3.92.94 \(\int \frac {e^{3 x} (2500 x^3-4375 x^4)+e^{1+x} (2500 x^3+625 x^4)}{16 e^5+16 e^{10 x}+80 e^{4+2 x}+160 e^{3+4 x}+160 e^{2+6 x}+80 e^{1+8 x}} \, dx\)

Optimal. Leaf size=25 \[ e^{e^4}+\frac {625 e^x x^4}{16 \left (e+e^{2 x}\right )^4} \]

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Rubi [F]  time = 2.90, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \begin {gather*} \int \frac {e^{3 x} \left (2500 x^3-4375 x^4\right )+e^{1+x} \left (2500 x^3+625 x^4\right )}{16 e^5+16 e^{10 x}+80 e^{4+2 x}+160 e^{3+4 x}+160 e^{2+6 x}+80 e^{1+8 x}} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Int[(E^(3*x)*(2500*x^3 - 4375*x^4) + E^(1 + x)*(2500*x^3 + 625*x^4))/(16*E^5 + 16*E^(10*x) + 80*E^(4 + 2*x) +
160*E^(3 + 4*x) + 160*E^(2 + 6*x) + 80*E^(1 + 8*x)),x]

[Out]

(625*E^(-1 + x)*x^3)/(24*(E + E^(2*x))^3) + (3125*E^(-2 + x)*x^3)/(96*(E + E^(2*x))^2) + (3125*E^(-3 + x)*x^3)
/(64*(E + E^(2*x))) - (4375*E^(-1 + x)*x^4)/(96*(E + E^(2*x))^3) - (21875*E^(-2 + x)*x^4)/(384*(E + E^(2*x))^2
) - (21875*E^(-3 + x)*x^4)/(256*(E + E^(2*x))) + (3125*x^3*ArcTan[E^(-1/2 + x)])/(64*E^(7/2)) - (21875*x^4*Arc
Tan[E^(-1/2 + x)])/(256*E^(7/2)) - (((9375*I)/128)*x^2*PolyLog[2, (-I)*E^(-1/2 + x)])/E^(7/2) + (((21875*I)/12
8)*x^3*PolyLog[2, (-I)*E^(-1/2 + x)])/E^(7/2) + (((9375*I)/128)*x^2*PolyLog[2, I*E^(-1/2 + x)])/E^(7/2) - (((2
1875*I)/128)*x^3*PolyLog[2, I*E^(-1/2 + x)])/E^(7/2) + (((9375*I)/64)*x*PolyLog[3, (-I)*E^(-1/2 + x)])/E^(7/2)
- (((65625*I)/128)*x^2*PolyLog[3, (-I)*E^(-1/2 + x)])/E^(7/2) - (((9375*I)/64)*x*PolyLog[3, I*E^(-1/2 + x)])/
E^(7/2) + (((65625*I)/128)*x^2*PolyLog[3, I*E^(-1/2 + x)])/E^(7/2) - (((9375*I)/64)*PolyLog[4, (-I)*E^(-1/2 +
x)])/E^(7/2) + (((65625*I)/64)*x*PolyLog[4, (-I)*E^(-1/2 + x)])/E^(7/2) + (((9375*I)/64)*PolyLog[4, I*E^(-1/2
+ x)])/E^(7/2) - (((65625*I)/64)*x*PolyLog[4, I*E^(-1/2 + x)])/E^(7/2) - (((65625*I)/64)*PolyLog[5, (-I)*E^(-1
/2 + x)])/E^(7/2) + (((65625*I)/64)*PolyLog[5, I*E^(-1/2 + x)])/E^(7/2) - (625*Defer[Int][(E^(-1 + x)*x^2)/(E
+ E^(2*x))^3, x])/8 - (3125*Defer[Int][(E^(-2 + x)*x^2)/(E + E^(2*x))^2, x])/32 - (9375*Defer[Int][(E^(-3 + x)
*x^2)/(E + E^(2*x)), x])/64 + (4375*Defer[Int][(E^(-1 + x)*x^3)/(E + E^(2*x))^3, x])/24 + (21875*Defer[Int][(E
^(-2 + x)*x^3)/(E + E^(2*x))^2, x])/96 + (21875*Defer[Int][(E^(-3 + x)*x^3)/(E + E^(2*x)), x])/64 + (625*Defer
[Int][(E^(1 + x)*x^4)/(E + E^(2*x))^5, x])/2

