3.5.48 \(\int (f x)^m (d+e x^r)^3 (a+b \log (c x^n))^p \, dx\) [448]

Optimal. Leaf size=480 \[ \frac {d^3 e^{-\frac {a (1+m)}{b n}} (f x)^{1+m} \left (c x^n\right )^{-\frac {1+m}{n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{f (1+m)}+\frac {3 d^2 e e^{-\frac {a (1+m+r)}{b n}} x^{1+r} (f x)^m \left (c x^n\right )^{-\frac {1+m+r}{n}} \Gamma \left (1+p,-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+r}+\frac {3 d e^2 e^{-\frac {a (1+m+2 r)}{b n}} x^{1+2 r} (f x)^m \left (c x^n\right )^{-\frac {1+m+2 r}{n}} \Gamma \left (1+p,-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+2 r}+\frac {e^3 e^{-\frac {a (1+m+3 r)}{b n}} x^{1+3 r} (f x)^m \left (c x^n\right )^{-\frac {1+m+3 r}{n}} \Gamma \left (1+p,-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+3 r} \]

[Out]

d^3*(f*x)^(1+m)*GAMMA(1+p,-(1+m)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m)/b/n)/f/(1+m)/((c*x^n)^((1+
m)/n))/((-(1+m)*(a+b*ln(c*x^n))/b/n)^p)+3*d^2*e*x^(1+r)*(f*x)^m*GAMMA(1+p,-(1+m+r)*(a+b*ln(c*x^n))/b/n)*(a+b*l
n(c*x^n))^p/exp(a*(1+m+r)/b/n)/(1+m+r)/((c*x^n)^((1+m+r)/n))/((-(1+m+r)*(a+b*ln(c*x^n))/b/n)^p)+3*d*e^2*x^(1+2
*r)*(f*x)^m*GAMMA(1+p,-(1+m+2*r)*(a+b*ln(c*x^n))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m+2*r)/b/n)/(1+m+2*r)/((c*x^n
)^((1+m+2*r)/n))/((-(1+m+2*r)*(a+b*ln(c*x^n))/b/n)^p)+e^3*x^(1+3*r)*(f*x)^m*GAMMA(1+p,-(1+m+3*r)*(a+b*ln(c*x^n
))/b/n)*(a+b*ln(c*x^n))^p/exp(a*(1+m+3*r)/b/n)/(1+m+3*r)/((c*x^n)^((1+m+3*r)/n))/((-(1+m+3*r)*(a+b*ln(c*x^n))/
b/n)^p)

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Rubi [A]
time = 0.45, antiderivative size = 480, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 4, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.148, Rules used = {2395, 2347, 2212, 20} \begin {gather*} \frac {d^3 (f x)^{m+1} e^{-\frac {a (m+1)}{b n}} \left (c x^n\right )^{-\frac {m+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{f (m+1)}+\frac {3 d^2 e x^{r+1} (f x)^m e^{-\frac {a (m+r+1)}{b n}} \left (c x^n\right )^{-\frac {m+r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+r+1}+\frac {3 d e^2 x^{2 r+1} (f x)^m e^{-\frac {a (m+2 r+1)}{b n}} \left (c x^n\right )^{-\frac {m+2 r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+2 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+2 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+2 r+1}+\frac {e^3 x^{3 r+1} (f x)^m e^{-\frac {a (m+3 r+1)}{b n}} \left (c x^n\right )^{-\frac {m+3 r+1}{n}} \left (a+b \log \left (c x^n\right )\right )^p \left (-\frac {(m+3 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p} \text {Gamma}\left (p+1,-\frac {(m+3 r+1) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )}{m+3 r+1} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n])^p,x]

[Out]

(d^3*(f*x)^(1 + m)*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m))/(
b*n))*f*(1 + m)*(c*x^n)^((1 + m)/n)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + (3*d^2*e*x^(1 + r)*(f*x)^m*Ga
mma[1 + p, -(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + r))/(b*n))*(1 + m
+ r)*(c*x^n)^((1 + m + r)/n)*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p) + (3*d*e^2*x^(1 + 2*r)*(f*x)^m*Gam
ma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + 2*r))/(b*n))*(1 +
m + 2*r)*(c*x^n)^((1 + m + 2*r)/n)*(-(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b*n)))^p) + (e^3*x^(1 + 3*r)*(f*x)^
m*Gamma[1 + p, -(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n))]*(a + b*Log[c*x^n])^p)/(E^((a*(1 + m + 3*r))/(b*n))
*(1 + m + 3*r)*(c*x^n)^((1 + m + 3*r)/n)*(-(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n)))^p)

