\(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx\) [188]
Optimal result
Integrand size = 46, antiderivative size = 205 \[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b}-\frac {j x^4}{4 b}-\frac {\left (b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\left (b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(b d+a h) \text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {(b f+a j) \log \left (a-b x^4\right )}{4 b^2}
\]
[Out]
-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*a
rctanh(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)
Rubi [A] (verified)
Time = 0.21 (sec) , antiderivative size = 205, normalized size of antiderivative = 1.00, number of
steps used = 13, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.196, Rules used = {1899, 1901, 1181, 211, 214,
1833, 1824, 649, 266} \[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {\arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\text {arctanh}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right ) (a h+b d)}{2 \sqrt {a} b^{3/2}}-\frac {(a j+b f) \log \left (a-b x^4\right )}{4 b^2}-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b}-\frac {j x^4}{4 b}
\]
[In]
Int[(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),x]
[Out]
-((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)*ArcT
an[(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)
Rule 211
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]/a)*ArcTan[x/Rt[a/b, 2]], x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]
Rule 214
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x/Rt[-a/b, 2]], x] /; FreeQ[{a, b},
x] && NegQ[a/b]
Rule 266
Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]
Rule 649
Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] && !NiceSqrtQ[(-a)*c]
Rule 1181
Int[((d_) + (e_.)*(x_)^2)/((a_) + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[(-a)*c, 2]}, Dist[e/2 + c*(d/(2*q))
, Int[1/(-q + c*x^2), x], x] + Dist[e/2 - c*(d/(2*q)), Int[1/(q + c*x^2), x], x]] /; FreeQ[{a, c, d, e}, x] &&
NeQ[c*d^2 - a*e^2, 0] && PosQ[(-a)*c]
Rule 1824
Int[(Pq_)*((a_) + (b_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[Pq*(a + b*x^2)^p, x], x] /; FreeQ[{a,
b}, x] && PolyQ[Pq, x] && IGtQ[p, -2]
Rule 1833
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/(m + 1), Subst[Int[SubstFor[x^(m + 1)
, Pq, x]*(a + b*x^Simplify[n/(m + 1)])^p, x], x, x^(m + 1)], x] /; FreeQ[{a, b, m, n, p}, x] && NeQ[m, -1] &&
IGtQ[Simplify[n/(m + 1)], 0] && PolyQ[Pq, x^(m + 1)]
Rule 1899
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], j, k}, Int[Sum[x^j*Sum[Coeff[P
q, 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)]
Rule 1901
Int[(Pq_)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Int[ExpandIntegrand[Pq/(a + b*x^n), x], x] /; FreeQ[{a, b}, x
] && PolyQ[Pq, x] && IntegerQ[n]
Rubi steps \begin{align*}
\text {integral}& = \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 \\ & = \int \frac {c+e x^2+g x^4+i x^6}{a-b x^4} \, dx+\int \frac {x \left (d+f x^2+h x^4+j x^6\right )}{a-b x^4} \, dx \\ & = \frac {1}{2} \text {Subst}\left (\int \frac {d+f x+h x^2+j x^3}{a-b x^2} \, dx,x,x^2\right )+\int \left (-\frac {g}{b}-\frac {i x^2}{b}+\frac {b c+a g+(b e+a i) x^2}{b \left (a-b x^4\right )}\right ) \, dx \\ & = -\frac {g x}{b}-\frac {i x^3}{3 b}+\frac {1}{2} \text {Subst}\left (\int \left (-\frac {h}{b}-\frac {j x}{b}+\frac {b d+a h+(b f+a j) x}{b \left (a-b x^2\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {b c+a g+(b e+a i) x^2}{a-b x^4} \, dx}{b} \\ & = -\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b}-\frac {j x^4}{4 b}+\frac {\text {Subst}\left (\int \frac {b d+a h+(b f+a j) x}{a-b x^2} \, dx,x,x^2\right )}{2 b}+\frac {\left (b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{2 b}+\frac {\left (b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}+a i\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{2 b} \\ & = -\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 ) \tan ^{-1}\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 ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(b d+a h) \text {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{2 b}+\frac {(b f+a j) \text {Subst}\left (\int \frac {x}{a-b x^2} \, dx,x,x^2\right )}{2 b} \\ & = -\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 ) \tan ^{-1}\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 ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(b d+a h) \tanh ^{-1}\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} \\
