Integrand size = 23, antiderivative size = 394 \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=-\frac {2 d^2 p x}{e^3}+\frac {2 a p x}{3 b e}+\frac {d p x^2}{2 e^2}-\frac {2 p x^3}{9 e}+\frac {2 \sqrt {a} d^2 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^3}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2} e}+\frac {d^3 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^4}+\frac {d^3 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}+\frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}-\frac {d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^4} \] Output:
-2*d^2*p*x/e^3+2/3*a*p*x/b/e+1/2*d*p*x^2/e^2-2/9*p*x^3/e+2*a^(1/2)*d^2*p*a rctan(b^(1/2)*x/a^(1/2))/b^(1/2)/e^3-2/3*a^(3/2)*p*arctan(b^(1/2)*x/a^(1/2 ))/b^(3/2)/e+d^3*p*ln(e*((-a)^(1/2)-b^(1/2)*x)/(b^(1/2)*d+(-a)^(1/2)*e))*l n(e*x+d)/e^4+d^3*p*ln(-e*((-a)^(1/2)+b^(1/2)*x)/(b^(1/2)*d-(-a)^(1/2)*e))* ln(e*x+d)/e^4+d^2*x*ln(c*(b*x^2+a)^p)/e^3+1/3*x^3*ln(c*(b*x^2+a)^p)/e-1/2* d*(b*x^2+a)*ln(c*(b*x^2+a)^p)/b/e^2-d^3*ln(e*x+d)*ln(c*(b*x^2+a)^p)/e^4+d^ 3*p*polylog(2,b^(1/2)*(e*x+d)/(b^(1/2)*d-(-a)^(1/2)*e))/e^4+d^3*p*polylog( 2,b^(1/2)*(e*x+d)/(b^(1/2)*d+(-a)^(1/2)*e))/e^4
Time = 0.36 (sec) , antiderivative size = 370, normalized size of antiderivative = 0.94 \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {-36 d^2 e p x+\frac {12 a e^3 p x}{b}-4 e^3 p x^3+\frac {36 \sqrt {a} d^2 e p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b}}-\frac {12 a^{3/2} e^3 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}+18 d^3 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+18 d^3 p \log \left (\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{-\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)+18 d^2 e x \log \left (c \left (a+b x^2\right )^p\right )+6 e^3 x^3 \log \left (c \left (a+b x^2\right )^p\right )-18 d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )+9 d e^2 \left (p x^2-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}\right )+18 d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )+18 d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{18 e^4} \] Input:
Integrate[(x^3*Log[c*(a + b*x^2)^p])/(d + e*x),x]
Output:
(-36*d^2*e*p*x + (12*a*e^3*p*x)/b - 4*e^3*p*x^3 + (36*Sqrt[a]*d^2*e*p*ArcT an[(Sqrt[b]*x)/Sqrt[a]])/Sqrt[b] - (12*a^(3/2)*e^3*p*ArcTan[(Sqrt[b]*x)/Sq rt[a]])/b^(3/2) + 18*d^3*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqr t[-a]*e)]*Log[d + e*x] + 18*d^3*p*Log[(e*(Sqrt[-a] + Sqrt[b]*x))/(-(Sqrt[b ]*d) + Sqrt[-a]*e)]*Log[d + e*x] + 18*d^2*e*x*Log[c*(a + b*x^2)^p] + 6*e^3 *x^3*Log[c*(a + b*x^2)^p] - 18*d^3*Log[d + e*x]*Log[c*(a + b*x^2)^p] + 9*d *e^2*(p*x^2 - ((a + b*x^2)*Log[c*(a + b*x^2)^p])/b) + 18*d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)] + 18*d^3*p*PolyLog[2, (Sqrt[ b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/(18*e^4)
Time = 1.23 (sec) , antiderivative size = 394, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {2916, 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.
