Integrand size = 20, antiderivative size = 119 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {2 \sqrt {a} \sqrt {b} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b d^2+a e^2}-\frac {2 b d p \log (d+e x)}{e \left (b d^2+a e^2\right )}+\frac {b d p \log \left (a+b x^2\right )}{e \left (b d^2+a e^2\right )}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)} \] Output:
2*a^(1/2)*b^(1/2)*p*arctan(b^(1/2)*x/a^(1/2))/(a*e^2+b*d^2)-2*b*d*p*ln(e*x +d)/e/(a*e^2+b*d^2)+b*d*p*ln(b*x^2+a)/e/(a*e^2+b*d^2)-ln(c*(b*x^2+a)^p)/e/ (e*x+d)
Time = 0.08 (sec) , antiderivative size = 137, normalized size of antiderivative = 1.15 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {2 \sqrt {a} \sqrt {b} e p (d+e x) \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )-2 b d p (d+e x) \log (d+e x)+b d^2 p \log \left (a+b x^2\right )+b d e p x \log \left (a+b x^2\right )-b d^2 \log \left (c \left (a+b x^2\right )^p\right )-a e^2 \log \left (c \left (a+b x^2\right )^p\right )}{e \left (b d^2+a e^2\right ) (d+e x)} \] Input:
Integrate[Log[c*(a + b*x^2)^p]/(d + e*x)^2,x]
Output:
(2*Sqrt[a]*Sqrt[b]*e*p*(d + e*x)*ArcTan[(Sqrt[b]*x)/Sqrt[a]] - 2*b*d*p*(d + e*x)*Log[d + e*x] + b*d^2*p*Log[a + b*x^2] + b*d*e*p*x*Log[a + b*x^2] - b*d^2*Log[c*(a + b*x^2)^p] - a*e^2*Log[c*(a + b*x^2)^p])/(e*(b*d^2 + a*e^2 )*(d + e*x))
Time = 0.44 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {2913, 587, 16, 452, 218, 240}
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.
| \(\displaystyle \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx\) |
| \(\Big \downarrow \) 2913 |
| \(\displaystyle \frac {2 b p \int \frac {x}{(d+e x) \left (b x^2+a\right )}dx}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 587 |
| \(\displaystyle \frac {2 b p \left (\frac {\int \frac {a e+b d x}{b x^2+a}dx}{a e^2+b d^2}-\frac {d e \int \frac {1}{d+e x}dx}{a e^2+b d^2}\right )}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 16 |
| \(\displaystyle \frac {2 b p \left (\frac {\int \frac {a e+b d x}{b x^2+a}dx}{a e^2+b d^2}-\frac {d \log (d+e x)}{a e^2+b d^2}\right )}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 452 |
| \(\displaystyle \frac {2 b p \left (\frac {b d \int \frac {x}{b x^2+a}dx+a e \int \frac {1}{b x^2+a}dx}{a e^2+b d^2}-\frac {d \log (d+e x)}{a e^2+b d^2}\right )}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 218 |
| \(\displaystyle \frac {2 b p \left (\frac {b d \int \frac {x}{b x^2+a}dx+\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}}{a e^2+b d^2}-\frac {d \log (d+e x)}{a e^2+b d^2}\right )}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
| \(\Big \downarrow \) 240 |
| \(\displaystyle \frac {2 b p \left (\frac {\frac {\sqrt {a} e \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}+\frac {1}{2} d \log \left (a+b x^2\right )}{a e^2+b d^2}-\frac {d \log (d+e x)}{a e^2+b d^2}\right )}{e}-\frac {\log \left (c \left (a+b x^2\right )^p\right )}{e (d+e x)}\) |
Input:
Int[Log[c*(a + b*x^2)^p]/(d + e*x)^2,x]
Output:
(2*b*p*(-((d*Log[d + e*x])/(b*d^2 + a*e^2)) + ((Sqrt[a]*e*ArcTan[(Sqrt[b]* x)/Sqrt[a]])/Sqrt[b] + (d*Log[a + b*x^2])/2)/(b*d^2 + a*e^2)))/e - Log[c*( a + b*x^2)^p]/(e*(d + e*x))
Int[(c_.)/((a_.) + (b_.)*(x_)), x_Symbol] :> Simp[c*(Log[RemoveContent[a + b*x, x]]/b), x] /; FreeQ[{a, b, c}, x]
