128-1344 x-4480 x^4)+e^{48 x+40 x^4} (4-48 x-160 x^4)}{x^5} \, dx\)

Optimal. Leaf size=28 \[ 5-\frac {\left (4+e^{x+5 \left (x+x^4\right )}\right )^8}{x^4}+x-x^2 \]

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Rubi [B]  time = 0.64, antiderivative size = 291, normalized size of antiderivative = 10.39, number of steps used = 12, number of rules used = 2, integrand size = 190, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.011, Rules used = {14, 2288} \begin {gather*} -\frac {65536}{x^4}-x^2-\frac {114688 e^{2 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {57344 e^{3 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {17920 e^{4 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {3584 e^{5 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {448 e^{6 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {32 e^{7 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {e^{8 x \left (5 x^3+6\right )} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}-\frac {131072 e^{5 x^4+6 x} \left (10 x^4+3 x\right )}{\left (10 x^3+3\right ) x^5}+x \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(262144 + x^5 - 2*x^6 + E^(12*x + 10*x^4)*(458752 - 1376256*x - 4587520*x^4) + E^(18*x + 15*x^4)*(229376 -
1032192*x - 3440640*x^4) + E^(6*x + 5*x^4)*(524288 - 786432*x - 2621440*x^4) + E^(24*x + 20*x^4)*(71680 - 430
080*x - 1433600*x^4) + E^(30*x + 25*x^4)*(14336 - 107520*x - 358400*x^4) + E^(36*x + 30*x^4)*(1792 - 16128*x -
53760*x^4) + E^(42*x + 35*x^4)*(128 - 1344*x - 4480*x^4) + E^(48*x + 40*x^4)*(4 - 48*x - 160*x^4))/x^5,x]

[Out]

-65536/x^4 + x - x^2 - (114688*E^(2*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (57344*E^(3*x*(6 + 5*x
^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (17920*E^(4*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (358
4*E^(5*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (448*E^(6*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 +
10*x^3)) - (32*E^(7*x*(6 + 5*x^3))*(3*x + 10*x^4))/(x^5*(3 + 10*x^3)) - (E^(8*x*(6 + 5*x^3))*(3*x + 10*x^4))/(
x^5*(3 + 10*x^3)) - (131072*E^(6*x + 5*x^4)*(3*x + 10*x^4))/(x^5*(3 + 10*x^3))

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 2288

Int[(y_.)*(F_)^(u_)*((v_) + (w_)), x_Symbol] :> With[{z = (v*y)/(Log[F]*D[u, x])}, Simp[F^u*z, x] /; EqQ[D[z,
x], w*y]] /; FreeQ[F, x]

