Optimal. Leaf size=193 \[ \frac {e^{-\frac {A (1+m)}{B n}} (a+b x) (g (a+b x))^m (i (c+d x))^{-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right ) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \left (-\frac {(1+m) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )^{-p}}{(b c-a d) i^2 (1+m) (c+d x)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.31, antiderivative size = 193, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 50, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.080, Rules used = {2573, 2563,
2347, 2212} \begin {gather*} \frac {(a+b x) e^{-\frac {A (m+1)}{B n}} (g (a+b x))^m (i (c+d x))^{-m} \left (e (a+b x)^n (c+d x)^{-n}\right )^{-\frac {m+1}{n}} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p \left (-\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2212
Rule 2347
Rule 2563
Rule 2573
Rubi steps
\begin {align*} \int (226 c+226 d x)^{-2-m} (a g+b g x)^m \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx &=\int (226 c+226 d x)^{-2-m} (a g+b g x)^m \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx\\ \end {align*}
________________________________________________________________________________________
Mathematica [F]
time = 0.30, size = 0, normalized size = 0.00 \begin {gather*} \int (a g+b g x)^m (c i+d i x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 1.33, size = 0, normalized size = 0.00 \[\int \left (b g x +a g \right )^{m} \left (d i x +c i \right )^{-2-m} \left (A +B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )\right )^{p}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: RuntimeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a\,g+b\,g\,x\right )}^m\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{{\left (c\,i+d\,i\,x\right )}^{m+2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________