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

Optimal. Leaf size=266 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {x \left (-a g+11 b c+10 b d x+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

1/12*x*(b*f*x^3+b*e*x^2+b*d*x+a*g+b*c)/a/b/(-b*x^4+a)^3+1/384*x*(45*b*e*x^2+60*b*d*x-7*a*g+77*b*c)/a^3/b/(-b*x
^4+a)+1/96*(8*a*f+x*(9*b*e*x^2+10*b*d*x-a*g+11*b*c))/a^2/b/(-b*x^4+a)^2+5/32*d*arctanh(x^2*b^(1/2)/a^(1/2))/a^
(7/2)/b^(1/2)+1/256*arctan(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g-15*e*a^(1/2)*b^(1/2))/a^(15/4)/b^(5/4)+1/256*arcta
nh(b^(1/4)*x/a^(1/4))*(77*b*c-7*a*g+15*e*a^(1/2)*b^(1/2))/a^(15/4)/b^(5/4)

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Rubi [A]  time = 0.32, antiderivative size = 266, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.258, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac {x \left (-a g+11 b c+10 b d x+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x \left (a g+b c+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*c + a*g + b*d*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b*c - a*g) + 60*b*d*x + 45*b*e*
x^2))/(384*a^3*b*(a - b*x^4)) + (8*a*f + x*(11*b*c - a*g + 10*b*d*x + 9*b*e*x^2))/(96*a^2*b*(a - b*x^4)^2) + (
(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + ((77*b*c + 15*Sq
rt[a]*Sqrt[b]*e - 7*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqr
t[a]])/(32*a^(7/2)*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 1854

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Module[{q = Expon[Pq, x], i}, Simp[((a*Coeff[Pq, x, q] -
b*x*ExpandToSum[Pq - Coeff[Pq, x, q]*x^q, x])*(a + b*x^n)^(p + 1))/(a*b*n*(p + 1)), x] + Dist[1/(a*n*(p + 1))
, Int[Sum[(n*(p + 1) + i + 1)*Coeff[Pq, x, i]*x^i, {i, 0, q - 1}]*(a + b*x^n)^(p + 1), x], x] /; q == n - 1] /
; FreeQ[{a, b}, x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1]

Rule 1855

Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> -Simp[(x*Pq*(a + b*x^n)^(p + 1))/(a*n*(p + 1)), x] + Di
st[1/(a*n*(p + 1)), Int[ExpandToSum[n*(p + 1)*Pq + D[x*Pq, x], x]*(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b},
x] && PolyQ[Pq, x] && IGtQ[n, 0] && LtQ[p, -1] && LtQ[Expon[Pq, x], n - 1]

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 )^4} \, dx &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {\int \frac {11 b c-a g+10 b d x+9 b e x^2+8 b f x^3}{\left (a-b x^4\right )^3} \, dx}{12 a b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac {\int \frac {-7 (11 b c-a g)-60 b d x-45 b e x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2 b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\int \frac {21 (11 b c-a g)+120 b d x+45 b e x^2}{a-b x^4} \, dx}{384 a^3 b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\int \left (\frac {120 b d x}{a-b x^4}+\frac {21 (11 b c-a g)+45 b e x^2}{a-b x^4}\right ) \, dx}{384 a^3 b}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\int \frac {21 (11 b c-a g)+45 b e x^2}{a-b x^4} \, dx}{384 a^3 b}+\frac {(5 d) \int \frac {x}{a-b x^4} \, dx}{16 a^3}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3}-\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e-7 a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^{7/2} \sqrt {b}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e-7 a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^{7/2} \sqrt {b}}\\ &=\frac {x \left (b c+a g+b d x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}+\frac {x \left (7 (11 b c-a g)+60 b d x+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac {8 a f+x \left (11 b c-a g+10 b d x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac {\left (77 b c-15 \sqrt {a} \sqrt {b} e-7 a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{5/4}}+\frac {\left (77 b c+15 \sqrt {a} \sqrt {b} e-7 a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{5/4}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.39, size = 313, normalized size = 1.18 \[ \frac {\frac {128 a^{11/4} \sqrt [4]{b} (a (f+g x)+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}+\frac {16 a^{7/4} \sqrt [4]{b} x (-a g+11 b c+b x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac {4 a^{3/4} \sqrt [4]{b} x (-7 a g+77 b c+15 b x (4 d+3 e x))}{a-b x^4}-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (40 \sqrt [4]{a} b^{3/4} d+15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (-40 \sqrt [4]{a} b^{3/4} d+15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )+120 \sqrt [4]{a} b^{3/4} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )+6 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt {a} \sqrt {b} e-7 a g+77 b c\right )}{1536 a^{15/4} b^{5/4}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a^(3/4)*b^(1/4)*x*(77*b*c - 7*a*g + 15*b*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^(7/4)*b^(1/4)*x*(11*b*c - a
*g + b*x*(10*d + 9*e*x)))/(a - b*x^4)^2 + (128*a^(11/4)*b^(1/4)*(a*(f + g*x) + b*x*(c + x*(d + e*x))))/(a - b*
x^4)^3 + 6*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^(1/4)] - 3*(77*b*c + 40*a^(1/4)*b^(3/4
)*d + 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*Log[a^(1/4) - b^(1/4)*x] + 3*(77*b*c - 40*a^(1/4)*b^(3/4)*d + 15*Sqrt[a]*S
qrt[b]*e - 7*a*g)*Log[a^(1/4) + b^(1/4)*x] + 120*a^(1/4)*b^(3/4)*d*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*
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)^4,x, algorithm="fricas")