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \left (-\frac {262144 e^{6 x+5 x^4} \left (-2+3 x+10 x^4\right )}{x^5}-\frac {458752 e^{2 x \left (6+5 x^3\right )} \left (-1+3 x+10 x^4\right )}{x^5}-\frac {71680 e^{4 x \left (6+5 x^3\right )} \left (-1+6 x+20 x^4\right )}{x^5}-\frac {114688 e^{3 x \left (6+5 x^3\right )} \left (-2+9 x+30 x^4\right )}{x^5}-\frac {1792 e^{6 x \left (6+5 x^3\right )} \left (-1+9 x+30 x^4\right )}{x^5}-\frac {4 e^{8 x \left (6+5 x^3\right )} \left (-1+12 x+40 x^4\right )}{x^5}-\frac {7168 e^{5 x \left (6+5 x^3\right )} \left (-2+15 x+50 x^4\right )}{x^5}-\frac {64 e^{7 x \left (6+5 x^3\right )} \left (-2+21 x+70 x^4\right )}{x^5}+\frac {262144+x^5-2 x^6}{x^5}\right ) \, dx\\ &=-\left (4 \int \frac {e^{8 x \left (6+5 x^3\right )} \left (-1+12 x+40 x^4\right )}{x^5} \, dx\right )-64 \int \frac {e^{7 x \left (6+5 x^3\right )} \left (-2+21 x+70 x^4\right )}{x^5} \, dx-1792 \int \frac {e^{6 x \left (6+5 x^3\right )} \left (-1+9 x+30 x^4\right )}{x^5} \, dx-7168 \int \frac {e^{5 x \left (6+5 x^3\right )} \left (-2+15 x+50 x^4\right )}{x^5} \, dx-71680 \int \frac {e^{4 x \left (6+5 x^3\right )} \left (-1+6 x+20 x^4\right )}{x^5} \, dx-114688 \int \frac {e^{3 x \left (6+5 x^3\right )} \left (-2+9 x+30 x^4\right )}{x^5} \, dx-262144 \int \frac {e^{6 x+5 x^4} \left (-2+3 x+10 x^4\right )}{x^5} \, dx-458752 \int \frac {e^{2 x \left (6+5 x^3\right )} \left (-1+3 x+10 x^4\right )}{x^5} \, dx+\int \frac {262144+x^5-2 x^6}{x^5} \, dx\\ &=-\frac {114688 e^{2 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {57344 e^{3 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {17920 e^{4 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {3584 e^{5 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {448 e^{6 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {32 e^{7 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {e^{8 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {131072 e^{6 x+5 x^4} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}+\int \left (1+\frac {262144}{x^5}-2 x\right ) \, dx\\ &=-\frac {65536}{x^4}+x-x^2-\frac {114688 e^{2 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {57344 e^{3 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {17920 e^{4 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {3584 e^{5 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {448 e^{6 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {32 e^{7 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {e^{8 x \left (6+5 x^3\right )} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}-\frac {131072 e^{6 x+5 x^4} \left (3 x+10 x^4\right )}{x^5 \left (3+10 x^3\right )}\\ \end {aligned} \end {gather*}

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Mathematica [B]  time = 0.19, size = 124, normalized size = 4.43 \begin {gather*} -\frac {65536+114688 e^{2 x \left (6+5 x^3\right )}+57344 e^{3 x \left (6+5 x^3\right )}+17920 e^{4 x \left (6+5 x^3\right )}+3584 e^{5 x \left (6+5 x^3\right )}+448 e^{6 x \left (6+5 x^3\right )}+32 e^{7 x \left (6+5 x^3\right )}+e^{8 x \left (6+5 x^3\right )}+131072 e^{6 x+5 x^4}-x^5+x^6}{x^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(262144 + x^5 - 2*x^6 + E^(12*x + 10*x^4)*(458752 - 1376256*x - 4587520*x^4) + E^(18*x + 15*x^4)*(22
9376 - 1032192*x - 3440640*x^4) + E^(6*x + 5*x^4)*(524288 - 786432*x - 2621440*x^4) + E^(24*x + 20*x^4)*(71680
- 430080*x - 1433600*x^4) + E^(30*x + 25*x^4)*(14336 - 107520*x - 358400*x^4) + E^(36*x + 30*x^4)*(1792 - 161
28*x - 53760*x^4) + E^(42*x + 35*x^4)*(128 - 1344*x - 4480*x^4) + E^(48*x + 40*x^4)*(4 - 48*x - 160*x^4))/x^5,
x]

[Out]

-((65536 + 114688*E^(2*x*(6 + 5*x^3)) + 57344*E^(3*x*(6 + 5*x^3)) + 17920*E^(4*x*(6 + 5*x^3)) + 3584*E^(5*x*(6
+ 5*x^3)) + 448*E^(6*x*(6 + 5*x^3)) + 32*E^(7*x*(6 + 5*x^3)) + E^(8*x*(6 + 5*x^3)) + 131072*E^(6*x + 5*x^4) -
x^5 + x^6)/x^4)

