\(\int \frac {x^3 (a+b \log (c x^n))}{d+e x^r} \, dx\) [406]

Optimal result

Integrand size = 23, antiderivative size = 23 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\text {Int}\left (\frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r},x\right ) \]

[Out]

Unintegrable(x^3*(a+b*ln(c*x^n))/(d+e*x^r),x)

Rubi [N/A]

Not integrable

Time = 0.04 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00, number of steps used = 0, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \]

[In]

Int[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

Defer[Int][(x^3*(a + b*Log[c*x^n]))/(d + e*x^r), x]

Rubi steps \begin{align*} \text {integral}& = \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(87\) vs. \(2(26)=52\).

Time = 0.09 (sec) , antiderivative size = 87, normalized size of antiderivative = 3.78 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\frac {x^4 \left (-b n \, _3F_2\left (1,\frac {4}{r},\frac {4}{r};1+\frac {4}{r},1+\frac {4}{r};-\frac {e x^r}{d}\right )+4 \operatorname {Hypergeometric2F1}\left (1,\frac {4}{r},\frac {4+r}{r},-\frac {e x^r}{d}\right ) \left (a+b \log \left (c x^n\right )\right )\right )}{16 d} \]

[In]

Integrate[(x^3*(a + b*Log[c*x^n]))/(d + e*x^r),x]

[Out]

(x^4*(-(b*n*HypergeometricPFQ[{1, 4/r, 4/r}, {1 + 4/r, 1 + 4/r}, -((e*x^r)/d)]) + 4*Hypergeometric2F1[1, 4/r,
(4 + r)/r, -((e*x^r)/d)]*(a + b*Log[c*x^n])))/(16*d)

Maple [N/A]

Not integrable

Time = 0.06 (sec) , antiderivative size = 23, normalized size of antiderivative = 1.00

\[\int \frac {x^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )}{d +e \,x^{r}}d x\]

[In]

int(x^3*(a+b*ln(c*x^n))/(d+e*x^r),x)

[Out]

int(x^3*(a+b*ln(c*x^n))/(d+e*x^r),x)

Fricas [N/A]

Not integrable

Time = 0.26 (sec) , antiderivative size = 29, normalized size of antiderivative = 1.26 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{r} + d} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e*x^r),x, algorithm="fricas")

[Out]

integral((b*x^3*log(c*x^n) + a*x^3)/(e*x^r + d), x)

Sympy [N/A]

Not integrable

Time = 6.41 (sec) , antiderivative size = 20, normalized size of antiderivative = 0.87 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x^{3} \left (a + b \log {\left (c x^{n} \right )}\right )}{d + e x^{r}}\, dx \]

[In]

integrate(x**3*(a+b*ln(c*x**n))/(d+e*x**r),x)

[Out]

Integral(x**3*(a + b*log(c*x**n))/(d + e*x**r), x)

Maxima [N/A]

Not integrable

Time = 0.23 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{r} + d} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e*x^r),x, algorithm="maxima")

[Out]

integrate((b*log(c*x^n) + a)*x^3/(e*x^r + d), x)

Giac [N/A]

Not integrable

Time = 0.31 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int { \frac {{\left (b \log \left (c x^{n}\right ) + a\right )} x^{3}}{e x^{r} + d} \,d x } \]

[In]

integrate(x^3*(a+b*log(c*x^n))/(d+e*x^r),x, algorithm="giac")

[Out]

integrate((b*log(c*x^n) + a)*x^3/(e*x^r + d), x)

Mupad [N/A]

Not integrable

Time = 0.48 (sec) , antiderivative size = 25, normalized size of antiderivative = 1.09 \[ \int \frac {x^3 \left (a+b \log \left (c x^n\right )\right )}{d+e x^r} \, dx=\int \frac {x^3\,\left (a+b\,\ln \left (c\,x^n\right )\right )}{d+e\,x^r} \,d x \]

[In]

int((x^3*(a + b*log(c*x^n)))/(d + e*x^r),x)

[Out]

int((x^3*(a + b*log(c*x^n)))/(d + e*x^r), x)