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/R t[a/b, 2]], x] /; FreeQ[{a, b}, x] && PosQ[a/b]
Int[(x_)/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[Log[RemoveContent[a + b*x ^2, x]]/(2*b), x] /; FreeQ[{a, b}, x]
Int[((c_) + (d_.)*(x_))/((a_) + (b_.)*(x_)^2), x_Symbol] :> Simp[c Int[1/ (a + b*x^2), x], x] + Simp[d Int[x/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c^2 + a*d^2, 0]
Int[(x_.)/(((c_) + (d_.)*(x_))*((a_) + (b_.)*(x_)^2)), x_Symbol] :> Simp[(- c)*(d/(b*c^2 + a*d^2)) Int[1/(c + d*x), x], x] + Simp[1/(b*c^2 + a*d^2) Int[(a*d + b*c*x)/(a + b*x^2), x], x] /; FreeQ[{a, b, c, d}, x] && NeQ[b*c ^2 + a*d^2, 0]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))*((f_.) + (g_. )*(x_))^(r_.), x_Symbol] :> Simp[(f + g*x)^(r + 1)*((a + b*Log[c*(d + e*x^n )^p])/(g*(r + 1))), x] - Simp[b*e*n*(p/(g*(r + 1))) Int[x^(n - 1)*((f + g *x)^(r + 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d, e, f, g, n, p, r}, x ] && (IGtQ[r, 0] || RationalQ[n]) && NeQ[r, -1]
Time = 1.96 (sec) , antiderivative size = 99, normalized size of antiderivative = 0.83
| method | result | size |
| parts | \(-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e \left (e x +d \right )}+\frac {2 p b \left (-\frac {d \ln \left (e x +d \right )}{a \,e^{2}+b \,d^{2}}+\frac {\frac {d \ln \left (b \,x^{2}+a \right )}{2}+\frac {e a \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{\sqrt {a b}}}{a \,e^{2}+b \,d^{2}}\right )}{e}\) | \(99\) |
| risch | \(-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right )}{e \left (e x +d \right )}+\frac {i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right ) e^{2}-i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right ) e^{2}+i \pi a {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3} e^{2}+i \pi b \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )+i \pi b \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}-i \pi a \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} e^{2}-i \pi b \,d^{2} {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )-i \pi b \,d^{2} \operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,e^{4}+b \,d^{2} e^{2}\right ) \textit {\_Z}^{2}-2 b d e p \textit {\_Z} +b \,p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a \,e^{4}-b \,d^{2} e^{2}\right ) \textit {\_R}^{2}-b d e p \textit {\_R} +2 b \,p^{2}\right ) x +4 a d \,e^{3} \textit {\_R}^{2}-a \,e^{2} p \textit {\_R} \right )\right ) a \,e^{4} x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,e^{4}+b \,d^{2} e^{2}\right ) \textit {\_Z}^{2}-2 b d e p \textit {\_Z} +b \,p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a \,e^{4}-b \,d^{2} e^{2}\right ) \textit {\_R}^{2}-b d e p \textit {\_R} +2 b \,p^{2}\right ) x +4 a d \,e^{3} \textit {\_R}^{2}-a \,e^{2} p \textit {\_R} \right )\right ) b \,d^{2} e^{2} x -4 \ln \left (e x +d \right ) b d e p x +2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,e^{4}+b \,d^{2} e^{2}\right ) \textit {\_Z}^{2}-2 b d e p \textit {\_Z} +b \,p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a \,e^{4}-b \,d^{2} e^{2}\right ) \textit {\_R}^{2}-b d e p \textit {\_R} +2 b \,p^{2}\right ) x +4 a d \,e^{3} \textit {\_R}^{2}-a \,e^{2} p \textit {\_R} \right )\right ) a d \,e^{3}+2 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\left (a \,e^{4}+b \,d^{2} e^{2}\right ) \textit {\_Z}^{2}-2 b d e p \textit {\_Z} +b \,p^{2}\right )}{\sum }\textit {\_R} \ln \left (\left (\left (3 a \,e^{4}-b \,d^{2} e^{2}\right ) \textit {\_R}^{2}-b d e p \textit {\_R} +2 b \,p^{2}\right ) x +4 a d \,e^{3} \textit {\_R}^{2}-a \,e^{2} p \textit {\_R} \right )\right ) b \,d^{3} e -4 \ln \left (e x +d \right ) b \,d^{2} p -2 \ln \left (c \right ) a \,e^{2}-2 d^{2} b \ln \left (c \right )}{2 \left (e x +d \right ) e \left (a \,e^{2}+b \,d^{2}\right )}\) | \(755\) |
