Optimal. Leaf size=40 \[ -\frac {x^{-((n+p) (1+q))} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.05, antiderivative size = 40, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2005, 2039}
\begin {gather*} -\frac {x^{-((q+1) (n+p))} \left (a x^n+b x^{n+p}\right )^{q+1}}{a p (q+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 2005
Rule 2039
Rubi steps
\begin {align*} \int x^{-1-n q-p (1+q)} \left (x^n \left (a+b x^p\right )\right )^q \, dx &=\int x^{-1-n q-p (1+q)} \left (a x^n+b x^{n+p}\right )^q \, dx\\ &=-\frac {x^{-(n+p) (1+q)} \left (a x^n+b x^{n+p}\right )^{1+q}}{a p (1+q)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.07, size = 38, normalized size = 0.95 \begin {gather*} -\frac {x^{-((n+p) (1+q))} \left (x^n \left (a+b x^p\right )\right )^{1+q}}{a p (1+q)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.18, size = 0, normalized size = 0.00 \[\int x^{-1-q n -p \left (q +1\right )} \left (x^{n} \left (a +b \,x^{p}\right )\right )^{q}\, dx\]
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 [A]
time = 3.47, size = 64, normalized size = 1.60 \begin {gather*} -\frac {{\left (b x x^{-{\left (n + p\right )} q - p - 1} x^{p} + a x x^{-{\left (n + p\right )} q - p - 1}\right )} {\left (b x^{n} x^{p} + a x^{n}\right )}^{q}}{a p q + a p} \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]
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]
________________________________________________________________________________________
Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int \frac {{\left (x^n\,\left (a+b\,x^p\right )\right )}^q}{x^{n\,q+p\,\left (q+1\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________