| \(\displaystyle \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx\) |
| \(\Big \downarrow \) 2916 |
| \(\displaystyle \int \left (-\frac {d^3 \log \left (c \left (a+b x^2\right )^p\right )}{e^3 (d+e x)}+\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{e}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{3 b^{3/2} e}+\frac {2 \sqrt {a} d^2 p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^3}-\frac {d^3 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^4}+\frac {d^2 x \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d \left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e^2}+\frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{3 e}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^4}+\frac {d^3 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^4}+\frac {2 a p x}{3 b e}-\frac {2 d^2 p x}{e^3}+\frac {d p x^2}{2 e^2}-\frac {2 p x^3}{9 e}\) |
Input:
Int[(x^3*Log[c*(a + b*x^2)^p])/(d + e*x),x]
Output:
(-2*d^2*p*x)/e^3 + (2*a*p*x)/(3*b*e) + (d*p*x^2)/(2*e^2) - (2*p*x^3)/(9*e) + (2*Sqrt[a]*d^2*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*e^3) - (2*a^(3/2 )*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(3*b^(3/2)*e) + (d^3*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + e*x])/e^4 + (d^3*p*Log[-((e *(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e^4 + (d ^2*x*Log[c*(a + b*x^2)^p])/e^3 + (x^3*Log[c*(a + b*x^2)^p])/(3*e) - (d*(a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b*e^2) - (d^3*Log[d + e*x]*Log[c*(a + b* x^2)^p])/e^4 + (d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a] *e)])/e^4 + (d^3*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e) ])/e^4
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_.) + (g_.)*(x_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log [c*(d + e*x^n)^p])^q, x^m*(f + g*x)^r, x], x] /; FreeQ[{a, b, c, d, e, f, g , n, p, q}, x] && IntegerQ[m] && IntegerQ[r]
Time = 2.11 (sec) , antiderivative size = 412, normalized size of antiderivative = 1.05
| method | result | size |
| parts | \(\frac {x^{3} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{3 e}-\frac {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right ) x^{2} d}{2 e^{2}}+\frac {d^{2} x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{4}}-\frac {2 p b \left (\frac {d^{3} \left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{e^{2}}+\frac {-\frac {2 \left (e x +d \right ) a \,e^{2}-11 \left (e x +d \right ) b \,d^{2}+\frac {7 d \left (e x +d \right )^{2} b}{2}-\frac {2 \left (e x +d \right )^{3} b}{3}}{b^{2}}+\frac {a \,e^{2} \left (\frac {3 d \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2}+\frac {\left (2 a \,e^{2}-6 b \,d^{2}\right ) \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{e \sqrt {a b}}\right )}{b^{2}}}{6 e^{2}}\right )}{e^{2}}\) | \(412\) |
| risch | \(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x^{3}}{3 e}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d \,x^{2}}{2 e^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x \,d^{2}}{e^{3}}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d^{3} \ln \left (e x +d \right )}{e^{4}}-\frac {2 p \,x^{3}}{9 e}+\frac {d p \,x^{2}}{2 e^{2}}-\frac {2 d^{2} p x}{e^{3}}-\frac {49 p \,d^{3}}{18 e^{4}}+\frac {2 a p x}{3 b e}+\frac {2 p d a}{3 b \,e^{2}}-\frac {p a d \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2 b \,e^{2}}-\frac {2 p \,a^{2} \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{3 b e \sqrt {a b}}+\frac {2 p a \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right ) d^{2}}{e^{3} \sqrt {a b}}+\frac {p \,d^{3} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{4}}+\frac {p \,d^{3} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{4}}+\frac {p \,d^{3} \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{4}}+\frac {p \,d^{3} \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{4}}+\left (\frac {i \pi \,\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}-\frac {i \pi \,\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 )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{3} e^{2} x^{3}-\frac {1}{2} e \,x^{2} d +d^{2} x}{e^{3}}-\frac {d^{3} \ln \left (e x +d \right )}{e^{4}}\right )\) | \(610\) |
