Optimal. Leaf size=61 \[ \frac {x^{\frac {(1-n) (p+1)}{p}} \left (a x^{-\frac {1-n}{p}}+b x^{n-\frac {1-n}{p}}\right )^{p+1}}{b n (p+1)} \]
________________________________________________________________________________________
Rubi [A] time = 0.02, antiderivative size = 61, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {1979, 2000} \begin {gather*} \frac {x^{\frac {(1-n) (p+1)}{p}} \left (a x^{-\frac {1-n}{p}}+b x^{n-\frac {1-n}{p}}\right )^{p+1}}{b n (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
Rule 1979
Rule 2000
Rubi steps
\begin {align*} \int \left (x^{\frac {-1+n}{p}} \left (a+b x^n\right )\right )^p \, dx &=\int \left (b x^{n+\frac {-1+n}{p}}+a x^{\frac {-1+n}{p}}\right )^p \, dx\\ &=\frac {x^{\frac {(1-n) (1+p)}{p}} \left (b x^{n-\frac {1-n}{p}}+a x^{-\frac {1-n}{p}}\right )^{1+p}}{b n (1+p)}\\ \end {align*}
________________________________________________________________________________________
Mathematica [A] time = 0.03, size = 45, normalized size = 0.74 \begin {gather*} \frac {x^{1-n} \left (a+b x^n\right ) \left (x^{\frac {n-1}{p}} \left (a+b x^n\right )\right )^p}{b n (p+1)} \end {gather*}
Antiderivative was successfully verified.
[In]
[Out]
________________________________________________________________________________________
IntegrateAlgebraic [F] time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^{\frac {-1+n}{p}} \left (a+b x^n\right )\right )^p \, dx \end {gather*}
Verification is not applicable to the result.
[In]
[Out]
________________________________________________________________________________________
fricas [A] time = 0.43, size = 54, normalized size = 0.89 \begin {gather*} \frac {{\left (b x x^{n} + a x\right )} {\left (b x^{n} x^{\frac {n - 1}{p}} + a x^{\frac {n - 1}{p}}\right )}^{p}}{{\left (b n p + b n\right )} x^{n}} \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*} \int \left ({\left (b x^{n} + a\right )} x^{\frac {n - 1}{p}}\right )^{p}\,{d x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [F] time = 0.73, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (b \,x^{n}+a \right ) x^{\frac {n -1}{p}}\right )^{p}\, dx \end {gather*}
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*} \int \left ({\left (b x^{n} + a\right )} x^{\frac {n - 1}{p}}\right )^{p}\,{d x} \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 {\left (x^{\frac {n-1}{p}}\,\left (a+b\,x^n\right )\right )}^p \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (x^{\frac {n - 1}{p}} \left (a + b x^{n}\right )\right )^{p}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________