\end{align*}
Mathematica [A] (verified)
Time = 0.31 (sec) , antiderivative size = 318, normalized size of antiderivative = 1.55
\[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\frac {-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3-3 b^{3/4} j x^4+\frac {6 \left (b^{3/2} c-\sqrt {a} b e+a \sqrt {b} g-a^{3/2} i\right ) \arctan \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{3/4}}-\frac {3 \left (b^{3/2} c+\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g+a^{5/4} \sqrt [4]{b} h+a^{3/2} i\right ) \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right )}{a^{3/4}}+\frac {3 \left (b^{3/2} c-\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g-a^{5/4} \sqrt [4]{b} h+a^{3/2} i\right ) \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right )}{a^{3/4}}+\frac {3 \sqrt [4]{b} (b d+a h) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {a}}-\frac {3 (b f+a j) \log \left (a-b x^4\right )}{\sqrt [4]{b}}}{12 b^{7/4}}
\]
[In]
Integrate[(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),x]
[Out]
(-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*S
qrt[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))
Maple [C] (verified)
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 |
\(-\frac {\frac {1}{4} j \,x^{4}+\frac {1}{3} i \,x^{3}+\frac {1}{2} h \,x^{2}+g x}{b}+\frac {\frac {\left (a g +b c \right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \left (\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )+2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 a}+\frac {\left (a h +b d \right ) \ln \left (\frac {a +x^{2} \sqrt {a b}}{a -x^{2} \sqrt {a b}}\right )}{4 \sqrt {a b}}-\frac {\left (a i +b e \right ) \left (2 \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )-\ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )\right )}{4 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}-\frac {\left (a j +b f \right ) \ln \left (-b \,x^{4}+a \right )}{4 b}}{b}\) |
\(210\) |
| | |
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[In]
int((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x,method=_RETURNVERBOSE)
[Out]
-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))
Fricas [F(-1)]
Timed out. \[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Timed out}
\]
[In]
integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="fricas")
[Out]
Timed out
Sympy [F(-1)]
Timed out. \[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Timed out}
\]
[In]
integrate((j*x**7+i*x**6+h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a),x)
[Out]
Timed out
Maxima [A] (verification not implemented)
none
Time = 0.29 (sec) , antiderivative size = 257, normalized size of antiderivative = 1.25
\[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {3 \, j x^{4} + 4 \, i x^{3} + 6 \, h x^{2} + 12 \, g x}{12 \, b} + \frac {\frac {2 \, {\left (b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g - a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b^{\frac {3}{2}} d - \sqrt {a} b f + a \sqrt {b} h - a^{\frac {3}{2}} j\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} d + \sqrt {a} b f + a \sqrt {b} h + a^{\frac {3}{2}} j\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} c + \sqrt {a} b e + a \sqrt {b} g + a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, b}
\]
[In]
integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="maxima")
[Out]
