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

Optimal. Leaf size=293 \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b*c - a*g) + 12*(5*b*d
- a*h)*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) + (8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*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*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + ((5
*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2))

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Rubi [A]  time = 0.430552, antiderivative size = 293, normalized size of antiderivative = 1., number of steps used = 10, number of rules used = 8, integrand size = 36, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.222, Rules used = {1858, 1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac{\tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{\tanh ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )}{256 a^{15/4} b^{5/4}}+\frac{(5 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}+\frac{x \left (2 x (5 b d-a h)-a g+11 b c+9 b e x^2\right )+8 a f}{96 a^2 b \left (a-b x^4\right )^2}+\frac{x \left (7 (11 b c-a g)+12 x (5 b d-a h)+45 b e x^2\right )}{384 a^3 b \left (a-b x^4\right )}+\frac{x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(12*a*b*(a - b*x^4)^3) + (x*(7*(11*b*c - a*g) + 12*(5*b*d
- a*h)*x + 45*b*e*x^2))/(384*a^3*b*(a - b*x^4)) + (8*a*f + x*(11*b*c - a*g + 2*(5*b*d - a*h)*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*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(5/4)) + ((5
*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(32*a^(7/2)*b^(3/2))

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+h x^5}{\left (a-b x^4\right )^4} \, dx &=\frac{x \left (b c+a g+(b d+a h) x+b e x^2+b f x^3\right )}{12 a b \left (a-b x^4\right )^3}-\frac{\int \frac{-b (11 b c-a g)-2 b (5 b d-a h) x-9 b^2 e x^2-8 b^2 f x^3}{\left (a-b x^4\right )^3} \, dx}{12 a b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) 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+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}+\frac{\int \frac{7 b (11 b c-a g)+12 b (5 b d-a h) x+45 b^2 e x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) 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)+12 (5 b d-a h) 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+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c-a g)-24 b (5 b d-a h) x-45 b^2 e x^2}{a-b x^4} \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) 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)+12 (5 b d-a h) 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+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \left (-\frac{24 b (5 b d-a h) x}{a-b x^4}+\frac{-21 b (11 b c-a g)-45 b^2 e x^2}{a-b x^4}\right ) \, dx}{384 a^3 b^2}\\ &=\frac{x \left (b c+a g+(b d+a h) 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)+12 (5 b d-a h) 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+2 (5 b d-a h) x+9 b e x^2\right )}{96 a^2 b \left (a-b x^4\right )^2}-\frac{\int \frac{-21 b (11 b c-a g)-45 b^2 e x^2}{a-b x^4} \, dx}{384 a^3 b^2}+\frac{(5 b d-a h) \int \frac{x}{a-b x^4} \, dx}{16 a^3 b}\\ &=\frac{x \left (b c+a g+(b d+a h) 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)+12 (5 b d-a h) 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+2 (5 b d-a h) 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 ) \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{(5 b d-a h) \operatorname{Subst}\left (\int \frac{1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3 b}\\ &=\frac{x \left (b c+a g+(b d+a h) 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)+12 (5 b d-a h) 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+2 (5 b d-a h) 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 b d-a h) \tanh ^{-1}\left (\frac{\sqrt{b} x^2}{\sqrt{a}}\right )}{32 a^{7/2} b^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.39516, size = 360, normalized size = 1.23 \[ \frac{-3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (-8 a^{5/4} h+15 \sqrt{a} b^{3/4} e+40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (8 a^{5/4} h+15 \sqrt{a} b^{3/4} e-40 \sqrt [4]{a} b d-7 a \sqrt [4]{b} g+77 b^{5/4} c\right )+\frac{128 a^{11/4} \sqrt{b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^3}+\frac{16 a^{7/4} \sqrt{b} x (-a (g+2 h x)+11 b c+b x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac{4 a^{3/4} \sqrt{b} x \left (-7 a g-12 a h x+77 b c+60 b d x+45 b e x^2\right )}{a-b x^4}+6 \sqrt [4]{b} \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-15 \sqrt{a} \sqrt{b} e-7 a g+77 b c\right )-24 \sqrt [4]{a} (a h-5 b d) \log \left (\sqrt{a}+\sqrt{b} x^2\right )}{1536 a^{15/4} b^{3/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a^(3/4)*Sqrt[b]*x*(77*b*c - 7*a*g + 60*b*d*x - 12*a*h*x + 45*b*e*x^2))/(a - b*x^4) + (16*a^(7/4)*Sqrt[b]*x
*(11*b*c + b*x*(10*d + 9*e*x) - a*(g + 2*h*x)))/(a - b*x^4)^2 + (128*a^(11/4)*Sqrt[b]*(b*x*(c + x*(d + e*x)) +
a*(f + x*(g + h*x))))/(a - b*x^4)^3 + 6*b^(1/4)*(77*b*c - 15*Sqrt[a]*Sqrt[b]*e - 7*a*g)*ArcTan[(b^(1/4)*x)/a^
(1/4)] - 3*(77*b^(5/4)*c + 40*a^(1/4)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a*b^(1/4)*g - 8*a^(5/4)*h)*Log[a^(1/4) -
b^(1/4)*x] + 3*(77*b^(5/4)*c - 40*a^(1/4)*b*d + 15*Sqrt[a]*b^(3/4)*e - 7*a*b^(1/4)*g + 8*a^(5/4)*h)*Log[a^(1/4
) + b^(1/4)*x] - 24*a^(1/4)*(-5*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(1536*a^(15/4)*b^(3/2))