[Out]

Timed out

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giac [A]  time = 0.19, size = 442, normalized size = 1.66 \[ -\frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g + 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} + \frac {\sqrt {2} {\left (77 \, b^{2} c - 7 \, a b g - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 7 \, a b^{2} g x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 18 \, a^{2} b g x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 21 \, a^{3} g x + 32 \, a^{3} f}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b} \]

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)^4,x, algorithm="giac")

[Out]

-1/512*sqrt(2)*(77*b^2*c - 7*a*b*g - 40*sqrt(2)*(-a*b^3)^(1/4)*b*d + 15*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^3) - 1/512*sqrt(2)*(77*b^2*c - 7*a*b*g + 40*sqrt(2)*
(-a*b^3)^(1/4)*b*d - 15*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^3) - 1/1024*sqrt(2)*(77*b^2*c - 7*a*b*g - 15*sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt
(-a/b))/((-a*b^3)^(3/4)*a^3) + 1/1024*sqrt(2)*(77*b^2*c - 7*a*b*g - 15*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a
/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^3) - 1/384*(45*b^3*x^11*e + 60*b^3*d*x^10 + 77*b^3*c*x^9 - 7*a*b^2*g
*x^9 - 126*a*b^2*x^7*e - 160*a*b^2*d*x^6 - 198*a*b^2*c*x^5 + 18*a^2*b*g*x^5 + 113*a^2*b*x^3*e + 132*a^2*b*d*x^
2 + 153*a^2*b*c*x + 21*a^3*g*x + 32*a^3*f)/((b*x^4 - a)^3*a^3*b)

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maple [A]  time = 0.06, size = 368, normalized size = 1.38 \[ -\frac {5 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{3}}-\frac {15 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{3} b}-\frac {7 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{4}}+\frac {-\frac {15 b^{2} e \,x^{11}}{128 a^{3}}-\frac {5 b^{2} d \,x^{10}}{32 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {7 \left (a g -11 b c \right ) b \,x^{9}}{384 a^{3}}+\frac {5 b d \,x^{6}}{12 a^{2}}-\frac {113 e \,x^{3}}{384 a}-\frac {3 \left (a g -11 b c \right ) x^{5}}{64 a^{2}}-\frac {11 d \,x^{2}}{32 a}-\frac {f}{12 b}-\frac {\left (7 a g +51 b c \right ) x}{128 a b}}{\left (b \,x^{4}-a \right )^{3}} \]

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)^4,x)

[Out]