Input:
int(ln(c*(b*x^2+a)^p)/(e*x+d)^2,x,method=_RETURNVERBOSE)
Output:
-ln(c*(b*x^2+a)^p)/e/(e*x+d)+2*p*b/e*(-d/(a*e^2+b*d^2)*ln(e*x+d)+1/(a*e^2+ b*d^2)*(1/2*d*ln(b*x^2+a)+e*a/(a*b)^(1/2)*arctan(b*x/(a*b)^(1/2))))
Time = 0.10 (sec) , antiderivative size = 261, normalized size of antiderivative = 2.19 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\left [\frac {{\left (e^{2} p x + d e p\right )} \sqrt {-a b} \log \left (\frac {b x^{2} + 2 \, \sqrt {-a b} x - a}{b x^{2} + a}\right ) + {\left (b d e p x - a e^{2} p\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b d e p x + b d^{2} p\right )} \log \left (e x + d\right ) - {\left (b d^{2} + a e^{2}\right )} \log \left (c\right )}{b d^{3} e + a d e^{3} + {\left (b d^{2} e^{2} + a e^{4}\right )} x}, \frac {2 \, {\left (e^{2} p x + d e p\right )} \sqrt {a b} \arctan \left (\frac {\sqrt {a b} x}{a}\right ) + {\left (b d e p x - a e^{2} p\right )} \log \left (b x^{2} + a\right ) - 2 \, {\left (b d e p x + b d^{2} p\right )} \log \left (e x + d\right ) - {\left (b d^{2} + a e^{2}\right )} \log \left (c\right )}{b d^{3} e + a d e^{3} + {\left (b d^{2} e^{2} + a e^{4}\right )} x}\right ] \] Input:
integrate(log(c*(b*x^2+a)^p)/(e*x+d)^2,x, algorithm="fricas")
Output:
[((e^2*p*x + d*e*p)*sqrt(-a*b)*log((b*x^2 + 2*sqrt(-a*b)*x - a)/(b*x^2 + a )) + (b*d*e*p*x - a*e^2*p)*log(b*x^2 + a) - 2*(b*d*e*p*x + b*d^2*p)*log(e* x + d) - (b*d^2 + a*e^2)*log(c))/(b*d^3*e + a*d*e^3 + (b*d^2*e^2 + a*e^4)* x), (2*(e^2*p*x + d*e*p)*sqrt(a*b)*arctan(sqrt(a*b)*x/a) + (b*d*e*p*x - a* e^2*p)*log(b*x^2 + a) - 2*(b*d*e*p*x + b*d^2*p)*log(e*x + d) - (b*d^2 + a* e^2)*log(c))/(b*d^3*e + a*d*e^3 + (b*d^2*e^2 + a*e^4)*x)]
Exception generated. \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\text {Exception raised: AttributeError} \] Input:
integrate(ln(c*(b*x**2+a)**p)/(e*x+d)**2,x)
Output:
Exception raised: AttributeError >> 'NoneType' object has no attribute 'pr imitive'
Time = 0.11 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.91 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {{\left (\frac {2 \, a e \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b d^{2} + a e^{2}\right )} \sqrt {a b}} + \frac {d \log \left (b x^{2} + a\right )}{b d^{2} + a e^{2}} - \frac {2 \, d \log \left (e x + d\right )}{b d^{2} + a e^{2}}\right )} b p}{e} - \frac {\log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{{\left (e x + d\right )} e} \] Input:
integrate(log(c*(b*x^2+a)^p)/(e*x+d)^2,x, algorithm="maxima")
Output:
(2*a*e*arctan(b*x/sqrt(a*b))/((b*d^2 + a*e^2)*sqrt(a*b)) + d*log(b*x^2 + a )/(b*d^2 + a*e^2) - 2*d*log(e*x + d)/(b*d^2 + a*e^2))*b*p/e - log((b*x^2 + a)^p*c)/((e*x + d)*e)