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 2212

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

Rule 2347

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)/n)*x)*(a + b*x)^p, x], x, Log[c*x^n]], x] /; FreeQ[{a, b, c, d, m, n, p}
, x]

Rule 2395

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^(r_.))^(q_.), x_Symbol]
:> With[{u = ExpandIntegrand[(a + b*Log[c*x^n])^p, (f*x)^m*(d + e*x^r)^q, x]}, Int[u, x] /; SumQ[u]] /; FreeQ[
{a, b, c, d, e, f, m, n, p, q, r}, x] && IntegerQ[q] && (GtQ[q, 0] || (IGtQ[p, 0] && IntegerQ[m] && IntegerQ[r
]))

Rubi steps

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

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Mathematica [A]
time = 1.52, size = 408, normalized size = 0.85 \begin {gather*} x^{-m} (f x)^m \left (a+b \log \left (c x^n\right )\right )^p \left (\frac {d^3 e^{-\frac {(1+m) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m}+e \left (\frac {3 d^2 e^{-\frac {(1+m+r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+r}+e \left (\frac {3 d e^{-\frac {(1+m+2 r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+2 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+2 r}+\frac {e e^{-\frac {(1+m+3 r) \left (a-b n \log (x)+b \log \left (c x^n\right )\right )}{b n}} \Gamma \left (1+p,-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right ) \left (-\frac {(1+m+3 r) \left (a+b \log \left (c x^n\right )\right )}{b n}\right )^{-p}}{1+m+3 r}\right )\right )\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(f*x)^m*(d + e*x^r)^3*(a + b*Log[c*x^n])^p,x]

[Out]

((f*x)^m*(a + b*Log[c*x^n])^p*((d^3*Gamma[1 + p, -(((1 + m)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m)*(a - b*n
*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m)*(-(((1 + m)*(a + b*Log[c*x^n]))/(b*n)))^p) + e*((3*d^2*Gamma[1 + p, -(
((1 + m + r)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m + r)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + r)
*(-(((1 + m + r)*(a + b*Log[c*x^n]))/(b*n)))^p) + e*((3*d*Gamma[1 + p, -(((1 + m + 2*r)*(a + b*Log[c*x^n]))/(b
*n))])/(E^(((1 + m + 2*r)*(a - b*n*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + 2*r)*(-(((1 + m + 2*r)*(a + b*Log[c
*x^n]))/(b*n)))^p) + (e*Gamma[1 + p, -(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n))])/(E^(((1 + m + 3*r)*(a - b*n
*Log[x] + b*Log[c*x^n]))/(b*n))*(1 + m + 3*r)*(-(((1 + m + 3*r)*(a + b*Log[c*x^n]))/(b*n)))^p)))))/x^m

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Maple [F]
time = 0.10, size = 0, normalized size = 0.00 \[\int \left (f x \right )^{m} \left (d +e \,x^{r}\right )^{3} \left (a +b \ln \left (c \,x^{n}\right )\right )^{p}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(d+e*x^r)^3*(a+b*ln(c*x^n))^p,x)

[Out]

int((f*x)^m*(d+e*x^r)^3*(a+b*ln(c*x^n))^p,x)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x)^m*(d+e*x^r)^3*(a+b*log(c*x^n))^p,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError >> ECL says: In function CAR, the value of the first argument is  0which is not
of the expected type LIST

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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*(d+e*x^r)^3*(a+b*log(c*x^n))^p,x, algorithm="fricas")

[Out]

integral((3*d^2*x^r*e + d^3 + 3*d*x^(2*r)*e^2 + x^(3*r)*e^3)*(f*x)^m*(b*log(c*x^n) + a)^p, x)

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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((f*x)**m*(d+e*x**r)**3*(a+b*ln(c*x**n))**p,x)

[Out]

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

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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*(d+e*x^r)^3*(a+b*log(c*x^n))^p,x, algorithm="giac")

[Out]

integrate((x^r*e + d)^3*(f*x)^m*(b*log(c*x^n) + a)^p, x)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int {\left (f\,x\right )}^m\,{\left (d+e\,x^r\right )}^3\,{\left (a+b\,\ln \left (c\,x^n\right )\right )}^p \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x)^m*(d + e*x^r)^3*(a + b*log(c*x^n))^p,x)

[Out]

int((f*x)^m*(d + e*x^r)^3*(a + b*log(c*x^n))^p, x)

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