(-15/128/a^3*b^2*e*x^11-5/32/a^3*b^2*d*x^10+7/384*(a*g-11*b*c)/a^3*b*x^9+21/64/a^2*b*e*x^7+5/12/a^2*b*d*x^6-3/
64/a^2*(a*g-11*b*c)*x^5-113/384/a*e*x^3-11/32/a*d*x^2-1/128*(7*a*g+51*b*c)/a/b*x-1/12/b*f)/(b*x^4-a)^3-7/256/a
^3/b*(a/b)^(1/4)*arctan(1/(a/b)^(1/4)*x)*g+77/256*(a/b)^(1/4)/a^4*c*arctan(1/(a/b)^(1/4)*x)-7/512/a^3/b*(a/b)^
(1/4)*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))*g+77/512*(a/b)^(1/4)/a^4*c*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))-5/64/
(a*b)^(1/2)/a^3*d*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))-15/256/(a/b)^(1/4)/a^3/b*e*arctan(1/(a/b)^(1/4)
*x)+15/512/(a/b)^(1/4)/a^3/b*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))

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maxima [A]  time = 3.18, size = 345, normalized size = 1.30 \[ -\frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} - 126 \, a b^{2} e x^{7} - 160 \, a b^{2} d x^{6} + 7 \, {\left (11 \, b^{3} c - a b^{2} g\right )} x^{9} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} - 18 \, {\left (11 \, a b^{2} c - a^{2} b g\right )} x^{5} + 32 \, a^{3} f + 3 \, {\left (51 \, a^{2} b c + 7 \, a^{3} g\right )} x}{384 \, {\left (a^{3} b^{4} x^{12} - 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} - a^{6} b\right )}} + \frac {\frac {40 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a}} - \frac {40 \, \sqrt {b} d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a}} + \frac {2 \, {\left (77 \, b^{\frac {3}{2}} c - 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, b^{\frac {3}{2}} c + 15 \, \sqrt {a} b e - 7 \, a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3} b} \]

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)^4,x, algorithm="maxima")

[Out]

-1/384*(45*b^3*e*x^11 + 60*b^3*d*x^10 - 126*a*b^2*e*x^7 - 160*a*b^2*d*x^6 + 7*(11*b^3*c - a*b^2*g)*x^9 + 113*a
^2*b*e*x^3 + 132*a^2*b*d*x^2 - 18*(11*a*b^2*c - a^2*b*g)*x^5 + 32*a^3*f + 3*(51*a^2*b*c + 7*a^3*g)*x)/(a^3*b^4
*x^12 - 3*a^4*b^3*x^8 + 3*a^5*b^2*x^4 - a^6*b) + 1/512*(40*sqrt(b)*d*log(sqrt(b)*x^2 + sqrt(a))/sqrt(a) - 40*s
qrt(b)*d*log(sqrt(b)*x^2 - sqrt(a))/sqrt(a) + 2*(77*b^(3/2)*c - 15*sqrt(a)*b*e - 7*a*sqrt(b)*g)*arctan(sqrt(b)
*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - (77*b^(3/2)*c + 15*sqrt(a)*b*e - 7*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^3*b)

