3.187 \(\int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+i x^6}{a-b x^4} \, dx\)

Optimal. Leaf size=188 \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(a h+b d) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {f \log \left (a-b x^4\right )}{4 b}-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b} \]

[Out]

-g*x/b-1/2*h*x^2/b-1/3*i*x^3/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*e+a*i-(a*g+b*c)*b^(1/2)/a^(1/2))/a^(1/4)/b^(7/4)+1/2*arctanh(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)

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Rubi [A]  time = 0.33, antiderivative size = 188, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 9, integrand size = 41, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.220, Rules used = {1885, 1819, 1810, 635, 208, 260, 1887, 1167, 205} \[ -\frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\frac {\sqrt {b} (a g+b c)}{\sqrt {a}}+a i+b e\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {(a h+b d) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{2 \sqrt {a} b^{3/2}}-\frac {f \log \left (a-b x^4\right )}{4 b}-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {i x^3}{3 b} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(a - b*x^4),x]

[Out]

-((g*x)/b) - (h*x^2)/(2*b) - (i*x^3)/(3*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)) - (f*Log[a - b*x^4])/(4*b)

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 260

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 635

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 1167

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 1810

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 1819

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 1885

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 1887

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*} \int \frac {c+d x+e x^2+f x^3+g x^4+h x^5+187 x^6}{a-b x^4} \, dx &=\int \left (\frac {x \left (d+f x^2+h x^4\right )}{a-b x^4}+\frac {c+e x^2+g x^4+187 x^6}{a-b x^4}\right ) \, dx\\ &=\int \frac {x \left (d+f x^2+h x^4\right )}{a-b x^4} \, dx+\int \frac {c+e x^2+g x^4+187 x^6}{a-b x^4} \, dx\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {d+f x+h x^2}{a-b x^2} \, dx,x,x^2\right )+\int \left (-\frac {g}{b}-\frac {187 x^2}{b}+\frac {b c+a g+(187 a+b e) x^2}{b \left (a-b x^4\right )}\right ) \, dx\\ &=-\frac {g x}{b}-\frac {187 x^3}{3 b}+\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {h}{b}+\frac {b d+a h+b f x}{b \left (a-b x^2\right )}\right ) \, dx,x,x^2\right )+\frac {\int \frac {b c+a g+(187 a+b e) x^2}{a-b x^4} \, dx}{b}\\ &=-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {187 x^3}{3 b}+\frac {\operatorname {Subst}\left (\int \frac {b d+a h+b f x}{a-b x^2} \, dx,x,x^2\right )}{2 b}+\frac {\left (187 a+b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{2 b}+\frac {\left (187 a+b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\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 {187 x^3}{3 b}-\frac {\left (187 a+b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\left (187 a+b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {1}{2} f \operatorname {Subst}\left (\int \frac {x}{a-b x^2} \, dx,x,x^2\right )+\frac {(b d+a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{2 b}\\ &=-\frac {g x}{b}-\frac {h x^2}{2 b}-\frac {187 x^3}{3 b}-\frac {\left (187 a+b e-\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{2 \sqrt [4]{a} b^{7/4}}+\frac {\left (187 a+b e+\frac {\sqrt {b} (b c+a g)}{\sqrt {a}}\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 {f \log \left (a-b x^4\right )}{4 b}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 301, normalized size = 1.60 \[ \frac {-\frac {3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (a^{5/4} \sqrt [4]{b} h+a^{3/2} i+\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g+b^{3/2} c\right )}{a^{3/4}}+\frac {3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-a^{5/4} \sqrt [4]{b} h+a^{3/2} i-\sqrt [4]{a} b^{5/4} d+\sqrt {a} b e+a \sqrt {b} g+b^{3/2} c\right )}{a^{3/4}}+\frac {6 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (a^{3/2} (-i)-\sqrt {a} b e+a \sqrt {b} g+b^{3/2} c\right )}{a^{3/4}}-3 b^{3/4} f \log \left (a-b x^4\right )+\frac {3 \sqrt [4]{b} (a h+b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {a}}-12 b^{3/4} g x-6 b^{3/4} h x^2-4 b^{3/4} i x^3}{12 b^{7/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(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 + (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*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^(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[Sq
rt[a] + Sqrt[b]*x^2])/Sqrt[a] - 3*b^(3/4)*f*Log[a - b*x^4])/(12*b^(7/4))

