\(\int \frac {x^2 \log (c (a+b x^2)^p)}{d+e x} \, dx\) [227]

Optimal result

Integrand size = 23, antiderivative size = 313 \[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}-\frac {2 \sqrt {a} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^3} \] Output:

2*d*p*x/e^2-1/2*p*x^2/e-2*a^(1/2)*d*p*arctan(b^(1/2)*x/a^(1/2))/b^(1/2)/e^ 
2-d^2*p*ln(e*((-a)^(1/2)-b^(1/2)*x)/(b^(1/2)*d+(-a)^(1/2)*e))*ln(e*x+d)/e^ 
3-d^2*p*ln(-e*((-a)^(1/2)+b^(1/2)*x)/(b^(1/2)*d-(-a)^(1/2)*e))*ln(e*x+d)/e 
^3-d*x*ln(c*(b*x^2+a)^p)/e^2+1/2*(b*x^2+a)*ln(c*(b*x^2+a)^p)/b/e+d^2*ln(e* 
x+d)*ln(c*(b*x^2+a)^p)/e^3-d^2*p*polylog(2,b^(1/2)*(e*x+d)/(b^(1/2)*d-(-a) 
^(1/2)*e))/e^3-d^2*p*polylog(2,b^(1/2)*(e*x+d)/(b^(1/2)*d+(-a)^(1/2)*e))/e 
^3
 

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.99 \[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {2 d p x}{e^2}-\frac {2 \sqrt {a} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}-\frac {d^2 p \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {b} d+\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d^2 p \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right ) \log (d+e x)}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {p x^2-\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{b}}{2 e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^3} \] Input:

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

Output:

(2*d*p*x)/e^2 - (2*Sqrt[a]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]])/(Sqrt[b]*e^2) 
- (d^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[-a]*e)]*Log[d + 
e*x])/e^3 - (d^2*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[b]*d - Sqrt[-a]* 
e))]*Log[d + e*x])/e^3 - (d*x*Log[c*(a + b*x^2)^p])/e^2 + (d^2*Log[d + e*x 
]*Log[c*(a + b*x^2)^p])/e^3 - (p*x^2 - ((a + b*x^2)*Log[c*(a + b*x^2)^p])/ 
b)/(2*e) - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[-a]*e)] 
)/e^3 - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]*e)])/e 
^3
 

Rubi [A] (verified)

Time = 1.00 (sec) , antiderivative size = 313, 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.

\(\displaystyle \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx\)

\(\Big \downarrow \) 2916

\(\displaystyle \int \left (\frac {d^2 \log \left (c \left (a+b x^2\right )^p\right )}{e^2 (d+e x)}-\frac {d \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {x \log \left (c \left (a+b x^2\right )^p\right )}{e}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 \sqrt {a} d p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{\sqrt {b} e^2}+\frac {d^2 \log (d+e x) \log \left (c \left (a+b x^2\right )^p\right )}{e^3}-\frac {d x \log \left (c \left (a+b x^2\right )^p\right )}{e^2}+\frac {\left (a+b x^2\right ) \log \left (c \left (a+b x^2\right )^p\right )}{2 b e}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \operatorname {PolyLog}\left (2,\frac {\sqrt {b} (d+e x)}{\sqrt {b} d+\sqrt {-a} e}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (\frac {e \left (\sqrt {-a}-\sqrt {b} x\right )}{\sqrt {-a} e+\sqrt {b} d}\right )}{e^3}-\frac {d^2 p \log (d+e x) \log \left (-\frac {e \left (\sqrt {-a}+\sqrt {b} x\right )}{\sqrt {b} d-\sqrt {-a} e}\right )}{e^3}+\frac {2 d p x}{e^2}-\frac {p x^2}{2 e}\)

Input:

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

Output:

(2*d*p*x)/e^2 - (p*x^2)/(2*e) - (2*Sqrt[a]*d*p*ArcTan[(Sqrt[b]*x)/Sqrt[a]] 
)/(Sqrt[b]*e^2) - (d^2*p*Log[(e*(Sqrt[-a] - Sqrt[b]*x))/(Sqrt[b]*d + Sqrt[ 
-a]*e)]*Log[d + e*x])/e^3 - (d^2*p*Log[-((e*(Sqrt[-a] + Sqrt[b]*x))/(Sqrt[ 
b]*d - Sqrt[-a]*e))]*Log[d + e*x])/e^3 - (d*x*Log[c*(a + b*x^2)^p])/e^2 + 
((a + b*x^2)*Log[c*(a + b*x^2)^p])/(2*b*e) + (d^2*Log[d + e*x]*Log[c*(a + 
b*x^2)^p])/e^3 - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d - Sqrt[- 
a]*e)])/e^3 - (d^2*p*PolyLog[2, (Sqrt[b]*(d + e*x))/(Sqrt[b]*d + Sqrt[-a]* 
e)])/e^3
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2916

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]
 

Maple [A] (verified)

