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

Optimal. Leaf size=241 \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

[Out]

1/8*x*(b*c+a*g+(a*h+b*d)*x+b*e*x^2+b*f*x^3)/a/b/(-b*x^4+a)^2+1/32*(4*a*f+x*(7*b*c-a*g+2*(-a*h+3*b*d)*x+5*b*e*x
^2))/a^2/b/(-b*x^4+a)+1/16*(-a*h+3*b*d)*arctanh(x^2*b^(1/2)/a^(1/2))/a^(5/2)/b^(3/2)+1/64*arctan(b^(1/4)*x/a^(
1/4))*(21*b*c-3*a*g-5*e*a^(1/2)*b^(1/2))/a^(11/4)/b^(5/4)+1/64*arctanh(b^(1/4)*x/a^(1/4))*(21*b*c-3*a*g+5*e*a^
(1/2)*b^(1/2))/a^(11/4)/b^(5/4)

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Rubi [A]  time = 0.34, antiderivative size = 241, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, integrand size = 36, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {1858, 1854, 1876, 275, 208, 1167, 205} \[ \frac {\tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {\tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )}{64 a^{11/4} b^{5/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}+\frac {x \left (2 x (3 b d-a h)-a g+7 b c+5 b e x^2\right )+4 a f}{32 a^2 b \left (a-b x^4\right )}+\frac {x \left (x (a h+b d)+a g+b c+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2} \]

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)^3,x]

[Out]

(x*(b*c + a*g + (b*d + a*h)*x + b*e*x^2 + b*f*x^3))/(8*a*b*(a - b*x^4)^2) + (4*a*f + x*(7*b*c - a*g + 2*(3*b*d
- a*h)*x + 5*b*e*x^2))/(32*a^2*b*(a - b*x^4)) + ((21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[(b^(1/4)*x)/a^
(1/4)])/(64*a^(11/4)*b^(5/4)) + ((21*b*c + 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTanh[(b^(1/4)*x)/a^(1/4)])/(64*a^(1
1/4)*b^(5/4)) + ((3*b*d - a*h)*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*b^(3/2))

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 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+h x^5}{\left (a-b x^4\right )^3} \, dx &=\frac {x \left (b c+a g+(b d+a h) x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}-\frac {\int \frac {-b (7 b c-a g)-2 b (3 b d-a h) x-5 b^2 e x^2-4 b^2 f x^3}{\left (a-b x^4\right )^2} \, dx}{8 a b^2}\\ &=\frac {x \left (b c+a g+(b d+a h) x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)+4 b (3 b d-a h) x+5 b^2 e x^2}{a-b x^4} \, dx}{32 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 )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \left (\frac {4 b (3 b d-a h) x}{a-b x^4}+\frac {3 b (7 b c-a g)+5 b^2 e x^2}{a-b x^4}\right ) \, dx}{32 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 )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\int \frac {3 b (7 b c-a g)+5 b^2 e x^2}{a-b x^4} \, dx}{32 a^2 b^2}+\frac {(3 b d-a h) \int \frac {x}{a-b x^4} \, dx}{8 a^2 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}-\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^{5/2} \sqrt {b}}+\frac {(3 b d-a h) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2 b}\\ &=\frac {x \left (b c+a g+(b d+a h) x+b e x^2+b f x^3\right )}{8 a b \left (a-b x^4\right )^2}+\frac {4 a f+x \left (7 b c-a g+2 (3 b d-a h) x+5 b e x^2\right )}{32 a^2 b \left (a-b x^4\right )}+\frac {\left (21 b c-5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {\left (21 b c+5 \sqrt {a} \sqrt {b} e-3 a g\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{5/4}}+\frac {(3 b d-a h) \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} b^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.42, size = 309, normalized size = 1.28 \[ \frac {\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (4 a^{5/4} h-5 \sqrt {a} b^{3/4} e-12 \sqrt [4]{a} b d+3 a \sqrt [4]{b} g-21 b^{5/4} c\right )+\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (4 a^{5/4} h+5 \sqrt {a} b^{3/4} e-12 \sqrt [4]{a} b d-3 a \sqrt [4]{b} g+21 b^{5/4} c\right )+\frac {16 a^{7/4} \sqrt {b} (a (f+x (g+h x))+b x (c+x (d+e x)))}{\left (a-b x^4\right )^2}+\frac {4 a^{3/4} \sqrt {b} x (-a (g+2 h x)+7 b c+b x (6 d+5 e x))}{a-b x^4}+2 \sqrt [4]{b} \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (-5 \sqrt {a} \sqrt {b} e-3 a g+21 b c\right )-4 \sqrt [4]{a} (a h-3 b d) \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{128 a^{11/4} b^{3/2}} \]

