3.72.24 \(\int \frac {2^{5 x/4} (100+(125 x-75 x^2+25 e^3 x^2) \log (2))}{800-960 x+288 x^2+32 e^6 x^2+e^3 (320 x-192 x^2)} \, dx\)

Optimal. Leaf size=23 \[ \frac {5\ 2^{-3+\frac {5 x}{4}}}{-3+e^3+\frac {5}{x}} \]

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Rubi [B]  time = 0.09, antiderivative size = 57, normalized size of antiderivative = 2.48, number of steps used = 2, number of rules used = 2, integrand size = 63, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.032, Rules used = {6, 2288} \begin {gather*} \frac {5\ 2^{\frac {5 x}{4}-3} \left (e^3 x^2-3 x^2+5 x\right )}{\left (9+e^6\right ) x^2+2 e^3 \left (5 x-3 x^2\right )-30 x+25} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(2^((5*x)/4)*(100 + (125*x - 75*x^2 + 25*E^3*x^2)*Log[2]))/(800 - 960*x + 288*x^2 + 32*E^6*x^2 + E^3*(320*
x - 192*x^2)),x]

[Out]

(5*2^(-3 + (5*x)/4)*(5*x - 3*x^2 + E^3*x^2))/(25 - 30*x + (9 + E^6)*x^2 + 2*E^3*(5*x - 3*x^2))

Rule 6

Int[(u_.)*((w_.) + (a_.)*(v_) + (b_.)*(v_))^(p_.), x_Symbol] :> Int[u*((a + b)*v + w)^p, x] /; FreeQ[{a, b}, x
] &&  !FreeQ[v, x]

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 \frac {2^{5 x/4} \left (100+\left (125 x-75 x^2+25 e^3 x^2\right ) \log (2)\right )}{800-960 x+\left (288+32 e^6\right ) x^2+e^3 \left (320 x-192 x^2\right )} \, dx\\ &=\frac {5\ 2^{-3+\frac {5 x}{4}} \left (5 x-3 x^2+e^3 x^2\right )}{25-30 x+\left (9+e^6\right ) x^2+2 e^3 \left (5 x-3 x^2\right )}\\ \end {aligned} \end {gather*}

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Mathematica [A]  time = 0.05, size = 39, normalized size = 1.70 \begin {gather*} \frac {5\ 2^{-3+\frac {5 x}{4}} x \left (\left (-3+e^3\right ) x \log (2)+\log (32)\right )}{\left (5+\left (-3+e^3\right ) x\right )^2 \log (2)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(2^((5*x)/4)*(100 + (125*x - 75*x^2 + 25*E^3*x^2)*Log[2]))/(800 - 960*x + 288*x^2 + 32*E^6*x^2 + E^3
*(320*x - 192*x^2)),x]

[Out]

(5*2^(-3 + (5*x)/4)*x*((-3 + E^3)*x*Log[2] + Log[32]))/((5 + (-3 + E^3)*x)^2*Log[2])

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fricas [A]  time = 0.58, size = 19, normalized size = 0.83 \begin {gather*} \frac {5 \cdot 2^{\frac {5}{4} \, x} x}{8 \, {\left (x e^{3} - 3 \, x + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2*exp(3)-75*x^2+125*x)*log(2)+100)*exp(5/4*x*log(2))/(32*x^2*exp(3)^2+(-192*x^2+320*x)*exp(3)
+288*x^2-960*x+800),x, algorithm="fricas")

[Out]

5/8*2^(5/4*x)*x/(x*e^3 - 3*x + 5)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {25 \, {\left ({\left (x^{2} e^{3} - 3 \, x^{2} + 5 \, x\right )} \log \relax (2) + 4\right )} 2^{\frac {5}{4} \, x}}{32 \, {\left (x^{2} e^{6} + 9 \, x^{2} - 2 \, {\left (3 \, x^{2} - 5 \, x\right )} e^{3} - 30 \, x + 25\right )}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2*exp(3)-75*x^2+125*x)*log(2)+100)*exp(5/4*x*log(2))/(32*x^2*exp(3)^2+(-192*x^2+320*x)*exp(3)
+288*x^2-960*x+800),x, algorithm="giac")

[Out]

integrate(25/32*((x^2*e^3 - 3*x^2 + 5*x)*log(2) + 4)*2^(5/4*x)/(x^2*e^6 + 9*x^2 - 2*(3*x^2 - 5*x)*e^3 - 30*x +
25), x)

