Integrand size = 36, antiderivative size = 165 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=-\frac {g x}{b}-\frac {h x^2}{2 b}+\frac {\left (b c-\sqrt {a} \sqrt {b} e+a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/4}}+\frac {\left (b c+\sqrt {a} \sqrt {b} e+a g\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 a^{3/4} b^{5/4}}+\frac {(b d+a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {f \log \left (a-b x^4\right )}{4 b} \]
-g*x/b-1/2*h*x^2/b-1/4*f*ln(-b*x^4+a)/b+1/2*(a*h+b*d)*arctanh(x^2*b^(1/2)/ a^(1/2))/b^(3/2)/a^(1/2)+1/2*arctan(b^(1/4)*x/a^(1/4))*(b*c+a*g-e*a^(1/2)* b^(1/2))/a^(3/4)/b^(5/4)+1/2*arctanh(b^(1/4)*x/a^(1/4))*(b*c+a*g+e*a^(1/2) *b^(1/2))/a^(3/4)/b^(5/4)
Time = 0.24 (sec) , antiderivative size = 256, normalized size of antiderivative = 1.55 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=\frac {-4 a^{3/4} \sqrt {b} g x-2 a^{3/4} \sqrt {b} h x^2+2 \sqrt [4]{b} \left (b c-\sqrt {a} \sqrt {b} e+a g\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )-\left (b^{5/4} c+\sqrt [4]{a} b d+\sqrt {a} b^{3/4} e+a \sqrt [4]{b} g+a^{5/4} h\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )+\left (b^{5/4} c-\sqrt [4]{a} b d+\sqrt {a} b^{3/4} e+a \sqrt [4]{b} g-a^{5/4} h\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )+\sqrt [4]{a} (b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )-a^{3/4} \sqrt {b} f \log \left (a-b x^4\right )}{4 a^{3/4} b^{3/2}} \]
(-4*a^(3/4)*Sqrt[b]*g*x - 2*a^(3/4)*Sqrt[b]*h*x^2 + 2*b^(1/4)*(b*c - Sqrt[ a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - (b^(5/4)*c + a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g + a^(5/4)*h)*Log[a^(1/4) - b^(1/4)*x] + (b^(5/4)*c - a^(1/4)*b*d + Sqrt[a]*b^(3/4)*e + a*b^(1/4)*g - a^(5/4)*h)*L og[a^(1/4) + b^(1/4)*x] + a^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2] - a^(3/4)*Sqrt[b]*f*Log[a - b*x^4])/(4*a^(3/4)*b^(3/2))
Time = 0.43 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.056, Rules used = {2424, 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.
\(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx\) |
\(\Big \downarrow \) 2424 |
\(\displaystyle \int \left (\frac {c+e x^2+g x^4}{a-b x^4}+\frac {x \left (d+f x^2+h x^4\right )}{a-b x^4}\right )dx\) |
\(\Big \downarrow \) 2009 |
\(\displaystyle \frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt {a} \sqrt {b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e+a g+b c\right )}{2 a^{3/4} b^{5/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+b d)}{2 \sqrt {a} b^{3/2}}-\frac {f \log \left (a-b x^4\right )}{4 b}-\frac {g x}{b}-\frac {h x^2}{2 b}\) |
-((g*x)/b) - (h*x^2)/(2*b) + ((b*c - Sqrt[a]*Sqrt[b]*e + a*g)*ArcTan[(b^(1 /4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + ((b*c + Sqrt[a]*Sqrt[b]*e + a*g)*Ar cTanh[(b^(1/4)*x)/a^(1/4)])/(2*a^(3/4)*b^(5/4)) + ((b*d + a*h)*ArcTanh[(Sq rt[b]*x^2)/Sqrt[a]])/(2*Sqrt[a]*b^(3/2)) - (f*Log[a - b*x^4])/(4*b)
3.2.86.3.1 Defintions of rubi rules used
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[Pq, x, j + k*(n/2)]*x^(k*(n/2)), {k, 0, 2 *((q - j)/n) + 1}]*(a + b*x^n)^p, {j, 0, n/2 - 1}], x]] /; FreeQ[{a, b, p}, x] && PolyQ[Pq, x] && IGtQ[n/2, 0] && !PolyQ[Pq, x^(n/2)]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 1.58 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.45
method | result | size |
risch | \(-\frac {h \,x^{2}}{2 b}-\frac {g x}{b}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (b c +a g +\left (a h +b d \right ) \textit {\_R} +\textit {\_R}^{2} b e +\textit {\_R}^{3} b f \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(74\) |
default | \(-\frac {\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (a h +b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {e \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {f \ln \left (-b \,x^{4}+a \right )}{4}}{b}\) | \(180\) |
-1/2*h*x^2/b-g*x/b-1/4/b^2*sum((b*c+a*g+(a*h+b*d)*_R+_R^2*b*e+_R^3*b*f)/_R ^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=\text {Timed out} \]
Time = 0.30 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.35 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=-\frac {h x^{2} + 2 \, g x}{2 \, b} + \frac {\frac {2 \, {\left (b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b^{\frac {3}{2}} d - \sqrt {a} b f + a \sqrt {b} h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} d + \sqrt {a} b f + a \sqrt {b} h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} c + \sqrt {a} b e + a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, b} \]
-1/2*(h*x^2 + 2*g*x)/b + 1/4*(2*(b^(3/2)*c - sqrt(a)*b*e + a*sqrt(b)*g)*ar ctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt( b)) + (b^(3/2)*d - sqrt(a)*b*f + a*sqrt(b)*h)*log(sqrt(b)*x^2 + sqrt(a))/( sqrt(a)*b) - (b^(3/2)*d + sqrt(a)*b*f + a*sqrt(b)*h)*log(sqrt(b)*x^2 - sqr t(a))/(sqrt(a)*b) - (b^(3/2)*c + sqrt(a)*b*e + a*sqrt(b)*g)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sq rt(sqrt(a)*sqrt(b))*sqrt(b)))/b
Leaf count of result is larger than twice the leaf count of optimal. 338 vs. \(2 (123) = 246\).