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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giac [B]  time = 0.21, size = 541, normalized size = 2.88 \[ \frac {1}{8} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \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 )}{b^{4}} - \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{b^{4}}\right )} + \frac {1}{8} \, i {\left (\frac {2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \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 )}{b^{4}} + \frac {\sqrt {2} \left (-a b^{3}\right )^{\frac {3}{4}} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{b^{4}}\right )} - \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 {2 \, b^{2} i x^{3} + 3 \, b^{2} h x^{2} + 6 \, b^{2} g x}{6 \, b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

1/8*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/b^4 - sqrt(2)*(-
a*b^3)^(3/4)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/b^4) + 1/8*i*(2*sqrt(2)*(-a*b^3)^(3/4)*arctan(1/2*
sqrt(2)*(2*x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/b^4 + sqrt(2)*(-a*b^3)^(3/4)*log(x^2 - sqrt(2)*x*(-a/b)^(1/
4) + sqrt(-a/b))/b^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) - 1/4*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*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) + 1/8*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 - 1/6*(2*b^2*i*x^3 + 3*b^2
*h*x^2 + 6*b^2*g*x)/b^3

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maple [B]  time = 0.05, size = 367, normalized size = 1.95 \[ -\frac {i \,x^{3}}{3 b}-\frac {a h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{4 \sqrt {a b}\, b}-\frac {h \,x^{2}}{2 b}-\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{4 \sqrt {a b}}-\frac {a i \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {a i \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b^{2}}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 a}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 a}-\frac {e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}+\frac {e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 \left (\frac {a}{b}\right )^{\frac {1}{4}} b}-\frac {f \ln \left (b \,x^{4}-a \right )}{4 b}-\frac {g x}{b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{2 b}+\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{4 b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((i*x^6+h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a),x)

[Out]

-1/3*i*x^3/b-1/2/b*h*x^2-1/b*g*x+1/4*(a/b)^(1/4)/b*g*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/4*(a/b)^(1/4)/a*c*l
n((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/2*(a/b)^(1/4)/b*g*arctan(1/(a/b)^(1/4)*x)+1/2*(a/b)^(1/4)/a*c*arctan(1/(a
/b)^(1/4)*x)-1/4/(a*b)^(1/2)*a/b*h*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))-1/4/(a*b)^(1/2)*d*ln(((a*b)^(1
/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))+1/4/b^2/(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*a*i+1/4/(a/b)^(1/4)/b
*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-1/2/b^2/(a/b)^(1/4)*arctan(1/(a/b)^(1/4)*x)*a*i-1/2/(a/b)^(1/4)/b*e*arc
tan(1/(a/b)^(1/4)*x)-1/4/b*f*ln(b*x^4-a)

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maxima [A]  time = 3.03, size = 240, normalized size = 1.28 \[ -\frac {2 \, i x^{3} + 3 \, h x^{2} + 6 \, g x}{6 \, b} + \frac {\frac {2 \, {\left (b^{\frac {3}{2}} c - \sqrt {a} b e + a \sqrt {b} g - a^{\frac {3}{2}} i\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {{\left (b^{\frac {3}{2}} d - \sqrt {a} b f + a \sqrt {b} h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} d + \sqrt {a} b f + a \sqrt {b} h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} b} - \frac {{\left (b^{\frac {3}{2}} c + \sqrt {a} b e + a \sqrt {b} g + a^{\frac {3}{2}} i\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{4 \, b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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]

-1/6*(2*i*x^3 + 3*h*x^2 + 6*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*sqrt(b)*h)*lo
g(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 - sqrt(a))/(sqr
t(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