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)^3,x]

[Out]

((4*a^(3/4)*Sqrt[b]*x*(7*b*c + b*x*(6*d + 5*e*x) - a*(g + 2*h*x)))/(a - b*x^4) + (16*a^(7/4)*Sqrt[b]*(b*x*(c +
x*(d + e*x)) + a*(f + x*(g + h*x))))/(a - b*x^4)^2 + 2*b^(1/4)*(21*b*c - 5*Sqrt[a]*Sqrt[b]*e - 3*a*g)*ArcTan[
(b^(1/4)*x)/a^(1/4)] + (-21*b^(5/4)*c - 12*a^(1/4)*b*d - 5*Sqrt[a]*b^(3/4)*e + 3*a*b^(1/4)*g + 4*a^(5/4)*h)*Lo
g[a^(1/4) - b^(1/4)*x] + (21*b^(5/4)*c - 12*a^(1/4)*b*d + 5*Sqrt[a]*b^(3/4)*e - 3*a*b^(1/4)*g + 4*a^(5/4)*h)*L
og[a^(1/4) + b^(1/4)*x] - 4*a^(1/4)*(-3*b*d + a*h)*Log[Sqrt[a] + Sqrt[b]*x^2])/(128*a^(11/4)*b^(3/2))

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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((h*x^5+g*x^4+f*x^3+e*x^2+d*x+c)/(-b*x^4+a)^3,x, algorithm="fricas")

[Out]

Timed out

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giac [B]  time = 0.21, size = 440, normalized size = 1.83 \[ -\frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 4 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} a h - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 3 \, a b g - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b^{2} x^{7} e + 6 \, b^{2} d x^{6} - 2 \, a b h x^{6} + 7 \, b^{2} c x^{5} - a b g x^{5} - 9 \, a b x^{3} e - 10 \, a b d x^{2} - 2 \, a^{2} h x^{2} - 11 \, a b c x - 3 \, a^{2} g x - 4 \, a^{2} f}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2} b} \]

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)^3,x, algorithm="giac")

[Out]

-1/128*sqrt(2)*(21*b^2*c - 3*a*b*g - 12*sqrt(2)*(-a*b^3)^(1/4)*b*d + 4*sqrt(2)*(-a*b^3)^(1/4)*a*h + 5*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^2) - 1/128*sqrt(2)*(21
*b^2*c - 3*a*b*g + 12*sqrt(2)*(-a*b^3)^(1/4)*b*d - 4*sqrt(2)*(-a*b^3)^(1/4)*a*h - 5*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^2) - 1/256*sqrt(2)*(21*b^2*c - 3*a*b*g -
5*sqrt(-a*b)*b*e)*log(x^2 + sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) + 1/256*sqrt(2)*(21*b^2
*c - 3*a*b*g - 5*sqrt(-a*b)*b*e)*log(x^2 - sqrt(2)*x*(-a/b)^(1/4) + sqrt(-a/b))/((-a*b^3)^(3/4)*a^2) - 1/32*(5
*b^2*x^7*e + 6*b^2*d*x^6 - 2*a*b*h*x^6 + 7*b^2*c*x^5 - a*b*g*x^5 - 9*a*b*x^3*e - 10*a*b*d*x^2 - 2*a^2*h*x^2 -
11*a*b*c*x - 3*a^2*g*x - 4*a^2*f)/((b*x^4 - a)^2*a^2*b)

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maple [A]  time = 0.06, size = 389, normalized size = 1.61 \[ \frac {h \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a b}-\frac {3 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a^{2}}-\frac {5 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{2} b}-\frac {3 \left (\frac {a}{b}\right )^{\frac {1}{4}} g \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{3}}-\frac {\frac {5 b e \,x^{7}}{32 a^{2}}-\frac {\left (a h -3 b d \right ) x^{6}}{16 a^{2}}-\frac {9 e \,x^{3}}{32 a}-\frac {\left (a g -7 b c \right ) x^{5}}{32 a^{2}}-\frac {\left (a h +5 b d \right ) x^{2}}{16 a b}-\frac {f}{8 b}-\frac {\left (3 a g +11 b c \right ) x}{32 a b}}{\left (b \,x^{4}-a \right )^{2}} \]

