3.31 \(\int \frac{(d x^m+e \log ^{-1+q}(c x^n)) (a x^m+b \log ^q(c x^n))^3}{x} \, dx\)

Optimal. Leaf size=331 \[ -\frac{a^2 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{b^2 x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} (a e m-b d n q) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{3 a b 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^3 x^{4 m} (a e m-b d n q)}{4 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]

[Out]

-(a^3*(a*e*m - b*d*n*q)*x^(4*m))/(4*b*m*n*q) - (b^2*(a*e*m - b*d*n*q)*x^m*Gamma[1 + 3*q, -((m*Log[c*x^n])/n)]*
Log[c*x^n]^(3*q))/(m*n*q*(c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)) - (3*2^(-1 - 2*q)*a*b*(a*e*m - b*d*n*q)*x^
(2*m)*Gamma[1 + 2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2*q))/(m*n*q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*
q)) - (a^2*(a*e*m - b*d*n*q)*x^(3*m)*Gamma[1 + q, (-3*m*Log[c*x^n])/n]*Log[c*x^n]^q)/(3^q*m*n*q*(c*x^n)^((3*m)
/n)*(-((m*Log[c*x^n])/n))^q) + (e*(a*x^m + b*Log[c*x^n]^q)^4)/(4*b*n*q)

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Rubi [A]  time = 0.503742, antiderivative size = 331, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 4, integrand size = 40, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1, Rules used = {2545, 6742, 2310, 2181} \[ -\frac{a^2 3^{-q} x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q} (a e m-b d n q) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{b^2 x^m \left (c x^n\right )^{-\frac{m}{n}} \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} (a e m-b d n q) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{3 a b 2^{-2 q-1} x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q} (a e m-b d n q) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )}{m n q}-\frac{a^3 x^{4 m} (a e m-b d n q)}{4 b m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q} \]

Antiderivative was successfully verified.

[In]

Int[((d*x^m + e*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q)^3)/x,x]

[Out]

-(a^3*(a*e*m - b*d*n*q)*x^(4*m))/(4*b*m*n*q) - (b^2*(a*e*m - b*d*n*q)*x^m*Gamma[1 + 3*q, -((m*Log[c*x^n])/n)]*
Log[c*x^n]^(3*q))/(m*n*q*(c*x^n)^(m/n)*(-((m*Log[c*x^n])/n))^(3*q)) - (3*2^(-1 - 2*q)*a*b*(a*e*m - b*d*n*q)*x^
(2*m)*Gamma[1 + 2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2*q))/(m*n*q*(c*x^n)^((2*m)/n)*(-((m*Log[c*x^n])/n))^(2*
q)) - (a^2*(a*e*m - b*d*n*q)*x^(3*m)*Gamma[1 + q, (-3*m*Log[c*x^n])/n]*Log[c*x^n]^q)/(3^q*m*n*q*(c*x^n)^((3*m)
/n)*(-((m*Log[c*x^n])/n))^q) + (e*(a*x^m + b*Log[c*x^n]^q)^4)/(4*b*n*q)

Rule 2545

Int[((Log[(c_.)*(x_)^(n_.)]^(q_)*(b_.) + (a_.)*(x_)^(m_.))^(p_.)*(Log[(c_.)*(x_)^(n_.)]^(r_.)*(e_.) + (d_.)*(x
_)^(m_.)))/(x_), x_Symbol] :> Simp[(e*(a*x^m + b*Log[c*x^n]^q)^(p + 1))/(b*n*q*(p + 1)), x] - Dist[(a*e*m - b*
d*n*q)/(b*n*q), Int[x^(m - 1)*(a*x^m + b*Log[c*x^n]^q)^p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p, q, r}, x] &
& EqQ[r, q - 1] && NeQ[p, -1] && NeQ[a*e*m - b*d*n*q, 0]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rule 2310

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_)*((d_.)*(x_))^(m_.), x_Symbol] :> Dist[(d*x)^(m + 1)/(d*n*(c*x^n
)^((m + 1)/n)), Subst[Int[E^(((m + 1)*x)/n)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2181

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))*((c_.) + (d_.)*(x_))^(m_), x_Symbol] :> -Simp[(F^(g*(e - (c*f)/d))*(c +
d*x)^FracPart[m]*Gamma[m + 1, (-((f*g*Log[F])/d))*(c + d*x)])/(d*(-((f*g*Log[F])/d))^(IntPart[m] + 1)*(-((f*g*
Log[F]*(c + d*x))/d))^FracPart[m]), x] /; FreeQ[{F, c, d, e, f, g, m}, x] &&  !IntegerQ[m]

