Integrand size = 24, antiderivative size = 220 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {a}{b n}} (e f-d g) \sqrt {\pi } (d+e x) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac {2 e^{-\frac {2 a}{b n}} g \sqrt {2 \pi } (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
2*(-d*g+e*f)*(e*x+d)*erfi((a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*Pi^ (1/2)/b^(3/2)/e^2/exp(a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(1/n))+2*g*(e*x+d)^2*e rfi(2^(1/2)*(a+b*ln(c*(e*x+d)^n))^(1/2)/b^(1/2)/n^(1/2))*2^(1/2)*Pi^(1/2)/ b^(3/2)/e^2/exp(2*a/b/n)/n^(3/2)/((c*(e*x+d)^n)^(2/n))-2*(e*x+d)*(g*x+f)/b /e/n/(a+b*ln(c*(e*x+d)^n))^(1/2)
Time = 0.49 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.54 \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\frac {2 e^{-\frac {2 a}{b n}} (d+e x) \left (c (d+e x)^n\right )^{-2/n} \left (-2 d e^{\frac {a}{b n}} g \sqrt {\pi } \left (c (d+e x)^n\right )^{\frac {1}{n}} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+g \sqrt {2 \pi } (d+e x) \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right ) \sqrt {a+b \log \left (c (d+e x)^n\right )}+\sqrt {b} e^{\frac {a}{b n}} \sqrt {n} \left (c (d+e x)^n\right )^{\frac {1}{n}} \left (-e e^{\frac {a}{b n}} \left (c (d+e x)^n\right )^{\frac {1}{n}} (f+g x)+(e f+d g) \Gamma \left (\frac {1}{2},-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}\right ) \sqrt {-\frac {a+b \log \left (c (d+e x)^n\right )}{b n}}\right )\right )}{b^{3/2} e^2 n^{3/2} \sqrt {a+b \log \left (c (d+e x)^n\right )}} \]
(2*(d + e*x)*(-2*d*E^(a/(b*n))*g*Sqrt[Pi]*(c*(d + e*x)^n)^n^(-1)*Erfi[Sqrt [a + b*Log[c*(d + e*x)^n]]/(Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n ]] + g*Sqrt[2*Pi]*(d + e*x)*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n]])/ (Sqrt[b]*Sqrt[n])]*Sqrt[a + b*Log[c*(d + e*x)^n]] + Sqrt[b]*E^(a/(b*n))*Sq rt[n]*(c*(d + e*x)^n)^n^(-1)*(-(e*E^(a/(b*n))*(c*(d + e*x)^n)^n^(-1)*(f + g*x)) + (e*f + d*g)*Gamma[1/2, -((a + b*Log[c*(d + e*x)^n])/(b*n))]*Sqrt[- ((a + b*Log[c*(d + e*x)^n])/(b*n))])))/(b^(3/2)*e^2*E^((2*a)/(b*n))*n^(3/2 )*(c*(d + e*x)^n)^(2/n)*Sqrt[a + b*Log[c*(d + e*x)^n]])
Time = 1.00 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.45, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {2847, 2836, 2737, 2611, 2633, 2848, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx\) |
| \(\Big \downarrow \) 2847 |
| \(\displaystyle -\frac {2 (e f-d g) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b e n}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2836 |
| \(\displaystyle -\frac {2 (e f-d g) \int \frac {1}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d(d+e x)}{b e^2 n}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2737 |
| \(\displaystyle -\frac {2 (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int \frac {\left (c (d+e x)^n\right )^{\frac {1}{n}}}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}d\log \left (c (d+e x)^n\right )}{b e^2 n^2}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2611 |
| \(\displaystyle -\frac {4 (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \int e^{\frac {a+b \log \left (c (d+e x)^n\right )}{b n}-\frac {a}{b n}}d\sqrt {a+b \log \left (c (d+e x)^n\right )}}{b^2 e^2 n^2}+\frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2633 |
| \(\displaystyle \frac {4 \int \frac {f+g x}{\sqrt {a+b \log \left (c (d+e x)^n\right )}}dx}{b n}-\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2848 |
