3.172 \(\int \frac {c+d x+e x^2+f x^3+g x^4}{(a-b x^4)^2} \, dx\)

Optimal. Leaf size=172 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt {a} \sqrt {b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

[Out]

1/4*x*(b*f*x^3+b*e*x^2+b*d*x+a*g+b*c)/a/b/(-b*x^4+a)+1/4*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^(3/2)/b^(1/2)+1/8*ar
ctan(b^(1/4)*x/a^(1/4))*(3*b*c-a*g-e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)+1/8*arctanh(b^(1/4)*x/a^(1/4))*(3*b*c-a*
g+e*a^(1/2)*b^(1/2))/a^(7/4)/b^(5/4)

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Rubi [A]  time = 0.16, antiderivative size = 172, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1858, 1876, 275, 208, 1167, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-\sqrt {a} \sqrt {b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e-a g+3 b c\right )}{8 a^{7/4} b^{5/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(4*a*b*(a - b*x^4)) + ((3*b*c - Sqrt[a]*Sqrt[b]*e - a*g)*ArcTan[(b
^(1/4)*x)/a^(1/4)])/(8*a^(7/4)*b^(5/4)) + ((3*b*c + Sqrt[a]*Sqrt[b]*e - a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(8*
a^(7/4)*b^(5/4)) + (d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(4*a^(3/2)*Sqrt[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 275

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> With[{k = GCD[m + 1, n]}, Dist[1/k, Subst[Int[x^((m
+ 1)/k - 1)*(a + b*x^(n/k))^p, x], x, x^k], x] /; k != 1] /; FreeQ[{a, b, p}, x] && IGtQ[n, 0] && IntegerQ[m]

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 1858

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> With[{q = Expon[Pq, x]}, Module[{Q = PolynomialQuotient
[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x], R = PolynomialRemainder[b^(Floor[(q - 1)/n] + 1)*Pq, a + b*x^n, x
]}, Dist[1/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), Int[(a + b*x^n)^(p + 1)*ExpandToSum[a*n*(p + 1)*Q + n*(p +
1)*R + D[x*R, x], x], x], x] - Simp[(x*R*(a + b*x^n)^(p + 1))/(a*n*(p + 1)*b^(Floor[(q - 1)/n] + 1)), x]] /; G
eQ[q, n]] /; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1876

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

Rubi steps

\begin {align*} \int \frac {c+d x+e x^2+f x^3+g x^4}{\left (a-b x^4\right )^2} \, dx &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\int \frac {3 b c-a g+2 b d x+b e x^2}{a-b x^4} \, dx}{4 a b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\int \left (\frac {2 b d x}{a-b x^4}+\frac {3 b c-a g+b e x^2}{a-b x^4}\right ) \, dx}{4 a b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\int \frac {3 b c-a g+b e x^2}{a-b x^4} \, dx}{4 a b}+\frac {d \int \frac {x}{a-b x^4} \, dx}{2 a}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {d \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{4 a}-\frac {\left (3 b c-\sqrt {a} \sqrt {b} e-a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a^{3/2} \sqrt {b}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e-a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{8 a^{3/2} \sqrt {b}}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{4 a b \left (a-b x^4\right )}+\frac {\left (3 b c-\sqrt {a} \sqrt {b} e-a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{5/4}}+\frac {\left (3 b c+\sqrt {a} \sqrt {b} e-a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 a^{7/4} b^{5/4}}+\frac {d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{4 a^{3/2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.43, size = 221, normalized size = 1.28 \[ \frac {\frac {4 a^{3/4} \sqrt [4]{b} (a (f+g x)+b x (c+x (d+e x)))}{a-b x^4}-\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (2 \sqrt [4]{a} b^{3/4} d+\sqrt {a} \sqrt {b} e-a g+3 b c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-2 \sqrt [4]{a} b^{3/4} d+\sqrt {a} \sqrt {b} e-a g+3 b c\right )+2 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )-2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} \sqrt {b} e+a g-3 b c\right )}{16 a^{7/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]

Integrate[(c + d*x + e*x^2 + f*x^3 + g*x^4)/(a - b*x^4)^2,x]

[Out]

((4*a^(3/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*x^4) - 2*(-3*b*c + Sqrt[a]*Sqrt[b]*e + a*g)*
ArcTan[(b^(1/4)*x)/a^(1/4)] - (3*b*c + 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) - b^(1/4)*x]
+ (3*b*c - 2*a^(1/4)*b^(3/4)*d + Sqrt[a]*Sqrt[b]*e - a*g)*Log[a^(1/4) + b^(1/4)*x] + 2*a^(1/4)*b^(3/4)*d*Log[
Sqrt[a] + Sqrt[b]*x^2])/(16*a^(7/4)*b^(5/4))

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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((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.18, size = 344, normalized size = 2.00 \[ -\frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g + 2 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} + \frac {\sqrt {2} {\left (3 \, b^{2} c - a b g - \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{32 \, \left (-a b^{3}\right )^{\frac {3}{4}} a} - \frac {b x^{3} e + b d x^{2} + b c x + a g x + a f}{4 \, {\left (b x^{4} - a\right )} a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="giac")