Rubi steps

\begin{align*} \int \frac{\left (d x^m+e \log ^{-1+q}\left (c x^n\right )\right ) \left (a x^m+b \log ^q\left (c x^n\right )\right )^3}{x} \, dx &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int x^{-1+m} \left (a x^m+b \log ^q\left (c x^n\right )\right )^3 \, dx\\ &=\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\left (-d+\frac{a e m}{b n q}\right ) \int \left (a^3 x^{-1+4 m}+3 a^2 b x^{-1+3 m} \log ^q\left (c x^n\right )+3 a b^2 x^{-1+2 m} \log ^{2 q}\left (c x^n\right )+b^3 x^{-1+m} \log ^{3 q}\left (c x^n\right )\right ) \, dx\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\left (3 a^2 b \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+3 m} \log ^q\left (c x^n\right ) \, dx-\left (b^3 \left (-d+\frac{a e m}{b n q}\right )\right ) \int x^{-1+m} \log ^{3 q}\left (c x^n\right ) \, dx-\frac{(3 a b (a e m-b d n q)) \int x^{-1+2 m} \log ^{2 q}\left (c x^n\right ) \, dx}{n q}\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}-\frac{\left (3 a^2 b \left (-d+\frac{a e m}{b n q}\right ) x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{3 m x}{n}} x^q \, dx,x,\log \left (c x^n\right )\right )}{n}-\frac{\left (3 a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{2 m x}{n}} x^{2 q} \, dx,x,\log \left (c x^n\right )\right )}{n^2 q}-\frac{\left (b^3 \left (-d+\frac{a e m}{b n q}\right ) x^m \left (c x^n\right )^{-\frac{m}{n}}\right ) \operatorname{Subst}\left (\int e^{\frac{m x}{n}} x^{3 q} \, dx,x,\log \left (c x^n\right )\right )}{n}\\ &=\frac{a^3 \left (d-\frac{a e m}{b n q}\right ) x^{4 m}}{4 m}-\frac{b^2 (a e m-b d n q) x^m \left (c x^n\right )^{-\frac{m}{n}} \Gamma \left (1+3 q,-\frac{m \log \left (c x^n\right )}{n}\right ) \log ^{3 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q}}{m n q}-\frac{3\ 2^{-1-2 q} a b (a e m-b d n q) x^{2 m} \left (c x^n\right )^{-\frac{2 m}{n}} \Gamma \left (1+2 q,-\frac{2 m \log \left (c x^n\right )}{n}\right ) \log ^{2 q}\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-2 q}}{m n q}-\frac{3^{-q} a^2 (a e m-b d n q) x^{3 m} \left (c x^n\right )^{-\frac{3 m}{n}} \Gamma \left (1+q,-\frac{3 m \log \left (c x^n\right )}{n}\right ) \log ^q\left (c x^n\right ) \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-q}}{m n q}+\frac{e \left (a x^m+b \log ^q\left (c x^n\right )\right )^4}{4 b n q}\\ \end{align*}

Mathematica [A]  time = 1.55066, size = 445, normalized size = 1.34 \[ \frac{3^{-q} 4^{-q-1} \left (c x^n\right )^{-\frac{3 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^{-3 q} \left (\left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (4^q \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (4 a^2 b d n q x^{3 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q+1,-\frac{3 m \log \left (c x^n\right )}{n}\right )-4 a^3 e m q x^{3 m} \log ^q\left (c x^n\right ) \text{Gamma}\left (q,-\frac{3 m \log \left (c x^n\right )}{n}\right )+3^q \left (c x^n\right )^{\frac{3 m}{n}} \left (-\frac{m \log \left (c x^n\right )}{n}\right )^q \left (a^3 d n q x^{4 m}+b^3 e m \log ^{4 q}\left (c x^n\right )\right )\right )-4 a^2 b e m 3^{q+1} q x^{2 m} \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q,-\frac{2 m \log \left (c x^n\right )}{n}\right )+2 a b^2 d n 3^{q+1} q x^{2 m} \left (c x^n\right )^{m/n} \log ^{2 q}\left (c x^n\right ) \text{Gamma}\left (2 q+1,-\frac{2 m \log \left (c x^n\right )}{n}\right )\right )+a b^2 e m \left (-12^{q+1}\right ) q x^m \left (c x^n\right )^{\frac{2 m}{n}} \log ^{3 q}\left (c x^n\right ) \text{Gamma}\left (3 q,-\frac{m \log \left (c x^n\right )}{n}\right )+b^3 d n 3^q 4^{q+1} q x^m \left (c x^n\right )^{\frac{2 m}{n}} \log ^{3 q}\left (c x^n\right ) \text{Gamma}\left (3 q+1,-\frac{m \log \left (c x^n\right )}{n}\right )\right )}{m n q} \]

Antiderivative was successfully verified.

