\(\int \frac {1}{(a+b \log (c (d+e x)^n))^{7/2}} \, dx\) [30]

Optimal result

Integrand size = 18, antiderivative size = 192 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{7/2}} \, dx=\frac {8 e^{-\frac {a}{b n}} \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{15 b^{7/2} e n^{7/2}}-\frac {2 (d+e x)}{5 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}}-\frac {4 (d+e x)}{15 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {8 (d+e x)}{15 b^3 e n^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]

[Out]

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

Rubi [A] (verified)

Time = 0.11 (sec) , antiderivative size = 192, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {2436, 2334, 2337, 2211, 2235} \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{7/2}} \, dx=\frac {8 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{15 b^{7/2} e n^{7/2}}-\frac {8 (d+e x)}{15 b^3 e n^3 \sqrt {a+b \log \left (c (d+e x)^n\right )}}-\frac {4 (d+e x)}{15 b^2 e n^2 \left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}}-\frac {2 (d+e x)}{5 b e n \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \]

[In]

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

[Out]

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

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

Mathematica [A] (verified)

Time = 0.14 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.06 \[ \int \frac {1}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{7/2}} \, dx=-\frac {2 e^{-\frac {a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-1/n} \left (-4 b^2 n^2 \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \left (-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right )^{5/2}+e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (4 a^2+2 a b n+3 b^2 n^2+2 b (4 a+b n) \log \left (c (d+e x)^n\right )+4 b^2 \log ^2\left (c (d+e x)^n\right )\right )\right )}{15 b^3 e n^3 \left (a+b \log \left (c (d+e x)^n\right )\right )^{5/2}} \]

[In]

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

[Out]

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

Maple [F]

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

[In]

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

[Out]

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

Fricas [F(-2)]

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

[In]

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

[Out]

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

Sympy [F]

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

[In]

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

[Out]

Integral((a + b*log(c*(d + e*x)**n))**(-7/2), x)

Maxima [F]

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

[In]

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

[Out]

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

Giac [F]

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

[In]

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

[Out]

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

Mupad [F(-1)]

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

[In]

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

[Out]

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