3.3.39 \(\int (e x)^q (a+b \log (c (d x^m)^n)) \, dx\) [239]

Optimal. Leaf size=51 \[ -\frac {b m n (e x)^{1+q}}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)} \]

[Out]

-b*m*n*(e*x)^(1+q)/e/(1+q)^2+(e*x)^(1+q)*(a+b*ln(c*(d*x^m)^n))/e/(1+q)

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Rubi [A]
time = 0.03, antiderivative size = 51, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {2341, 2495} \begin {gather*} \frac {(e x)^{q+1} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (q+1)}-\frac {b m n (e x)^{q+1}}{e (q+1)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(e*x)^q*(a + b*Log[c*(d*x^m)^n]),x]

[Out]

-((b*m*n*(e*x)^(1 + q))/(e*(1 + q)^2)) + ((e*x)^(1 + q)*(a + b*Log[c*(d*x^m)^n]))/(e*(1 + q))

Rule 2341

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))*((d_.)*(x_))^(m_.), x_Symbol] :> Simp[(d*x)^(m + 1)*((a + b*Log[c*x^
n])/(d*(m + 1))), x] - Simp[b*n*((d*x)^(m + 1)/(d*(m + 1)^2)), x] /; FreeQ[{a, b, c, d, m, n}, x] && NeQ[m, -1
]

Rule 2495

Int[((a_.) + Log[(c_.)*((d_.)*((e_.) + (f_.)*(x_))^(m_.))^(n_)]*(b_.))^(p_.)*(u_.), x_Symbol] :> Subst[Int[u*(
a + b*Log[c*d^n*(e + f*x)^(m*n)])^p, x], c*d^n*(e + f*x)^(m*n), c*(d*(e + f*x)^m)^n] /; FreeQ[{a, b, c, d, e,
f, m, n, p}, x] &&  !IntegerQ[n] &&  !(EqQ[d, 1] && EqQ[m, 1]) && IntegralFreeQ[IntHide[u*(a + b*Log[c*d^n*(e
+ f*x)^(m*n)])^p, x]]

Rubi steps

\begin {align*} \int (e x)^q \left (a+b \log \left (c \left (d x^m\right )^n\right )\right ) \, dx &=\text {Subst}\left (\int (e x)^q \left (a+b \log \left (c d^n x^{m n}\right )\right ) \, dx,c d^n x^{m n},c \left (d x^m\right )^n\right )\\ &=-\frac {b m n (e x)^{1+q}}{e (1+q)^2}+\frac {(e x)^{1+q} \left (a+b \log \left (c \left (d x^m\right )^n\right )\right )}{e (1+q)}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 37, normalized size = 0.73 \begin {gather*} \frac {x (e x)^q \left (a-b m n+a q+b (1+q) \log \left (c \left (d x^m\right )^n\right )\right )}{(1+q)^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(e*x)^q*(a + b*Log[c*(d*x^m)^n]),x]

[Out]

(x*(e*x)^q*(a - b*m*n + a*q + b*(1 + q)*Log[c*(d*x^m)^n]))/(1 + q)^2

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Maple [F]
time = 0.03, size = 0, normalized size = 0.00 \[\int \left (e x \right )^{q} \left (a +b \ln \left (c \left (d \,x^{m}\right )^{n}\right )\right )\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^q*(a+b*ln(c*(d*x^m)^n)),x)

[Out]

int((e*x)^q*(a+b*ln(c*(d*x^m)^n)),x)

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Maxima [A]
time = 0.28, size = 63, normalized size = 1.24 \begin {gather*} -\frac {b m n x e^{\left (q \log \left (x\right ) + q\right )}}{{\left (q + 1\right )}^{2}} + \frac {\left (x e\right )^{q + 1} b e^{\left (-1\right )} \log \left (\left (d x^{m}\right )^{n} c\right )}{q + 1} + \frac {\left (x e\right )^{q + 1} a e^{\left (-1\right )}}{q + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n)),x, algorithm="maxima")

[Out]

-b*m*n*x*e^(q*log(x) + q)/(q + 1)^2 + (x*e)^(q + 1)*b*e^(-1)*log((d*x^m)^n*c)/(q + 1) + (x*e)^(q + 1)*a*e^(-1)
/(q + 1)

