3.11.55 \(\int x^{-1+m} (a+b x^n)^{-1+p} (a m+b (m+n p) x^n) \, dx\) [1055]

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 = 0.01, antiderivative size = 13, normalized size of antiderivative = 1.00, 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 = {460} \begin {gather*} x^m \left (a+b x^n\right )^p \end {gather*}

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 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 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*}

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Mathematica [A]
time = 0.14, size = 13, normalized size = 1.00 \begin {gather*} x^m \left (a+b x^n\right )^p \end {gather*}

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

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Maple [F]
time = 0.08, size = 0, normalized size = 0.00 \[\int x^{-1+m} \left (a +b \,x^{n}\right )^{-1+p} \left (a m +b \left (n p +m \right ) x^{n}\right )\, dx\]

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 = 0.42, size = 16, normalized size = 1.23 \begin {gather*} e^{\left (p \log \left (b x^{n} + a\right ) + m \log \left (x\right )\right )} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 32 vs. \(2 (13) = 26\).
time = 1.38, size = 32, normalized size = 2.46 \begin {gather*} {\left (b x x^{m - 1} x^{n} + a x x^{m - 1}\right )} {\left (b x^{n} + a\right )}^{p - 1} \end {gather*}

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(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \end {gather*}

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]

Exception raised: SystemError >> excessive stack use: stack is 3005 deep

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 70 vs. \(2 (13) = 26\).
time = 0.57, size = 70, normalized size = 5.38 \begin {gather*} 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 {gather*}

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))

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Mupad [B]
time = 4.81, size = 25, normalized size = 1.92 \begin {gather*} \left (a\,x^m+b\,x^{m+n}\right )\,{\left (a+b\,x^n\right )}^{p-1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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