Integrand size = 28, antiderivative size = 86 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}} \] Output:
-4*b*f^(1/2)*p*q*arctanh(f^(1/2)*(h*x+g)^(1/2)/(-e*h+f*g)^(1/2))/h/(-e*h+f *g)^(1/2)-2*(a+b*ln(c*(d*(f*x+e)^p)^q))/h/(h*x+g)^(1/2)
Time = 0.18 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.98 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\frac {-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{\sqrt {f g-e h}}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{\sqrt {g+h x}}}{h} \] Input:
Integrate[(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(3/2),x]
Output:
((-4*b*Sqrt[f]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/Sqrt[ f*g - e*h] - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/Sqrt[g + h*x])/h
Time = 0.59 (sec) , antiderivative size = 86, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.143, Rules used = {2895, 2842, 73, 221}
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.
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx\) |
\(\Big \downarrow \) 2895 |
\(\displaystyle \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}}dx\) |
\(\Big \downarrow \) 2842 |
\(\displaystyle \frac {2 b f p q \int \frac {1}{(e+f x) \sqrt {g+h x}}dx}{h}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}\) |
\(\Big \downarrow \) 73 |
\(\displaystyle \frac {4 b f p q \int \frac {1}{e+\frac {f (g+h x)}{h}-\frac {f g}{h}}d\sqrt {g+h x}}{h^2}-\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}\) |
\(\Big \downarrow \) 221 |
\(\displaystyle -\frac {2 \left (a+b \log \left (c \left (d (e+f x)^p\right )^q\right )\right )}{h \sqrt {g+h x}}-\frac {4 b \sqrt {f} p q \text {arctanh}\left (\frac {\sqrt {f} \sqrt {g+h x}}{\sqrt {f g-e h}}\right )}{h \sqrt {f g-e h}}\) |
Input:
Int[(a + b*Log[c*(d*(e + f*x)^p)^q])/(g + h*x)^(3/2),x]
Output:
(-4*b*Sqrt[f]*p*q*ArcTanh[(Sqrt[f]*Sqrt[g + h*x])/Sqrt[f*g - e*h]])/(h*Sqr t[f*g - e*h]) - (2*(a + b*Log[c*(d*(e + f*x)^p)^q]))/(h*Sqrt[g + h*x])
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x /Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))*((f_.) + (g_.)*(x_ ))^(q_.), x_Symbol] :> Simp[(f + g*x)^(q + 1)*((a + b*Log[c*(d + e*x)^n])/( g*(q + 1))), x] - Simp[b*e*(n/(g*(q + 1))) Int[(f + g*x)^(q + 1)/(d + e*x ), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, q}, x] && NeQ[e*f - d*g, 0] && NeQ[q, -1]
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]]
\[\int \frac {a +b \ln \left (c \left (d \left (f x +e \right )^{p}\right )^{q}\right )}{\left (h x +g \right )^{\frac {3}{2}}}d x\]
Input:
int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x)
Output:
int((a+b*ln(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x)
Time = 0.10 (sec) , antiderivative size = 222, normalized size of antiderivative = 2.58 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\left [\frac {2 \, {\left ({\left (b h p q x + b g p q\right )} \sqrt {\frac {f}{f g - e h}} \log \left (\frac {f h x + 2 \, f g - e h - 2 \, {\left (f g - e h\right )} \sqrt {h x + g} \sqrt {\frac {f}{f g - e h}}}{f x + e}\right ) - {\left (b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}, \frac {2 \, {\left (2 \, {\left (b h p q x + b g p q\right )} \sqrt {-\frac {f}{f g - e h}} \arctan \left (\sqrt {h x + g} \sqrt {-\frac {f}{f g - e h}}\right ) - {\left (b p q \log \left (f x + e\right ) + b q \log \left (d\right ) + b \log \left (c\right ) + a\right )} \sqrt {h x + g}\right )}}{h^{2} x + g h}\right ] \] Input:
integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x, algorithm="fricas" )
Output:
[2*((b*h*p*q*x + b*g*p*q)*sqrt(f/(f*g - e*h))*log((f*h*x + 2*f*g - e*h - 2 *(f*g - e*h)*sqrt(h*x + g)*sqrt(f/(f*g - e*h)))/(f*x + e)) - (b*p*q*log(f* x + e) + b*q*log(d) + b*log(c) + a)*sqrt(h*x + g))/(h^2*x + g*h), 2*(2*(b* h*p*q*x + b*g*p*q)*sqrt(-f/(f*g - e*h))*arctan(sqrt(h*x + g)*sqrt(-f/(f*g - e*h))) - (b*p*q*log(f*x + e) + b*q*log(d) + b*log(c) + a)*sqrt(h*x + g)) /(h^2*x + g*h)]
\[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\int \frac {a + b \log {\left (c \left (d \left (e + f x\right )^{p}\right )^{q} \right )}}{\left (g + h x\right )^{\frac {3}{2}}}\, dx \] Input:
integrate((a+b*ln(c*(d*(f*x+e)**p)**q))/(h*x+g)**(3/2),x)
Output:
Integral((a + b*log(c*(d*(e + f*x)**p)**q))/(g + h*x)**(3/2), x)
Exception generated. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\text {Exception raised: ValueError} \] Input:
integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x, algorithm="maxima" )
Output:
Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'assume' command before evaluation *may* help (example of legal syntax is 'assume(e*h-f*g>0)', see `assume?` for m ore detail
Time = 0.13 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.28 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\frac {4 \, b f p q \arctan \left (\frac {\sqrt {h x + g} f}{\sqrt {-f^{2} g + e f h}}\right )}{\sqrt {-f^{2} g + e f h} h} - \frac {2 \, b p q \log \left ({\left (h x + g\right )} f - f g + e h\right )}{\sqrt {h x + g} h} + \frac {2 \, {\left (b p q \log \left (h\right ) - b q \log \left (d\right ) - b \log \left (c\right ) - a\right )}}{\sqrt {h x + g} h} \] Input:
integrate((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x, algorithm="giac")
Output:
4*b*f*p*q*arctan(sqrt(h*x + g)*f/sqrt(-f^2*g + e*f*h))/(sqrt(-f^2*g + e*f* h)*h) - 2*b*p*q*log((h*x + g)*f - f*g + e*h)/(sqrt(h*x + g)*h) + 2*(b*p*q* log(h) - b*q*log(d) - b*log(c) - a)/(sqrt(h*x + g)*h)
Timed out. \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\int \frac {a+b\,\ln \left (c\,{\left (d\,{\left (e+f\,x\right )}^p\right )}^q\right )}{{\left (g+h\,x\right )}^{3/2}} \,d x \] Input:
int((a + b*log(c*(d*(e + f*x)^p)^q))/(g + h*x)^(3/2),x)
Output:
int((a + b*log(c*(d*(e + f*x)^p)^q))/(g + h*x)^(3/2), x)
Time = 0.17 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.60 \[ \int \frac {a+b \log \left (c \left (d (e+f x)^p\right )^q\right )}{(g+h x)^{3/2}} \, dx=\frac {4 \sqrt {f}\, \sqrt {h x +g}\, \sqrt {e h -f g}\, \mathit {atan} \left (\frac {\sqrt {h x +g}\, f}{\sqrt {f}\, \sqrt {e h -f g}}\right ) b p q -2 \,\mathrm {log}\left (\frac {d^{q} \left (f h x +e h \right )^{p q} c}{h^{p q}}\right ) b e h +2 \,\mathrm {log}\left (\frac {d^{q} \left (f h x +e h \right )^{p q} c}{h^{p q}}\right ) b f g -2 a e h +2 a f g}{\sqrt {h x +g}\, h \left (e h -f g \right )} \] Input:
int((a+b*log(c*(d*(f*x+e)^p)^q))/(h*x+g)^(3/2),x)
Output:
(2*(2*sqrt(f)*sqrt(g + h*x)*sqrt(e*h - f*g)*atan((sqrt(g + h*x)*f)/(sqrt(f )*sqrt(e*h - f*g)))*b*p*q - log((d**q*(e*h + f*h*x)**(p*q)*c)/h**(p*q))*b* e*h + log((d**q*(e*h + f*h*x)**(p*q)*c)/h**(p*q))*b*f*g - a*e*h + a*f*g))/ (sqrt(g + h*x)*h*(e*h - f*g))