3.1024 \(\int (e x)^m (a+b x^n)^p (a (1+m)+b (1+m+n+n p) x^n) \, dx\)

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 = 0.0148312, 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, Rules used = {449} \[ \frac{(e x)^{m+1} \left (a+b x^n\right )^{p+1}}{e} \]

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

Rule 449

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*}

Mathematica [C]  time = 0.12954, size = 110, normalized size = 5. \[ x (e x)^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (\frac{b x^n (m+n p+n+1) \, _2F_1\left (\frac{m+n+1}{n},-p;\frac{m+2 n+1}{n};-\frac{b x^n}{a}\right )}{m+n+1}+a \, _2F_1\left (\frac{m+1}{n},-p;\frac{m+n+1}{n};-\frac{b x^n}{a}\right )\right ) \]

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)^p*(a*Hypergeometric2F1[(1 + m)/n, -p, (1 + m + n)/n, -((b*x^n)/a)] + (b*(1 + m + n + n*
p)*x^n*Hypergeometric2F1[(1 + m + n)/n, -p, (1 + m + 2*n)/n, -((b*x^n)/a)])/(1 + m + n)))/(1 + (b*x^n)/a)^p

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Maple [F]  time = 0.07, size = 0, normalized size = 0. \begin{align*} \int \left ( ex \right ) ^{m} \left ( a+b{x}^{n} \right ) ^{p} \left ( a \left ( 1+m \right ) +b \left ( pn+m+n+1 \right ){x}^{n} \right ) \, dx \end{align*}

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 = 1.19691, size = 49, normalized size = 2.23 \begin{align*}{\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} \end{align*}

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]

(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 = 1.03107, size = 107, normalized size = 4.86 \begin{align*}{\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} \end{align*}

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

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Sympy [B]  time = 9.47734, size = 39, normalized size = 1.77 \begin{align*} 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} \end{align*}

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 [A]  time = 1.09913, size = 51, normalized size = 2.32 \begin{align*}{\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{align*}

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