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)^3,x)

[Out]

-(5/32/a^2*b*e*x^7-1/16*(a*h-3*b*d)/a^2*x^6-1/32*(a*g-7*b*c)/a^2*x^5-9/32/a*e*x^3-1/16*(a*h+5*b*d)/a/b*x^2-1/3
2*(3*a*g+11*b*c)/a/b*x-1/8/b*f)/(b*x^4-a)^2-3/64*(a/b)^(1/4)/a^2/b*g*arctan(1/(a/b)^(1/4)*x)+21/64*(a/b)^(1/4)
/a^3*c*arctan(1/(a/b)^(1/4)*x)-3/128*(a/b)^(1/4)/a^2/b*g*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+21/128*(a/b)^(1/4
)/a^3*c*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))+1/32/a/b/(a*b)^(1/2)*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))*
h-3/32/(a*b)^(1/2)/a^2*d*ln(((a*b)^(1/2)*x^2-a)/(-(a*b)^(1/2)*x^2-a))-5/64/(a/b)^(1/4)/a^2/b*e*arctan(1/(a/b)^
(1/4)*x)+5/128/(a/b)^(1/4)/a^2/b*e*ln((x+(a/b)^(1/4))/(x-(a/b)^(1/4)))

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maxima [A]  time = 2.96, size = 316, normalized size = 1.31 \[ -\frac {5 \, b^{2} e x^{7} + 2 \, {\left (3 \, b^{2} d - a b h\right )} x^{6} - 9 \, a b e x^{3} + {\left (7 \, b^{2} c - a b g\right )} x^{5} - 4 \, a^{2} f - 2 \, {\left (5 \, a b d + a^{2} h\right )} x^{2} - {\left (11 \, a b c + 3 \, a^{2} g\right )} x}{32 \, {\left (a^{2} b^{3} x^{8} - 2 \, a^{3} b^{2} x^{4} + a^{4} b\right )}} + \frac {\frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {4 \, {\left (3 \, b d - a h\right )} \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, b^{\frac {3}{2}} c - 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, b^{\frac {3}{2}} c + 5 \, \sqrt {a} b e - 3 \, a \sqrt {b} g\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2} b} \]

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)^3,x, algorithm="maxima")

[Out]

-1/32*(5*b^2*e*x^7 + 2*(3*b^2*d - a*b*h)*x^6 - 9*a*b*e*x^3 + (7*b^2*c - a*b*g)*x^5 - 4*a^2*f - 2*(5*a*b*d + a^
2*h)*x^2 - (11*a*b*c + 3*a^2*g)*x)/(a^2*b^3*x^8 - 2*a^3*b^2*x^4 + a^4*b) + 1/128*(4*(3*b*d - a*h)*log(sqrt(b)*
x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 4*(3*b*d - a*h)*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b)) + 2*(21*b^(3/2
)*c - 5*sqrt(a)*b*e - 3*a*sqrt(b)*g)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sq
rt(b)) - (21*b^(3/2)*c + 5*sqrt(a)*b*e - 3*a*sqrt(b)*g)*log((sqrt(b)*x - sqrt(sqrt(a)*sqrt(b)))/(sqrt(b)*x + s
qrt(sqrt(a)*sqrt(b))))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqrt(b)))/(a^2*b)