Input:
int(x^3*ln(c*(b*x^2+a)^p)/(e*x+d),x,method=_RETURNVERBOSE)
Output:
1/3*x^3*ln(c*(b*x^2+a)^p)/e-1/2*ln(c*(b*x^2+a)^p)/e^2*x^2*d+d^2*x*ln(c*(b* x^2+a)^p)/e^3-d^3*ln(e*x+d)*ln(c*(b*x^2+a)^p)/e^4-2*p*b/e^2*(d^3/e^2*(-1/2 *ln(e*x+d)*(ln((e*(-a*b)^(1/2)-(e*x+d)*b+b*d)/(e*(-a*b)^(1/2)+b*d))+ln((e* (-a*b)^(1/2)+(e*x+d)*b-b*d)/(e*(-a*b)^(1/2)-b*d)))/b-1/2*(dilog((e*(-a*b)^ (1/2)-(e*x+d)*b+b*d)/(e*(-a*b)^(1/2)+b*d))+dilog((e*(-a*b)^(1/2)+(e*x+d)*b -b*d)/(e*(-a*b)^(1/2)-b*d)))/b)+1/6/e^2*(-1/b^2*(2*(e*x+d)*a*e^2-11*(e*x+d )*b*d^2+7/2*d*(e*x+d)^2*b-2/3*(e*x+d)^3*b)+a*e^2/b^2*(3/2*d*ln((e*x+d)^2*b -2*d*(e*x+d)*b+a*e^2+b*d^2)+(2*a*e^2-6*b*d^2)/e/(a*b)^(1/2)*arctan(1/2*(2* (e*x+d)*b-2*b*d)/e/(a*b)^(1/2)))))
\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="fricas")
Output:
integral(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)
\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {x^{3} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{d + e x}\, dx \] Input:
integrate(x**3*ln(c*(b*x**2+a)**p)/(e*x+d),x)
Output:
Integral(x**3*log(c*(a + b*x**2)**p)/(d + e*x), x)
\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="maxima")
Output:
integrate(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)
\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{3} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:
integrate(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x, algorithm="giac")
Output:
integrate(x^3*log((b*x^2 + a)^p*c)/(e*x + d), x)
Timed out. \[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {x^3\,\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}{d+e\,x} \,d x \] Input:
int((x^3*log(c*(a + b*x^2)^p))/(d + e*x),x)
Output:
int((x^3*log(c*(a + b*x^2)^p))/(d + e*x), x)
\[ \int \frac {x^3 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {-24 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) a \,e^{3} p^{2}+72 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) b \,d^{2} e \,p^{2}-36 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}{b e \,x^{3}+b d \,x^{2}+a e x +a d}d x \right ) a \,b^{2} d^{3} e p +36 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) x}{b e \,x^{3}+b d \,x^{2}+a e x +a d}d x \right ) b^{3} d^{4} p -9 {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} b^{2} d^{3}-18 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a b d \,e^{2} p +36 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{2} d^{2} e p x -18 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{2} d \,e^{2} p \,x^{2}+12 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b^{2} e^{3} p \,x^{3}+24 a b \,e^{3} p^{2} x -72 b^{2} d^{2} e \,p^{2} x +18 b^{2} d \,e^{2} p^{2} x^{2}-8 b^{2} e^{3} p^{2} x^{3}}{36 b^{2} e^{4} p} \] Input:
int(x^3*log(c*(b*x^2+a)^p)/(e*x+d),x)
Output:
( - 24*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*a*e**3*p**2 + 72*sqrt (b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*b*d**2*e*p**2 - 36*int(log((a + b*x**2)**p*c)/(a*d + a*e*x + b*d*x**2 + b*e*x**3),x)*a*b**2*d**3*e*p + 36* int((log((a + b*x**2)**p*c)*x)/(a*d + a*e*x + b*d*x**2 + b*e*x**3),x)*b**3 *d**4*p - 9*log((a + b*x**2)**p*c)**2*b**2*d**3 - 18*log((a + b*x**2)**p*c )*a*b*d*e**2*p + 36*log((a + b*x**2)**p*c)*b**2*d**2*e*p*x - 18*log((a + b *x**2)**p*c)*b**2*d*e**2*p*x**2 + 12*log((a + b*x**2)**p*c)*b**2*e**3*p*x* *3 + 24*a*b*e**3*p**2*x - 72*b**2*d**2*e*p**2*x + 18*b**2*d*e**2*p**2*x**2 - 8*b**2*e**3*p**2*x**3)/(36*b**2*e**4*p)