3.11.24 \(\int (e x)^m (a+b x^n)^p (a (1+m)+b (1+m+n+n p) x^n) \, dx\) [1024]

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

[Out]

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

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Rubi [A]
time = 0.02, antiderivative size = 22, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 33, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.030, Rules used = {460} \begin {gather*} \frac {(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \end {gather*}

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

Rule 460

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[c*(e*x)^(m +
1)*((a + b*x^n)^(p + 1)/(a*e*(m + 1))), x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && NeQ[b*c - a*d, 0] && EqQ[
a*d*(m + 1) - b*c*(m + n*(p + 1) + 1), 0] && NeQ[m, -1]

Rubi steps

\begin {align*} \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 &=\frac {(e x)^{1+m} \left (a+b x^n\right )^{1+p}}{e}\\ \end {align*}

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Mathematica [A]
time = 0.16, size = 18, normalized size = 0.82 \begin {gather*} x (e x)^m \left (a+b x^n\right )^{1+p} \end {gather*}

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 = 0.09, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{m} \left (a +b \,x^{n}\right )^{p} \left (a \left (1+m \right )+b \left (n p +m +n +1\right ) x^{n}\right )\, dx\]

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 = 0.35, size = 35, normalized size = 1.59 \begin {gather*} {\left (b x e^{\left (m \log \left (x\right ) + n \log \left (x\right ) + m\right )} + a x e^{\left (m \log \left (x\right ) + m\right )}\right )} {\left (b x^{n} + a\right )}^{p} \end {gather*}

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, algorithm="maxima")

[Out]

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

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Fricas [A]
time = 1.69, size = 34, normalized size = 1.55 \begin {gather*} {\left (b x x^{n} e^{\left (m \log \left (x\right ) + m\right )} + a x e^{\left (m \log \left (x\right ) + m\right )}\right )} {\left (b x^{n} + a\right )}^{p} \end {gather*}

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, algorithm="fricas")

[Out]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 36 vs. \(2 (17) = 34\).
time = 2.26, size = 36, normalized size = 1.64 \begin {gather*} a x \left (e x\right )^{m} \left (a + b x^{n}\right )^{p} + b x x^{n} \left (e x\right )^{m} \left (a + b x^{n}\right )^{p} \end {gather*}

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

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Giac [A]
time = 0.94, size = 38, normalized size = 1.73 \begin {gather*} {\left (b x^{n} + a\right )}^{p} b x x^{m} x^{n} e^{m} + {\left (b x^{n} + a\right )}^{p} a x x^{m} e^{m} \end {gather*}

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, algorithm="giac")

[Out]

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

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Mupad [B]
time = 4.94, size = 31, normalized size = 1.41 \begin {gather*} \left (a\,x\,{\left (e\,x\right )}^m+b\,x^{n+1}\,{\left (e\,x\right )}^m\right )\,{\left (a+b\,x^n\right )}^p \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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