Optimal. Leaf size=96 \[ -\frac {b^2 (1+n+n p) x \left (a-b x^{n/2}\right )^{1+p} \left (a+b x^{n/2}\right )^{1+p} \left (-\frac {a^2 d n (1+p)}{b^2 (1+n+n p)}+d x^n\right )^{-\frac {1+n+n p}{n}}}{a^4 d n (1+p)} \]
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Rubi [A]
time = 0.09, antiderivative size = 104, normalized size of antiderivative = 1.08, number of steps
used = 2, number of rules used = 2, integrand size = 76, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.026, Rules used = {533, 389}
\begin {gather*} -\frac {b^2 x (n p+n+1) \left (a^2-b^2 x^n\right ) \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (d x^n-\frac {a^2 d n (p+1)}{b^2 (n p+n+1)}\right )^{-\frac {n p+n+1}{n}}}{a^4 d n (p+1)} \end {gather*}
Antiderivative was successfully verified.
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Rule 389
Rule 533
Rubi steps
\begin {align*} \int \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx &=\left (\left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (a^2-b^2 x^n\right )^{-p}\right ) \int \left (a^2-b^2 x^n\right )^p \left (\frac {a^2 d (1+p)}{b^2 \left (1+\frac {-1-2 n-n p}{n}\right )}+d x^n\right )^{\frac {-1-2 n-n p}{n}} \, dx\\ &=-\frac {b^2 (1+n+n p) x \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (a^2-b^2 x^n\right ) \left (-\frac {a^2 d n (1+p)}{b^2 (1+n+n p)}+d x^n\right )^{-\frac {1+n+n p}{n}}}{a^4 d n (1+p)}\\ \end {align*}
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Mathematica [A]
time = 0.90, size = 103, normalized size = 1.07 \begin {gather*} -\frac {b^2 (1+n+n p) x \left (a-b x^{n/2}\right )^p \left (a+b x^{n/2}\right )^p \left (d \left (-\frac {a^2 n (1+p)}{b^2 (1+n+n p)}+x^n\right )\right )^{-\frac {1+n+n p}{n}} \left (a^2-b^2 x^n\right )}{a^4 d n (1+p)} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.39, size = 0, normalized size = 0.00 \[\int \left (a -b \,x^{\frac {n}{2}}\right )^{p} \left (a +b \,x^{\frac {n}{2}}\right )^{p} \left (\frac {a^{2} d \left (1+p \right )}{b^{2} \left (1+\frac {-n p -2 n -1}{n}\right )}+d \,x^{n}\right )^{\frac {-n p -2 n -1}{n}}\, dx\]
Verification of antiderivative is not currently implemented for this CAS.
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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.
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Fricas [A]
time = 3.27, size = 180, normalized size = 1.88 \begin {gather*} \frac {{\left ({\left (b^{4} n p + b^{4} n + b^{4}\right )} x x^{2 \, n} - {\left (2 \, a^{2} b^{2} n p + 2 \, a^{2} b^{2} n + a^{2} b^{2}\right )} x x^{n} + {\left (a^{4} n p + a^{4} n\right )} x\right )} {\left (b x^{\frac {1}{2} \, n} + a\right )}^{p} {\left (-b x^{\frac {1}{2} \, n} + a\right )}^{p}}{{\left (a^{4} n p + a^{4} n\right )} \left (-\frac {a^{2} d n p + a^{2} d n - {\left (b^{2} d n p + b^{2} d n + b^{2} d\right )} x^{n}}{b^{2} n p + b^{2} n + b^{2}}\right )^{\frac {n p + 2 \, n + 1}{n}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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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.
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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.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {{\left (a+b\,x^{n/2}\right )}^p\,{\left (a-b\,x^{n/2}\right )}^p}{{\left (d\,x^n-\frac {a^2\,d\,\left (p+1\right )}{b^2\,\left (\frac {2\,n+n\,p+1}{n}-1\right )}\right )}^{\frac {2\,n+n\,p+1}{n}}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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