\(\int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx\) [4000]

Optimal result

Integrand size = 37, antiderivative size = 15 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {1}{25} e^x \log ^2(-7+4 x) \]

[Out]

1/25*exp(x)*ln(4*x-7)^2

Rubi [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(38\) vs. \(2(15)=30\).

Time = 0.09 (sec) , antiderivative size = 38, normalized size of antiderivative = 2.53, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.054, Rules used = {6873, 2326} \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {e^x \log (4 x-7) (7 \log (4 x-7)-4 x \log (4 x-7))}{25 (7-4 x)} \]

[In]

Int[(8*E^x*Log[-7 + 4*x] + E^x*(-7 + 4*x)*Log[-7 + 4*x]^2)/(-175 + 100*x),x]

[Out]

(E^x*Log[-7 + 4*x]*(7*Log[-7 + 4*x] - 4*x*Log[-7 + 4*x]))/(25*(7 - 4*x))

Rule 2326

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]

Rule 6873

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

Rubi steps \begin{align*} \text {integral}& = \int \frac {e^x \log (-7+4 x) (-8+7 \log (-7+4 x)-4 x \log (-7+4 x))}{175-100 x} \, dx \\ & = \frac {e^x \log (-7+4 x) (7 \log (-7+4 x)-4 x \log (-7+4 x))}{25 (7-4 x)} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 15, normalized size of antiderivative = 1.00 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {1}{25} e^x \log ^2(-7+4 x) \]

[In]

Integrate[(8*E^x*Log[-7 + 4*x] + E^x*(-7 + 4*x)*Log[-7 + 4*x]^2)/(-175 + 100*x),x]

[Out]

(E^x*Log[-7 + 4*x]^2)/25

Maple [A] (verified)

Time = 1.32 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.87

method	result	size
norman	\(\frac {{\mathrm e}^{x} \ln \left (4 x -7\right )^{2}}{25}\)	\(13\)
risch	\(\frac {{\mathrm e}^{x} \ln \left (4 x -7\right )^{2}}{25}\)	\(13\)
parallelrisch	\(\frac {{\mathrm e}^{x} \ln \left (4 x -7\right )^{2}}{25}\)	\(13\)

[In]

int(((4*x-7)*exp(x)*ln(4*x-7)^2+8*exp(x)*ln(4*x-7))/(100*x-175),x,method=_RETURNVERBOSE)

[Out]

1/25*exp(x)*ln(4*x-7)^2

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {1}{25} \, e^{x} \log \left (4 \, x - 7\right )^{2} \]

[In]

integrate(((4*x-7)*exp(x)*log(4*x-7)^2+8*exp(x)*log(4*x-7))/(100*x-175),x, algorithm="fricas")

[Out]

1/25*e^x*log(4*x - 7)^2

Sympy [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {e^{x} \log {\left (4 x - 7 \right )}^{2}}{25} \]

[In]

integrate(((4*x-7)*exp(x)*ln(4*x-7)**2+8*exp(x)*ln(4*x-7))/(100*x-175),x)

[Out]

exp(x)*log(4*x - 7)**2/25

Maxima [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {1}{25} \, e^{x} \log \left (4 \, x - 7\right )^{2} \]

[In]

integrate(((4*x-7)*exp(x)*log(4*x-7)^2+8*exp(x)*log(4*x-7))/(100*x-175),x, algorithm="maxima")

[Out]

1/25*e^x*log(4*x - 7)^2

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {1}{25} \, e^{x} \log \left (4 \, x - 7\right )^{2} \]

[In]

integrate(((4*x-7)*exp(x)*log(4*x-7)^2+8*exp(x)*log(4*x-7))/(100*x-175),x, algorithm="giac")

[Out]

1/25*e^x*log(4*x - 7)^2

Mupad [B] (verification not implemented)

Time = 9.43 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.80 \[ \int \frac {8 e^x \log (-7+4 x)+e^x (-7+4 x) \log ^2(-7+4 x)}{-175+100 x} \, dx=\frac {{\mathrm {e}}^x\,{\ln \left (4\,x-7\right )}^2}{25} \]

[In]

int((8*exp(x)*log(4*x - 7) + exp(x)*log(4*x - 7)^2*(4*x - 7))/(100*x - 175),x)

[Out]

(exp(x)*log(4*x - 7)^2)/25