Integrand size = 18, antiderivative size = 18 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {208 a p^3 x}{9 b}-\frac {16 p^3 x^3}{27}-\frac {208 a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{3/2}}+\frac {32 i a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{3/2}}+\frac {64 a^{3/2} p^3 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {32 a p^2 x \log \left (c \left (a+b x^2\right )^p\right )}{3 b}+\frac {8}{9} p^2 x^3 \log \left (c \left (a+b x^2\right )^p\right )+\frac {32 a^{3/2} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{3/2}}+\frac {2 a p x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b}-\frac {2}{3} p x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )+\frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )+\frac {32 i a^{3/2} p^3 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{3/2}}-\frac {2 a^2 p \text {Int}\left (\frac {\log ^2\left (c \left (a+b x^2\right )^p\right )}{a+b x^2},x\right )}{b} \]
208/9*a*p^3*x/b-16/27*p^3*x^3-208/9*a^(3/2)*p^3*arctan(x*b^(1/2)/a^(1/2))/ b^(3/2)+32/3*I*a^(3/2)*p^3*arctan(x*b^(1/2)/a^(1/2))^2/b^(3/2)-32/3*a*p^2* x*ln(c*(b*x^2+a)^p)/b+8/9*p^2*x^3*ln(c*(b*x^2+a)^p)+32/3*a^(3/2)*p^2*arcta n(x*b^(1/2)/a^(1/2))*ln(c*(b*x^2+a)^p)/b^(3/2)+2*a*p*x*ln(c*(b*x^2+a)^p)^2 /b-2/3*p*x^3*ln(c*(b*x^2+a)^p)^2+1/3*x^3*ln(c*(b*x^2+a)^p)^3+64/3*a^(3/2)* p^3*arctan(x*b^(1/2)/a^(1/2))*ln(2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))/b^(3/2)+ 32/3*I*a^(3/2)*p^3*polylog(2,1-2*a^(1/2)/(a^(1/2)+I*x*b^(1/2)))/b^(3/2)-2* a^2*p*Unintegrable(ln(c*(b*x^2+a)^p)^2/(b*x^2+a),x)/b
Both result and optimal contain complex but leaf count is larger than twice the leaf count of optimal. \(909\) vs. \(2(380)=760\).
Time = 3.13 (sec) , antiderivative size = 909, normalized size of antiderivative = 50.50 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\frac {2 a p x \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b}-\frac {2 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2}{b^{3/2}}+p x^3 \log \left (a+b x^2\right ) \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2+\frac {1}{3} x^3 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )^2 \left (-2 p-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right )+3 p^2 \left (-p \log \left (a+b x^2\right )+\log \left (c \left (a+b x^2\right )^p\right )\right ) \left (\frac {1}{3} x^3 \log ^2\left (a+b x^2\right )-\frac {4 \left (9 i a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2+3 a^{3/2} \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \left (-8+6 \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )+3 \log \left (a+b x^2\right )\right )+\sqrt {b} x \left (24 a-2 b x^2+\left (-9 a+3 b x^2\right ) \log \left (a+b x^2\right )\right )+9 i a^{3/2} \operatorname {PolyLog}\left (2,\frac {i \sqrt {a}+\sqrt {b} x}{-i \sqrt {a}+\sqrt {b} x}\right )\right )}{27 b^{3/2}}\right )+\frac {p^3 \left (416 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right )+\frac {2}{3} \sqrt {-a} b x^2 \left (624 a-16 b x^2+\left (-288 a+24 b x^2\right ) \log \left (a+b x^2\right )+18 \left (3 a-b x^2\right ) \log ^2\left (a+b x^2\right )+9 b x^2 \log ^3\left (a+b x^2\right )\right )+36 \sqrt {-a} a^{3/2} \sqrt {\frac {b x^2}{a+b x^2}} \left (8 \sqrt {a} \, _4F_3\left (\frac {1}{2},\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\log \left (a+b x^2\right ) \left (4 \sqrt {a} \, _3F_2\left (\frac {1}{2},\frac {1}{2},\frac {1}{2};\frac {3}{2},\frac {3}{2};\frac {a}{a+b x^2}\right )+\sqrt {a+b x^2} \arcsin \left (\frac {\sqrt {a}}{\sqrt {a+b x^2}}\right ) \log \left (a+b x^2\right )\right )\right )-48 a^2 \left (4 \sqrt {b x^2} \text {arctanh}\left (\frac {\sqrt {b x^2}}{\sqrt {-a}}\right ) \left (\log \left (a+b x^2\right )-\log \left (1+\frac {b x^2}{a}\right )\right )-\sqrt {-a} \sqrt {-\frac {b x^2}{a}} \left (\log ^2\left (1+\frac {b x^2}{a}\right )-4 \log \left (1+\frac {b x^2}{a}\right ) \log \left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )+2 \log ^2\left (\frac {1}{2} \left (1+\sqrt {-\frac {b x^2}{a}}\right )\right )-4 \operatorname {PolyLog}\left (2,\frac {1}{2}-\frac {1}{2} \sqrt {-\frac {b x^2}{a}}\right )\right )\right )\right )}{18 \sqrt {-a} b^2 x} \]
(2*a*p*x*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2)/b - (2*a^(3/2)*p* ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2 )/b^(3/2) + p*x^3*Log[a + b*x^2]*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^ p])^2 + (x^3*(-(p*Log[a + b*x^2]) + Log[c*(a + b*x^2)^p])^2*(-2*p - p*Log[ a + b*x^2] + Log[c*(a + b*x^2)^p]))/3 + 3*p^2*(-(p*Log[a + b*x^2]) + Log[c *(a + b*x^2)^p])*((x^3*Log[a + b*x^2]^2)/3 - (4*((9*I)*a^(3/2)*ArcTan[(Sqr t[b]*x)/Sqrt[a]]^2 + 3*a^(3/2)*ArcTan[(Sqrt[b]*x)/Sqrt[a]]*(-8 + 6*Log[(2* Sqrt[a])/(Sqrt[a] + I*Sqrt[b]*x)] + 3*Log[a + b*x^2]) + Sqrt[b]*x*(24*a - 2*b*x^2 + (-9*a + 3*b*x^2)*Log[a + b*x^2]) + (9*I)*a^(3/2)*PolyLog[2, (I*S qrt[a] + Sqrt[b]*x)/((-I)*Sqrt[a] + Sqrt[b]*x)]))/(27*b^(3/2))) + (p^3*(41 6*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + b*x^2)]*Sqrt[a + b*x^2]*ArcSin[Sqrt[a ]/Sqrt[a + b*x^2]] + (2*Sqrt[-a]*b*x^2*(624*a - 16*b*x^2 + (-288*a + 24*b* x^2)*Log[a + b*x^2] + 18*(3*a - b*x^2)*Log[a + b*x^2]^2 + 9*b*x^2*Log[a + b*x^2]^3))/3 + 36*Sqrt[-a]*a^(3/2)*Sqrt[(b*x^2)/(a + b*x^2)]*(8*Sqrt[a]*Hy pergeometricPFQ[{1/2, 1/2, 1/2, 1/2}, {3/2, 3/2, 3/2}, a/(a + b*x^2)] + Lo g[a + b*x^2]*(4*Sqrt[a]*HypergeometricPFQ[{1/2, 1/2, 1/2}, {3/2, 3/2}, a/( a + b*x^2)] + Sqrt[a + b*x^2]*ArcSin[Sqrt[a]/Sqrt[a + b*x^2]]*Log[a + b*x^ 2])) - 48*a^2*(4*Sqrt[b*x^2]*ArcTanh[Sqrt[b*x^2]/Sqrt[-a]]*(Log[a + b*x^2] - Log[1 + (b*x^2)/a]) - Sqrt[-a]*Sqrt[-((b*x^2)/a)]*(Log[1 + (b*x^2)/a]^2 - 4*Log[1 + (b*x^2)/a]*Log[(1 + Sqrt[-((b*x^2)/a)])/2] + 2*Log[(1 + Sq...
Not integrable
Time = 0.97 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 0, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.000, Rules used = {2907, 2926, 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.
