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

Optimal. Leaf size=220 \[ \frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]

[Out]

1/96*x*(9*e*x^2+10*d*x+11*c)/a^2/(-b*x^4+a)^2+1/384*x*(45*e*x^2+60*d*x+77*c)/a^3/(-b*x^4+a)+1/12*(a*f+b*x*(e*x
^2+d*x+c))/a/b/(-b*x^4+a)^3+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)
)*(-15*e*a^(1/2)+77*c*b^(1/2))/a^(15/4)/b^(3/4)+1/256*arctanh(b^(1/4)*x/a^(1/4))*(15*e*a^(1/2)+77*c*b^(1/2))/a
^(15/4)/b^(3/4)

________________________________________________________________________________________

Rubi [A]  time = 0.19, antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1854, 1855, 1876, 275, 208, 1167, 205} \[ \frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac {\left (15 \sqrt {a} e+77 \sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {5 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{32 a^{7/2} \sqrt {b}}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(11*c + 10*d*x + 9*e*x^2))/(96*a^2*(a - b*x^4)^2) + (x*(77*c + 60*d*x + 45*e*x^2))/(384*a^3*(a - b*x^4)) +
(a*f + b*x*(c + d*x + e*x^2))/(12*a*b*(a - b*x^4)^3) + ((77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/
4)])/(256*a^(15/4)*b^(3/4)) + ((77*Sqrt[b]*c + 15*Sqrt[a]*e)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(256*a^(15/4)*b^(3/
4)) + (5*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[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 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}{\left (a-b x^4\right )^4} \, dx &=\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \frac {-11 c-10 d x-9 e x^2}{\left (a-b x^4\right )^3} \, dx}{12 a}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac {\int \frac {77 c+60 d x+45 e x^2}{\left (a-b x^4\right )^2} \, dx}{96 a^2}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \frac {-231 c-120 d x-45 e x^2}{a-b x^4} \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \left (-\frac {120 d x}{a-b x^4}+\frac {-231 c-45 e x^2}{a-b x^4}\right ) \, dx}{384 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}-\frac {\int \frac {-231 c-45 e x^2}{a-b x^4} \, dx}{384 a^3}+\frac {(5 d) \int \frac {x}{a-b x^4} \, dx}{16 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac {(5 d) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{32 a^3}-\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}-15 e\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3}+\frac {\left (\frac {77 \sqrt {b} c}{\sqrt {a}}+15 e\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{256 a^3}\\ &=\frac {x \left (11 c+10 d x+9 e x^2\right )}{96 a^2 \left (a-b x^4\right )^2}+\frac {x \left (77 c+60 d x+45 e x^2\right )}{384 a^3 \left (a-b x^4\right )}+\frac {a f+b x \left (c+d x+e x^2\right )}{12 a b \left (a-b x^4\right )^3}+\frac {\left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\frac {\left (77 \sqrt {b} c+15 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{256 a^{15/4} b^{3/4}}+\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 = 0.50, size = 286, normalized size = 1.30 \[ \frac {-\frac {3 \log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt {b} c+40 \sqrt {a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac {3 \log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (15 a^{3/4} e+77 \sqrt [4]{a} \sqrt {b} c-40 \sqrt {a} \sqrt [4]{b} d\right )}{b^{3/4}}-\frac {128 a^3 (a f+b x (c+x (d+e x)))}{b \left (b x^4-a\right )^3}+\frac {16 a^2 x (11 c+x (10 d+9 e x))}{\left (a-b x^4\right )^2}+\frac {6 \sqrt [4]{a} \left (77 \sqrt {b} c-15 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {4 a x (77 c+15 x (4 d+3 e x))}{a-b x^4}+\frac {120 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{1536 a^4} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*x*(77*c + 15*x*(4*d + 3*e*x)))/(a - b*x^4) + (16*a^2*x*(11*c + x*(10*d + 9*e*x)))/(a - b*x^4)^2 - (128*a
^3*(a*f + b*x*(c + x*(d + e*x))))/(b*(-a + b*x^4)^3) + (6*a^(1/4)*(77*Sqrt[b]*c - 15*Sqrt[a]*e)*ArcTan[(b^(1/4
)*x)/a^(1/4)])/b^(3/4) - (3*(77*a^(1/4)*Sqrt[b]*c + 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) - b^(1/4)
*x])/b^(3/4) + (3*(77*a^(1/4)*Sqrt[b]*c - 40*Sqrt[a]*b^(1/4)*d + 15*a^(3/4)*e)*Log[a^(1/4) + b^(1/4)*x])/b^(3/
4) + (120*Sqrt[a]*d*Log[Sqrt[a] + Sqrt[b]*x^2])/Sqrt[b])/(1536*a^4)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________