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maple [A]  time = 0.19, size = 20, normalized size = 0.87

method	result	size
risch	\(\frac {5 x 2^{\frac {5 x}{4}}}{8 \left (x \,{\mathrm e}^{3}-3 x +5\right )}\)	\(20\)
gosper	\(\frac {5 x \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{8 \left (x \,{\mathrm e}^{3}-3 x +5\right )}\)	\(21\)
norman	\(\frac {5 x \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{8 \left (x \,{\mathrm e}^{3}-3 x +5\right )}\)	\(21\)
derivativedivides	\(\frac {-\frac {125 \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}} \ln \relax (2)^{2}}{8 \left ({\mathrm e}^{3}-3\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}-\frac {125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{32 \left ({\mathrm e}^{3}-3\right )^{2}}+\frac {3125 \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}} \ln \relax (2)^{3}}{32 \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}+\frac {125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{32 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {3125 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{128 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {375 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{32 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {30 \ln \relax (2) {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144}+\frac {9375 \ln \relax (2)^{3} {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \left ({\mathrm e}^{3}-3\right ) \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}-\frac {750 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {1125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {9375 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{8 \left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {20 \,{\mathrm e}^{3} \ln \relax (2) {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \,{\mathrm e}^{6}-192 \,{\mathrm e}^{3}+288}-\frac {3125 \,{\mathrm e}^{3} \ln \relax (2)^{3} {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \left ({\mathrm e}^{3}-3\right ) \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}+\frac {125 \,{\mathrm e}^{6} \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}-\frac {3125 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{8 \left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}}{\ln \relax (2)}\)	\(723\)
default	\(\frac {-\frac {125 \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}} \ln \relax (2)^{2}}{8 \left ({\mathrm e}^{3}-3\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}-\frac {125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{32 \left ({\mathrm e}^{3}-3\right )^{2}}+\frac {3125 \,{\mathrm e}^{\frac {5 x \ln \relax (2)}{4}} \ln \relax (2)^{3}}{32 \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}+\frac {125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{32 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {3125 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{128 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {375 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{32 \left (-{\mathrm e}^{6}+6 \,{\mathrm e}^{3}-9\right ) \left ({\mathrm e}^{3}-3\right )}-\frac {30 \ln \relax (2) {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144}+\frac {9375 \ln \relax (2)^{3} {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \left ({\mathrm e}^{3}-3\right ) \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}-\frac {750 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {1125 \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {9375 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{8 \left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}+\frac {20 \,{\mathrm e}^{3} \ln \relax (2) {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \,{\mathrm e}^{6}-192 \,{\mathrm e}^{3}+288}-\frac {3125 \,{\mathrm e}^{3} \ln \relax (2)^{3} {\mathrm e}^{\frac {5 x \ln \relax (2)}{4}}}{32 \left ({\mathrm e}^{3}-3\right ) \left ({\mathrm e}^{6}-6 \,{\mathrm e}^{3}+9\right ) \left (5 \,{\mathrm e}^{3} \ln \relax (2) x +25 \ln \relax (2)-15 x \ln \relax (2)\right )}+\frac {125 \,{\mathrm e}^{6} \ln \relax (2)^{2} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right )}{\left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}-\frac {3125 \ln \relax (2)^{3} {\mathrm e}^{-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}} \expIntegralEi \left (1, -\frac {5 x \ln \relax (2)}{4}-\frac {25 \ln \relax (2)}{4 \left ({\mathrm e}^{3}-3\right )}\right ) {\mathrm e}^{3}}{8 \left ({\mathrm e}^{3}-3\right )^{2} \left (16 \,{\mathrm e}^{6}-96 \,{\mathrm e}^{3}+144\right )}}{\ln \relax (2)}\)	\(723\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((25*x^2*exp(3)-75*x^2+125*x)*ln(2)+100)*exp(5/4*x*ln(2))/(32*x^2*exp(3)^2+(-192*x^2+320*x)*exp(3)+288*x^2
-960*x+800),x,method=_RETURNVERBOSE)

[Out]

5/8*x*2^(5/4*x)/(x*exp(3)-3*x+5)

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maxima [A]  time = 0.49, size = 18, normalized size = 0.78 \begin {gather*} \frac {5 \cdot 2^{\frac {5}{4} \, x} x}{8 \, {\left (x {\left (e^{3} - 3\right )} + 5\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x^2*exp(3)-75*x^2+125*x)*log(2)+100)*exp(5/4*x*log(2))/(32*x^2*exp(3)^2+(-192*x^2+320*x)*exp(3)
+288*x^2-960*x+800),x, algorithm="maxima")

[Out]

5/8*2^(5/4*x)*x/(x*(e^3 - 3) + 5)

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mupad [B]  time = 0.33, size = 20, normalized size = 0.87 \begin {gather*} \frac {20\,2^{\frac {5\,x}{4}}\,x}{32\,x\,{\mathrm {e}}^3-96\,x+160} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((exp((5*x*log(2))/4)*(log(2)*(125*x + 25*x^2*exp(3) - 75*x^2) + 100))/(exp(3)*(320*x - 192*x^2) - 960*x +
32*x^2*exp(6) + 288*x^2 + 800),x)

[Out]

(20*2^((5*x)/4)*x)/(32*x*exp(3) - 96*x + 160)

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sympy [A]  time = 0.21, size = 24, normalized size = 1.04 \begin {gather*} \frac {5 x e^{\frac {5 x \log {\relax (2 )}}{4}}}{- 24 x + 8 x e^{3} + 40} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(((25*x**2*exp(3)-75*x**2+125*x)*ln(2)+100)*exp(5/4*x*ln(2))/(32*x**2*exp(3)**2+(-192*x**2+320*x)*exp
(3)+288*x**2-960*x+800),x)

[Out]

5*x*exp(5*x*log(2)/4)/(-24*x + 8*x*exp(3) + 40)

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