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mupad [B]  time = 5.07, size = 3810, normalized size = 20.27 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x + e*x^2 + f*x^3 + g*x^4 + h*x^5 + i*x^6)/(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^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 + a*b^3*c*f^2 + a*b^3*d^2*g - a*b^3*c^2*i + 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*g*i - 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*f*h*i)/b^2 - root(256*a^3*b^7*z^4 + 256*a^3*b^6*f*z^3 - 64*a^4*b^4*g*i*z^2 - 64*a^3*b^5*e*g*z^2 - 64
*a^3*b^5*d*h*z^2 - 64*a^3*b^5*c*i*z^2 - 64*a^2*b^6*c*e*z^2 - 32*a^4*b^4*h^2*z^2 + 96*a^3*b^5*f^2*z^2 - 32*a^2*
b^6*d^2*z^2 - 32*a^4*b^3*f*g*i*z + 32*a^4*b^3*e*h*i*z - 32*a^3*b^4*e*f*g*z - 32*a^3*b^4*d*f*h*z + 32*a^3*b^4*d
*e*i*z + 32*a^3*b^4*c*g*h*z - 32*a^3*b^4*c*f*i*z - 32*a^2*b^5*c*e*f*z + 32*a^2*b^5*c*d*g*z + 16*a^5*b^2*h*i^2*
z + 16*a^4*b^3*g^2*h*z - 16*a^4*b^3*f*h^2*z + 16*a^4*b^3*d*i^2*z + 16*a^3*b^4*e^2*h*z + 16*a^3*b^4*d*g^2*z + 1
6*a^2*b^5*c^2*h*z - 16*a^2*b^5*d^2*f*z + 16*a^2*b^5*d*e^2*z + 16*a*b^6*c^2*d*z + 16*a^3*b^4*f^3*z + 8*a^4*b^2*
e*f*h*i - 8*a^4*b^2*d*g*h*i - 8*a^3*b^3*d*e*g*h + 8*a^3*b^3*d*e*f*i + 8*a^3*b^3*c*f*g*h + 8*a^3*b^3*c*e*g*i -
8*a^3*b^3*c*d*h*i + 8*a^2*b^4*c*d*f*g - 8*a^2*b^4*c*d*e*h - 4*a^4*b^2*f^2*g*i + 4*a^4*b^2*f*g^2*h + 4*a^4*b^2*
e*g^2*i - 4*a^4*b^2*e*g*h^2 - 4*a^4*b^2*c*h^2*i - 4*a^3*b^3*d^2*g*i + 4*a^4*b^2*d*f*i^2 + 4*a^4*b^2*c*g*i^2 +
4*a^3*b^3*e^2*f*h - 4*a^3*b^3*e*f^2*g - 4*a^3*b^3*d*f^2*h - 4*a^3*b^3*c*f^2*i + 4*a^3*b^3*d*f*g^2 + 4*a^2*b^4*
c^2*f*h + 4*a^2*b^4*c^2*e*i - 4*a^3*b^3*c*e*h^2 - 4*a^2*b^4*d^2*e*g - 4*a^2*b^4*c*d^2*i + 4*a^2*b^4*d*e^2*f +
4*a^2*b^4*c*e^2*g - 4*a^2*b^4*c*e*f^2 - 4*a^5*b*g*h^2*i + 4*a^5*b*f*h*i^2 + 4*a*b^5*c^2*d*f - 4*a*b^5*c*d^2*e
- 4*a^5*b*e*i^3 - 4*a*b^5*c^3*g - 6*a^4*b^2*e^2*i^2 - 2*a^4*b^2*f^2*h^2 + 6*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2*g^
2 + 2*a^3*b^3*c^2*i^2 - 6*a^2*b^4*c^2*g^2 - 2*a^2*b^4*d^2*f^2 + 2*a^5*b*g^2*i^2 - 4*a^3*b^3*e^3*i + 4*a^4*b^2*
d*h^3 + 4*a^2*b^4*d^3*h - 4*a^3*b^3*c*g^3 + 2*a*b^5*c^2*e^2 + a^3*b^3*f^4 + a^5*b*h^4 + a*b^5*d^4 - a^4*b^2*g^