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mupad [B]  time = 5.66, size = 1056, normalized size = 3.97 \[ \left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (68719476736\,a^{15}\,b^5\,z^4-1211105280\,a^8\,b^4\,c\,e\,z^2+110100480\,a^9\,b^3\,e\,g\,z^2-838860800\,a^8\,b^4\,d^2\,z^2-88309760\,a^5\,b^3\,c\,d\,g\,z+485703680\,a^4\,b^4\,c^2\,d\,z+4014080\,a^6\,b^2\,d\,g^2\,z+18432000\,a^5\,b^3\,d\,e^2\,z+672000\,a^2\,b^2\,d^2\,e\,g-485100\,a^2\,b^2\,c\,e^2\,g-7392000\,a\,b^3\,c\,d^2\,e+12782924\,a\,b^3\,c^3\,g+105644\,a^3\,b\,c\,g^3-1743126\,a^2\,b^2\,c^2\,g^2+22050\,a^3\,b\,e^2\,g^2+2668050\,a\,b^3\,c^2\,e^2-50625\,a^2\,b^2\,e^4+2560000\,a\,b^3\,d^4-2401\,a^4\,g^4-35153041\,b^4\,c^4,z,k\right )\,\left (\mathrm {root}\left (68719476736\,a^{15}\,b^5\,z^4-1211105280\,a^8\,b^4\,c\,e\,z^2+110100480\,a^9\,b^3\,e\,g\,z^2-838860800\,a^8\,b^4\,d^2\,z^2-88309760\,a^5\,b^3\,c\,d\,g\,z+485703680\,a^4\,b^4\,c^2\,d\,z+4014080\,a^6\,b^2\,d\,g^2\,z+18432000\,a^5\,b^3\,d\,e^2\,z+672000\,a^2\,b^2\,d^2\,e\,g-485100\,a^2\,b^2\,c\,e^2\,g-7392000\,a\,b^3\,c\,d^2\,e+12782924\,a\,b^3\,c^3\,g+105644\,a^3\,b\,c\,g^3-1743126\,a^2\,b^2\,c^2\,g^2+22050\,a^3\,b\,e^2\,g^2+2668050\,a\,b^3\,c^2\,e^2-50625\,a^2\,b^2\,e^4+2560000\,a\,b^3\,d^4-2401\,a^4\,g^4-35153041\,b^4\,c^4,z,k\right )\,\left (\frac {20185088\,a^7\,b^3\,c-1835008\,a^8\,b^2\,g}{2097152\,a^9}-\frac {5\,b^3\,d\,x}{a^2}\right )+\frac {x\,\left (1568\,a^5\,b\,g^2-34496\,a^4\,b^2\,c\,g+7200\,a^4\,b^2\,e^2+189728\,a^3\,b^3\,c^2\right )}{131072\,a^9}-\frac {75\,b^2\,d\,e}{256\,a^5}\right )-\frac {-735\,a^2\,e\,g^2+16170\,a\,b\,c\,e\,g-11200\,a\,b\,d^2\,g+3375\,a\,b\,e^3-88935\,b^2\,c^2\,e+123200\,b^2\,c\,d^2}{2097152\,a^9}-\frac {x\,\left (4000\,b^2\,d^3-5775\,c\,e\,b^2\,d+525\,a\,e\,g\,b\,d\right )}{131072\,a^9}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^5\,z^4-1211105280\,a^8\,b^4\,c\,e\,z^2+110100480\,a^9\,b^3\,e\,g\,z^2-838860800\,a^8\,b^4\,d^2\,z^2-88309760\,a^5\,b^3\,c\,d\,g\,z+485703680\,a^4\,b^4\,c^2\,d\,z+4014080\,a^6\,b^2\,d\,g^2\,z+18432000\,a^5\,b^3\,d\,e^2\,z+672000\,a^2\,b^2\,d^2\,e\,g-485100\,a^2\,b^2\,c\,e^2\,g-7392000\,a\,b^3\,c\,d^2\,e+12782924\,a\,b^3\,c^3\,g+105644\,a^3\,b\,c\,g^3-1743126\,a^2\,b^2\,c^2\,g^2+22050\,a^3\,b\,e^2\,g^2+2668050\,a\,b^3\,c^2\,e^2-50625\,a^2\,b^2\,e^4+2560000\,a\,b^3\,d^4-2401\,a^4\,g^4-35153041\,b^4\,c^4,z,k\right )\right )+\frac {\frac {f}{12\,b}+\frac {11\,d\,x^2}{32\,a}+\frac {113\,e\,x^3}{384\,a}-\frac {3\,x^5\,\left (11\,b\,c-a\,g\right )}{64\,a^2}+\frac {7\,b\,x^9\,\left (11\,b\,c-a\,g\right )}{384\,a^3}+\frac {x\,\left (51\,b\,c+7\,a\,g\right )}{128\,a\,b}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {15\,b^2\,e\,x^{11}}{128\,a^3}-\frac {5\,b\,d\,x^6}{12\,a^2}-\frac {21\,b\,e\,x^7}{64\,a^2}}{a^3-3\,a^2\,b\,x^4+3\,a\,b^2\,x^8-b^3\,x^{12}} \]

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)^4,x)

