3.239 \(\int \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t \, dx\)

Optimal. Leaf size=36 \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]

[Out]

(x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t)/(1 + m*r + n*s + p*t)

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Rubi [A]  time = 0.0323343, antiderivative size = 36, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 2, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091 \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x]

[Out]

(x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t)/(1 + m*r + n*s + p*t)

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Rubi in Sympy [A]  time = 20.9521, size = 60, normalized size = 1.67 \[ \frac{x^{- m r} x^{- n s} x^{- p t} x^{m r + n s + p t + 1} \left (a x^{m}\right )^{r} \left (b x^{n}\right )^{s} \left (c x^{p}\right )^{t}}{m r + n s + p t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**m)**r*(b*x**n)**s*(c*x**p)**t,x)

[Out]

x**(-m*r)*x**(-n*s)*x**(-p*t)*x**(m*r + n*s + p*t + 1)*(a*x**m)**r*(b*x**n)**s*(
c*x**p)**t/(m*r + n*s + p*t + 1)

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Mathematica [A]  time = 0.016587, size = 36, normalized size = 1. \[ \frac{x \left (a x^m\right )^r \left (b x^n\right )^s \left (c x^p\right )^t}{m r+n s+p t+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x]

[Out]

(x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t)/(1 + m*r + n*s + p*t)

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Maple [A]  time = 0.003, size = 37, normalized size = 1. \[{\frac{x \left ( a{x}^{m} \right ) ^{r} \left ( b{x}^{n} \right ) ^{s} \left ( c{x}^{p} \right ) ^{t}}{mr+ns+pt+1}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x)

[Out]

x*(a*x^m)^r*(b*x^n)^s*(c*x^p)^t/(m*r+n*s+p*t+1)

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Maxima [A]  time = 0.756761, size = 59, normalized size = 1.64 \[ \frac{a^{r} b^{s} c^{t} x e^{\left (r \log \left (x^{m}\right ) + s \log \left (x^{n}\right ) + t \log \left (x^{p}\right )\right )}}{m r + n s + p t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="maxima")

[Out]

a^r*b^s*c^t*x*e^(r*log(x^m) + s*log(x^n) + t*log(x^p))/(m*r + n*s + p*t + 1)

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Fricas [A]  time = 0.299202, size = 59, normalized size = 1.64 \[ \frac{x e^{\left (m r \log \left (x\right ) + n s \log \left (x\right ) + p t \log \left (x\right ) + r \log \left (a\right ) + s \log \left (b\right ) + t \log \left (c\right )\right )}}{m r + n s + p t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="fricas")

[Out]

x*e^(m*r*log(x) + n*s*log(x) + p*t*log(x) + r*log(a) + s*log(b) + t*log(c))/(m*r
+ n*s + p*t + 1)

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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*x**m)**r*(b*x**n)**s*(c*x**p)**t,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.265416, size = 59, normalized size = 1.64 \[ \frac{x e^{\left (m r{\rm ln}\left (x\right ) + n s{\rm ln}\left (x\right ) + p t{\rm ln}\left (x\right ) + r{\rm ln}\left (a\right ) + s{\rm ln}\left (b\right ) + t{\rm ln}\left (c\right )\right )}}{m r + n s + p t + 1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((a*x^m)^r*(b*x^n)^s*(c*x^p)^t,x, algorithm="giac")

[Out]

x*e^(m*r*ln(x) + n*s*ln(x) + p*t*ln(x) + r*ln(a) + s*ln(b) + t*ln(c))/(m*r + n*s
+ p*t + 1)