3.227 \(\int (a g+b g x)^{-2-m} (c i+d i x)^m (A+B \log (e (a+b x)^n (c+d x)^{-n}))^p \, dx\)

Optimal. Leaf size=194 \[ -\frac {(a+b x) e^{\frac {A (m+1)}{B n}} (g (a+b x))^{-m-2} (i (c+d x))^{m+2} \left (e (a+b x)^n (c+d x)^{-n}\right )^{\frac {m+1}{n}} \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )^p \left (\frac {(m+1) \left (B \log \left (e (a+b x)^n (c+d x)^{-n}\right )+A\right )}{B n}\right )^{-p} \Gamma \left (p+1,\frac {(m+1) \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )}{B n}\right )}{i^2 (m+1) (c+d x) (b c-a d)} \]

[Out]

-exp(A*(1+m)/B/n)*(b*x+a)*(g*(b*x+a))^(-2-m)*(i*(d*x+c))^(2+m)*(e*(b*x+a)^n/((d*x+c)^n))^((1+m)/n)*GAMMA(1+p,(
1+m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/n)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p/(-a*d+b*c)/i^2/(1+m)/(d*x+c)/(
((1+m)*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))/B/n)^p)

________________________________________________________________________________________

Rubi [F]  time = 0.71, antiderivative size = 0, normalized size of antiderivative = 0.00, number of steps used = 0, number of rules used = 0, integrand size = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {} \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Int[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p,x]

[Out]

Defer[Int][(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p, x]

Rubi steps

\begin {align*} \int (227 c+227 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx &=\int (227 c+227 d x)^m (a g+b g x)^{-2-m} \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.41, size = 0, normalized size = 0.00 \[ \int (a g+b g x)^{-2-m} (c i+d i x)^m \left (A+B \log \left (e (a+b x)^n (c+d x)^{-n}\right )\right )^p \, dx \]

Verification is Not applicable to the result.

[In]

Integrate[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p,x]

[Out]

Integrate[(a*g + b*g*x)^(-2 - m)*(c*i + d*i*x)^m*(A + B*Log[(e*(a + b*x)^n)/(c + d*x)^n])^p, x]

________________________________________________________________________________________

fricas [F]  time = 0.82, size = 0, normalized size = 0.00 \[ {\rm integral}\left ({\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="fricas")

[Out]

integral((b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p, x)

________________________________________________________________________________________

giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: RuntimeError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="giac")

[Out]