[Out]

symsum(log(- root(68719476736*a^15*b^5*z^4 - 1211105280*a^8*b^4*c*e*z^2 + 110100480*a^9*b^3*e*g*z^2 - 83886080
0*a^8*b^4*d^2*z^2 - 88309760*a^5*b^3*c*d*g*z + 485703680*a^4*b^4*c^2*d*z + 4014080*a^6*b^2*d*g^2*z + 18432000*
a^5*b^3*d*e^2*z + 672000*a^2*b^2*d^2*e*g - 485100*a^2*b^2*c*e^2*g - 7392000*a*b^3*c*d^2*e + 12782924*a*b^3*c^3
*g + 105644*a^3*b*c*g^3 - 1743126*a^2*b^2*c^2*g^2 + 22050*a^3*b*e^2*g^2 + 2668050*a*b^3*c^2*e^2 - 50625*a^2*b^
2*e^4 + 2560000*a*b^3*d^4 - 2401*a^4*g^4 - 35153041*b^4*c^4, z, k)*(root(68719476736*a^15*b^5*z^4 - 1211105280
*a^8*b^4*c*e*z^2 + 110100480*a^9*b^3*e*g*z^2 - 838860800*a^8*b^4*d^2*z^2 - 88309760*a^5*b^3*c*d*g*z + 48570368
0*a^4*b^4*c^2*d*z + 4014080*a^6*b^2*d*g^2*z + 18432000*a^5*b^3*d*e^2*z + 672000*a^2*b^2*d^2*e*g - 485100*a^2*b
^2*c*e^2*g - 7392000*a*b^3*c*d^2*e + 12782924*a*b^3*c^3*g + 105644*a^3*b*c*g^3 - 1743126*a^2*b^2*c^2*g^2 + 220
50*a^3*b*e^2*g^2 + 2668050*a*b^3*c^2*e^2 - 50625*a^2*b^2*e^4 + 2560000*a*b^3*d^4 - 2401*a^4*g^4 - 35153041*b^4
*c^4, z, k)*((20185088*a^7*b^3*c - 1835008*a^8*b^2*g)/(2097152*a^9) - (5*b^3*d*x)/a^2) + (x*(1568*a^5*b*g^2 +
189728*a^3*b^3*c^2 + 7200*a^4*b^2*e^2 - 34496*a^4*b^2*c*g))/(131072*a^9) - (75*b^2*d*e)/(256*a^5)) - (123200*b
^2*c*d^2 - 88935*b^2*c^2*e - 735*a^2*e*g^2 + 3375*a*b*e^3 - 11200*a*b*d^2*g + 16170*a*b*c*e*g)/(2097152*a^9) -
(x*(4000*b^2*d^3 - 5775*b^2*c*d*e + 525*a*b*d*e*g))/(131072*a^9))*root(68719476736*a^15*b^5*z^4 - 1211105280*
a^8*b^4*c*e*z^2 + 110100480*a^9*b^3*e*g*z^2 - 838860800*a^8*b^4*d^2*z^2 - 88309760*a^5*b^3*c*d*g*z + 485703680
*a^4*b^4*c^2*d*z + 4014080*a^6*b^2*d*g^2*z + 18432000*a^5*b^3*d*e^2*z + 672000*a^2*b^2*d^2*e*g - 485100*a^2*b^
2*c*e^2*g - 7392000*a*b^3*c*d^2*e + 12782924*a*b^3*c^3*g + 105644*a^3*b*c*g^3 - 1743126*a^2*b^2*c^2*g^2 + 2205
0*a^3*b*e^2*g^2 + 2668050*a*b^3*c^2*e^2 - 50625*a^2*b^2*e^4 + 2560000*a*b^3*d^4 - 2401*a^4*g^4 - 35153041*b^4*
c^4, z, k), k, 1, 4) + (f/(12*b) + (11*d*x^2)/(32*a) + (113*e*x^3)/(384*a) - (3*x^5*(11*b*c - a*g))/(64*a^2) +
(7*b*x^9*(11*b*c - a*g))/(384*a^3) + (x*(51*b*c + 7*a*g))/(128*a*b) + (5*b^2*d*x^10)/(32*a^3) + (15*b^2*e*x^1
1)/(128*a^3) - (5*b*d*x^6)/(12*a^2) - (21*b*e*x^7)/(64*a^2))/(a^3 - b^3*x^12 - 3*a^2*b*x^4 + 3*a*b^2*x^8)

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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)**4,x)

[Out]

Timed out

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