\(\int \frac {1}{(a+b \log (c (d (e+f x)^m)^n))^{7/2}} \, dx\) [417]

Optimal result

Integrand size = 22, antiderivative size = 237 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\frac {8 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{15 b^{7/2} f m^{7/2} n^{7/2}}-\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \]

[Out]

-2/5*(f*x+e)/b/f/m/n/(a+b*ln(c*(d*(f*x+e)^m)^n))^(5/2)-4/15*(f*x+e)/b^2/f/m^2/n^2/(a+b*ln(c*(d*(f*x+e)^m)^n))^
(3/2)+8/15*(f*x+e)*erfi((a+b*ln(c*(d*(f*x+e)^m)^n))^(1/2)/b^(1/2)/m^(1/2)/n^(1/2))*Pi^(1/2)/b^(7/2)/exp(a/b/m/
n)/f/m^(7/2)/n^(7/2)/((c*(d*(f*x+e)^m)^n)^(1/m/n))-8/15*(f*x+e)/b^3/f/m^3/n^3/(a+b*ln(c*(d*(f*x+e)^m)^n))^(1/2
)

Rubi [A] (verified)

Time = 0.28 (sec) , antiderivative size = 237, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {2436, 2334, 2337, 2211, 2235, 2495} \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\frac {8 \sqrt {\pi } (e+f x) e^{-\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{15 b^{7/2} f m^{7/2} n^{7/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}} \]

[In]

Int[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-7/2),x]

[Out]

(8*Sqrt[Pi]*(e + f*x)*Erfi[Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]]/(Sqrt[b]*Sqrt[m]*Sqrt[n])])/(15*b^(7/2)*E^(a/(
b*m*n))*f*m^(7/2)*n^(7/2)*(c*(d*(e + f*x)^m)^n)^(1/(m*n))) - (2*(e + f*x))/(5*b*f*m*n*(a + b*Log[c*(d*(e + f*x
)^m)^n])^(5/2)) - (4*(e + f*x))/(15*b^2*f*m^2*n^2*(a + b*Log[c*(d*(e + f*x)^m)^n])^(3/2)) - (8*(e + f*x))/(15*
b^3*f*m^3*n^3*Sqrt[a + b*Log[c*(d*(e + f*x)^m)^n]])

Rule 2211

Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] :> Dist[2/d, Subst[Int[F^(g*(e - c*(
f/d)) + f*g*(x^2/d)), x], x, Sqrt[c + d*x]], x] /; FreeQ[{F, c, d, e, f, g}, x] &&  !TrueQ[$UseGamma]

Rule 2235

Int[(F_)^((a_.) + (b_.)*((c_.) + (d_.)*(x_))^2), x_Symbol] :> Simp[F^a*Sqrt[Pi]*(Erfi[(c + d*x)*Rt[b*Log[F], 2
]]/(2*d*Rt[b*Log[F], 2])), x] /; FreeQ[{F, a, b, c, d}, x] && PosQ[b]

Rule 2334

Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[x*((a + b*Log[c*x^n])^(p + 1)/(b*n*(p + 1)))
, x] - Dist[1/(b*n*(p + 1)), Int[(a + b*Log[c*x^n])^(p + 1), x], x] /; FreeQ[{a, b, c, n}, x] && LtQ[p, -1] &&
IntegerQ[2*p]

