3.1055 \(\int x^{-1+m} (a+b x^n)^{-1+p} (a m+b (m+n p) x^n) \, dx\)

Optimal. Leaf size=13 \[ x^m \left (a+b x^n\right )^p \]

[Out]

x^m*(a + b*x^n)^p

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Rubi [A]  time = 0.0138087, antiderivative size = 13, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 31, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.032, Rules used = {449} \[ x^m \left (a+b x^n\right )^p \]

Antiderivative was successfully verified.

[In]

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

[Out]

x^m*(a + b*x^n)^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]

Rubi steps

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

Mathematica [C]  time = 0.155663, size = 107, normalized size = 8.23 \[ \frac{x^m \left (a+b x^n\right )^p \left (\frac{b x^n}{a}+1\right )^{-p} \left (b x^n (m+n p) \, _2F_1\left (\frac{m+n}{n},1-p;\frac{m}{n}+2;-\frac{b x^n}{a}\right )+a (m+n) \, _2F_1\left (\frac{m}{n},1-p;\frac{m+n}{n};-\frac{b x^n}{a}\right )\right )}{a (m+n)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^m*(a + b*x^n)^p*(a*(m + n)*Hypergeometric2F1[m/n, 1 - p, (m + n)/n, -((b*x^n)/a)] + b*(m + n*p)*x^n*Hyperge
ometric2F1[(m + n)/n, 1 - p, 2 + m/n, -((b*x^n)/a)]))/(a*(m + n)*(1 + (b*x^n)/a)^p)

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Maple [F]  time = 0.075, size = 0, normalized size = 0. \begin{align*} \int{x}^{-1+m} \left ( a+b{x}^{n} \right ) ^{-1+p} \left ( am+b \left ( pn+m \right ){x}^{n} \right ) \, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.15387, size = 22, normalized size = 1.69 \begin{align*} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

e^(p*log(b*x^n + a) + m*log(x))

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Fricas [B]  time = 1.01659, size = 77, normalized size = 5.92 \begin{align*}{\left (b x x^{m - 1} x^{n} + a x x^{m - 1}\right )}{\left (b x^{n} + a\right )}^{p - 1} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.60004, size = 95, normalized size = 7.31 \begin{align*} b x x^{n} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right ) - \log \left (b x^{n} + a\right ) - \log \left (x\right )\right )} + a x e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right ) - \log \left (b x^{n} + a\right ) - \log \left (x\right )\right )} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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