-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)*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*s
qrt(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
Giac [B] (verification not implemented)
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
\[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=-\frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d - \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b^{2} d + \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a b h - \sqrt {-a b} b^{2} e + \sqrt {-a b} a b i\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} + \frac {\sqrt {2} {\left (b^{3} c + a b^{2} g - \sqrt {-a b} b^{2} e - \sqrt {-a b} a b i\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{8 \, \left (-a b^{3}\right )^{\frac {3}{4}} b} - \frac {{\left (b f + a j\right )} \log \left ({\left | b x^{4} - a \right |}\right )}{4 \, b^{2}} - \frac {3 \, b^{3} j x^{4} + 4 \, b^{3} i x^{3} + 6 \, b^{3} h x^{2} + 12 \, b^{3} g x}{12 \, b^{4}}
\]
[In]
integrate((j*x^7+i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x, algorithm="giac")
[Out]
-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*sq
rt(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
Mupad [B] (verification not implemented)
Time = 9.56 (sec) , antiderivative size = 5673, normalized size of antiderivative = 27.67
\[
\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6+j x^7}{a-b x^4} \, dx=\text {Too large to display}
\]
[In]
int((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),x)
[Out]
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 - 32*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*a^4*b^3*d*f*h*j + 8*a^4*b^3*d*e*i*j + 8*a^4*b^3*c*g*h*j - 8*a^4
*b^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*d*e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e*g*i - 8*a^3*b^4*c*e*f
*j - 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j + 8*a^2*b^5*c*d*f*g - 8*a^2*b^5*c*d*e*h + 4*a^5*b^2*g^2*h*j - 4*a^5
*b^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*f*h*i^2 + 4*a^5*b^2*d*i^2*j + 4*a^4*b^3*e^2*h*j - 4*a^5*b^2*e*g*j
^2 - 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 - 4*a^4*b^3*f^2*g*i + 4*a^4*b^3*f*g^2*h + 4*a^4*b^3*e*g^2*i + 4*a^4
*b^3*d*g^2*j + 4*a^3*b^4*c^2*h*j - 4*a^4*b^3*e*g*h^2 - 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2*g*i - 4*a^3*b^4*d^2*f
*j + 4*a^4*b^3*d*f*i^2 + 4*a^4*b^3*c*g*i^2 + 4*a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j - 4*a^4*b^3*c*e*j^2 - 4*a^3
*b^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*c*f^2*i + 4*a^3*b^4*d*f*g^2 + 4*a^2*b^5*c^2*f*h + 4*a^2*b^5*c^2*e
*i + 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 - 4*a^2*b^5*d^2*e*g - 4*a^2*b^5*c*d^2*i + 4*a^2*b^5*d*e^2*f + 4*a^2
*b^5*c*e^2*g - 4*a^2*b^5*c*e*f^2 + 4*a^6*b*h*i^2*j - 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f - 4*a*b^6*c*d^2*e + 4*a
^6*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^2 + 2*a^5*b^2*g^2*i^2 - 6*a^4*b^3*e^2*i^2 - 2*a^4*b^3*f^2*h^2 - 2
*a^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 - 6*a^2*b^5*c^2*g^2 - 2*a^2*b^5*d
^2*f^2 - 2*a^6*b*h^2*j^2 + 4*a^4*b^3*f^3*j - 4*a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i + 4*a^4*b^3*d*h^3 + 4*a^2*b^5*d
^3*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^5*b^2*h^4 + a^3*b^4*f^4 + a*b^6*d^4 + a^7*j^4 - a^4*b^3*g^4 - a^2
*b^5*e^4 - a^6*b*i^4 - b^7*c^4, z, m)*((8*a*b^4*c*f - 8*a*b^4*d*e + 8*a^2*b^3*c*j - 8*a^2*b^3*d*i - 8*a^2*b^3*
e*h + 8*a^2*b^3*f*g + 8*a^3*b^2*g*j - 8*a^3*b^2*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 - 32*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 + 1
6*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*a^4*b^3*d*f*h*j + 8*a^4*b^3*d*e*i*j + 8*a
^4*b^3*c*g*h*j - 8*a^4*b^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*d*e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e
*g*i - 8*a^3*b^4*c*e*f*j - 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j + 8*a^2*b^5*c*d*f*g - 8*a^2*b^5*c*d*e*h + 4*a