Rubi steps

\begin {gather*} \begin {aligned} \text {integral} &=\int \frac {625 e^x x^3 \left (e (4+x)-e^{2 x} (-4+7 x)\right )}{16 \left (e+e^{2 x}\right )^5} \, dx\\ &=\frac {625}{16} \int \frac {e^x x^3 \left (e (4+x)-e^{2 x} (-4+7 x)\right )}{\left (e+e^{2 x}\right )^5} \, dx\\ &=\frac {625}{16} \int \left (\frac {8 e^{1+x} x^4}{\left (e+e^{2 x}\right )^5}-\frac {e^x x^3 (-4+7 x)}{\left (e+e^{2 x}\right )^4}\right ) \, dx\\ &=-\left (\frac {625}{16} \int \frac {e^x x^3 (-4+7 x)}{\left (e+e^{2 x}\right )^4} \, dx\right )+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx\\ &=-\left (\frac {625}{16} \int \left (-\frac {4 e^x x^3}{\left (e+e^{2 x}\right )^4}+\frac {7 e^x x^4}{\left (e+e^{2 x}\right )^4}\right ) \, dx\right )+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx\\ &=\frac {625}{4} \int \frac {e^x x^3}{\left (e+e^{2 x}\right )^4} \, dx-\frac {4375}{16} \int \frac {e^x x^4}{\left (e+e^{2 x}\right )^4} \, dx+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx-\frac {1875}{4} \int x^2 \left (\frac {e^{-1+x}}{6 \left (e+e^{2 x}\right )^3}+\frac {5 e^{-2+x}}{24 \left (e+e^{2 x}\right )^2}+\frac {5 e^{-3+x}}{16 \left (e+e^{2 x}\right )}+\frac {5 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{16 e^{7/2}}\right ) \, dx+\frac {4375}{4} \int x^3 \left (\frac {e^{-1+x}}{6 \left (e+e^{2 x}\right )^3}+\frac {5 e^{-2+x}}{24 \left (e+e^{2 x}\right )^2}+\frac {5 e^{-3+x}}{16 \left (e+e^{2 x}\right )}+\frac {5 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{16 e^{7/2}}\right ) \, dx\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx-\frac {1875}{4} \int \left (\frac {e^{-1+x} x^2}{6 \left (e+e^{2 x}\right )^3}+\frac {5 e^{-2+x} x^2}{24 \left (e+e^{2 x}\right )^2}+\frac {5 e^{-3+x} x^2}{16 \left (e+e^{2 x}\right )}+\frac {5 x^2 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{16 e^{7/2}}\right ) \, dx+\frac {4375}{4} \int \left (\frac {e^{-1+x} x^3}{6 \left (e+e^{2 x}\right )^3}+\frac {5 e^{-2+x} x^3}{24 \left (e+e^{2 x}\right )^2}+\frac {5 e^{-3+x} x^3}{16 \left (e+e^{2 x}\right )}+\frac {5 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{16 e^{7/2}}\right ) \, dx\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}-\frac {625}{8} \int \frac {e^{-1+x} x^2}{\left (e+e^{2 x}\right )^3} \, dx-\frac {3125}{32} \int \frac {e^{-2+x} x^2}{\left (e+e^{2 x}\right )^2} \, dx-\frac {9375}{64} \int \frac {e^{-3+x} x^2}{e+e^{2 x}} \, dx+\frac {4375}{24} \int \frac {e^{-1+x} x^3}{\left (e+e^{2 x}\right )^3} \, dx+\frac {21875}{96} \int \frac {e^{-2+x} x^3}{\left (e+e^{2 x}\right )^2} \, dx+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx+\frac {21875}{64} \int \frac {e^{-3+x} x^3}{e+e^{2 x}} \, dx-\frac {9375 \int x^2 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}+\frac {21875 \int x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}-\frac {625}{8} \int \frac {e^{-1+x} x^2}{\left (e+e^{2 x}\right )^3} \, dx-\frac {3125}{32} \int \frac {e^{-2+x} x^2}{\left (e+e^{2 x}\right )^2} \, dx-\frac {9375}{64} \int \frac {e^{-3+x} x^2}{e+e^{2 x}} \, dx+\frac {4375}{24} \int \frac {e^{-1+x} x^3}{\left (e+e^{2 x}\right )^3} \, dx+\frac {21875}{96} \int \frac {e^{-2+x} x^3}{\left (e+e^{2 x}\right )^2} \, dx+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx+\frac {21875}{64} \int \frac {e^{-3+x} x^3}{e+e^{2 x}} \, dx-\frac {(9375 i) \int x^2 \log \left (1-i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}+\frac {(9375 i) \int x^2 \log \left (1+i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}+\frac {(21875 i) \int x^3 \log \left (1-i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}-\frac {(21875 i) \int x^3 \log \left (1+i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}-\frac {9375 i x^2 \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}+\frac {21875 i x^3 \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}+\frac {9375 i x^2 \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}-\frac {21875 i x^3 \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}-\frac {625}{8} \int \frac {e^{-1+x} x^2}{\left (e+e^{2 x}\right )^3} \, dx-\frac {3125}{32} \int \frac {e^{-2+x} x^2}{\left (e+e^{2 x}\right )^2} \, dx-\frac {9375}{64} \int \frac {e^{-3+x} x^2}{e+e^{2 x}} \, dx+\frac {4375}{24} \int \frac {e^{-1+x} x^3}{\left (e+e^{2 x}\right )^3} \, dx+\frac {21875}{96} \int \frac {e^{-2+x} x^3}{\left (e+e^{2 x}\right )^2} \, dx+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx+\frac {21875}{64} \int \frac {e^{-3+x} x^3}{e+e^{2 x}} \, dx+\frac {(9375 i) \int x \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}-\frac {(9375 i) \int x \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}-\frac {(65625 i) \int x^2 \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}+\frac {(65625 i) \int x^2 \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right ) \, dx}{128 e^{7/2}}\\ &=\frac {625 e^{-1+x} x^3}{24 \left (e+e^{2 x}\right )^3}+\frac {3125 e^{-2+x} x^3}{96 \left (e+e^{2 x}\right )^2}+\frac {3125 e^{-3+x} x^3}{64 \left (e+e^{2 x}\right )}-\frac {4375 e^{-1+x} x^4}{96 \left (e+e^{2 x}\right )^3}-\frac {21875 e^{-2+x} x^4}{384 \left (e+e^{2 x}\right )^2}-\frac {21875 e^{-3+x} x^4}{256 \left (e+e^{2 x}\right )}+\frac {3125 x^3 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {21875 x^4 \tan ^{-1}\left (e^{-\frac {1}{2}+x}\right )}{256 e^{7/2}}-\frac {9375 i x^2 \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}+\frac {21875 i x^3 \text {Li}_2\left (-i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}+\frac {9375 i x^2 \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}-\frac {21875 i x^3 \text {Li}_2\left (i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}+\frac {9375 i x \text {Li}_3\left (-i e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}-\frac {65625 i x^2 \text {Li}_3\left (-i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}-\frac {9375 i x \text {Li}_3\left (i e^{-\frac {1}{2}+x}\right )}{64 e^{7/2}}+\frac {65625 i x^2 \text {Li}_3\left (i e^{-\frac {1}{2}+x}\right )}{128 e^{7/2}}-\frac {625}{8} \int \frac {e^{-1+x} x^2}{\left (e+e^{2 x}\right )^3} \, dx-\frac {3125}{32} \int \frac {e^{-2+x} x^2}{\left (e+e^{2 x}\right )^2} \, dx-\frac {9375}{64} \int \frac {e^{-3+x} x^2}{e+e^{2 x}} \, dx+\frac {4375}{24} \int \frac {e^{-1+x} x^3}{\left (e+e^{2 x}\right )^3} \, dx+\frac {21875}{96} \int \frac {e^{-2+x} x^3}{\left (e+e^{2 x}\right )^2} \, dx+\frac {625}{2} \int \frac {e^{1+x} x^4}{\left (e+e^{2 x}\right )^5} \, dx+\frac {21875}{64} \int \frac {e^{-3+x} x^3}{e+e^{2 x}} \, dx-\frac {(9375 i) \int \text {Li}_3\left (-i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}+\frac {(9375 i) \int \text {Li}_3\left (i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}+\frac {(65625 i) \int x \text {Li}_3\left (-i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}-\frac {(65625 i) \int x \text {Li}_3\left (i e^{-\frac {1}{2}+x}\right ) \, dx}{64 e^{7/2}}\\ &=\text {Rest of rules removed due to large latex content} \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.18, size = 19, normalized size = 0.76 \begin {gather*} \frac {625 e^x x^4}{16 \left (e+e^{2 x}\right )^4} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(E^(3*x)*(2500*x^3 - 4375*x^4) + E^(1 + x)*(2500*x^3 + 625*x^4))/(16*E^5 + 16*E^(10*x) + 80*E^(4 + 2
*x) + 160*E^(3 + 4*x) + 160*E^(2 + 6*x) + 80*E^(1 + 8*x)),x]

