3.1.63 \(\int (f x)^m \log (c (d+e x^n)^p) \, dx\) [63]

Optimal. Leaf size=87 \[ -\frac {e n p x^{1+n} (f x)^m \, _2F_1\left (1,\frac {1+m+n}{n};\frac {1+m+2 n}{n};-\frac {e x^n}{d}\right )}{d (1+m) (1+m+n)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)} \]

[Out]

-e*n*p*x^(1+n)*(f*x)^m*hypergeom([1, (1+m+n)/n],[(1+m+2*n)/n],-e*x^n/d)/d/(1+m)/(1+m+n)+(f*x)^(1+m)*ln(c*(d+e*
x^n)^p)/f/(1+m)

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Rubi [A]
time = 0.03, antiderivative size = 87, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 18, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {2505, 20, 371} \begin {gather*} \frac {(f x)^{m+1} \log \left (c \left (d+e x^n\right )^p\right )}{f (m+1)}-\frac {e n p x^{n+1} (f x)^m \, _2F_1\left (1,\frac {m+n+1}{n};\frac {m+2 n+1}{n};-\frac {e x^n}{d}\right )}{d (m+1) (m+n+1)} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*Log[c*(d + e*x^n)^p],x]

[Out]

-((e*n*p*x^(1 + n)*(f*x)^m*Hypergeometric2F1[1, (1 + m + n)/n, (1 + m + 2*n)/n, -((e*x^n)/d)])/(d*(1 + m)*(1 +
m + n))) + ((f*x)^(1 + m)*Log[c*(d + e*x^n)^p])/(f*(1 + m))

Rule 20

Int[(u_.)*((a_.)*(v_))^(m_)*((b_.)*(v_))^(n_), x_Symbol] :> Dist[b^IntPart[n]*((b*v)^FracPart[n]/(a^IntPart[n]
*(a*v)^FracPart[n])), Int[u*(a*v)^(m + n), x], x] /; FreeQ[{a, b, m, n}, x] &&  !IntegerQ[m] &&  !IntegerQ[n]
&&  !IntegerQ[m + n]

Rule 371

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^p*((c*x)^(m + 1)/(c*(m + 1)))*Hyperg
eometric2F1[-p, (m + 1)/n, (m + 1)/n + 1, (-b)*(x^n/a)], x] /; FreeQ[{a, b, c, m, n, p}, x] &&  !IGtQ[p, 0] &&
(ILtQ[p, 0] || GtQ[a, 0])

Rule 2505

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.)*(x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m +
1)*((a + b*Log[c*(d + e*x^n)^p])/(f*(m + 1))), x] - Dist[b*e*n*(p/(f*(m + 1))), Int[x^(n - 1)*((f*x)^(m + 1)/
(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && NeQ[m, -1]

Rubi steps

\begin {align*} \int (f x)^m \log \left (c \left (d+e x^n\right )^p\right ) \, dx &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {(e n p) \int \frac {x^{-1+n} (f x)^{1+m}}{d+e x^n} \, dx}{f (1+m)}\\ &=\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}-\frac {\left (e n p x^{-m} (f x)^m\right ) \int \frac {x^{m+n}}{d+e x^n} \, dx}{1+m}\\ &=-\frac {e n p x^{1+n} (f x)^m \, _2F_1\left (1,\frac {1+m+n}{n};\frac {1+m+2 n}{n};-\frac {e x^n}{d}\right )}{d (1+m) (1+m+n)}+\frac {(f x)^{1+m} \log \left (c \left (d+e x^n\right )^p\right )}{f (1+m)}\\ \end {align*}

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Mathematica [A]
time = 0.03, size = 77, normalized size = 0.89 \begin {gather*} \frac {x (f x)^m \left (-e n p x^n \, _2F_1\left (1,\frac {1+m+n}{n};\frac {1+m+2 n}{n};-\frac {e x^n}{d}\right )+d (1+m+n) \log \left (c \left (d+e x^n\right )^p\right )\right )}{d (1+m) (1+m+n)} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^m*Log[c*(d + e*x^n)^p],x]

[Out]

(x*(f*x)^m*(-(e*n*p*x^n*Hypergeometric2F1[1, (1 + m + n)/n, (1 + m + 2*n)/n, -((e*x^n)/d)]) + d*(1 + m + n)*Lo
g[c*(d + e*x^n)^p]))/(d*(1 + m)*(1 + m + n))

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Maple [F]
time = 0.06, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \ln \left (c \left (d +e \,x^{n}\right )^{p}\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*ln(c*(d+e*x^n)^p),x)

[Out]

int((f*x)^m*ln(c*(d+e*x^n)^p),x)

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*log(c*(d+e*x^n)^p),x, algorithm="maxima")

[Out]

d*f^m*n*p*integrate(x^m/(d*(m + 1) + (m + 1)*e^(n*log(x) + 1)), x) + (f^m*(m + 1)*x*x^m*log((d + e^(n*log(x) +
1))^p) - (f^m*n*p - f^m*(m + 1)*log(c))*x*x^m)/(m^2 + 2*m + 1)

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Fricas [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*log(c*(d+e*x^n)^p),x, algorithm="fricas")

[Out]

integral((f*x)^m*log((x^n*e + d)^p*c), x)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \left (f x\right )^{m} \log {\left (c \left (d + e x^{n}\right )^{p} \right )}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)**m*ln(c*(d+e*x**n)**p),x)

[Out]

Integral((f*x)**m*log(c*(d + e*x**n)**p), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*log(c*(d+e*x^n)^p),x, algorithm="giac")

[Out]

integrate((f*x)^m*log((x^n*e + d)^p*c), x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \ln \left (c\,{\left (d+e\,x^n\right )}^p\right )\,{\left (f\,x\right )}^m \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(log(c*(d + e*x^n)^p)*(f*x)^m,x)

[Out]

int(log(c*(d + e*x^n)^p)*(f*x)^m, x)

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