[In]

Integrate[((d*x^m + e*Log[c*x^n]^(-1 + q))*(a*x^m + b*Log[c*x^n]^q)^3)/x,x]

[Out]

(4^(-1 - q)*(-(12^(1 + q)*a*b^2*e*m*q*x^m*(c*x^n)^((2*m)/n)*Gamma[3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q))
+ 3^q*4^(1 + q)*b^3*d*n*q*x^m*(c*x^n)^((2*m)/n)*Gamma[1 + 3*q, -((m*Log[c*x^n])/n)]*Log[c*x^n]^(3*q) + (-((m*L
og[c*x^n])/n))^q*(-4*3^(1 + q)*a^2*b*e*m*q*x^(2*m)*(c*x^n)^(m/n)*Gamma[2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2
*q) + 2*3^(1 + q)*a*b^2*d*n*q*x^(2*m)*(c*x^n)^(m/n)*Gamma[1 + 2*q, (-2*m*Log[c*x^n])/n]*Log[c*x^n]^(2*q) + 4^q
*(-((m*Log[c*x^n])/n))^q*(-4*a^3*e*m*q*x^(3*m)*Gamma[q, (-3*m*Log[c*x^n])/n]*Log[c*x^n]^q + 4*a^2*b*d*n*q*x^(3
*m)*Gamma[1 + q, (-3*m*Log[c*x^n])/n]*Log[c*x^n]^q + 3^q*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^q*(a^3*d*n*q*
x^(4*m) + b^3*e*m*Log[c*x^n]^(4*q))))))/(3^q*m*n*q*(c*x^n)^((3*m)/n)*(-((m*Log[c*x^n])/n))^(3*q))

________________________________________________________________________________________

Maple [F]  time = 7.961, size = 0, normalized size = 0. \begin{align*} \int{\frac{ \left ( d{x}^{m}+e \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{-1+q} \right ) \left ( a{x}^{m}+b \left ( \ln \left ( c{x}^{n} \right ) \right ) ^{q} \right ) ^{3}}{x}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d*x^m+e*ln(c*x^n)^(-1+q))*(a*x^m+b*ln(c*x^n)^q)^3/x,x)

[Out]

int((d*x^m+e*ln(c*x^n)^(-1+q))*(a*x^m+b*ln(c*x^n)^q)^3/x,x)

________________________________________________________________________________________

Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (\frac{a^{3} e x^{3 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{3} d x^{4 \, m} +{\left (b^{3} d x^{m} + b^{3} e \log \left (c x^{n}\right )^{q - 1}\right )} \log \left (c x^{n}\right )^{3 \, q} + 3 \,{\left (a b^{2} e x^{m} \log \left (c x^{n}\right )^{q - 1} + a b^{2} d x^{2 \, m}\right )} \log \left (c x^{n}\right )^{2 \, q} + 3 \,{\left (a^{2} b e x^{2 \, m} \log \left (c x^{n}\right )^{q - 1} + a^{2} b d x^{3 \, m}\right )} \log \left (c x^{n}\right )^{q}}{x}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="fricas")

[Out]

integral((a^3*e*x^(3*m)*log(c*x^n)^(q - 1) + a^3*d*x^(4*m) + (b^3*d*x^m + b^3*e*log(c*x^n)^(q - 1))*log(c*x^n)
^(3*q) + 3*(a*b^2*e*x^m*log(c*x^n)^(q - 1) + a*b^2*d*x^(2*m))*log(c*x^n)^(2*q) + 3*(a^2*b*e*x^(2*m)*log(c*x^n)
^(q - 1) + a^2*b*d*x^(3*m))*log(c*x^n)^q)/x, x)

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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((d*x**m+e*ln(c*x**n)**(-1+q))*(a*x**m+b*ln(c*x**n)**q)**3/x,x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{{\left (a x^{m} + b \log \left (c x^{n}\right )^{q}\right )}^{3}{\left (d x^{m} + e \log \left (c x^{n}\right )^{q - 1}\right )}}{x}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x^m+e*log(c*x^n)^(-1+q))*(a*x^m+b*log(c*x^n)^q)^3/x,x, algorithm="giac")

[Out]

integrate((a*x^m + b*log(c*x^n)^q)^3*(d*x^m + e*log(c*x^n)^(q - 1))/x, x)