Integrand size = 46, antiderivative size = 205 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b}-\frac {j x^4}{4 b}-\frac {\left (b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(b d+a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {(b f+a j) \log \left (a-b x^4\right )}{4 b^2} \]
-g*x/b-1/2*h*x^2/b-1/3*i*x^3/b-1/4*j*x^4/b-1/4*(a*j+b*f)*ln(-b*x^4+a)/b^2+ 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*e+a*i-(a*g+b*c)*b^(1/2)/a^(1/2))/a^(1/4)/b^(7/4)+1/2*arc tanh(b^(1/4)*x/a^(1/4))*(b*e+a*i+(a*g+b*c)*b^(1/2)/a^(1/2))/a^(1/4)/b^(7/4 )
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.55 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\frac {-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3-3 b^{3/4} j x^4+\frac {6 \left (b^{3/2} c-\sqrt {a} b e+a \sqrt {b} g-a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {3 \left (b^{3/2} c+\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g+a^{5/4} \sqrt [4]{b} h+a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{a^{3/4}}+\frac {3 \left (b^{3/2} c-\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g-a^{5/4} \sqrt [4]{b} h+a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{a^{3/4}}+\frac {3 \sqrt [4]{b} (b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {a}}-\frac {3 (b f+a j) \log \left (a-b x^4\right )}{\sqrt [4]{b}}}{12 b^{7/4}} \]
(-12*b^(3/4)*g*x - 6*b^(3/4)*h*x^2 - 4*b^(3/4)*i*x^3 - 3*b^(3/4)*j*x^4 + ( 6*(b^(3/2)*c - Sqrt[a]*b*e + a*Sqrt[b]*g - a^(3/2)*i)*ArcTan[(b^(1/4)*x)/a ^(1/4)])/a^(3/4) - (3*(b^(3/2)*c + a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqr t[b]*g + a^(5/4)*b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) - b^(1/4)*x])/a^(3/4) + (3*(b^(3/2)*c - a^(1/4)*b^(5/4)*d + Sqrt[a]*b*e + a*Sqrt[b]*g - a^(5/4)* b^(1/4)*h + a^(3/2)*i)*Log[a^(1/4) + b^(1/4)*x])/a^(3/4) + (3*b^(1/4)*(b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[a] - (3*(b*f + a*j)*Log[a - b*x^4 ])/b^(1/4))/(12*b^(7/4))
Time = 0.50 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, 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.
| \(\displaystyle \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx\) |
| \(\Big \downarrow \) 2424 |
| \(\displaystyle \int \left (\frac {c+e x^2+g x^4+i x^6}{a-b x^4}+\frac {x \left (d+f x^2+h x^4+j x^6\right )}{a-b x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+b d)}{2 \sqrt {a} b^{3/2}}-\frac {(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b}-\frac {j x^4}{4 b}\) |
-((g*x)/b) - (h*x^2)/(2*b) - (i*x^3)/(3*b) - (j*x^4)/(4*b) - ((b*e - (Sqrt [b]*(b*c + a*g))/Sqrt[a] + a*i)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(2*a^(1/4)*b^ (7/4)) + ((b*e + (Sqrt[b]*(b*c + a*g))/Sqrt[a] + a*i)*ArcTanh[(b^(1/4)*x)/ a^(1/4)])/(2*a^(1/4)*b^(7/4)) + ((b*d + a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a] ])/(2*Sqrt[a]*b^(3/2)) - ((b*f + a*j)*Log[a - b*x^4])/(4*b^2)
3.2.88.3.1 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 = 1.57 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.50
| method | result | size |
| risch | \(-\frac {j \,x^{4}}{4 b}-\frac {i \,x^{3}}{3 b}-\frac {h \,x^{2}}{2 b}-\frac {g x}{b}-\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{4} b -a \right )}{\sum }\frac {\left (b c +a g +\left (a h +b d \right ) \textit {\_R} +\left (a i +b e \right ) \textit {\_R}^{2}+\left (a j +b f \right ) \textit {\_R}^{3}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{4 b^{2}}\) | \(102\) |
| default | \(-\frac {\frac {1}{4} j \,x^{4}+\frac {1}{3} i \,x^{3}+\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (a h +b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (a i +b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\left (a j +b f \right ) \ln \left (-b \,x^{4}+a \right )}{4 b}}{b}\) | \(210\) |
-1/4*j*x^4/b-1/3*i*x^3/b-1/2*h*x^2/b-g*x/b-1/4/b^2*sum((b*c+a*g+(a*h+b*d)* _R+(a*i+b*e)*_R^2+(a*j+b*f)*_R^3)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b-a))
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Timed out} \]
Timed out. \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Timed out} \]
Time = 0.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.25 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {3 \, j x^{4} + 4 \, i x^{3} + 6 \, h x^{2} + 12 \, g x}{12 \, b} + \frac {\frac {2 \, {\left (b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g - a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b^{\frac {3}{2}} d - \sqrt {a} b f + a \sqrt {b} h - a^{\frac {3}{2}} j\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} d + \sqrt {a} b f + a \sqrt {b} h + a^{\frac {3}{2}} j\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} c + \sqrt {a} b e + a \sqrt {b} g + a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, b} \]
-1/12*(3*j*x^4 + 4*i*x^3 + 6*h*x^2 + 12*g*x)/b + 1/4*(2*(b^(3/2)*c - sqrt( a)*b*e + a*sqrt(b)*g - a^(3/2)*i)*arctan(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*sqr t(b)*h - a^(3/2)*j)*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*b) - (b^(3/2)*d + sqrt(a)*b*f + a*sqrt(b)*h + a^(3/2)*j)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a) *b) - (b^(3/2)*c + sqrt(a)*b*e + a*sqrt(b)*g + a^(3/2)*i)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt (sqrt(a)*sqrt(b))*sqrt(b)))/b
Leaf count of result is larger than twice the leaf count of optimal. 445 vs. \(2 (159) = 318\).
