Optimal. Leaf size=45 \[ \frac{e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]
[Out]
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Rubi [A] time = 0.839767, antiderivative size = 45, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 86, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.012 \[ \frac{e (h x)^{m+1} \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c h (m+1)} \]
Antiderivative was successfully verified.
[In] Int[(h*x)^m*(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + m + n + n*p)*x^n)/(a*c*(1 + m)) + (b*d*e*(1 + m + 2*n + 2*n*p)*x^(2*n))/(a*c*(1 + m))),x]
[Out]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((h*x)**m*(a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+m+n+1)*x**n/a/c/(1+m)+b*d*e*(2*n*p+m+2*n+1)*x**(2*n)/a/c/(1+m)),x)
[Out]
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Mathematica [A] time = 0.423018, 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.
[In] Integrate[(h*x)^m*(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + m + n + n*p)*x^n)/(a*c*(1 + m)) + (b*d*e*(1 + m + 2*n + 2*n*p)*x^(2*n))/(a*c*(1 + m))),x]
[Out]
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Maple [C] time = 0.605, size = 136, normalized size = 3. \[{\frac{ \left ( a+b{x}^{n} \right ) ^{p} \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) ex \left ( c+d{x}^{n} \right ) ^{p}}{ac \left ( 1+m \right ) }{{\rm e}^{{\frac{m \left ( -i \left ({\it csgn} \left ( ihx \right ) \right ) ^{3}\pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ih \right ) \pi +i \left ({\it csgn} \left ( ihx \right ) \right ) ^{2}{\it csgn} \left ( ix \right ) \pi -i{\it csgn} \left ( ihx \right ){\it csgn} \left ( ih \right ){\it csgn} \left ( ix \right ) \pi +2\,\ln \left ( h \right ) +2\,\ln \left ( x \right ) \right ) }{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((h*x)^m*(a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+m+n+1)*x^n/a/c/(1+m)+b*d*e*(2*n*p+m+2*n+1)*x^(2*n)/a/c/(1+m)),x)
[Out]
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Maxima [A] time = 2.11185, size = 124, normalized size = 2.76 \[ \frac{{\left (a c e h^{m} x x^{m} + b d e h^{m} x e^{\left (m \log \left (x\right ) + 2 \, n \log \left (x\right )\right )} +{\left (b c e h^{m} + a d e h^{m}\right )} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c{\left (m + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + m + 2*n + 1)*b*d*e*x^(2*n)/(a*c*(m + 1)) + e + (b*c + a*d)*(n*p + m + n + 1)*e*x^n/(a*c*(m + 1)))*(b*x^n + a)^p*(d*x^n + c)^p*(h*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.254125, size = 119, normalized size = 2.64 \[ \frac{{\left (b d e x x^{2 \, n} e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )} + a c e x e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )} +{\left (b c + a d\right )} e x x^{n} e^{\left (m \log \left (h\right ) + m \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{a c m + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + m + 2*n + 1)*b*d*e*x^(2*n)/(a*c*(m + 1)) + e + (b*c + a*d)*(n*p + m + n + 1)*e*x^n/(a*c*(m + 1)))*(b*x^n + a)^p*(d*x^n + c)^p*(h*x)^m,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((h*x)**m*(a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+m+n+1)*x**n/a/c/(1+m)+b*d*e*(2*n*p+m+2*n+1)*x**(2*n)/a/c/(1+m)),x)
[Out]
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GIAC/XCAS [A] time = 0.274291, size = 244, normalized size = 5.42 \[ \frac{b d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + 2 \, n{\rm ln}\left (x\right ) + 1\right )} + b c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + m{\rm ln}\left (h\right ) + m{\rm ln}\left (x\right ) + 1\right )}}{a c m + a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + m + 2*n + 1)*b*d*e*x^(2*n)/(a*c*(m + 1)) + e + (b*c + a*d)*(n*p + m + n + 1)*e*x^n/(a*c*(m + 1)))*(b*x^n + a)^p*(d*x^n + c)^p*(h*x)^m,x, algorithm="giac")
[Out]