4 - a^2*b^4*e^4 - a^6*i^4 - b^6*c^4, z, l)*(root(256*a^3*b^7*z^4 + 256*a^3*b^6*f*z^3 - 64*a^4*b^4*g*i*z^2 - 64
*a^3*b^5*e*g*z^2 - 64*a^3*b^5*d*h*z^2 - 64*a^3*b^5*c*i*z^2 - 64*a^2*b^6*c*e*z^2 - 32*a^4*b^4*h^2*z^2 + 96*a^3*
b^5*f^2*z^2 - 32*a^2*b^6*d^2*z^2 - 32*a^4*b^3*f*g*i*z + 32*a^4*b^3*e*h*i*z - 32*a^3*b^4*e*f*g*z - 32*a^3*b^4*d
*f*h*z + 32*a^3*b^4*d*e*i*z + 32*a^3*b^4*c*g*h*z - 32*a^3*b^4*c*f*i*z - 32*a^2*b^5*c*e*f*z + 32*a^2*b^5*c*d*g*
z + 16*a^5*b^2*h*i^2*z + 16*a^4*b^3*g^2*h*z - 16*a^4*b^3*f*h^2*z + 16*a^4*b^3*d*i^2*z + 16*a^3*b^4*e^2*h*z + 1
6*a^3*b^4*d*g^2*z + 16*a^2*b^5*c^2*h*z - 16*a^2*b^5*d^2*f*z + 16*a^2*b^5*d*e^2*z + 16*a*b^6*c^2*d*z + 16*a^3*b
^4*f^3*z + 8*a^4*b^2*e*f*h*i - 8*a^4*b^2*d*g*h*i - 8*a^3*b^3*d*e*g*h + 8*a^3*b^3*d*e*f*i + 8*a^3*b^3*c*f*g*h +
8*a^3*b^3*c*e*g*i - 8*a^3*b^3*c*d*h*i + 8*a^2*b^4*c*d*f*g - 8*a^2*b^4*c*d*e*h - 4*a^4*b^2*f^2*g*i + 4*a^4*b^2
*f*g^2*h + 4*a^4*b^2*e*g^2*i - 4*a^4*b^2*e*g*h^2 - 4*a^4*b^2*c*h^2*i - 4*a^3*b^3*d^2*g*i + 4*a^4*b^2*d*f*i^2 +
4*a^4*b^2*c*g*i^2 + 4*a^3*b^3*e^2*f*h - 4*a^3*b^3*e*f^2*g - 4*a^3*b^3*d*f^2*h - 4*a^3*b^3*c*f^2*i + 4*a^3*b^3
*d*f*g^2 + 4*a^2*b^4*c^2*f*h + 4*a^2*b^4*c^2*e*i - 4*a^3*b^3*c*e*h^2 - 4*a^2*b^4*d^2*e*g - 4*a^2*b^4*c*d^2*i +
4*a^2*b^4*d*e^2*f + 4*a^2*b^4*c*e^2*g - 4*a^2*b^4*c*e*f^2 - 4*a^5*b*g*h^2*i + 4*a^5*b*f*h*i^2 + 4*a*b^5*c^2*d
*f - 4*a*b^5*c*d^2*e - 4*a^5*b*e*i^3 - 4*a*b^5*c^3*g - 6*a^4*b^2*e^2*i^2 - 2*a^4*b^2*f^2*h^2 + 6*a^3*b^3*d^2*h
^2 + 2*a^3*b^3*e^2*g^2 + 2*a^3*b^3*c^2*i^2 - 6*a^2*b^4*c^2*g^2 - 2*a^2*b^4*d^2*f^2 + 2*a^5*b*g^2*i^2 - 4*a^3*b
^3*e^3*i + 4*a^4*b^2*d*h^3 + 4*a^2*b^4*d^3*h - 4*a^3*b^3*c*g^3 + 2*a*b^5*c^2*e^2 + a^3*b^3*f^4 + a^5*b*h^4 + a
*b^5*d^4 - a^4*b^2*g^4 - a^2*b^4*e^4 - a^6*i^4 - b^6*c^4, z, l)*((16*a^2*b^4*g + 16*a*b^5*c)/b^2 - (x*(16*a^2*
b^3*h + 16*a*b^4*d))/b) - (8*a*b^4*d*e - 8*a*b^4*c*f + 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*h*i)/b^2 + (x*(4*b^4*c^2 + 4*a*b^3*e^2 + 4*a^3*b*i^2 + 4*a^2*b^2*g^2 + 8*a*b^3*c*g - 8*a*b^3*d*f + 8*a^2*b^
2*e*i - 8*a^2*b^2*f*h))/b) - (x*(b^3*d^3 + a^3*h^3 + b^3*c^2*f + a^3*f*i^2 - 2*b^3*c*d*e - 2*a^3*g*h*i - a*b^2