Time = 0.27 (sec) , antiderivative size = 338, normalized size of antiderivative = 2.05 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{2} c + a b g - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + a b g + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (b^{2} c + a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (b^{2} c + a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}}} - \frac {f \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b} - \frac {b h x^{2} + 2 \, b g x}{2 \, b^{2}} \]
-1/4*sqrt(2)*(b^2*c + a*b*g - sqrt(2)*(-a*b^3)^(1/4)*b*d - sqrt(2)*(-a*b^3 )^(1/4)*a*h + sqrt(-a*b)*b*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/ 4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/4*sqrt(2)*(b^2*c + a*b*g + sqrt(2)*(- a*b^3)^(1/4)*b*d + sqrt(2)*(-a*b^3)^(1/4)*a*h - sqrt(-a*b)*b*e)*arctan(1/2 *sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(-a*b^3)^(3/4) - 1/8*s qrt(2)*(b^2*c + a*b*g - sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4) + 1/8*sqrt(2)*(b^2*c + a*b*g - sqrt(-a*b)*b*e) *log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(-a*b^3)^(3/4) - 1/4*f*log (abs(b*x^4 - a))/b - 1/2*(b*h*x^2 + 2*b*g*x)/b^2
Time = 9.92 (sec) , antiderivative size = 2478, normalized size of antiderivative = 15.02 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5}{a-b x^4} \, dx=\text {Too large to display} \]
symsum(log(- root(256*a^3*b^6*z^4 + 256*a^3*b^5*f*z^3 - 64*a^3*b^4*e*g*z^2 - 64*a^3*b^4*d*h*z^2 - 64*a^2*b^5*c*e*z^2 - 32*a^4*b^3*h^2*z^2 + 96*a^3*b ^4*f^2*z^2 - 32*a^2*b^5*d^2*z^2 - 32*a^3*b^3*e*f*g*z - 32*a^3*b^3*d*f*h*z + 32*a^3*b^3*c*g*h*z - 32*a^2*b^4*c*e*f*z + 32*a^2*b^4*c*d*g*z + 16*a^4*b^ 2*g^2*h*z - 16*a^4*b^2*f*h^2*z + 16*a^3*b^3*e^2*h*z + 16*a^3*b^3*d*g^2*z + 16*a^2*b^4*c^2*h*z - 16*a^2*b^4*d^2*f*z + 16*a^2*b^4*d*e^2*z + 16*a*b^5*c ^2*d*z + 16*a^3*b^3*f^3*z - 8*a^3*b^2*d*e*g*h + 8*a^3*b^2*c*f*g*h + 8*a^2* b^3*c*d*f*g - 8*a^2*b^3*c*d*e*h + 4*a^3*b^2*e^2*f*h - 4*a^3*b^2*e*f^2*g - 4*a^3*b^2*d*f^2*h + 4*a^3*b^2*d*f*g^2 + 4*a^2*b^3*c^2*f*h - 4*a^3*b^2*c*e* h^2 - 4*a^2*b^3*d^2*e*g + 4*a^2*b^3*d*e^2*f + 4*a^2*b^3*c*e^2*g - 4*a^2*b^ 3*c*e*f^2 + 4*a^4*b*f*g^2*h - 4*a^4*b*e*g*h^2 + 4*a*b^4*c^2*d*f - 4*a*b^4* c*d^2*e + 4*a^4*b*d*h^3 - 4*a*b^4*c^3*g + 6*a^3*b^2*d^2*h^2 + 2*a^3*b^2*e^ 2*g^2 - 6*a^2*b^3*c^2*g^2 - 2*a^2*b^3*d^2*f^2 - 2*a^4*b*f^2*h^2 + 4*a^2*b^ 3*d^3*h - 4*a^3*b^2*c*g^3 + 2*a*b^4*c^2*e^2 + a^3*b^2*f^4 + a*b^4*d^4 + a^ 5*h^4 - a^2*b^3*e^4 - a^4*b*g^4 - b^5*c^4, z, k)*((8*a*b^3*c*f - 8*a*b^3*d *e - 8*a^2*b^2*e*h + 8*a^2*b^2*f*g)/b + root(256*a^3*b^6*z^4 + 256*a^3*b^5 *f*z^3 - 64*a^3*b^4*e*g*z^2 - 64*a^3*b^4*d*h*z^2 - 64*a^2*b^5*c*e*z^2 - 32 *a^4*b^3*h^2*z^2 + 96*a^3*b^4*f^2*z^2 - 32*a^2*b^5*d^2*z^2 - 32*a^3*b^3*e* f*g*z - 32*a^3*b^3*d*f*h*z + 32*a^3*b^3*c*g*h*z - 32*a^2*b^4*c*e*f*z + 32* a^2*b^4*c*d*g*z + 16*a^4*b^2*g^2*h*z - 16*a^4*b^2*f*h^2*z + 16*a^3*b^3*...