\(\displaystyle \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx\) |
\(\Big \downarrow \) 2907 |
\(\displaystyle \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-2 b p \int \frac {x^4 \log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^2+a}dx\) |
\(\Big \downarrow \) 2926 |
\(\displaystyle \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-2 b p \int \left (\frac {x^2 \log ^2\left (c \left (b x^2+a\right )^p\right )}{b}+\frac {a^2 \log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2 \left (b x^2+a\right )}-\frac {a \log ^2\left (c \left (b x^2+a\right )^p\right )}{b^2}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {1}{3} x^3 \log ^3\left (c \left (a+b x^2\right )^p\right )-2 b p \left (\frac {a^2 \int \frac {\log ^2\left (c \left (b x^2+a\right )^p\right )}{b x^2+a}dx}{b^2}-\frac {16 a^{3/2} p \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (c \left (a+b x^2\right )^p\right )}{3 b^{5/2}}-\frac {16 i a^{3/2} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )^2}{3 b^{5/2}}+\frac {104 a^{3/2} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{9 b^{5/2}}-\frac {32 a^{3/2} p^2 \arctan \left (\frac {\sqrt {b} x}{\sqrt {a}}\right ) \log \left (\frac {2 \sqrt {a}}{\sqrt {a}+i \sqrt {b} x}\right )}{3 b^{5/2}}-\frac {16 i a^{3/2} p^2 \operatorname {PolyLog}\left (2,1-\frac {2 \sqrt {a}}{i \sqrt {b} x+\sqrt {a}}\right )}{3 b^{5/2}}-\frac {a x \log ^2\left (c \left (a+b x^2\right )^p\right )}{b^2}+\frac {16 a p x \log \left (c \left (a+b x^2\right )^p\right )}{3 b^2}-\frac {104 a p^2 x}{9 b^2}+\frac {x^3 \log ^2\left (c \left (a+b x^2\right )^p\right )}{3 b}-\frac {4 p x^3 \log \left (c \left (a+b x^2\right )^p\right )}{9 b}+\frac {8 p^2 x^3}{27 b}\right )\) |
3.1.98.3.1 Defintions of rubi rules used
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_)*((f_.)*( x_))^(m_.), x_Symbol] :> Simp[(f*x)^(m + 1)*((a + b*Log[c*(d + e*x^n)^p])^q /(f*(m + 1))), x] - Simp[b*e*n*p*(q/(f^n*(m + 1))) Int[(f*x)^(m + n)*((a + b*Log[c*(d + e*x^n)^p])^(q - 1)/(d + e*x^n)), x], x] /; FreeQ[{a, b, c, d , e, f, m, p}, x] && IGtQ[q, 1] && IntegerQ[n] && NeQ[m, -1]
Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_)^(n_))^(p_.)]*(b_.))^(q_.)*(x_)^(m _.)*((f_) + (g_.)*(x_)^(s_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[(a + b *Log[c*(d + e*x^n)^p])^q, x^m*(f + g*x^s)^r, x], x] /; FreeQ[{a, b, c, d, e , f, g, m, n, p, q, r, s}, x] && IGtQ[q, 0] && IntegerQ[m] && IntegerQ[r] & & IntegerQ[s]
Not integrable
Time = 0.04 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00
\[\int x^{2} {\ln \left (c \left (b \,x^{2}+a \right )^{p}\right )}^{3}d x\]
Not integrable
Time = 0.30 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
Not integrable
Time = 3.88 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int x^{2} \log {\left (c \left (a + b x^{2}\right )^{p} \right )}^{3}\, dx \]
Not integrable
Time = 1.12 (sec) , antiderivative size = 123, normalized size of antiderivative = 6.83 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
1/3*p^3*x^3*log(b*x^2 + a)^3 + integrate((b*x^4*log(c)^3 + a*x^2*log(c)^3 - ((2*p^3 - 3*p^2*log(c))*b*x^4 - 3*a*p^2*x^2*log(c))*log(b*x^2 + a)^2 + 3 *(b*p*x^4*log(c)^2 + a*p*x^2*log(c)^2)*log(b*x^2 + a))/(b*x^2 + a), x)
Not integrable
Time = 0.38 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int { x^{2} \log \left ({\left (b x^{2} + a\right )}^{p} c\right )^{3} \,d x } \]
Not integrable
Time = 1.28 (sec) , antiderivative size = 20, normalized size of antiderivative = 1.11 \[ \int x^2 \log ^3\left (c \left (a+b x^2\right )^p\right ) \, dx=\int x^2\,{\ln \left (c\,{\left (b\,x^2+a\right )}^p\right )}^3 \,d x \]