Integrand size = 15, antiderivative size = 241 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\frac {x (c+d x)}{4 a \left (a+b x^4\right )}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}-\frac {3 c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} \sqrt [4]{b}}-\frac {3 c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} \sqrt [4]{b}} \]
1/4*x*(d*x+c)/a/(b*x^4+a)+3/16*c*arctan(-1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7 /4)/b^(1/4)*2^(1/2)+3/16*c*arctan(1+b^(1/4)*x*2^(1/2)/a^(1/4))/a^(7/4)/b^( 1/4)*2^(1/2)-3/32*c*ln(-a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/2))/a^( 7/4)/b^(1/4)*2^(1/2)+3/32*c*ln(a^(1/4)*b^(1/4)*x*2^(1/2)+a^(1/2)+x^2*b^(1/ 2))/a^(7/4)/b^(1/4)*2^(1/2)+1/4*d*arctan(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1 /2)
Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 0.93 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\frac {\frac {8 a^{3/4} x (c+d x)}{a+b x^4}-\frac {2 \left (3 \sqrt {2} \sqrt [4]{b} c+4 \sqrt [4]{a} d\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}+\frac {2 \left (3 \sqrt {2} \sqrt [4]{b} c-4 \sqrt [4]{a} d\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{\sqrt {b}}-\frac {3 \sqrt {2} c \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}+\frac {3 \sqrt {2} c \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{\sqrt [4]{b}}}{32 a^{7/4}} \]
((8*a^(3/4)*x*(c + d*x))/(a + b*x^4) - (2*(3*Sqrt[2]*b^(1/4)*c + 4*a^(1/4) *d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] + (2*(3*Sqrt[2]*b^(1/ 4)*c - 4*a^(1/4)*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/Sqrt[b] - (3* Sqrt[2]*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^(1/4) + (3*Sqrt[2]*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/b^( 1/4))/(32*a^(7/4))
Time = 0.39 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.02, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {2394, 25, 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.
\(\displaystyle \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx\) |
\(\Big \downarrow \) 2394 |
\(\displaystyle \frac {x (c+d x)}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 c+2 d x}{b x^4+a}dx}{4 a}\) |
\(\Big \downarrow \) 25 |
\(\displaystyle \frac {\int \frac {3 c+2 d x}{b x^4+a}dx}{4 a}+\frac {x (c+d x)}{4 a \left (a+b x^4\right )}\) |
\(\Big \downarrow \) 2415 |
\(\displaystyle \frac {\int \left (\frac {3 c}{b x^4+a}+\frac {2 d x}{b x^4+a}\right )dx}{4 a}+\frac {x (c+d x)}{4 a \left (a+b x^4\right )}\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {-\frac {3 c \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {3 c \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right )}{2 \sqrt {2} a^{3/4} \sqrt [4]{b}}-\frac {3 c \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {3 c \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} \sqrt [4]{b}}+\frac {d \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{4 a}+\frac {x (c+d x)}{4 a \left (a+b x^4\right )}\) |
(x*(c + d*x))/(4*a*(a + b*x^4)) + ((d*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt [a]*Sqrt[b]) - (3*c*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^ (3/4)*b^(1/4)) + (3*c*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]* a^(3/4)*b^(1/4)) - (3*c*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]* x^2])/(4*Sqrt[2]*a^(3/4)*b^(1/4)) + (3*c*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^( 1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(1/4)))/(4*a)
3.2.18.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[1/(a*n*(p + 1)) Int[ExpandToSum[n *(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b}, x ] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]
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 = 1.48 (sec) , antiderivative size = 66, normalized size of antiderivative = 0.27
method | result | size |
risch | \(\frac {\frac {d \,x^{2}}{4 a}+\frac {c x}{4 a}}{b \,x^{4}+a}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b +a \right )}{\sum }\frac {\left (2 \textit {\_R} d +3 c \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b a}\) | \(66\) |
default | \(c \left (\frac {x}{4 a \left (b \,x^{4}+a \right )}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}\right )+d \left (\frac {x^{2}}{4 a \left (b \,x^{4}+a \right )}+\frac {\arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{4 a \sqrt {a b}}\right )\) | \(163\) |