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Fricas [A]
time = 0.37, size = 69, normalized size = 1.35 \begin {gather*} \frac {{\left ({\left (b q + b\right )} x \log \left (c\right ) + {\left (b n q + b n\right )} x \log \left (d\right ) + {\left (b m n q + b m n\right )} x \log \left (x\right ) - {\left (b m n - a q - a\right )} x\right )} e^{\left (q \log \left (x\right ) + q\right )}}{q^{2} + 2 \, q + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n)),x, algorithm="fricas")

[Out]

((b*q + b)*x*log(c) + (b*n*q + b*n)*x*log(d) + (b*m*n*q + b*m*n)*x*log(x) - (b*m*n - a*q - a)*x)*e^(q*log(x) +
q)/(q^2 + 2*q + 1)

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Sympy [A]
time = 4.27, size = 110, normalized size = 2.16 \begin {gather*} a \left (\begin {cases} 0^{q} x & \text {for}\: e = 0 \\\frac {\begin {cases} \frac {\left (e x\right )^{q + 1}}{q + 1} & \text {for}\: q \neq -1 \\\log {\left (e x \right )} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) - b m n \left (\begin {cases} 0^{q} x & \text {for}\: \left (e = 0 \wedge q \neq -1\right ) \vee e = 0 \\\frac {\begin {cases} \frac {e x \left (e x\right )^{q}}{q + 1} & \text {for}\: q \neq -1 \\\log {\left (x \right )} & \text {otherwise} \end {cases}}{e q + e} & \text {for}\: q > -\infty \wedge q < \infty \wedge q \neq -1 \\\frac {\log {\left (e x \right )}^{2}}{2 e} & \text {otherwise} \end {cases}\right ) + b \left (\begin {cases} 0^{q} x & \text {for}\: e = 0 \\\frac {\begin {cases} \frac {\left (e x\right )^{q + 1}}{q + 1} & \text {for}\: q \neq -1 \\\log {\left (e x \right )} & \text {otherwise} \end {cases}}{e} & \text {otherwise} \end {cases}\right ) \log {\left (c \left (d x^{m}\right )^{n} \right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)**q*(a+b*ln(c*(d*x**m)**n)),x)

[Out]

a*Piecewise((0**q*x, Eq(e, 0)), (Piecewise(((e*x)**(q + 1)/(q + 1), Ne(q, -1)), (log(e*x), True))/e, True)) -
b*m*n*Piecewise((0**q*x, Eq(e, 0) | (Eq(e, 0) & Ne(q, -1))), (Piecewise((e*x*(e*x)**q/(q + 1), Ne(q, -1)), (lo
g(x), True))/(e*q + e), (q > -oo) & (q < oo) & Ne(q, -1)), (log(e*x)**2/(2*e), True)) + b*Piecewise((0**q*x, E
q(e, 0)), (Piecewise(((e*x)**(q + 1)/(q + 1), Ne(q, -1)), (log(e*x), True))/e, True))*log(c*(d*x**m)**n)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 111 vs. \(2 (51) = 102\).
time = 4.96, size = 111, normalized size = 2.18 \begin {gather*} \frac {b m n q x x^{q} e^{q} \log \left (x\right )}{q^{2} + 2 \, q + 1} + \frac {b m n x x^{q} e^{q} \log \left (x\right )}{q^{2} + 2 \, q + 1} - \frac {b m n x x^{q} e^{q}}{q^{2} + 2 \, q + 1} + \frac {b n x x^{q} e^{q} \log \left (d\right )}{q + 1} + \frac {b x x^{q} e^{q} \log \left (c\right )}{q + 1} + \frac {a x x^{q} e^{q}}{q + 1} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x)^q*(a+b*log(c*(d*x^m)^n)),x, algorithm="giac")

[Out]

b*m*n*q*x*x^q*e^q*log(x)/(q^2 + 2*q + 1) + b*m*n*x*x^q*e^q*log(x)/(q^2 + 2*q + 1) - b*m*n*x*x^q*e^q/(q^2 + 2*q
+ 1) + b*n*x*x^q*e^q*log(d)/(q + 1) + b*x*x^q*e^q*log(c)/(q + 1) + a*x*x^q*e^q/(q + 1)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int {\left (e\,x\right )}^q\,\left (a+b\,\ln \left (c\,{\left (d\,x^m\right )}^n\right )\right ) \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x)^q*(a + b*log(c*(d*x^m)^n)),x)

[Out]

int((e*x)^q*(a + b*log(c*(d*x^m)^n)), x)

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