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

[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])

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Rubi [A]  time = 0.320272, antiderivative size = 266, normalized size of antiderivative = 1., 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 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 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 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

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 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 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 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]

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*}

Mathematica [A]  time = 0.323845, 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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Maple [A]  time = 0.012, size = 368, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}-{\frac{5\,{b}^{2}d{x}^{10}}{32\,{a}^{3}}}+{\frac{ \left ( 7\,ag-77\,bc \right ) b{x}^{9}}{384\,{a}^{3}}}+{\frac{21\,be{x}^{7}}{64\,{a}^{2}}}+{\frac{5\,bd{x}^{6}}{12\,{a}^{2}}}-{\frac{ \left ( 3\,ag-33\,bc \right ){x}^{5}}{64\,{a}^{2}}}-{\frac{113\,e{x}^{3}}{384\,a}}-{\frac{11\,d{x}^{2}}{32\,a}}-{\frac{ \left ( 7\,ag+51\,bc \right ) x}{128\,ab}}-{\frac{f}{12\,b}} \right ) }-{\frac{7\,g}{512\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }+{\frac{77\,c}{512\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{7\,g}{256\,{a}^{3}b}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }+{\frac{77\,c}{256\,{a}^{4}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ) }-{\frac{5\,d}{64\,{a}^{3}}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\frac{15\,e}{256\,{a}^{3}b}\arctan \left ({x{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+{\frac{15\,e}{512\,{a}^{3}b}\ln \left ({ \left ( x+\sqrt [4]{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}} \right ) ^{-1}} \right ){\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}} \end{align*}

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

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]

Exception raised: ValueError

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Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

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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Giac [B]  time = 1.09935, size = 662, normalized size = 2.49 \begin{align*} \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} 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 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d + 77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g + 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} 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 \, a^{4} b^{3}} + \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} + \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{3}} - \frac{\sqrt{2}{\left (77 \, \left (-a b^{3}\right )^{\frac{1}{4}} b^{2} c - 7 \, \left (-a b^{3}\right )^{\frac{1}{4}} a b g - 15 \, \left (-a b^{3}\right )^{\frac{3}{4}} e\right )} \log \left (x^{2} - \sqrt{2} x \left (-\frac{a}{b}\right )^{\frac{1}{4}} + \sqrt{-\frac{a}{b}}\right )}{1024 \, a^{4} b^{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} \end{align*}

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)*(40*sqrt(2)*sqrt(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c - 7*(-a*b^3)^(1/4)*a*b*g + 15*(-a*b^3)^(3
/4)*e)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b^3) + 1/512*sqrt(2)*(40*sqrt(2)*sqr
t(-a*b)*b^2*d + 77*(-a*b^3)^(1/4)*b^2*c - 7*(-a*b^3)^(1/4)*a*b*g + 15*(-a*b^3)^(3/4)*e)*arctan(1/2*sqrt(2)*(2*
x - sqrt(2)*(-a/b)^(1/4))/(-a/b)^(1/4))/(a^4*b^3) + 1/1024*sqrt(2)*(77*(-a*b^3)^(1/4)*b^2*c - 7*(-a*b^3)^(1/4)
*a*b*g - 15*(-a*b^3)^(3/4)*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/(a^4*b^3) - 1/1024*sqrt(2)*(77*(-
a*b^3)^(1/4)*b^2*c - 7*(-a*b^3)^(1/4)*a*b*g - 15*(-a*b^3)^(3/4)*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/
b))/(a^4*b^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)