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fricas [B]  time = 0.56, size = 109, normalized size = 3.89 \begin {gather*} -\frac {x^{6} - x^{5} + e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )} + 65536}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6*x)^7+(-53760*x^4-16128*x+1792
)*exp(5*x^4+6*x)^6+(-358400*x^4-107520*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^
4+(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752)*exp(5*x^4+6*x)^2+(-2621440*
x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+262144)/x^5,x, algorithm="fricas")

[Out]

-(x^6 - x^5 + e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 36*x) + 3584*e^(25*x^4 + 30*x) + 1792
0*e^(20*x^4 + 24*x) + 57344*e^(15*x^4 + 18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x) + 65536)/x^4

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giac [B]  time = 0.15, size = 109, normalized size = 3.89 \begin {gather*} -\frac {x^{6} - x^{5} + e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )} + 65536}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6*x)^7+(-53760*x^4-16128*x+1792
)*exp(5*x^4+6*x)^6+(-358400*x^4-107520*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^
4+(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752)*exp(5*x^4+6*x)^2+(-2621440*
x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+262144)/x^5,x, algorithm="giac")

[Out]

-(x^6 - x^5 + e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 36*x) + 3584*e^(25*x^4 + 30*x) + 1792
0*e^(20*x^4 + 24*x) + 57344*e^(15*x^4 + 18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x) + 65536)/x^4

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maple [B]  time = 0.15, size = 140, normalized size = 5.00

method	result	size
risch	\(-x^{2}+x -\frac {65536}{x^{4}}-\frac {{\mathrm e}^{8 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {32 \,{\mathrm e}^{7 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {448 \,{\mathrm e}^{6 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {3584 \,{\mathrm e}^{5 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {17920 \,{\mathrm e}^{4 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {57344 \,{\mathrm e}^{3 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {114688 \,{\mathrm e}^{2 x \left (5 x^{3}+6\right )}}{x^{4}}-\frac {131072 \,{\mathrm e}^{x \left (5 x^{3}+6\right )}}{x^{4}}\)	\(140\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6*x)^7+(-53760*x^4-16128*x+1792)*exp(
5*x^4+6*x)^6+(-358400*x^4-107520*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^4+(-34
40640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752)*exp(5*x^4+6*x)^2+(-2621440*x^4-78
6432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+262144)/x^5,x,method=_RETURNVERBOSE)

[Out]

-x^2+x-65536/x^4-1/x^4*exp(8*x*(5*x^3+6))-32/x^4*exp(7*x*(5*x^3+6))-448/x^4*exp(6*x*(5*x^3+6))-3584/x^4*exp(5*
x*(5*x^3+6))-17920/x^4*exp(4*x*(5*x^3+6))-57344/x^4*exp(3*x*(5*x^3+6))-114688/x^4*exp(2*x*(5*x^3+6))-131072*ex
p(x*(5*x^3+6))/x^4

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maxima [B]  time = 0.45, size = 112, normalized size = 4.00 \begin {gather*} -x^{2} + x - \frac {e^{\left (40 \, x^{4} + 48 \, x\right )} + 32 \, e^{\left (35 \, x^{4} + 42 \, x\right )} + 448 \, e^{\left (30 \, x^{4} + 36 \, x\right )} + 3584 \, e^{\left (25 \, x^{4} + 30 \, x\right )} + 17920 \, e^{\left (20 \, x^{4} + 24 \, x\right )} + 57344 \, e^{\left (15 \, x^{4} + 18 \, x\right )} + 114688 \, e^{\left (10 \, x^{4} + 12 \, x\right )} + 131072 \, e^{\left (5 \, x^{4} + 6 \, x\right )}}{x^{4}} - \frac {65536}{x^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x^4-48*x+4)*exp(5*x^4+6*x)^8+(-4480*x^4-1344*x+128)*exp(5*x^4+6*x)^7+(-53760*x^4-16128*x+1792
)*exp(5*x^4+6*x)^6+(-358400*x^4-107520*x+14336)*exp(5*x^4+6*x)^5+(-1433600*x^4-430080*x+71680)*exp(5*x^4+6*x)^
4+(-3440640*x^4-1032192*x+229376)*exp(5*x^4+6*x)^3+(-4587520*x^4-1376256*x+458752)*exp(5*x^4+6*x)^2+(-2621440*
x^4-786432*x+524288)*exp(5*x^4+6*x)-2*x^6+x^5+262144)/x^5,x, algorithm="maxima")

