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

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

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Rubi [A]  time = 0.01, 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 = {449} \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 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*}

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Mathematica [C]  time = 0.17, size = 110, normalized size = 5.00 \begin {gather*} 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 ) \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)^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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IntegrateAlgebraic [F]  time = 0.26, size = 0, normalized size = 0.00 \begin {gather*} \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 \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [A]  time = 0.41, size = 40, normalized size = 1.82 \begin {gather*} {\left (b x x^{n} e^{\left (m \log \relax (e) + m \log \relax (x)\right )} + a x e^{\left (m \log \relax (e) + m \log \relax (x)\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(e) + m*log(x)) + a*x*e^(m*log(e) + m*log(x)))*(b*x^n + a)^p

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giac [A]  time = 0.29, 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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maple [F]  time = 0.67, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (n p +m +n +1\right ) b \,x^{n}+\left (m +1\right ) a \right ) \left (e x \right )^{m} \left (b \,x^{n}+a \right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 0.95, size = 36, normalized size = 1.64 \begin {gather*} {\left (a e^{m} x x^{m} + b e^{m} x e^{\left (m \log \relax (x) + n \log \relax (x)\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]

(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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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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sympy [B]  time = 8.43, size = 39, normalized size = 1.77 \begin {gather*} 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 {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*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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