Rule 2337

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

Rule 2436

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

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*} \text {integral}& = \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n (e+f x)^{m n}\right )\right )^{7/2}} \, dx,c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \text {Subst}\left (\frac {\text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^{7/2}} \, dx,x,e+f x\right )}{f},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}+\text {Subst}\left (\frac {2 \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^{5/2}} \, dx,x,e+f x\right )}{5 b f m n},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}+\text {Subst}\left (\frac {4 \text {Subst}\left (\int \frac {1}{\left (a+b \log \left (c d^n x^{m n}\right )\right )^{3/2}} \, dx,x,e+f x\right )}{15 b^2 f m^2 n^2},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {8 \text {Subst}\left (\int \frac {1}{\sqrt {a+b \log \left (c d^n x^{m n}\right )}} \, dx,x,e+f x\right )}{15 b^3 f m^3 n^3},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {\left (8 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}}\right ) \text {Subst}\left (\int \frac {e^{\frac {x}{m n}}}{\sqrt {a+b x}} \, dx,x,\log \left (c d^n (e+f x)^{m n}\right )\right )}{15 b^3 f m^4 n^4},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = -\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}+\text {Subst}\left (\frac {\left (16 (e+f x) \left (c d^n (e+f x)^{m n}\right )^{-\frac {1}{m n}}\right ) \text {Subst}\left (\int e^{-\frac {a}{b m n}+\frac {x^2}{b m n}} \, dx,x,\sqrt {a+b \log \left (c d^n (e+f x)^{m n}\right )}\right )}{15 b^4 f m^4 n^4},c d^n (e+f x)^{m n},c \left (d (e+f x)^m\right )^n\right ) \\ & = \frac {8 e^{-\frac {a}{b m n}} \sqrt {\pi } (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}}{\sqrt {b} \sqrt {m} \sqrt {n}}\right )}{15 b^{7/2} f m^{7/2} n^{7/2}}-\frac {2 (e+f x)}{5 b f m n \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}}-\frac {4 (e+f x)}{15 b^2 f m^2 n^2 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{3/2}}-\frac {8 (e+f x)}{15 b^3 f m^3 n^3 \sqrt {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.28 (sec) , antiderivative size = 272, normalized size of antiderivative = 1.15 \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=-\frac {2 e^{-\frac {a}{b m n}} (e+f x) \left (c \left (d (e+f x)^m\right )^n\right )^{-\frac {1}{m n}} \left (-4 \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}\right ) \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^2 \sqrt {-\frac {a+b \log \left (c \left (d (e+f x)^m\right )^n\right )}{b m n}}+e^{\frac {a}{b m n}} \left (c \left (d (e+f x)^m\right )^n\right )^{\frac {1}{m n}} \left (4 a^2+2 a b m n+3 b^2 m^2 n^2+2 b (4 a+b m n) \log \left (c \left (d (e+f x)^m\right )^n\right )+4 b^2 \log ^2\left (c \left (d (e+f x)^m\right )^n\right )\right )\right )}{15 b^3 f m^3 n^3 \left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{5/2}} \]

[In]

Integrate[(a + b*Log[c*(d*(e + f*x)^m)^n])^(-7/2),x]

[Out]

(-2*(e + f*x)*(-4*Gamma[1/2, -((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n))]*(a + b*Log[c*(d*(e + f*x)^m)^n])^2*S
qrt[-((a + b*Log[c*(d*(e + f*x)^m)^n])/(b*m*n))] + E^(a/(b*m*n))*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(4*a^2 + 2*a*
b*m*n + 3*b^2*m^2*n^2 + 2*b*(4*a + b*m*n)*Log[c*(d*(e + f*x)^m)^n] + 4*b^2*Log[c*(d*(e + f*x)^m)^n]^2)))/(15*b
^3*E^(a/(b*m*n))*f*m^3*n^3*(c*(d*(e + f*x)^m)^n)^(1/(m*n))*(a + b*Log[c*(d*(e + f*x)^m)^n])^(5/2))

Maple [F]

\[\int \frac {1}{{\left (a +b \ln \left (c \left (d \left (f x +e \right )^{m}\right )^{n}\right )\right )}^{\frac {7}{2}}}d x\]

[In]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^(7/2),x)

[Out]

int(1/(a+b*ln(c*(d*(f*x+e)^m)^n))^(7/2),x)

Fricas [F(-2)]

Exception generated. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\text {Exception raised: TypeError} \]

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="fricas")

[Out]

Exception raised: TypeError >>  Error detected within library code:   integrate: implementation incomplete (co
nstant residues)

Sympy [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\text {Timed out} \]

[In]

integrate(1/(a+b*ln(c*(d*(f*x+e)**m)**n))**(7/2),x)

[Out]

Timed out

Maxima [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="maxima")

[Out]

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(-7/2), x)

Giac [F]

\[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int { \frac {1}{{\left (b \log \left (\left ({\left (f x + e\right )}^{m} d\right )^{n} c\right ) + a\right )}^{\frac {7}{2}}} \,d x } \]

[In]

integrate(1/(a+b*log(c*(d*(f*x+e)^m)^n))^(7/2),x, algorithm="giac")

[Out]

integrate((b*log(((f*x + e)^m*d)^n*c) + a)^(-7/2), x)

Mupad [F(-1)]

Timed out. \[ \int \frac {1}{\left (a+b \log \left (c \left (d (e+f x)^m\right )^n\right )\right )^{7/2}} \, dx=\int \frac {1}{{\left (a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^m\right )}^n\right )\right )}^{7/2}} \,d x \]

[In]

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^(7/2),x)

[Out]

int(1/(a + b*log(c*(d*(e + f*x)^m)^n))^(7/2), x)