Optimal. Leaf size=45 \[ \frac {e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]
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Rubi [A] time = 0.55, antiderivative size = 45, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 86, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.012, Rules used = {1848} \[ \frac {e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]
Antiderivative was successfully verified.
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Rule 1848
Rubi steps
\begin {align*} \int (h x)^m \left (a+b x^n\right )^p \left (c+d x^n\right )^p \left (e+\frac {(b c+a d) e (1+m+n+n p) x^n}{a c (1+m)}+\frac {b d e (1+m+2 n+2 n p) x^{2 n}}{a c (1+m)}\right ) \, dx &=\frac {e (h x)^{1+m} \left (a+b x^n\right )^{1+p} \left (c+d x^n\right )^{1+p}}{a c h (1+m)}\\ \end {align*}
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Mathematica [A] time = 0.89, size = 41, normalized size = 0.91 \[ \frac {e x (h x)^m \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c (m+1)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 88, normalized size = 1.96 \[ \frac {{\left (b d e x x^{2 \, n} e^{\left (m \log \relax (h) + m \log \relax (x)\right )} + a c e x e^{\left (m \log \relax (h) + m \log \relax (x)\right )} + {\left (b c + a d\right )} e x x^{n} e^{\left (m \log \relax (h) + m \log \relax (x)\right )}\right )} {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p}}{a c m + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.81, size = 155, normalized size = 3.44 \[ \frac {{\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b d x x^{2 \, n} e^{\left (m \log \relax (h) + m \log \relax (x) + 1\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} b c x x^{n} e^{\left (m \log \relax (h) + m \log \relax (x) + 1\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a d x x^{n} e^{\left (m \log \relax (h) + m \log \relax (x) + 1\right )} + {\left (b x^{n} + a\right )}^{p} {\left (d x^{n} + c\right )}^{p} a c x e^{\left (m \log \relax (h) + m \log \relax (x) + 1\right )}}{a c m + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [C] time = 0.50, size = 136, normalized size = 3.02 \[ \frac {\left (a d \,x^{n}+b c \,x^{n}+b d \,x^{2 n}+a c \right ) e x \left (b \,x^{n}+a \right )^{p} \left (d \,x^{n}+c \right )^{p} {\mathrm e}^{\frac {\left (-i \pi \,\mathrm {csgn}\left (i h \right ) \mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i h x \right )+i \pi \,\mathrm {csgn}\left (i h \right ) \mathrm {csgn}\left (i h x \right )^{2}+i \pi \,\mathrm {csgn}\left (i x \right ) \mathrm {csgn}\left (i h x \right )^{2}-i \pi \mathrm {csgn}\left (i h x \right )^{3}+2 \ln \relax (h )+2 \ln \relax (x )\right ) m}{2}}}{\left (m +1\right ) a c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 3.04, size = 92, normalized size = 2.04 \[ \frac {{\left (a c e h^{m} x x^{m} + b d e h^{m} x e^{\left (m \log \relax (x) + 2 \, n \log \relax (x)\right )} + {\left (b c e h^{m} + a d e h^{m}\right )} x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c {\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 5.64, size = 106, normalized size = 2.36 \[ {\left (c+d\,x^n\right )}^p\,\left (\frac {e\,x\,{\left (h\,x\right )}^m\,{\left (a+b\,x^n\right )}^p}{m+1}+\frac {e\,x\,x^n\,{\left (h\,x\right )}^m\,\left (a\,d+b\,c\right )\,{\left (a+b\,x^n\right )}^p}{a\,c\,\left (m+1\right )}+\frac {b\,d\,e\,x\,x^{2\,n}\,{\left (h\,x\right )}^m\,{\left (a+b\,x^n\right )}^p}{a\,c\,\left (m+1\right )}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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