*d*f^2 + a*b^2*e^2*f + 3*a*b^2*d^2*h + 3*a^2*b*d*h^2 + a^2*b*f*g^2 - a^2*b*f^2*h - 2*a*b^2*c*d*i - 2*a*b^2*c*e
*h + 2*a*b^2*c*f*g - 2*a*b^2*d*e*g - 2*a^2*b*c*h*i - 2*a^2*b*d*g*i + 2*a^2*b*e*f*i - 2*a^2*b*e*g*h))/b)*root(2
56*a^3*b^7*z^4 + 256*a^3*b^6*f*z^3 - 64*a^4*b^4*g*i*z^2 - 64*a^3*b^5*e*g*z^2 - 64*a^3*b^5*d*h*z^2 - 64*a^3*b^5
*c*i*z^2 - 64*a^2*b^6*c*e*z^2 - 32*a^4*b^4*h^2*z^2 + 96*a^3*b^5*f^2*z^2 - 32*a^2*b^6*d^2*z^2 - 32*a^4*b^3*f*g*
i*z + 32*a^4*b^3*e*h*i*z - 32*a^3*b^4*e*f*g*z - 32*a^3*b^4*d*f*h*z + 32*a^3*b^4*d*e*i*z + 32*a^3*b^4*c*g*h*z -
32*a^3*b^4*c*f*i*z - 32*a^2*b^5*c*e*f*z + 32*a^2*b^5*c*d*g*z + 16*a^5*b^2*h*i^2*z + 16*a^4*b^3*g^2*h*z - 16*a
^4*b^3*f*h^2*z + 16*a^4*b^3*d*i^2*z + 16*a^3*b^4*e^2*h*z + 16*a^3*b^4*d*g^2*z + 16*a^2*b^5*c^2*h*z - 16*a^2*b^
5*d^2*f*z + 16*a^2*b^5*d*e^2*z + 16*a*b^6*c^2*d*z + 16*a^3*b^4*f^3*z + 8*a^4*b^2*e*f*h*i - 8*a^4*b^2*d*g*h*i -
8*a^3*b^3*d*e*g*h + 8*a^3*b^3*d*e*f*i + 8*a^3*b^3*c*f*g*h + 8*a^3*b^3*c*e*g*i - 8*a^3*b^3*c*d*h*i + 8*a^2*b^4
*c*d*f*g - 8*a^2*b^4*c*d*e*h - 4*a^4*b^2*f^2*g*i + 4*a^4*b^2*f*g^2*h + 4*a^4*b^2*e*g^2*i - 4*a^4*b^2*e*g*h^2 -
4*a^4*b^2*c*h^2*i - 4*a^3*b^3*d^2*g*i + 4*a^4*b^2*d*f*i^2 + 4*a^4*b^2*c*g*i^2 + 4*a^3*b^3*e^2*f*h - 4*a^3*b^3
*e*f^2*g - 4*a^3*b^3*d*f^2*h - 4*a^3*b^3*c*f^2*i + 4*a^3*b^3*d*f*g^2 + 4*a^2*b^4*c^2*f*h + 4*a^2*b^4*c^2*e*i -
4*a^3*b^3*c*e*h^2 - 4*a^2*b^4*d^2*e*g - 4*a^2*b^4*c*d^2*i + 4*a^2*b^4*d*e^2*f + 4*a^2*b^4*c*e^2*g - 4*a^2*b^4
*c*e*f^2 - 4*a^5*b*g*h^2*i + 4*a^5*b*f*h*i^2 + 4*a*b^5*c^2*d*f - 4*a*b^5*c*d^2*e - 4*a^5*b*e*i^3 - 4*a*b^5*c^3
*g - 6*a^4*b^2*e^2*i^2 - 2*a^4*b^2*f^2*h^2 + 6*a^3*b^3*d^2*h^2 + 2*a^3*b^3*e^2*g^2 + 2*a^3*b^3*c^2*i^2 - 6*a^2
*b^4*c^2*g^2 - 2*a^2*b^4*d^2*f^2 + 2*a^5*b*g^2*i^2 - 4*a^3*b^3*e^3*i + 4*a^4*b^2*d*h^3 + 4*a^2*b^4*d^3*h - 4*a
^3*b^3*c*g^3 + 2*a*b^5*c^2*e^2 + a^3*b^3*f^4 + a^5*b*h^4 + a*b^5*d^4 - a^4*b^2*g^4 - a^2*b^4*e^4 - a^6*i^4 - b
^6*c^4, z, l), l, 1, 4) - (h*x^2)/(2*b) - (i*x^3)/(3*b) - (g*x)/b

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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

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