Time = 0.28 (sec) , antiderivative size = 445, normalized size of antiderivative = 2.17 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - \sqrt {-a b} b^{2} e + \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} + \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {{\left (b f + a j\right )} \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b^{2}} - \frac {3 \, b^{3} j x^{4} + 4 \, b^{3} i x^{3} + 6 \, b^{3} h x^{2} + 12 \, b^{3} g x}{12 \, b^{4}} \]
-1/4*sqrt(2)*(b^3*c + a*b^2*g - sqrt(2)*(-a*b^3)^(1/4)*b^2*d - sqrt(2)*(-a *b^3)^(1/4)*a*b*h - sqrt(-a*b)*b^2*e - sqrt(-a*b)*a*b*i)*arctan(1/2*sqrt(2 )*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*b) - 1/4*sqrt (2)*(b^3*c + a*b^2*g + sqrt(2)*(-a*b^3)^(1/4)*b^2*d + sqrt(2)*(-a*b^3)^(1/ 4)*a*b*h - sqrt(-a*b)*b^2*e + sqrt(-a*b)*a*b*i)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/((-a*b^3)^(3/4)*b) - 1/8*sqrt(2)*(b^3* c + a*b^2*g - sqrt(-a*b)*b^2*e - sqrt(-a*b)*a*b*i)*log(x^2 + sqrt(2)*x*(-a /b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*b) + 1/8*sqrt(2)*(b^3*c + a*b^2*g - sqrt(-a*b)*b^2*e - sqrt(-a*b)*a*b*i)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*b) - 1/4*(b*f + a*j)*log(abs(b*x^4 - a))/b^2 - 1/12*(3*b^3*j*x^4 + 4*b^3*i*x^3 + 6*b^3*h*x^2 + 12*b^3*g*x)/b^4
Time = 9.56 (sec) , antiderivative size = 5673, normalized size of antiderivative = 27.67 \[ \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Too large to display} \]
symsum(log(- (a^4*i^3 + a*b^3*e^3 + b^4*c*d^2 - b^4*c^2*e + a^4*g*j^2 + a^ 2*b^2*c*h^2 - a^2*b^2*e*g^2 + a^2*b^2*f^2*g + 3*a^2*b^2*e^2*i - 2*a^4*h*i* j + a*b^3*c*f^2 + a*b^3*d^2*g - a*b^3*c^2*i + a^3*b*c*j^2 + 3*a^3*b*e*i^2 + a^3*b*g*h^2 - a^3*b*g^2*i + 2*a^2*b^2*c*f*j - 2*a^2*b^2*c*g*i - 2*a^2*b^ 2*d*e*j - 2*a^2*b^2*d*f*i + 2*a^2*b^2*d*g*h - 2*a^2*b^2*e*f*h + 2*a*b^3*c* d*h - 2*a*b^3*c*e*g - 2*a*b^3*d*e*f - 2*a^3*b*d*i*j - 2*a^3*b*e*h*j + 2*a^ 3*b*f*g*j - 2*a^3*b*f*h*i)/b^2 - root(256*a^3*b^8*z^4 + 256*a^4*b^6*j*z^3 + 256*a^3*b^7*f*z^3 + 192*a^4*b^5*f*j*z^2 - 64*a^4*b^5*g*i*z^2 - 64*a^3*b^ 6*e*g*z^2 - 64*a^3*b^6*d*h*z^2 - 64*a^3*b^6*c*i*z^2 - 64*a^2*b^7*c*e*z^2 + 96*a^5*b^4*j^2*z^2 - 32*a^4*b^5*h^2*z^2 + 96*a^3*b^6*f^2*z^2 - 32*a^2*b^7 *d^2*z^2 - 32*a^5*b^3*g*i*j*z - 32*a^4*b^4*f*g*i*z + 32*a^4*b^4*e*h*i*z - 32*a^4*b^4*e*g*j*z - 32*a^4*b^4*d*h*j*z - 32*a^4*b^4*c*i*j*z - 32*a^3*b^5* e*f*g*z - 32*a^3*b^5*d*f*h*z + 32*a^3*b^5*d*e*i*z + 32*a^3*b^5*c*g*h*z - 3 2*a^3*b^5*c*f*i*z - 32*a^3*b^5*c*e*j*z - 32*a^2*b^6*c*e*f*z + 32*a^2*b^6*c *d*g*z - 16*a^5*b^3*h^2*j*z + 16*a^5*b^3*h*i^2*z + 48*a^5*b^3*f*j^2*z + 48 *a^4*b^4*f^2*j*z + 16*a^4*b^4*g^2*h*z - 16*a^4*b^4*f*h^2*z - 16*a^3*b^5*d^ 2*j*z + 16*a^4*b^4*d*i^2*z + 16*a^3*b^5*e^2*h*z + 16*a^3*b^5*d*g^2*z + 16* a^2*b^6*c^2*h*z - 16*a^2*b^6*d^2*f*z + 16*a^2*b^6*d*e^2*z + 16*a*b^7*c^2*d *z + 16*a^6*b^2*j^3*z + 16*a^3*b^5*f^3*z - 8*a^5*b^2*f*g*i*j + 8*a^5*b^2*e *h*i*j + 8*a^4*b^3*e*f*h*i - 8*a^4*b^3*e*f*g*j - 8*a^4*b^3*d*g*h*i - 8*...