3.3.28 \(\int x^m (a+b x^n)^p (a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}) \, dx\)

Optimal. Leaf size=18 \[ x^{m+q+1} \left (a+b x^n\right )^{p+1} \]

________________________________________________________________________________________

Rubi [A]  time = 0.04, antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 39, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.051, Rules used = {1584, 449} \begin {gather*} x^{m+q+1} \left (a+b x^n\right )^{p+1} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

x^(1 + m + q)*(a + b*x^n)^(1 + p)

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]

Rule 1584

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps

\begin {align*} \int x^m \left (a+b x^n\right )^p \left (a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}\right ) \, dx &=\int x^{m+q} \left (a+b x^n\right )^p \left (a (1+m+q)+b (1+m+n (1+p)+q) x^n\right ) \, dx\\ &=x^{1+m+q} \left (a+b x^n\right )^{1+p}\\ \end {align*}

________________________________________________________________________________________

Mathematica [C]  time = 0.22, size = 116, normalized size = 6.44 \begin {gather*} x^{m+q+1} \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+q+1) \, _2F_1\left (-p,\frac {m+n+q+1}{n};\frac {m+2 n+q+1}{n};-\frac {b x^n}{a}\right )}{m+n+q+1}+a \, _2F_1\left (-p,\frac {m+q+1}{n};\frac {m+n+q+1}{n};-\frac {b x^n}{a}\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

________________________________________________________________________________________

IntegrateAlgebraic [F]  time = 0.36, size = 0, normalized size = 0.00 \begin {gather*} \int x^m \left (a+b x^n\right )^p \left (a (1+m+q) x^q+b (1+m+n (1+p)+q) x^{n+q}\right ) \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

________________________________________________________________________________________

fricas [B]  time = 0.43, size = 43, normalized size = 2.39 \begin {gather*} {\left (b x x^{m} x^{n + q} + a x x^{m} x^{q}\right )} \left (\frac {b x^{n + q} + a x^{q}}{x^{q}}\right )^{p} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^n)^p*(a*(1+m+q)*x^q+b*(1+m+n*(1+p)+q)*x^(n+q)),x, algorithm="fricas")

[Out]

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

________________________________________________________________________________________

giac [B]  time = 0.37, size = 48, normalized size = 2.67 \begin {gather*} {\left (b x^{n} + a\right )}^{p} b x x^{n} e^{\left (m \log \relax (x) + q \log \relax (x)\right )} + {\left (b x^{n} + a\right )}^{p} a x e^{\left (m \log \relax (x) + q \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^n)^p*(a*(1+m+q)*x^q+b*(1+m+n*(1+p)+q)*x^(n+q)),x, algorithm="giac")

[Out]

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

________________________________________________________________________________________

maple [F]  time = 1.10, size = 0, normalized size = 0.00 \begin {gather*} \int \left (\left (m +q +1\right ) a \,x^{q}+\left (m +\left (p +1\right ) n +q +1\right ) b \,x^{n +q}\right ) x^{m} \left (b \,x^{n}+a \right )^{p}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

maxima [B]  time = 2.15, size = 37, normalized size = 2.06 \begin {gather*} {\left (a x x^{m} + b x e^{\left (m \log \relax (x) + n \log \relax (x)\right )}\right )} e^{\left (p \log \left (b x^{n} + a\right ) + q \log \relax (x)\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^m*(a+b*x^n)^p*(a*(1+m+q)*x^q+b*(1+m+n*(1+p)+q)*x^(n+q)),x, algorithm="maxima")

[Out]

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

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.06 \begin {gather*} \int x^m\,\left (a\,x^q\,\left (m+q+1\right )+b\,x^{n+q}\,\left (m+q+n\,\left (p+1\right )+1\right )\right )\,{\left (a+b\,x^n\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**m*(a+b*x**n)**p*(a*(1+m+q)*x**q+b*(1+m+n*(1+p)+q)*x**(n+q)),x)

[Out]

Timed out

________________________________________________________________________________________