Time = 1.77 (sec) , antiderivative size = 348, normalized size of antiderivative = 1.11

method	result	size
parts	\(\frac {x^{2} \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{2 e}-\frac {d x \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right ) \ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}{e^{3}}-\frac {2 p b \left (\frac {\left (e x +d \right )^{2}}{4 e b}-\frac {3 d \left (e x +d \right )}{2 b e}-\frac {e a \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{4 b^{2}}+\frac {a d \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{b \sqrt {a b}}-\frac {d^{2} \left (-\frac {\ln \left (e x +d \right ) \left (\ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )\right )}{2 b}-\frac {\operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )+\operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{2 b}\right )}{e}\right )}{e^{2}}\)	\(348\)
risch	\(\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) x^{2}}{2 e}-\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d x}{e^{2}}+\frac {\ln \left (\left (b \,x^{2}+a \right )^{p}\right ) d^{2} \ln \left (e x +d \right )}{e^{3}}-\frac {p \,d^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{3}}-\frac {p \,d^{2} \ln \left (e x +d \right ) \ln \left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{3}}-\frac {p \,d^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}-\left (e x +d \right ) b +b d}{e \sqrt {-a b}+b d}\right )}{e^{3}}-\frac {p \,d^{2} \operatorname {dilog}\left (\frac {e \sqrt {-a b}+\left (e x +d \right ) b -b d}{e \sqrt {-a b}-b d}\right )}{e^{3}}-\frac {p \,x^{2}}{2 e}+\frac {2 d p x}{e^{2}}+\frac {5 p \,d^{2}}{2 e^{3}}+\frac {p a \ln \left (\left (e x +d \right )^{2} b -2 d \left (e x +d \right ) b +a \,e^{2}+b \,d^{2}\right )}{2 b e}-\frac {2 p a d \arctan \left (\frac {2 \left (e x +d \right ) b -2 b d}{2 e \sqrt {a b}}\right )}{e^{2} \sqrt {a b}}+\left (\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2}}{2}-\frac {i \pi \,\operatorname {csgn}\left (i \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right ) \operatorname {csgn}\left (i c \right )}{2}-\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{3}}{2}+\frac {i \pi {\operatorname {csgn}\left (i c \left (b \,x^{2}+a \right )^{p}\right )}^{2} \operatorname {csgn}\left (i c \right )}{2}+\ln \left (c \right )\right ) \left (\frac {\frac {1}{2} e \,x^{2}-d x}{e^{2}}+\frac {d^{2} \ln \left (e x +d \right )}{e^{3}}\right )\)	\(504\)

Input:

int(x^2*ln(c*(b*x^2+a)^p)/(e*x+d),x,method=_RETURNVERBOSE)
 

Output:

1/2*x^2*ln(c*(b*x^2+a)^p)/e-d*x*ln(c*(b*x^2+a)^p)/e^2+d^2*ln(e*x+d)*ln(c*( 
b*x^2+a)^p)/e^3-2*p*b/e^2*(1/4/e/b*(e*x+d)^2-3/2/b/e*d*(e*x+d)-1/4*e*a/b^2 
*ln((e*x+d)^2*b-2*d*(e*x+d)*b+a*e^2+b*d^2)+a/b*d/(a*b)^(1/2)*arctan(1/2*(2 
*(e*x+d)*b-2*b*d)/e/(a*b)^(1/2))-d^2/e*(-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 
))
 

Fricas [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:

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

Output:

integral(x^2*log((b*x^2 + a)^p*c)/(e*x + d), x)
 

Sympy [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int \frac {x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}}{d + e x}\, dx \] Input:

integrate(x**2*ln(c*(b*x**2+a)**p)/(e*x+d),x)
 

Output:

Integral(x**2*log(c*(a + b*x**2)**p)/(d + e*x), x)
 

Maxima [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:

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

Output:

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

Giac [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\int { \frac {x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )}{e x + d} \,d x } \] Input:

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

Output:

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

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

\[ \int \frac {x^2 \log \left (c \left (a+b x^2\right )^p\right )}{d+e x} \, dx=\frac {-8 \sqrt {b}\, \sqrt {a}\, \mathit {atan} \left (\frac {b x}{\sqrt {b}\, \sqrt {a}}\right ) d e \,p^{2}+4 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}{b e \,x^{3}+b d \,x^{2}+a e x +a d}d x \right ) a b \,d^{2} e p -4 \left (\int \frac {\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) x}{b e \,x^{3}+b d \,x^{2}+a e x +a d}d x \right ) b^{2} d^{3} p +{\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right )}^{2} b \,d^{2}+2 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) a \,e^{2} p -4 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b d e p x +2 \,\mathrm {log}\left (\left (b \,x^{2}+a \right )^{p} c \right ) b \,e^{2} p \,x^{2}+8 b d e \,p^{2} x -2 b \,e^{2} p^{2} x^{2}}{4 b \,e^{3} p} \] Input:

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

Output:

( - 8*sqrt(b)*sqrt(a)*atan((b*x)/(sqrt(b)*sqrt(a)))*d*e*p**2 + 4*int(log(( 
a + b*x**2)**p*c)/(a*d + a*e*x + b*d*x**2 + b*e*x**3),x)*a*b*d**2*e*p - 4* 
int((log((a + b*x**2)**p*c)*x)/(a*d + a*e*x + b*d*x**2 + b*e*x**3),x)*b**2 
*d**3*p + log((a + b*x**2)**p*c)**2*b*d**2 + 2*log((a + b*x**2)**p*c)*a*e* 
*2*p - 4*log((a + b*x**2)**p*c)*b*d*e*p*x + 2*log((a + b*x**2)**p*c)*b*e** 
2*p*x**2 + 8*b*d*e*p**2*x - 2*b*e**2*p**2*x**2)/(4*b*e**3*p)