Optimal. Leaf size=69 \[ \frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (2,1+p;2+p;1+\frac {b x^{n-q}}{a}\right )}{a^2 (1+p) (n-q)} \]
[Out]
________________________________________________________________________________________
Rubi [A]
time = 0.06, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {2057, 272, 67}
\begin {gather*} \frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (a x^q+b x^n\right )^p \, _2F_1\left (2,p+1;p+2;\frac {b x^{n-q}}{a}+1\right )}{a^2 (p+1) (n-q)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 67
Rule 272
Rule 2057
Rubi steps
\begin {align*} \int x^{-1-n-(-1+p) q} \left (b x^n+a x^q\right )^p \, dx &=\left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \int x^{-1-n-(-1+p) q+p q} \left (a+b x^{n-q}\right )^p \, dx\\ &=\frac {\left (x^{-p q} \left (a+b x^{n-q}\right )^{-p} \left (b x^n+a x^q\right )^p\right ) \text {Subst}\left (\int \frac {(a+b x)^p}{x^2} \, dx,x,x^{n-q}\right )}{n-q}\\ &=\frac {b x^{-p q} \left (a+b x^{n-q}\right ) \left (b x^n+a x^q\right )^p \, _2F_1\left (2,1+p;2+p;1+\frac {b x^{n-q}}{a}\right )}{a^2 (1+p) (n-q)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A]
time = 0.14, size = 82, normalized size = 1.19 \begin {gather*} \frac {x^{-n+q-p q} \left (b x^n+a x^q\right )^p \left (1+\frac {a x^{-n+q}}{b}\right )^{-p} \, _2F_1\left (1-p,-p;2-p;-\frac {a x^{-n+q}}{b}\right )}{(-1+p) (n-q)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
Maple [F]
time = 0.20, size = 0, normalized size = 0.00 \[\int x^{-1-n -\left (-1+p \right ) q} \left (b \,x^{n}+a \,x^{q}\right )^{p}\, dx\]
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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 = -0.01 \begin {gather*} \int \frac {{\left (b\,x^n+a\,x^q\right )}^p}{x^{n+q\,\left (p-1\right )+1}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________