giac [B]  time = 0.19, size = 395, normalized size = 1.80 \[ -\frac {\sqrt {2} {\left (77 \, b^{2} c - 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c + 40 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 15 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {\sqrt {2} {\left (77 \, b^{2} c - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} + \frac {\sqrt {2} {\left (77 \, b^{2} c - 15 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{1024 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{3}} - \frac {45 \, b^{3} x^{11} e + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 126 \, a b^{2} x^{7} e - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 113 \, a^{2} b x^{3} e + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 32 \, a^{3} f}{384 \, {\left (b x^{4} - a\right )}^{3} a^{3} b} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 - 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 + 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) - 1/102
4*sqrt(2)*(77*b^2*c - 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 - 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 - 126*a*b^2*x^7*e - 160*a*b^2*d*x^6 - 198*a*b^2*c*x
^5 + 113*a^2*b*x^3*e + 132*a^2*b*d*x^2 + 153*a^2*b*c*x + 32*a^3*f)/((b*x^4 - a)^3*a^3*b)

________________________________________________________________________________________

maple [A]  time = 0.06, size = 280, normalized size = 1.27 \[ -\frac {5 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{64 \sqrt {a b}\, a^{3}}-\frac {15 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {15 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{3} b}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{256 a^{4}}+\frac {77 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{512 a^{4}}+\frac {-\frac {15 b^{2} e \,x^{11}}{128 a^{3}}-\frac {5 b^{2} d \,x^{10}}{32 a^{3}}-\frac {77 b^{2} c \,x^{9}}{384 a^{3}}+\frac {21 b e \,x^{7}}{64 a^{2}}+\frac {5 b d \,x^{6}}{12 a^{2}}+\frac {33 b c \,x^{5}}{64 a^{2}}-\frac {113 e \,x^{3}}{384 a}-\frac {11 d \,x^{2}}{32 a}-\frac {51 c x}{128 a}-\frac {f}{12 b}}{\left (b \,x^{4}-a \right )^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((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^10-77/384/a^3*b^2*c*x^9+21/64/a^2*b*e*x^7+5/12/a^2*b*d*x^6+33/64/a^2*
b*c*x^5-113/384/a*e*x^3-11/32/a*d*x^2-51/128/a*c*x-1/12*f/b)/(b*x^4-a)^3+77/512*(a/b)^(1/4)/a^4*c*ln((x+(a/b)^
(1/4))/(x-(a/b)^(1/4)))+77/256*(a/b)^(1/4)/a^4*c*arctan(1/(a/b)^(1/4)*x)-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*l
n((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))

________________________________________________________________________________________

maxima [A]  time = 3.06, size = 297, normalized size = 1.35 \[ -\frac {45 \, b^{3} e x^{11} + 60 \, b^{3} d x^{10} + 77 \, b^{3} c x^{9} - 126 \, a b^{2} e x^{7} - 160 \, a b^{2} d x^{6} - 198 \, a b^{2} c x^{5} + 113 \, a^{2} b e x^{3} + 132 \, a^{2} b d x^{2} + 153 \, a^{2} b c x + 32 \, a^{3} f}{384 \, {\left (a^{3} b^{4} x^{12} - 3 \, a^{4} b^{3} x^{8} + 3 \, a^{5} b^{2} x^{4} - a^{6} b\right )}} + \frac {\frac {40 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {40 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (77 \, \sqrt {b} c - 15 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (77 \, \sqrt {b} c + 15 \, \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{512 \, a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((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 + 77*b^3*c*x^9 - 126*a*b^2*e*x^7 - 160*a*b^2*d*x^6 - 198*a*b^2*c*x^5 + 1
13*a^2*b*e*x^3 + 132*a^2*b*d*x^2 + 153*a^2*b*c*x + 32*a^3*f)/(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*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 40*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sq
rt(b)) + 2*(77*sqrt(b)*c - 15*sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b)
)*sqrt(b)) - (77*sqrt(b)*c + 15*sqrt(a)*e)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + sqrt(sqrt(a)*s
qrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/a^3

