Integrand size = 17, antiderivative size = 318 \[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=-\frac {c \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b} d^3}-\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4} d^3}+\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3}-\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4} d^3} \]
-c*arctan((d*x+c)^2*b^(1/2)/a^(1/2))/d^3/a^(1/2)/b^(1/2)+1/8*ln(-a^(1/4)*b ^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+(d*x+c)^2*b^(1/2))*(a^(1/2)-b^(1/2)*c^2)/a^ (3/4)/b^(3/4)/d^3*2^(1/2)-1/8*ln(a^(1/4)*b^(1/4)*(d*x+c)*2^(1/2)+a^(1/2)+( d*x+c)^2*b^(1/2))*(a^(1/2)-b^(1/2)*c^2)/a^(3/4)/b^(3/4)/d^3*2^(1/2)+1/4*ar ctan(-1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))*(a^(1/2)+b^(1/2)*c^2)/a^(3/4)/b^( 3/4)/d^3*2^(1/2)+1/4*arctan(1+b^(1/4)*(d*x+c)*2^(1/2)/a^(1/4))*(a^(1/2)+b^ (1/2)*c^2)/a^(3/4)/b^(3/4)/d^3*2^(1/2)
Result contains higher order function than in optimal. Order 9 vs. order 3 in optimal.
Time = 0.02 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.33 \[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\frac {\text {RootSum}\left [a+b c^4+4 b c^3 d \text {$\#$1}+6 b c^2 d^2 \text {$\#$1}^2+4 b c d^3 \text {$\#$1}^3+b d^4 \text {$\#$1}^4\&,\frac {\log (x-\text {$\#$1}) \text {$\#$1}^2}{c^3+3 c^2 d \text {$\#$1}+3 c d^2 \text {$\#$1}^2+d^3 \text {$\#$1}^3}\&\right ]}{4 b d} \]
RootSum[a + b*c^4 + 4*b*c^3*d*#1 + 6*b*c^2*d^2*#1^2 + 4*b*c*d^3*#1^3 + b*d ^4*#1^4 & , (Log[x - #1]*#1^2)/(c^3 + 3*c^2*d*#1 + 3*c*d^2*#1^2 + d^3*#1^3 ) & ]/(4*b*d)
Time = 0.43 (sec) , antiderivative size = 307, normalized size of antiderivative = 0.97, number of steps used = 4, number of rules used = 3, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {896, 2415, 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.
| \(\displaystyle \int \frac {x^2}{a+b (c+d x)^4} \, dx\) |
| \(\Big \downarrow \) 896 |
| \(\displaystyle \frac {\int \frac {d^2 x^2}{b (c+d x)^4+a}d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\int \left (\frac {c^2+(c+d x)^2}{b (c+d x)^4+a}-\frac {2 c (c+d x)}{b (c+d x)^4+a}\right )d(c+d x)}{d^3}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {-\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}+\sqrt {b} c^2\right ) \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} (c+d x)}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {\left (\sqrt {a}-\sqrt {b} c^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} (c+d x)+\sqrt {a}+\sqrt {b} (c+d x)^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}-\frac {c \arctan \left (\frac {\sqrt {b} (c+d x)^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{d^3}\) |
(-((c*ArcTan[(Sqrt[b]*(c + d*x)^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[b])) - ((Sqrt[a ] + Sqrt[b]*c^2)*ArcTan[1 - (Sqrt[2]*b^(1/4)*(c + d*x))/a^(1/4)])/(2*Sqrt[ 2]*a^(3/4)*b^(3/4)) + ((Sqrt[a] + Sqrt[b]*c^2)*ArcTan[1 + (Sqrt[2]*b^(1/4) *(c + d*x))/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) + ((Sqrt[a] - Sqrt[b]*c^ 2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2]) /(4*Sqrt[2]*a^(3/4)*b^(3/4)) - ((Sqrt[a] - Sqrt[b]*c^2)*Log[Sqrt[a] + Sqrt [2]*a^(1/4)*b^(1/4)*(c + d*x) + Sqrt[b]*(c + d*x)^2])/(4*Sqrt[2]*a^(3/4)*b ^(3/4)))/d^3
3.2.11.3.1 Defintions of rubi rules used
Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coeff icient[v, x, 0], d = Coefficient[v, x, 1]}, Simp[1/d^(m + 1) Subst[Int[Si mplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> With[{v = Sum[x^ii*((Coeff [Pq, x, ii] + Coeff[Pq, x, n/2 + ii]*x^(n/2))/(a + b*x^n)), {ii, 0, n/2 - 1 }]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && Expon[Pq, x] < n
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.08 (sec) , antiderivative size = 97, normalized size of antiderivative = 0.31
| method | result | size |
| default | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(97\) |
| risch | \(\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,d^{4} \textit {\_Z}^{4}+4 b c \,d^{3} \textit {\_Z}^{3}+6 b \,c^{2} d^{2} \textit {\_Z}^{2}+4 b \,c^{3} d \textit {\_Z} +b \,c^{4}+a \right )}{\sum }\frac {\textit {\_R}^{2} \ln \left (x -\textit {\_R} \right )}{d^{3} \textit {\_R}^{3}+3 c \,d^{2} \textit {\_R}^{2}+3 c^{2} d \textit {\_R} +c^{3}}}{4 b d}\) | \(97\) |
1/4/b/d*sum(_R^2/(_R^3*d^3+3*_R^2*c*d^2+3*_R*c^2*d+c^3)*ln(x-_R),_R=RootOf (_Z^4*b*d^4+4*_Z^3*b*c*d^3+6*_Z^2*b*c^2*d^2+4*_Z*b*c^3*d+b*c^4+a))
Result contains complex when optimal does not.