[Out]

-1/16*sqrt(2)*(3*b^2*c - a*b*g - 2*sqrt(2)*(-a*b^3)^(1/4)*b*d + 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)*a) - 1/16*sqrt(2)*(3*b^2*c - a*b*g + 2*sqrt(2)*(-a*b^3)^(1/4)*
b*d - 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)*a) - 1/32*
sqrt(2)*(3*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)*a) +
1/32*sqrt(2)*(3*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
)*a) - 1/4*(b*x^3*e + b*d*x^2 + b*c*x + a*g*x + a*f)/((b*x^4 - a)*a*b)

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maple [B]  time = 0.05, size = 289, normalized size = 1.68 \[ -\frac {d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{8 \sqrt {a b}\, a}-\frac {e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}+\frac {e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 \left (\frac {a}{b}\right )^{\frac {1}{4}} a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a b}-\frac {\left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a b}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{8 a^{2}}+\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{16 a^{2}}+\frac {-\frac {e \,x^{3}}{4 a}-\frac {d \,x^{2}}{4 a}-\frac {f}{4 b}-\frac {\left (a g +b c \right ) x}{4 a b}}{b \,x^{4}-a} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-1/4/a*e*x^3-1/4/a*d*x^2-1/4*(a*g+b*c)/a/b*x-1/4/b*f)/(b*x^4-a)-1/8/b/a*(a/b)^(1/4)*arctan(1/(a/b)^(1/4)*x)*g
+3/8*(a/b)^(1/4)/a^2*c*arctan(1/(a/b)^(1/4)*x)-1/16/b/a*(a/b)^(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+3/16
*(a/b)^(1/4)/a^2*c*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-1/8/(a*b)^(1/2)/a*d*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2
)*x^2-a))-1/8/(a/b)^(1/4)/a/b*e*arctan(1/(a/b)^(1/4)*x)+1/16/(a/b)^(1/4)/a/b*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/
4)))

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maxima [A]  time = 3.11, size = 224, normalized size = 1.30 \[ -\frac {b e x^{3} + b d x^{2} + a f + {\left (b c + a g\right )} x}{4 \, {\left (a b^{2} x^{4} - a^{2} b\right )}} + \frac {\frac {2 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a}} - \frac {2 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a}} + \frac {2 \, {\left (3 \, b^{\frac {3}{2}} c - \sqrt {a} b e - a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (3 \, b^{\frac {3}{2}} c + \sqrt {a} b e - a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{16 \, a b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^2,x, algorithm="maxima")

[Out]

-1/4*(b*e*x^3 + b*d*x^2 + a*f + (b*c + a*g)*x)/(a*b^2*x^4 - a^2*b) + 1/16*(2*sqrt(b)*d*log(sqrt(b)*x^2 + sqrt(
a))/sqrt(a) - 2*sqrt(b)*d*log(sqrt(b)*x^2 - sqrt(a))/sqrt(a) + 2*(3*b^(3/2)*c - sqrt(a)*b*e - a*sqrt(b)*g)*arc
tan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (3*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)*sqrt(sqrt(a)*
sqrt(b))*sqrt(b)))/(a*b)

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mupad [B]  time = 5.56, size = 1393, normalized size = 8.10 \[ \left (\sum _{k=1}^4\ln \left (-\frac {-a^2\,e\,g^2+6\,a\,b\,c\,e\,g-4\,a\,b\,d^2\,g+a\,b\,e^3-9\,b^2\,c^2\,e+12\,b^2\,c\,d^2}{64\,a^3}-\frac {\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2-3072\,a^4\,b^4\,c\,e\,z^2-2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z+128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z+1152\,a^2\,b^4\,c^2\,d\,z+16\,a^2\,b^2\,d^2\,e\,g-12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3-54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4-81\,b^4\,c^4-a^2\,b^2\,e^4-a^4\,g^4,z,k\right )\,b\,\left (9\,b^2\,c^2\,x+a^2\,g^2\,x-\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2-3072\,a^4\,b^4\,c\,e\,z^2-2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z+128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z+1152\,a^2\,b^4\,c^2\,d\,z+16\,a^2\,b^2\,d^2\,e\,g-12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3-54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4-81\,b^4\,c^4-a^2\,b^2\,e^4-a^4\,g^4,z,k\right )\,a^3\,b\,g\,16+a\,b\,e^2\,x+\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2-3072\,a^4\,b^4\,c\,e\,z^2-2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z+128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z+1152\,a^2\,b^4\,c^2\,d\,z+16\,a^2\,b^2\,d^2\,e\,g-12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3-54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4-81\,b^4\,c^4-a^2\,b^2\,e^4-a^4\,g^4,z,k\right )\,a^2\,b^2\,c\,48-4\,a\,b\,d\,e-\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2-3072\,a^4\,b^4\,c\,e\,z^2-2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z+128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z+1152\,a^2\,b^4\,c^2\,d\,z+16\,a^2\,b^2\,d^2\,e\,g-12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3-54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4-81\,b^4\,c^4-a^2\,b^2\,e^4-a^4\,g^4,z,k\right )\,a^2\,b^2\,d\,x\,32-6\,a\,b\,c\,g\,x\right )}{a^2\,4}-\frac {b\,d\,x\,\left (2\,b\,d^2-3\,b\,c\,e+a\,e\,g\right )}{16\,a^3}\right )\,\mathrm {root}\left (65536\,a^7\,b^5\,z^4+1024\,a^5\,b^3\,e\,g\,z^2-3072\,a^4\,b^4\,c\,e\,z^2-2048\,a^4\,b^4\,d^2\,z^2-768\,a^3\,b^3\,c\,d\,g\,z+128\,a^4\,b^2\,d\,g^2\,z+128\,a^3\,b^3\,d\,e^2\,z+1152\,a^2\,b^4\,c^2\,d\,z+16\,a^2\,b^2\,d^2\,e\,g-12\,a^2\,b^2\,c\,e^2\,g-48\,a\,b^3\,c\,d^2\,e+108\,a\,b^3\,c^3\,g+12\,a^3\,b\,c\,g^3-54\,a^2\,b^2\,c^2\,g^2+2\,a^3\,b\,e^2\,g^2+18\,a\,b^3\,c^2\,e^2+16\,a\,b^3\,d^4-81\,b^4\,c^4-a^2\,b^2\,e^4-a^4\,g^4,z,k\right )\right )+\frac {\frac {f}{4\,b}+\frac {d\,x^2}{4\,a}+\frac {e\,x^3}{4\,a}+\frac {x\,\left (b\,c+a\,g\right )}{4\,a\,b}}{a-b\,x^4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