________________________________________________________________________________________

mupad [B]  time = 5.25, size = 880, normalized size = 4.00 \[ \left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (3375\,a\,e^3+123200\,b\,c\,d^2-88935\,b\,c^2\,e+64000\,b\,d^3\,x+{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,c\,20185088+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^4,z,k\right )\,a^4\,b\,e^2\,x\,115200-92400\,b\,c\,d\,e\,x+\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^4,z,k\right )\,a^3\,b^2\,c^2\,x\,3035648-{\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^4,z,k\right )}^2\,a^7\,b^2\,d\,x\,10485760-\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^4,z,k\right )\,a^4\,b\,d\,e\,614400\right )}{a^9\,2097152}\right )\,\mathrm {root}\left (68719476736\,a^{15}\,b^3\,z^4-1211105280\,a^8\,b^2\,c\,e\,z^2-838860800\,a^8\,b^2\,d^2\,z^2+485703680\,a^4\,b^2\,c^2\,d\,z+18432000\,a^5\,b\,d\,e^2\,z-7392000\,a\,b\,c\,d^2\,e+2668050\,a\,b\,c^2\,e^2+2560000\,a\,b\,d^4-35153041\,b^2\,c^4-50625\,a^2\,e^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 {51\,c\,x}{128\,a}+\frac {77\,b^2\,c\,x^9}{384\,a^3}+\frac {5\,b^2\,d\,x^{10}}{32\,a^3}+\frac {15\,b^2\,e\,x^{11}}{128\,a^3}-\frac {33\,b\,c\,x^5}{64\,a^2}-\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)/(a - b*x^4)^4,x)

[Out]

symsum(log(-(b*(3375*a*e^3 + 123200*b*c*d^2 - 88935*b*c^2*e + 64000*b*d^3*x + 20185088*root(68719476736*a^15*b
^3*z^4 - 1211105280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^2 + 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d
*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 - 35153041*b^2*c^4 - 50625*a^2*e^4, z, k)
^2*a^7*b^2*c + 115200*root(68719476736*a^15*b^3*z^4 - 1211105280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^2 +
485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*
d^4 - 35153041*b^2*c^4 - 50625*a^2*e^4, z, k)*a^4*b*e^2*x - 92400*b*c*d*e*x + 3035648*root(68719476736*a^15*b^
3*z^4 - 1211105280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^2 + 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*
e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 - 35153041*b^2*c^4 - 50625*a^2*e^4, z, k)*
a^3*b^2*c^2*x - 10485760*root(68719476736*a^15*b^3*z^4 - 1211105280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^
2 + 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a
*b*d^4 - 35153041*b^2*c^4 - 50625*a^2*e^4, z, k)^2*a^7*b^2*d*x - 614400*root(68719476736*a^15*b^3*z^4 - 121110
5280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^2 + 485703680*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 739200
0*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 - 35153041*b^2*c^4 - 50625*a^2*e^4, z, k)*a^4*b*d*e))/(2
097152*a^9))*root(68719476736*a^15*b^3*z^4 - 1211105280*a^8*b^2*c*e*z^2 - 838860800*a^8*b^2*d^2*z^2 + 48570368
0*a^4*b^2*c^2*d*z + 18432000*a^5*b*d*e^2*z - 7392000*a*b*c*d^2*e + 2668050*a*b*c^2*e^2 + 2560000*a*b*d^4 - 351
53041*b^2*c^4 - 50625*a^2*e^4, z, k), k, 1, 4) + (f/(12*b) + (11*d*x^2)/(32*a) + (113*e*x^3)/(384*a) + (51*c*x
)/(128*a) + (77*b^2*c*x^9)/(384*a^3) + (5*b^2*d*x^10)/(32*a^3) + (15*b^2*e*x^11)/(128*a^3) - (33*b*c*x^5)/(64*
a^2) - (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)

________________________________________________________________________________________

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

[Out]

Timed out

________________________________________________________________________________________