Time = 58.70 (sec) , antiderivative size = 61993, normalized size of antiderivative = 194.95 \[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\text {Too large to display} \]
Time = 1.60 (sec) , antiderivative size = 274, normalized size of antiderivative = 0.86 \[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\operatorname {RootSum} {\left (256 t^{4} a^{3} b^{3} d^{12} + 192 t^{2} a^{2} b^{2} c^{2} d^{6} + t \left (- 32 a^{2} b c d^{3} + 32 a b^{2} c^{5} d^{3}\right ) + a^{2} + 2 a b c^{4} + b^{2} c^{8}, \left ( t \mapsto t \log {\left (x + \frac {64 t^{3} a^{4} b^{2} d^{9} + 448 t^{3} a^{3} b^{3} c^{4} d^{9} + 160 t^{2} a^{3} b^{2} c^{3} d^{6} - 32 t^{2} a^{2} b^{3} c^{7} d^{6} + 60 t a^{3} b c^{2} d^{3} + 256 t a^{2} b^{2} c^{6} d^{3} + 4 t a b^{3} c^{10} d^{3} - 5 a^{3} c - 9 a^{2} b c^{5} - 3 a b^{2} c^{9} + b^{3} c^{13}}{a^{3} d - 33 a^{2} b c^{4} d - 33 a b^{2} c^{8} d + b^{3} c^{12} d} \right )} \right )\right )} \]
RootSum(256*_t**4*a**3*b**3*d**12 + 192*_t**2*a**2*b**2*c**2*d**6 + _t*(-3 2*a**2*b*c*d**3 + 32*a*b**2*c**5*d**3) + a**2 + 2*a*b*c**4 + b**2*c**8, La mbda(_t, _t*log(x + (64*_t**3*a**4*b**2*d**9 + 448*_t**3*a**3*b**3*c**4*d* *9 + 160*_t**2*a**3*b**2*c**3*d**6 - 32*_t**2*a**2*b**3*c**7*d**6 + 60*_t* a**3*b*c**2*d**3 + 256*_t*a**2*b**2*c**6*d**3 + 4*_t*a*b**3*c**10*d**3 - 5 *a**3*c - 9*a**2*b*c**5 - 3*a*b**2*c**9 + b**3*c**13)/(a**3*d - 33*a**2*b* c**4*d - 33*a*b**2*c**8*d + b**3*c**12*d))))
\[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\int { \frac {x^{2}}{{\left (d x + c\right )}^{4} b + a} \,d x } \]
\[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\int { \frac {x^{2}}{{\left (d x + c\right )}^{4} b + a} \,d x } \]
Time = 9.27 (sec) , antiderivative size = 625, normalized size of antiderivative = 1.97 \[ \int \frac {x^2}{a+b (c+d x)^4} \, dx=\sum _{k=1}^4\ln \left (-b\,d^4\,\left (a+b\,c^4+4\,b\,c^3\,d\,x+\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,b^2\,c^5\,d^3\,4+\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,b^2\,c^4\,d^4\,x\,4-\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,a\,b\,c\,d^3\,20-\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )\,a\,b\,d^4\,x\,4+{\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )}^2\,a\,b^2\,c^2\,d^6\,48+{\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right )}^2\,a\,b^2\,c\,d^7\,x\,32\right )\right )\,\mathrm {root}\left (256\,a^3\,b^3\,d^{12}\,z^4+192\,a^2\,b^2\,c^2\,d^6\,z^2+32\,a\,b^2\,c^5\,d^3\,z-32\,a^2\,b\,c\,d^3\,z+2\,a\,b\,c^4+b^2\,c^8+a^2,z,k\right ) \]
symsum(log(-b*d^4*(a + b*c^4 + 4*b*c^3*d*x + 4*root(256*a^3*b^3*d^12*z^4 + 192*a^2*b^2*c^2*d^6*z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c ^4 + b^2*c^8 + a^2, z, k)*b^2*c^5*d^3 + 4*root(256*a^3*b^3*d^12*z^4 + 192* a^2*b^2*c^2*d^6*z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^2*c^8 + a^2, z, k)*b^2*c^4*d^4*x - 20*root(256*a^3*b^3*d^12*z^4 + 192*a^ 2*b^2*c^2*d^6*z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^ 2*c^8 + a^2, z, k)*a*b*c*d^3 - 4*root(256*a^3*b^3*d^12*z^4 + 192*a^2*b^2*c ^2*d^6*z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^2*c^8 + a^2, z, k)*a*b*d^4*x + 48*root(256*a^3*b^3*d^12*z^4 + 192*a^2*b^2*c^2*d^6 *z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^2*c^8 + a^2, z, k)^2*a*b^2*c^2*d^6 + 32*root(256*a^3*b^3*d^12*z^4 + 192*a^2*b^2*c^2*d^6 *z^2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^2*c^8 + a^2, z, k)^2*a*b^2*c*d^7*x))*root(256*a^3*b^3*d^12*z^4 + 192*a^2*b^2*c^2*d^6*z^ 2 + 32*a*b^2*c^5*d^3*z - 32*a^2*b*c*d^3*z + 2*a*b*c^4 + b^2*c^8 + a^2, z, k), k, 1, 4)