^5*b^2*g^2*h*j - 4*a^5*b^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*f*h*i^2 + 4*a^5*b^2*d*i^2*j + 4*a^4*b^3*e^2
*h*j - 4*a^5*b^2*e*g*j^2 - 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 - 4*a^4*b^3*f^2*g*i + 4*a^4*b^3*f*g^2*h + 4*a
^4*b^3*e*g^2*i + 4*a^4*b^3*d*g^2*j + 4*a^3*b^4*c^2*h*j - 4*a^4*b^3*e*g*h^2 - 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2
*g*i - 4*a^3*b^4*d^2*f*j + 4*a^4*b^3*d*f*i^2 + 4*a^4*b^3*c*g*i^2 + 4*a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j - 4*a
^4*b^3*c*e*j^2 - 4*a^3*b^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*c*f^2*i + 4*a^3*b^4*d*f*g^2 + 4*a^2*b^5*c^2
*f*h + 4*a^2*b^5*c^2*e*i + 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 - 4*a^2*b^5*d^2*e*g - 4*a^2*b^5*c*d^2*i + 4*a
^2*b^5*d*e^2*f + 4*a^2*b^5*c*e^2*g - 4*a^2*b^5*c*e*f^2 + 4*a^6*b*h*i^2*j - 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f -
4*a*b^6*c*d^2*e + 4*a^6*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^2 + 2*a^5*b^2*g^2*i^2 - 6*a^4*b^3*e^2*i^2 -
2*a^4*b^3*f^2*h^2 - 2*a^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*b^4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 - 6*a^2*b^5
*c^2*g^2 - 2*a^2*b^5*d^2*f^2 - 2*a^6*b*h^2*j^2 + 4*a^4*b^3*f^3*j - 4*a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i + 4*a^4*b
^3*d*h^3 + 4*a^2*b^5*d^3*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^5*b^2*h^4 + a^3*b^4*f^4 + a*b^6*d^4 + a^7*j
^4 - a^4*b^3*g^4 - a^2*b^5*e^4 - a^6*b*i^4 - b^7*c^4, z, m)*((16*a^2*b^4*g + 16*a*b^5*c)/b^2 - (x*(16*a^2*b^4*
h + 16*a*b^5*d))/b^2) + (x*(4*b^5*c^2 + 4*a*b^4*e^2 + 4*a^2*b^3*g^2 + 4*a^3*b^2*i^2 + 8*a*b^4*c*g - 8*a*b^4*d*
f - 8*a^2*b^3*d*j + 8*a^2*b^3*e*i - 8*a^2*b^3*f*h - 8*a^3*b^2*h*j))/b^2) - (x*(b^4*d^3 + a^3*b*h^3 + b^4*c^2*f
- a^4*h*j^2 + a^4*i^2*j + 3*a^2*b^2*d*h^2 + a^2*b^2*f*g^2 - a^2*b^2*f^2*h + a^2*b^2*e^2*j - 2*b^4*c*d*e - a*b
^3*d*f^2 + a*b^3*e^2*f + 3*a*b^3*d^2*h + a*b^3*c^2*j - a^3*b*d*j^2 + a^3*b*f*i^2 + a^3*b*g^2*j + 2*a^2*b^2*c*g
*j - 2*a^2*b^2*c*h*i - 2*a^2*b^2*d*f*j - 2*a^2*b^2*d*g*i + 2*a^2*b^2*e*f*i - 2*a^2*b^2*e*g*h - 2*a*b^3*c*d*i -
2*a*b^3*c*e*h + 2*a*b^3*c*f*g - 2*a*b^3*d*e*g + 2*a^3*b*e*i*j - 2*a^3*b*f*h*j - 2*a^3*b*g*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 - 3
2*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 - 32*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*a^4*b^3*d*f*h*j + 8*a^4*b^3*d*e*i*j + 8*a^4*b^3*c*g*h*j - 8*a^4*b^3*c*f*i*j - 8*a^3*b^4*d*e*g*h + 8*a^3*b^4*
d*e*f*i + 8*a^3*b^4*c*f*g*h + 8*a^3*b^4*c*e*g*i - 8*a^3*b^4*c*e*f*j - 8*a^3*b^4*c*d*h*i + 8*a^3*b^4*c*d*g*j +
8*a^2*b^5*c*d*f*g - 8*a^2*b^5*c*d*e*h + 4*a^5*b^2*g^2*h*j - 4*a^5*b^2*g*h^2*i - 4*a^5*b^2*f*h^2*j + 4*a^5*b^2*
f*h*i^2 + 4*a^5*b^2*d*i^2*j + 4*a^4*b^3*e^2*h*j - 4*a^5*b^2*e*g*j^2 - 4*a^5*b^2*d*h*j^2 - 4*a^5*b^2*c*i*j^2 -
4*a^4*b^3*f^2*g*i + 4*a^4*b^3*f*g^2*h + 4*a^4*b^3*e*g^2*i + 4*a^4*b^3*d*g^2*j + 4*a^3*b^4*c^2*h*j - 4*a^4*b^3*
e*g*h^2 - 4*a^4*b^3*c*h^2*i - 4*a^3*b^4*d^2*g*i - 4*a^3*b^4*d^2*f*j + 4*a^4*b^3*d*f*i^2 + 4*a^4*b^3*c*g*i^2 +
4*a^3*b^4*e^2*f*h + 4*a^3*b^4*d*e^2*j - 4*a^4*b^3*c*e*j^2 - 4*a^3*b^4*e*f^2*g - 4*a^3*b^4*d*f^2*h - 4*a^3*b^4*
c*f^2*i + 4*a^3*b^4*d*f*g^2 + 4*a^2*b^5*c^2*f*h + 4*a^2*b^5*c^2*e*i + 4*a^2*b^5*c^2*d*j - 4*a^3*b^4*c*e*h^2 -
4*a^2*b^5*d^2*e*g - 4*a^2*b^5*c*d^2*i + 4*a^2*b^5*d*e^2*f + 4*a^2*b^5*c*e^2*g - 4*a^2*b^5*c*e*f^2 + 4*a^6*b*h*
i^2*j - 4*a^6*b*g*i*j^2 + 4*a*b^6*c^2*d*f - 4*a*b^6*c*d^2*e + 4*a^6*b*f*j^3 - 4*a*b^6*c^3*g + 6*a^5*b^2*f^2*j^
2 + 2*a^5*b^2*g^2*i^2 - 6*a^4*b^3*e^2*i^2 - 2*a^4*b^3*f^2*h^2 - 2*a^4*b^3*d^2*j^2 + 6*a^3*b^4*d^2*h^2 + 2*a^3*
b^4*e^2*g^2 + 2*a^3*b^4*c^2*i^2 - 6*a^2*b^5*c^2*g^2 - 2*a^2*b^5*d^2*f^2 - 2*a^6*b*h^2*j^2 + 4*a^4*b^3*f^3*j -
4*a^5*b^2*e*i^3 - 4*a^3*b^4*e^3*i + 4*a^4*b^3*d*h^3 + 4*a^2*b^5*d^3*h - 4*a^3*b^4*c*g^3 + 2*a*b^6*c^2*e^2 + a^
5*b^2*h^4 + a^3*b^4*f^4 + a*b^6*d^4 + a^7*j^4 - a^4*b^3*g^4 - a^2*b^5*e^4 - a^6*b*i^4 - b^7*c^4, z, m), m, 1,
4) - (h*x^2)/(2*b) - (i*x^3)/(3*b) - (j*x^4)/(4*b) - (g*x)/b