Time = 0.12 (sec) , antiderivative size = 121, normalized size of antiderivative = 1.02 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {b d p \log \left (b x^{2} + a\right )}{b d^{2} e + a e^{3}} - \frac {2 \, b d p \log \left (e x + d\right )}{b d^{2} e + a e^{3}} + \frac {2 \, a b p \arctan \left (\frac {b x}{\sqrt {a b}}\right )}{{\left (b d^{2} + a e^{2}\right )} \sqrt {a b}} - \frac {p \log \left (b x^{2} + a\right )}{e^{2} x + d e} - \frac {\log \left (c\right )}{e^{2} x + d e} \] Input:
integrate(log(c*(b*x^2+a)^p)/(e*x+d)^2,x, algorithm="giac")
Output:
b*d*p*log(b*x^2 + a)/(b*d^2*e + a*e^3) - 2*b*d*p*log(e*x + d)/(b*d^2*e + a *e^3) + 2*a*b*p*arctan(b*x/sqrt(a*b))/((b*d^2 + a*e^2)*sqrt(a*b)) - p*log( b*x^2 + a)/(e^2*x + d*e) - log(c)/(e^2*x + d*e)
Time = 26.91 (sec) , antiderivative size = 337, normalized size of antiderivative = 2.83 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {\ln \left (\frac {4\,b^3\,p^2\,x}{e}-\frac {p\,\left (b\,d+e\,\sqrt {-a\,b}\right )\,\left (2\,a\,b^2\,e\,p+2\,b^3\,d\,p\,x-\frac {2\,b^2\,e\,p\,\left (b\,d+e\,\sqrt {-a\,b}\right )\,\left (-b\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{b\,d^2\,e+a\,e^3}\right )}{b\,d^2\,e+a\,e^3}\right )\,\left (b\,d\,p+e\,p\,\sqrt {-a\,b}\right )}{b\,d^2\,e+a\,e^3}-\frac {\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{e\,\left (d+e\,x\right )}+\frac {\ln \left (\frac {4\,b^3\,p^2\,x}{e}-\frac {p\,\left (b\,d-e\,\sqrt {-a\,b}\right )\,\left (2\,a\,b^2\,e\,p+2\,b^3\,d\,p\,x-\frac {2\,b^2\,e\,p\,\left (b\,d-e\,\sqrt {-a\,b}\right )\,\left (-b\,x\,d^2+4\,a\,d\,e+3\,a\,x\,e^2\right )}{b\,d^2\,e+a\,e^3}\right )}{b\,d^2\,e+a\,e^3}\right )\,\left (b\,d\,p-e\,p\,\sqrt {-a\,b}\right )}{b\,d^2\,e+a\,e^3}-\frac {2\,b\,d\,p\,\ln \left (d+e\,x\right )}{b\,d^2\,e+a\,e^3} \] Input:
int(log(c*(a + b*x^2)^p)/(d + e*x)^2,x)
Output:
(log((4*b^3*p^2*x)/e - (p*(b*d + e*(-a*b)^(1/2))*(2*a*b^2*e*p + 2*b^3*d*p* x - (2*b^2*e*p*(b*d + e*(-a*b)^(1/2))*(4*a*d*e + 3*a*e^2*x - b*d^2*x))/(a* e^3 + b*d^2*e)))/(a*e^3 + b*d^2*e))*(b*d*p + e*p*(-a*b)^(1/2)))/(a*e^3 + b *d^2*e) - log(c*(a + b*x^2)^p)/(e*(d + e*x)) + (log((4*b^3*p^2*x)/e - (p*( b*d - e*(-a*b)^(1/2))*(2*a*b^2*e*p + 2*b^3*d*p*x - (2*b^2*e*p*(b*d - e*(-a *b)^(1/2))*(4*a*d*e + 3*a*e^2*x - b*d^2*x))/(a*e^3 + b*d^2*e)))/(a*e^3 + b *d^2*e))*(b*d*p - e*p*(-a*b)^(1/2)))/(a*e^3 + b*d^2*e) - (2*b*d*p*log(d + e*x))/(a*e^3 + b*d^2*e)
Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.50 \[ \int \frac {\log \left (c \left (a+b x^2\right )^p\right )}{(d+e x)^2} \, dx=\frac {2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) d^{2} e p +2 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) d \,e^{2} p x -\mathrm {log}\left (b \,x^{2}+a \right ) a d \,e^{2} p -\mathrm {log}\left (b \,x^{2}+a \right ) a \,e^{3} p x -2 \,\mathrm {log}\left (e x +d \right ) b \,d^{3} p -2 \,\mathrm {log}\left (e x +d \right ) b \,d^{2} e p x +\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a \,e^{3} x +\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b \,d^{2} e x}{d e \left (a \,e^{3} x +b \,d^{2} e x +a d \,e^{2}+b \,d^{3}\right )} \] Input:
int(log(c*(b*x^2+a)^p)/(e*x+d)^2,x)
Output:
(2*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*d**2*e*p + 2*sqrt(b)*sqrt (a)*atan((b*x)/(sqrt(b)*sqrt(a)))*d*e**2*p*x - log(a + b*x**2)*a*d*e**2*p - log(a + b*x**2)*a*e**3*p*x - 2*log(d + e*x)*b*d**3*p - 2*log(d + e*x)*b* d**2*e*p*x + log((a + b*x**2)**p*c)*a*e**3*x + log((a + b*x**2)**p*c)*b*d* *2*e*x)/(d*e*(a*d*e**2 + a*e**3*x + b*d**3 + b*d**2*e*x))