| \(\displaystyle \frac {4 \int \left (\frac {e f-d g}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}+\frac {g (d+e x)}{e \sqrt {a+b \log \left (c (d+e x)^n\right )}}\right )dx}{b n}-\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 \sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{b^{3/2} e^2 n^{3/2}}+\frac {4 \left (\frac {\sqrt {\pi } e^{-\frac {a}{b n}} (d+e x) (e f-d g) \left (c (d+e x)^n\right )^{-1/n} \text {erfi}\left (\frac {\sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}}+\frac {\sqrt {\frac {\pi }{2}} g e^{-\frac {2 a}{b n}} (d+e x)^2 \left (c (d+e x)^n\right )^{-2/n} \text {erfi}\left (\frac {\sqrt {2} \sqrt {a+b \log \left (c (d+e x)^n\right )}}{\sqrt {b} \sqrt {n}}\right )}{\sqrt {b} e^2 \sqrt {n}}\right )}{b n}-\frac {2 (d+e x) (f+g x)}{b e n \sqrt {a+b \log \left (c (d+e x)^n\right )}}\) |
(-2*(e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/(Sq rt[b]*Sqrt[n])])/(b^(3/2)*e^2*E^(a/(b*n))*n^(3/2)*(c*(d + e*x)^n)^n^(-1)) + (4*(((e*f - d*g)*Sqrt[Pi]*(d + e*x)*Erfi[Sqrt[a + b*Log[c*(d + e*x)^n]]/ (Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^(a/(b*n))*Sqrt[n]*(c*(d + e*x)^n)^n^(-1 )) + (g*Sqrt[Pi/2]*(d + e*x)^2*Erfi[(Sqrt[2]*Sqrt[a + b*Log[c*(d + e*x)^n] ])/(Sqrt[b]*Sqrt[n])])/(Sqrt[b]*e^2*E^((2*a)/(b*n))*Sqrt[n]*(c*(d + e*x)^n )^(2/n))))/(b*n) - (2*(d + e*x)*(f + g*x))/(b*e*n*Sqrt[a + b*Log[c*(d + e* x)^n]])
3.2.30.3.1 Defintions of rubi rules used
Int[(F_)^((g_.)*((e_.) + (f_.)*(x_)))/Sqrt[(c_.) + (d_.)*(x_)], x_Symbol] : > Simp[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]
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]
Int[((a_.) + Log[(c_.)*(x_)^(n_.)]*(b_.))^(p_), x_Symbol] :> Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] : > Simp[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]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Simp[(d + e*x)*(f + g*x)^q*((a + b*Log[c*(d + e *x)^n])^(p + 1)/(b*e*n*(p + 1))), x] + (-Simp[(q + 1)/(b*n*(p + 1)) Int[( f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^(p + 1), x], x] + Simp[q*((e*f - d*g) /(b*e*n*(p + 1))) Int[(f + g*x)^(q - 1)*(a + b*Log[c*(d + e*x)^n])^(p + 1 ), x], x]) /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0] && Lt Q[p, -1] && GtQ[q, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_)*((f_.) + (g_. )*(x_))^(q_.), x_Symbol] :> Int[ExpandIntegrand[(f + g*x)^q*(a + b*Log[c*(d + e*x)^n])^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p}, x] && NeQ[e*f - d*g, 0] && IGtQ[q, 0]
\[\int \frac {g x +f}{{\left (a +b \ln \left (c \left (e x +d \right )^{n}\right )\right )}^{\frac {3}{2}}}d x\]
Exception generated. \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\text {Exception raised: TypeError} \]
Exception raised: TypeError >> Error detected within library code: inte grate: implementation incomplete (constant residues)
\[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {f + g x}{\left (a + b \log {\left (c \left (d + e x\right )^{n} \right )}\right )^{\frac {3}{2}}}\, dx \]
\[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
\[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int { \frac {g x + f}{{\left (b \log \left ({\left (e x + d\right )}^{n} c\right ) + a\right )}^{\frac {3}{2}}} \,d x } \]
Timed out. \[ \int \frac {f+g x}{\left (a+b \log \left (c (d+e x)^n\right )\right )^{3/2}} \, dx=\int \frac {f+g\,x}{{\left (a+b\,\ln \left (c\,{\left (d+e\,x\right )}^n\right )\right )}^{3/2}} \,d x \]