symsum(log(- (12*b^2*c*d^2 - 9*b^2*c^2*e - a^2*e*g^2 + a*b*e^3 - 4*a*b*d^2*g + 6*a*b*c*e*g)/(64*a^3) - (root(6
5536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g*z^2 - 3072*a^4*b^4*c*e*z^2 - 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z +
128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d*e^2*z + 1152*a^2*b^4*c^2*d*z + 16*a^2*b^2*d^2*e*g - 12*a^2*b^2*c*e^2*g - 4
8*a*b^3*c*d^2*e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 - 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 +
16*a*b^3*d^4 - 81*b^4*c^4 - a^2*b^2*e^4 - a^4*g^4, z, k)*b*(9*b^2*c^2*x + a^2*g^2*x - 16*root(65536*a^7*b^5*z
^4 + 1024*a^5*b^3*e*g*z^2 - 3072*a^4*b^4*c*e*z^2 - 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z + 128*a^4*b^2*d*
g^2*z + 128*a^3*b^3*d*e^2*z + 1152*a^2*b^4*c^2*d*z + 16*a^2*b^2*d^2*e*g - 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*
e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 - 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4
- 81*b^4*c^4 - a^2*b^2*e^4 - a^4*g^4, z, k)*a^3*b*g + a*b*e^2*x + 48*root(65536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g
*z^2 - 3072*a^4*b^4*c*e*z^2 - 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z + 128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d
*e^2*z + 1152*a^2*b^4*c^2*d*z + 16*a^2*b^2*d^2*e*g - 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e + 108*a*b^3*c^3*g +
12*a^3*b*c*g^3 - 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 - 81*b^4*c^4 - a^2*b^
2*e^4 - a^4*g^4, z, k)*a^2*b^2*c - 4*a*b*d*e - 32*root(65536*a^7*b^5*z^4 + 1024*a^5*b^3*e*g*z^2 - 3072*a^4*b^4
*c*e*z^2 - 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z + 128*a^4*b^2*d*g^2*z + 128*a^3*b^3*d*e^2*z + 1152*a^2*b
^4*c^2*d*z + 16*a^2*b^2*d^2*e*g - 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e + 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 - 5
4*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 - 81*b^4*c^4 - a^2*b^2*e^4 - a^4*g^4, z,
k)*a^2*b^2*d*x - 6*a*b*c*g*x))/(4*a^2) - (b*d*x*(2*b*d^2 - 3*b*c*e + a*e*g))/(16*a^3))*root(65536*a^7*b^5*z^4
+ 1024*a^5*b^3*e*g*z^2 - 3072*a^4*b^4*c*e*z^2 - 2048*a^4*b^4*d^2*z^2 - 768*a^3*b^3*c*d*g*z + 128*a^4*b^2*d*g^
2*z + 128*a^3*b^3*d*e^2*z + 1152*a^2*b^4*c^2*d*z + 16*a^2*b^2*d^2*e*g - 12*a^2*b^2*c*e^2*g - 48*a*b^3*c*d^2*e
+ 108*a*b^3*c^3*g + 12*a^3*b*c*g^3 - 54*a^2*b^2*c^2*g^2 + 2*a^3*b*e^2*g^2 + 18*a*b^3*c^2*e^2 + 16*a*b^3*d^4 -
81*b^4*c^4 - a^2*b^2*e^4 - a^4*g^4, z, k), k, 1, 4) + (f/(4*b) + (d*x^2)/(4*a) + (e*x^3)/(4*a) + (x*(b*c + a*g
))/(4*a*b))/(a - b*x^4)

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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((g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**2,x)

[Out]

Timed out

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