Exception raised: RuntimeError >> An error occurred running a Giac command:INPUT:sage2OUTPUT:Simplification as
suming c near 0Simplification assuming c near 0Simplification assuming t_nostep near 0Simplification assuming
t_nostep near 0Simplification assuming a near 0Simplification assuming a near 0Simplification assuming c near
0Simplification assuming c near 0Simplification assuming c near 0Simplification assuming c near 0Simplificatio
n assuming t_nostep near 0Simplification assuming t_nostep near 0Simplification assuming a near 0Simplificatio
n assuming a near 0Simplification assuming c near 0Simplification assuming c near 0Simplification assuming c n
ear 0Simplification assuming c near 0Simplification assuming x near 0Simplification assuming x near 0Simplific
ation assuming a near 0Simplification assuming a near 0Simplification assuming c near 0Simplification assuming
c near 0Unable to divide, perhaps due to rounding error%%%{1,[0,0,5,5,0,2,2,3,3,0,0,2]%%%}+%%%{-2,[0,0,5,4,1,
3,1,3,3,0,0,2]%%%}+%%%{1,[0,0,5,3,2,4,0,3,3,0,0,2]%%%}+%%%{2,[0,0,4,5,0,1,3,3,3,0,0,2]%%%}+%%%{-1,[0,0,4,4,1,2
,2,3,3,0,0,2]%%%}+%%%{-4,[0,0,4,3,2,3,1,3,3,0,0,2]%%%}+%%%{3,[0,0,4,2,3,4,0,3,3,0,0,2]%%%}+%%%{1,[0,0,3,5,0,0,
4,3,3,0,0,2]%%%}+%%%{4,[0,0,3,4,1,1,3,3,3,0,0,2]%%%}+%%%{-8,[0,0,3,3,2,2,2,3,3,0,0,2]%%%}+%%%{3,[0,0,3,1,4,4,0
,3,3,0,0,2]%%%}+%%%{3,[0,0,2,4,1,0,4,3,3,0,0,2]%%%}+%%%{-8,[0,0,2,2,3,2,2,3,3,0,0,2]%%%}+%%%{4,[0,0,2,1,4,3,1,
3,3,0,0,2]%%%}+%%%{1,[0,0,2,0,5,4,0,3,3,0,0,2]%%%}+%%%{3,[0,0,1,3,2,0,4,3,3,0,0,2]%%%}+%%%{-4,[0,0,1,2,3,1,3,3
,3,0,0,2]%%%}+%%%{-1,[0,0,1,1,4,2,2,3,3,0,0,2]%%%}+%%%{2,[0,0,1,0,5,3,1,3,3,0,0,2]%%%}+%%%{1,[0,0,0,2,3,0,4,3,
3,0,0,2]%%%}+%%%{-2,[0,0,0,1,4,1,3,3,3,0,0,2]%%%}+%%%{1,[0,0,0,0,5,2,2,3,3,0,0,2]%%%} / %%%{1,[0,0,6,5,0,3,2,3
,3,0,0,2]%%%}+%%%{-2,[0,0,6,4,1,4,1,3,3,0,0,2]%%%}+%%%{1,[0,0,6,3,2,5,0,3,3,0,0,2]%%%}+%%%{3,[0,0,5,5,0,2,3,3,
3,0,0,2]%%%}+%%%{-3,[0,0,5,4,1,3,2,3,3,0,0,2]%%%}+%%%{-3,[0,0,5,3,2,4,1,3,3,0,0,2]%%%}+%%%{3,[0,0,5,2,3,5,0,3,
3,0,0,2]%%%}+%%%{3,[0,0,4,5,0,1,4,3,3,0,0,2]%%%}+%%%{3,[0,0,4,4,1,2,3,3,3,0,0,2]%%%}+%%%{-12,[0,0,4,3,2,3,2,3,
3,0,0,2]%%%}+%%%{3,[0,0,4,2,3,4,1,3,3,0,0,2]%%%}+%%%{3,[0,0,4,1,4,5,0,3,3,0,0,2]%%%}+%%%{1,[0,0,3,5,0,0,5,3,3,
0,0,2]%%%}+%%%{7,[0,0,3,4,1,1,4,3,3,0,0,2]%%%}+%%%{-8,[0,0,3,3,2,2,3,3,3,0,0,2]%%%}+%%%{-8,[0,0,3,2,3,3,2,3,3,
0,0,2]%%%}+%%%{7,[0,0,3,1,4,4,1,3,3,0,0,2]%%%}+%%%{1,[0,0,3,0,5,5,0,3,3,0,0,2]%%%}+%%%{3,[0,0,2,4,1,0,5,3,3,0,
0,2]%%%}+%%%{3,[0,0,2,3,2,1,4,3,3,0,0,2]%%%}+%%%{-12,[0,0,2,2,3,2,3,3,3,0,0,2]%%%}+%%%{3,[0,0,2,1,4,3,2,3,3,0,
0,2]%%%}+%%%{3,[0,0,2,0,5,4,1,3,3,0,0,2]%%%}+%%%{3,[0,0,1,3,2,0,5,3,3,0,0,2]%%%}+%%%{-3,[0,0,1,2,3,1,4,3,3,0,0
,2]%%%}+%%%{-3,[0,0,1,1,4,2,3,3,3,0,0,2]%%%}+%%%{3,[0,0,1,0,5,3,2,3,3,0,0,2]%%%}+%%%{1,[0,0,0,2,3,0,5,3,3,0,0,
2]%%%}+%%%{-2,[0,0,0,1,4,1,4,3,3,0,0,2]%%%}+%%%{1,[0,0,0,0,5,2,3,3,3,0,0,2]%%%} Error: Bad Argument Value

________________________________________________________________________________________

maple [F]  time = 3.24, size = 0, normalized size = 0.00 \[ \int \left (B \ln \left (e \left (b x +a \right )^{n} \left (d x +c \right )^{-n}\right )+A \right )^{p} \left (b g x +a g \right )^{-m -2} \left (d i x +c i \right )^{m}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*g*x+a*g)^(-m-2)*(d*i*x+c*i)^m*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,x)

[Out]

int((b*g*x+a*g)^(-m-2)*(d*i*x+c*i)^m*(A+B*ln(e*(b*x+a)^n/((d*x+c)^n)))^p,x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int {\left (b g x + a g\right )}^{-m - 2} {\left (d i x + c i\right )}^{m} {\left (B \log \left (\frac {{\left (b x + a\right )}^{n} e}{{\left (d x + c\right )}^{n}}\right ) + A\right )}^{p}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)^(-2-m)*(d*i*x+c*i)^m*(A+B*log(e*(b*x+a)^n/((d*x+c)^n)))^p,x, algorithm="maxima")

[Out]

integrate((b*g*x + a*g)^(-m - 2)*(d*i*x + c*i)^m*(B*log((b*x + a)^n*e/(d*x + c)^n) + A)^p, x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\left (c\,i+d\,i\,x\right )}^m\,{\left (A+B\,\ln \left (\frac {e\,{\left (a+b\,x\right )}^n}{{\left (c+d\,x\right )}^n}\right )\right )}^p}{{\left (a\,g+b\,g\,x\right )}^{m+2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((c*i + d*i*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(a*g + b*g*x)^(m + 2),x)

[Out]

int(((c*i + d*i*x)^m*(A + B*log((e*(a + b*x)^n)/(c + d*x)^n))^p)/(a*g + b*g*x)^(m + 2), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*g*x+a*g)**(-2-m)*(d*i*x+c*i)**m*(A+B*ln(e*(b*x+a)**n/((d*x+c)**n)))**p,x)

[Out]

Timed out

________________________________________________________________________________________