[Out]

(625*E^x*x^4)/(16*(E + E^(2*x))^4)

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fricas [B]  time = 0.94, size = 44, normalized size = 1.76 \begin {gather*} \frac {625 \, x^{4} e^{\left (x + 8\right )}}{16 \, {\left (e^{12} + e^{\left (8 \, x + 8\right )} + 4 \, e^{\left (6 \, x + 9\right )} + 6 \, e^{\left (4 \, x + 10\right )} + 4 \, e^{\left (2 \, x + 11\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4375*x^4+2500*x^3)*exp(x)^3+(625*x^4+2500*x^3)*exp(1)*exp(x))/(16*exp(x)^10+80*exp(1)*exp(x)^8+16
0*exp(1)^2*exp(x)^6+160*exp(1)^3*exp(x)^4+80*exp(1)^4*exp(x)^2+16*exp(1)^5),x, algorithm="fricas")

[Out]

625/16*x^4*e^(x + 8)/(e^12 + e^(8*x + 8) + 4*e^(6*x + 9) + 6*e^(4*x + 10) + 4*e^(2*x + 11))

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giac [A]  time = 0.19, size = 40, normalized size = 1.60 \begin {gather*} \frac {625 \, x^{4} e^{x}}{8 \, {\left (e^{4} + e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x + 1\right )} + 6 \, e^{\left (4 \, x + 2\right )} + 4 \, e^{\left (2 \, x + 3\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4375*x^4+2500*x^3)*exp(x)^3+(625*x^4+2500*x^3)*exp(1)*exp(x))/(16*exp(x)^10+80*exp(1)*exp(x)^8+16
0*exp(1)^2*exp(x)^6+160*exp(1)^3*exp(x)^4+80*exp(1)^4*exp(x)^2+16*exp(1)^5),x, algorithm="giac")

[Out]

625/8*x^4*e^x/(e^4 + e^(8*x) + 4*e^(6*x + 1) + 6*e^(4*x + 2) + 4*e^(2*x + 3))

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maple [A]  time = 0.10, size = 17, normalized size = 0.68

method	result	size
risch	\(\frac {625 x^{4} {\mathrm e}^{x}}{16 \left ({\mathrm e}^{2 x}+{\mathrm e}\right )^{4}}\)	\(17\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((-4375*x^4+2500*x^3)*exp(x)^3+(625*x^4+2500*x^3)*exp(1)*exp(x))/(16*exp(x)^10+80*exp(1)*exp(x)^8+160*exp(
1)^2*exp(x)^6+160*exp(1)^3*exp(x)^4+80*exp(1)^4*exp(x)^2+16*exp(1)^5),x,method=_RETURNVERBOSE)

[Out]

625/16*x^4*exp(x)/(exp(2*x)+exp(1))^4

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maxima [A]  time = 0.40, size = 40, normalized size = 1.60 \begin {gather*} \frac {625 \, x^{4} e^{x}}{16 \, {\left (e^{4} + e^{\left (8 \, x\right )} + 4 \, e^{\left (6 \, x + 1\right )} + 6 \, e^{\left (4 \, x + 2\right )} + 4 \, e^{\left (2 \, x + 3\right )}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4375*x^4+2500*x^3)*exp(x)^3+(625*x^4+2500*x^3)*exp(1)*exp(x))/(16*exp(x)^10+80*exp(1)*exp(x)^8+16
0*exp(1)^2*exp(x)^6+160*exp(1)^3*exp(x)^4+80*exp(1)^4*exp(x)^2+16*exp(1)^5),x, algorithm="maxima")

[Out]

625/16*x^4*e^x/(e^4 + e^(8*x) + 4*e^(6*x + 1) + 6*e^(4*x + 2) + 4*e^(2*x + 3))

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mupad [B]  time = 8.28, size = 44, normalized size = 1.76 \begin {gather*} \frac {625\,x^4\,{\mathrm {e}}^x}{16\,\left ({\mathrm {e}}^{8\,x}+{\mathrm {e}}^4+4\,{\mathrm {e}}^{2\,x+3}+6\,{\mathrm {e}}^{4\,x+2}+4\,{\mathrm {e}}^{6\,x+1}\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp(3*x)*(2500*x^3 - 4375*x^4) + exp(1)*exp(x)*(2500*x^3 + 625*x^4))/(16*exp(10*x) + 16*exp(5) + 80*exp(2
*x)*exp(4) + 160*exp(4*x)*exp(3) + 160*exp(6*x)*exp(2) + 80*exp(8*x)*exp(1)),x)

[Out]

(625*x^4*exp(x))/(16*(exp(8*x) + exp(4) + 4*exp(2*x + 3) + 6*exp(4*x + 2) + 4*exp(6*x + 1)))

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sympy [A]  time = 0.19, size = 49, normalized size = 1.96 \begin {gather*} \frac {625 x^{4} e^{x}}{16 e^{8 x} + 64 e e^{6 x} + 96 e^{2} e^{4 x} + 64 e^{3} e^{2 x} + 16 e^{4}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((-4375*x**4+2500*x**3)*exp(x)**3+(625*x**4+2500*x**3)*exp(1)*exp(x))/(16*exp(x)**10+80*exp(1)*exp(x
)**8+160*exp(1)**2*exp(x)**6+160*exp(1)**3*exp(x)**4+80*exp(1)**4*exp(x)**2+16*exp(1)**5),x)

[Out]

625*x**4*exp(x)/(16*exp(8*x) + 64*E*exp(6*x) + 96*exp(2)*exp(4*x) + 64*exp(3)*exp(2*x) + 16*exp(4))

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