(1/4*d/a*x^2+1/4*c/a*x)/(b*x^4+a)+1/16/b/a*sum((2*_R*d+3*c)/_R^3*ln(x-_R), _R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 1.49 (sec) , antiderivative size = 43065, normalized size of antiderivative = 178.69 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \]
Time = 0.59 (sec) , antiderivative size = 155, normalized size of antiderivative = 0.64 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{2} + 2048 t^{2} a^{4} b d^{2} - 1152 t a^{2} b c^{2} d + 16 a d^{4} + 81 b c^{4}, \left ( t \mapsto t \log {\left (x + \frac {- 32768 t^{3} a^{6} b d^{2} - 4608 t^{2} a^{4} b c^{2} d - 512 t a^{3} d^{4} - 1296 t a^{2} b c^{4} + 360 a c^{2} d^{3}}{192 a c d^{4} - 243 b c^{5}} \right )} \right )\right )} + \frac {c x + d x^{2}}{4 a^{2} + 4 a b x^{4}} \]
RootSum(65536*_t**4*a**7*b**2 + 2048*_t**2*a**4*b*d**2 - 1152*_t*a**2*b*c* *2*d + 16*a*d**4 + 81*b*c**4, Lambda(_t, _t*log(x + (-32768*_t**3*a**6*b*d **2 - 4608*_t**2*a**4*b*c**2*d - 512*_t*a**3*d**4 - 1296*_t*a**2*b*c**4 + 360*a*c**2*d**3)/(192*a*c*d**4 - 243*b*c**5)))) + (c*x + d*x**2)/(4*a**2 + 4*a*b*x**4)
Time = 0.29 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\frac {d x^{2} + c x}{4 \, {\left (a b x^{4} + a^{2}\right )}} + \frac {\frac {3 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} - \frac {3 \, \sqrt {2} c \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {1}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c - 4 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} c + 4 \, \sqrt {a} d\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {1}{4}}}}{32 \, a} \]
1/4*(d*x^2 + c*x)/(a*b*x^4 + a^2) + 1/32*(3*sqrt(2)*c*log(sqrt(b)*x^2 + sq rt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - 3*sqrt(2)*c*log(sqr t(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) + 2*(3*s qrt(2)*a^(1/4)*b^(1/4)*c - 4*sqrt(a)*d)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sqrt (b))*b^(1/4)) + 2*(3*sqrt(2)*a^(1/4)*b^(1/4)*c + 4*sqrt(a)*d)*arctan(1/2*s qrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^( 3/4)*sqrt(sqrt(a)*sqrt(b))*b^(1/4)))/a
Time = 0.28 (sec) , antiderivative size = 238, normalized size of antiderivative = 0.99 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b} - \frac {3 \, \sqrt {2} \left (a b^{3}\right )^{\frac {1}{4}} c \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{2} b} + \frac {d x^{2} + c x}{4 \, {\left (b x^{4} + a\right )} a} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} + \frac {\sqrt {2} {\left (2 \, \sqrt {2} \sqrt {a b} b d + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b c\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, a^{2} b^{2}} \]
3/32*sqrt(2)*(a*b^3)^(1/4)*c*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/ (a^2*b) - 3/32*sqrt(2)*(a*b^3)^(1/4)*c*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + s qrt(a/b))/(a^2*b) + 1/4*(d*x^2 + c*x)/((b*x^4 + a)*a) + 1/16*sqrt(2)*(2*sq rt(2)*sqrt(a*b)*b*d + 3*(a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x + sqrt( 2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^2) + 1/16*sqrt(2)*(2*sqrt(2)*sqrt(a*b) *b*d + 3*(a*b^3)^(1/4)*b*c)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4)) /(a/b)^(1/4))/(a^2*b^2)
Time = 9.46 (sec) , antiderivative size = 282, normalized size of antiderivative = 1.17 \[ \int \frac {c+d x}{\left (a+b x^4\right )^2} \, dx=\left (\sum _{k=1}^4\ln \left (\frac {b^2\,\left (3\,c\,d^2+2\,d^3\,x-{\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,c\,192+{\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )}^2\,a^3\,b\,d\,x\,128-\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )\,a\,b\,c^2\,x\,36\right )}{a^3\,16}\right )\,\mathrm {root}\left (65536\,a^7\,b^2\,z^4+2048\,a^4\,b\,d^2\,z^2-1152\,a^2\,b\,c^2\,d\,z+81\,b\,c^4+16\,a\,d^4,z,k\right )\right )+\frac {\frac {d\,x^2}{4\,a}+\frac {c\,x}{4\,a}}{b\,x^4+a} \]
symsum(log((b^2*(3*c*d^2 + 2*d^3*x - 192*root(65536*a^7*b^2*z^4 + 2048*a^4 *b*d^2*z^2 - 1152*a^2*b*c^2*d*z + 81*b*c^4 + 16*a*d^4, z, k)^2*a^3*b*c + 1 28*root(65536*a^7*b^2*z^4 + 2048*a^4*b*d^2*z^2 - 1152*a^2*b*c^2*d*z + 81*b *c^4 + 16*a*d^4, z, k)^2*a^3*b*d*x - 36*root(65536*a^7*b^2*z^4 + 2048*a^4* b*d^2*z^2 - 1152*a^2*b*c^2*d*z + 81*b*c^4 + 16*a*d^4, z, k)*a*b*c^2*x))/(1 6*a^3))*root(65536*a^7*b^2*z^4 + 2048*a^4*b*d^2*z^2 - 1152*a^2*b*c^2*d*z + 81*b*c^4 + 16*a*d^4, z, k), k, 1, 4) + ((d*x^2)/(4*a) + (c*x)/(4*a))/(a + b*x^4)