Optimal. Leaf size=31 \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]
[Out]
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Rubi [A] time = 0.313953, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 69, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.014 \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*x^(2*n))/(a*c)),x]
[Out]
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Rubi in Sympy [A] time = 75.3723, size = 26, normalized size = 0.84 \[ \frac{e x \left (a + b x^{n}\right )^{p + 1} \left (c + d x^{n}\right )^{p + 1}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+n+1)*x**n/a/c+b*d*e*(2*n*p+2*n+1)*x**(2*n)/a/c),x)
[Out]
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Mathematica [A] time = 0.292104, size = 31, normalized size = 1. \[ \frac{e x \left (a+b x^n\right )^{p+1} \left (c+d x^n\right )^{p+1}}{a c} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^n)^p*(c + d*x^n)^p*(e + ((b*c + a*d)*e*(1 + n + n*p)*x^n)/(a*c) + (b*d*e*(1 + 2*n + 2*n*p)*x^(2*n))/(a*c)),x]
[Out]
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Maple [A] time = 0.185, size = 52, normalized size = 1.7 \[{\frac{ \left ( a+b{x}^{n} \right ) ^{p} \left ( bd \left ({x}^{n} \right ) ^{2}+ad{x}^{n}+bc{x}^{n}+ac \right ) ex \left ( c+d{x}^{n} \right ) ^{p}}{ac}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((a+b*x^n)^p*(c+d*x^n)^p*(e+(a*d+b*c)*e*(n*p+n+1)*x^n/a/c+b*d*e*(2*n*p+2*n+1)*x^(2*n)/a/c),x)
[Out]
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Maxima [A] time = 1.93932, size = 80, normalized size = 2.58 \[ \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c e + a d e\right )} x x^{n}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + p \log \left (d x^{n} + c\right )\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.248956, size = 73, normalized size = 2.35 \[ \frac{{\left (b d e x x^{2 \, n} + a c e x +{\left (b c + a d\right )} e x x^{n}\right )}{\left (b x^{n} + a\right )}^{p}{\left (d x^{n} + c\right )}^{p}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,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((a+b*x**n)**p*(c+d*x**n)**p*(e+(a*d+b*c)*e*(n*p+n+1)*x**n/a/c+b*d*e*(2*n*p+2*n+1)*x**(2*n)/a/c),x)
[Out]
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GIAC/XCAS [A] time = 0.264626, size = 196, normalized size = 6.32 \[ \frac{b d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + 2 \, n{\rm ln}\left (x\right ) + 1\right )} + b c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a d x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + n{\rm ln}\left (x\right ) + 1\right )} + a c x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + p{\rm ln}\left (d e^{\left (n{\rm ln}\left (x\right )\right )} + c\right ) + 1\right )}}{a c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((2*n*p + 2*n + 1)*b*d*e*x^(2*n)/(a*c) + (b*c + a*d)*(n*p + n + 1)*e*x^n/(a*c) + e)*(b*x^n + a)^p*(d*x^n + c)^p,x, algorithm="giac")
[Out]