Optimal. Leaf size=95 \[ \frac{d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}-\frac{x^{-j (p+1)} (a d-b c (p+2)) \left (a x^j+b x^{j+n}\right )^{p+1}}{b^2 n (p+1) (p+2)} \]
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Rubi [A] time = 0.156137, antiderivative size = 95, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.062, Rules used = {2039, 2014} \[ \frac{d x^{n-j (p+1)} \left (a x^j+b x^{j+n}\right )^{p+1}}{b n (p+2)}-\frac{x^{-j (p+1)} (a d-b c (p+2)) \left (a x^j+b x^{j+n}\right )^{p+1}}{b^2 n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Rule 2039
Rule 2014
Rubi steps
\begin{align*} \int x^{-1+n-j p} \left (c+d x^n\right ) \left (a x^j+b x^{j+n}\right )^p \, dx &=\frac{d x^{n-j (1+p)} \left (a x^j+b x^{j+n}\right )^{1+p}}{b n (2+p)}-\left (-c+\frac{a d}{b (2+p)}\right ) \int x^{-1+n-j p} \left (a x^j+b x^{j+n}\right )^p \, dx\\ &=\frac{\left (c-\frac{a d}{b (2+p)}\right ) x^{-j (1+p)} \left (a x^j+b x^{j+n}\right )^{1+p}}{b n (1+p)}+\frac{d x^{n-j (1+p)} \left (a x^j+b x^{j+n}\right )^{1+p}}{b n (2+p)}\\ \end{align*}
Mathematica [A] time = 0.0685855, size = 63, normalized size = 0.66 \[ \frac{x^{-j p} \left (a+b x^n\right ) \left (x^j \left (a+b x^n\right )\right )^p \left (-a d+b c (p+2)+b d (p+1) x^n\right )}{b^2 n (p+1) (p+2)} \]
Antiderivative was successfully verified.
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Maple [F] time = 0.547, size = 0, normalized size = 0. \begin{align*} \int{x}^{-jp+n-1} \left ( c+d{x}^{n} \right ) \left ( a{x}^{j}+b{x}^{j+n} \right ) ^{p}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A] time = 1.47314, size = 151, normalized size = 1.59 \begin{align*} \frac{{\left (b x^{n} + a\right )} c e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{b n{\left (p + 1\right )}} + \frac{{\left (b^{2}{\left (p + 1\right )} x^{2 \, n} + a b p x^{n} - a^{2}\right )} d e^{\left (-j p \log \left (x\right ) + p \log \left (b x^{n} + a\right ) + p \log \left (x^{j}\right )\right )}}{{\left (p^{2} + 3 \, p + 2\right )} b^{2} n} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A] time = 2.4282, size = 300, normalized size = 3.16 \begin{align*} \frac{{\left ({\left (b^{2} d p + b^{2} d\right )} x x^{-j p + n - 1} x^{2 \, n} +{\left (2 \, b^{2} c +{\left (b^{2} c + a b d\right )} p\right )} x x^{-j p + n - 1} x^{n} +{\left (a b c p + 2 \, a b c - a^{2} d\right )} x x^{-j p + n - 1}\right )} \left (\frac{{\left (b x^{n} + a\right )} x^{j + n}}{x^{n}}\right )^{p}}{{\left (b^{2} n p^{2} + 3 \, b^{2} n p + 2 \, b^{2} n\right )} x^{n}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int{\left (d x^{n} + c\right )}{\left (b x^{j + n} + a x^{j}\right )}^{p} x^{-j p + n - 1}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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