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Maple [A]  time = 0.013, size = 434, normalized size = 1.5 \begin{align*}{\frac{1}{ \left ( b{x}^{4}-a \right ) ^{3}} \left ( -{\frac{15\,{b}^{2}e{x}^{11}}{128\,{a}^{3}}}+{\frac{ \left ( ah-5\,bd \right ) b{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{ \left ( ah-5\,bd \right ){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{ \left ( ah+11\,bd \right ){x}^{2}}{32\,ab}}-{\frac{ \left ( 7\,ag+51\,bc \right ) x}{128\,ab}}-{\frac{f}{12\,b}} \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{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{h}{64\,b{a}^{2}}\ln \left ({ \left ( -a+{x}^{2}\sqrt{ab} \right ) \left ( -a-{x}^{2}\sqrt{ab} \right ) ^{-1}} \right ){\frac{1}{\sqrt{ab}}}}-{\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((h*x^5+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+1/32*(a*h-5*b*d)/a^3*b*x^10+7/384*(a*g-11*b*c)/a^3*b*x^9+21/64/a^2*b*e*x^7-1/12/a^2*(a
*h-5*b*d)*x^6-3/64/a^2*(a*g-11*b*c)*x^5-113/384/a*e*x^3-1/32*(a*h+11*b*d)/a/b*x^2-1/128*(7*a*g+51*b*c)/a/b*x-1
/12*f/b)/(b*x^4-a)^3-7/256/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))-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)*l
n((x+(1/b*a)^(1/4))/(x-(1/b*a)^(1/4)))+1/64/b/a^2/(a*b)^(1/2)*ln((-a+x^2*(a*b)^(1/2))/(-a-x^2*(a*b)^(1/2)))*h-
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)*arctan(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((h*x^5+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((h*x^5+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((h*x**5+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 = 1.08415, size = 738, normalized size = 2.52 \begin{align*} -\frac{\sqrt{2}{\left (40 \, \sqrt{2} \sqrt{-a b} b^{2} d - 8 \, \sqrt{2} \sqrt{-a b} a b h - 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 - 8 \, \sqrt{2} \sqrt{-a b} a b h - 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} - 12 \, a b^{2} h 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} + 32 \, a^{2} b h 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} + 12 \, a^{3} h 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((h*x^5+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 - 8*sqrt(2)*sqrt(-a*b)*a*b*h - 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)*sqrt(-a*b)*b^2*d - 8*sqrt(2)*sqrt(-a*b)*a*b*h - 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^1
0 - 12*a*b^2*h*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 + 32*a^2*b*h*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 + 12*a^3*h*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)