3.855 \(\int (e x)^m \left (a+b x^n\right )^p \left (a (1+m)+b (1+m+n+n p) x^n\right ) \, dx\)

Optimal. Leaf size=22 \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \]

[Out]

((e*x)^(1 + m)*(a + b*x^n)^(1 + p))/e

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Rubi [A]  time = 0.0601766, antiderivative size = 22, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.03 \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \]

Antiderivative was successfully verified.

[In]  Int[(e*x)^m*(a + b*x^n)^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n),x]

[Out]

((e*x)^(1 + m)*(a + b*x^n)^(1 + p))/e

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Rubi in Sympy [A]  time = 8.50469, size = 17, normalized size = 0.77 \[ \frac{\left (e x\right )^{m + 1} \left (a + b x^{n}\right )^{p + 1}}{e} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((e*x)**m*(a+b*x**n)**p*(a*(1+m)+b*(n*p+m+n+1)*x**n),x)

[Out]

(e*x)**(m + 1)*(a + b*x**n)**(p + 1)/e

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Mathematica [A]  time = 0.0939755, size = 18, normalized size = 0.82 \[ x (e x)^m \left (a+b x^n\right )^{p+1} \]

Antiderivative was successfully verified.

[In]  Integrate[(e*x)^m*(a + b*x^n)^p*(a*(1 + m) + b*(1 + m + n + n*p)*x^n),x]

[Out]

x*(e*x)^m*(a + b*x^n)^(1 + p)

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Maple [F]  time = 0.103, size = 0, normalized size = 0. \[ \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( a \left ( 1+m \right ) +b \left ( np+m+n+1 \right ){x}^{n} \right ) \, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((e*x)^m*(a+b*x^n)^p*(a*(1+m)+b*(n*p+m+n+1)*x^n),x)

[Out]

int((e*x)^m*(a+b*x^n)^p*(a*(1+m)+b*(n*p+m+n+1)*x^n),x)

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Maxima [A]  time = 15.9588, size = 49, normalized size = 2.23 \[{\left (a e^{m} x x^{m} + b e^{m} x e^{\left (m \log \left (x\right ) + n \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*p + m + n + 1)*b*x^n + a*(m + 1))*(b*x^n + a)^p*(e*x)^m,x, algorithm="maxima")

[Out]

(a*e^m*x*x^m + b*e^m*x*e^(m*log(x) + n*log(x)))*(b*x^n + a)^p

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Fricas [A]  time = 0.235327, size = 54, normalized size = 2.45 \[{\left (b x x^{n} e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )} + a x e^{\left (m \log \left (e\right ) + m \log \left (x\right )\right )}\right )}{\left (b x^{n} + a\right )}^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*p + m + n + 1)*b*x^n + a*(m + 1))*(b*x^n + a)^p*(e*x)^m,x, algorithm="fricas")

[Out]

(b*x*x^n*e^(m*log(e) + m*log(x)) + a*x*e^(m*log(e) + m*log(x)))*(b*x^n + a)^p

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Sympy [A]  time = 15.7014, size = 39, normalized size = 1.77 \[ a e^{m} x x^{m} \left (a + b x^{n}\right )^{p} + b e^{m} x x^{m} x^{n} \left (a + b x^{n}\right )^{p} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((e*x)**m*(a+b*x**n)**p*(a*(1+m)+b*(n*p+m+n+1)*x**n),x)

[Out]

a*e**m*x*x**m*(a + b*x**n)**p + b*e**m*x*x**m*x**n*(a + b*x**n)**p

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GIAC/XCAS [A]  time = 0.220194, size = 66, normalized size = 3. \[ b x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) + n{\rm ln}\left (x\right ) + m\right )} + a x e^{\left (p{\rm ln}\left (b e^{\left (n{\rm ln}\left (x\right )\right )} + a\right ) + m{\rm ln}\left (x\right ) + m\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(((n*p + m + n + 1)*b*x^n + a*(m + 1))*(b*x^n + a)^p*(e*x)^m,x, algorithm="giac")

[Out]

b*x*e^(p*ln(b*e^(n*ln(x)) + a) + m*ln(x) + n*ln(x) + m) + a*x*e^(p*ln(b*e^(n*ln(
x)) + a) + m*ln(x) + m)