[Out]

-x^2 + x - (e^(40*x^4 + 48*x) + 32*e^(35*x^4 + 42*x) + 448*e^(30*x^4 + 36*x) + 3584*e^(25*x^4 + 30*x) + 17920*
e^(20*x^4 + 24*x) + 57344*e^(15*x^4 + 18*x) + 114688*e^(10*x^4 + 12*x) + 131072*e^(5*x^4 + 6*x))/x^4 - 65536/x
^4

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mupad [B]  time = 5.36, size = 26, normalized size = 0.93 \begin {gather*} -x\,\left (x-1\right )-\frac {{\left ({\mathrm {e}}^{5\,x^4+6\,x}+4\right )}^8}{x^4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(exp(48*x + 40*x^4)*(48*x + 160*x^4 - 4) + exp(42*x + 35*x^4)*(1344*x + 4480*x^4 - 128) + exp(36*x + 30*x
^4)*(16128*x + 53760*x^4 - 1792) + exp(30*x + 25*x^4)*(107520*x + 358400*x^4 - 14336) + exp(24*x + 20*x^4)*(43
0080*x + 1433600*x^4 - 71680) + exp(6*x + 5*x^4)*(786432*x + 2621440*x^4 - 524288) + exp(18*x + 15*x^4)*(10321
92*x + 3440640*x^4 - 229376) + exp(12*x + 10*x^4)*(1376256*x + 4587520*x^4 - 458752) - x^5 + 2*x^6 - 262144)/x
^5,x)

[Out]

- x*(x - 1) - (exp(6*x + 5*x^4) + 4)^8/x^4

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sympy [B]  time = 0.34, size = 134, normalized size = 4.79 \begin {gather*} - x^{2} + x - \frac {65536}{x^{4}} + \frac {- 131072 x^{28} e^{5 x^{4} + 6 x} - 114688 x^{28} e^{10 x^{4} + 12 x} - 57344 x^{28} e^{15 x^{4} + 18 x} - 17920 x^{28} e^{20 x^{4} + 24 x} - 3584 x^{28} e^{25 x^{4} + 30 x} - 448 x^{28} e^{30 x^{4} + 36 x} - 32 x^{28} e^{35 x^{4} + 42 x} - x^{28} e^{40 x^{4} + 48 x}}{x^{32}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-160*x**4-48*x+4)*exp(5*x**4+6*x)**8+(-4480*x**4-1344*x+128)*exp(5*x**4+6*x)**7+(-53760*x**4-16128
*x+1792)*exp(5*x**4+6*x)**6+(-358400*x**4-107520*x+14336)*exp(5*x**4+6*x)**5+(-1433600*x**4-430080*x+71680)*ex
p(5*x**4+6*x)**4+(-3440640*x**4-1032192*x+229376)*exp(5*x**4+6*x)**3+(-4587520*x**4-1376256*x+458752)*exp(5*x*
*4+6*x)**2+(-2621440*x**4-786432*x+524288)*exp(5*x**4+6*x)-2*x**6+x**5+262144)/x**5,x)

[Out]

-x**2 + x - 65536/x**4 + (-131072*x**28*exp(5*x**4 + 6*x) - 114688*x**28*exp(10*x**4 + 12*x) - 57344*x**28*exp
(15*x**4 + 18*x) - 17920*x**28*exp(20*x**4 + 24*x) - 3584*x**28*exp(25*x**4 + 30*x) - 448*x**28*exp(30*x**4 +
36*x) - 32*x**28*exp(35*x**4 + 42*x) - x**28*exp(40*x**4 + 48*x))/x**32

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