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mupad [B]  time = 5.73, size = 1687, normalized size = 7.00 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(f/(8*b) + (9*e*x^3)/(32*a) - (x^5*(7*b*c - a*g))/(32*a^2) - (x^6*(3*b*d - a*h))/(16*a^2) + (x*(11*b*c + 3*a*g
))/(32*a*b) + (x^2*(5*b*d + a*h))/(16*a*b) - (5*b*e*x^7)/(32*a^2))/(a^2 + b^2*x^8 - 2*a*b*x^4) + symsum(log(-
root(268435456*a^11*b^6*z^4 + 3145728*a^7*b^4*d*h*z^2 + 983040*a^7*b^4*e*g*z^2 - 6881280*a^6*b^5*c*e*z^2 - 524
288*a^8*b^3*h^2*z^2 - 4718592*a^6*b^5*d^2*z^2 + 258048*a^5*b^3*c*g*h*z - 774144*a^4*b^4*c*d*g*z - 18432*a^6*b^
2*g^2*h*z - 51200*a^5*b^3*e^2*h*z - 903168*a^4*b^4*c^2*h*z + 55296*a^5*b^3*d*g^2*z + 153600*a^4*b^4*d*e^2*z +
2709504*a^3*b^5*c^2*d*z - 5760*a^3*b^2*d*e*g*h + 40320*a^2*b^3*c*d*e*h + 8640*a^2*b^3*d^2*e*g - 6720*a^3*b^2*c
*e*h^2 - 6300*a^2*b^3*c*e^2*g + 960*a^4*b*e*g*h^2 - 60480*a*b^4*c*d^2*e - 3072*a^4*b*d*h^3 + 111132*a*b^4*c^3*
g + 13824*a^3*b^2*d^2*h^2 + 450*a^3*b^2*e^2*g^2 - 23814*a^2*b^3*c^2*g^2 - 27648*a^2*b^3*d^3*h + 2268*a^3*b^2*c
*g^3 + 22050*a*b^4*c^2*e^2 - 625*a^2*b^3*e^4 - 81*a^4*b*g^4 + 20736*a*b^4*d^4 + 256*a^5*h^4 - 194481*b^5*c^4,
z, k)*(root(268435456*a^11*b^6*z^4 + 3145728*a^7*b^4*d*h*z^2 + 983040*a^7*b^4*e*g*z^2 - 6881280*a^6*b^5*c*e*z^
2 - 524288*a^8*b^3*h^2*z^2 - 4718592*a^6*b^5*d^2*z^2 + 258048*a^5*b^3*c*g*h*z - 774144*a^4*b^4*c*d*g*z - 18432
*a^6*b^2*g^2*h*z - 51200*a^5*b^3*e^2*h*z - 903168*a^4*b^4*c^2*h*z + 55296*a^5*b^3*d*g^2*z + 153600*a^4*b^4*d*e
^2*z + 2709504*a^3*b^5*c^2*d*z - 5760*a^3*b^2*d*e*g*h + 40320*a^2*b^3*c*d*e*h + 8640*a^2*b^3*d^2*e*g - 6720*a^
3*b^2*c*e*h^2 - 6300*a^2*b^3*c*e^2*g + 960*a^4*b*e*g*h^2 - 60480*a*b^4*c*d^2*e - 3072*a^4*b*d*h^3 + 111132*a*b
^4*c^3*g + 13824*a^3*b^2*d^2*h^2 + 450*a^3*b^2*e^2*g^2 - 23814*a^2*b^3*c^2*g^2 - 27648*a^2*b^3*d^3*h + 2268*a^
3*b^2*c*g^3 + 22050*a*b^4*c^2*e^2 - 625*a^2*b^3*e^4 - 81*a^4*b*g^4 + 20736*a*b^4*d^4 + 256*a^5*h^4 - 194481*b^
5*c^4, z, k)*((344064*a^5*b^4*c - 49152*a^6*b^3*g)/(32768*a^6*b) - (x*(24576*a^5*b^4*d - 8192*a^6*b^3*h))/(409
6*a^6*b)) - (15360*a^3*b^3*d*e - 5120*a^4*b^2*e*h)/(32768*a^6*b) + (x*(7056*a^2*b^4*c^2 + 400*a^3*b^3*e^2 + 14
4*a^4*b^2*g^2 - 2016*a^3*b^3*c*g))/(4096*a^6*b)) - (125*a*b^2*e^3 + 3024*b^3*c*d^2 - 2205*b^3*c^2*e - 48*a^3*g
*h^2 - 432*a*b^2*d^2*g + 336*a^2*b*c*h^2 - 45*a^2*b*e*g^2 - 2016*a*b^2*c*d*h + 630*a*b^2*c*e*g + 288*a^2*b*d*g
*h)/(32768*a^6*b) - (x*(216*b^3*d^3 - 8*a^3*h^3 - 315*b^3*c*d*e - 216*a*b^2*d^2*h + 72*a^2*b*d*h^2 + 105*a*b^2
*c*e*h + 45*a*b^2*d*e*g - 15*a^2*b*e*g*h))/(4096*a^6*b))*root(268435456*a^11*b^6*z^4 + 3145728*a^7*b^4*d*h*z^2
+ 983040*a^7*b^4*e*g*z^2 - 6881280*a^6*b^5*c*e*z^2 - 524288*a^8*b^3*h^2*z^2 - 4718592*a^6*b^5*d^2*z^2 + 25804
8*a^5*b^3*c*g*h*z - 774144*a^4*b^4*c*d*g*z - 18432*a^6*b^2*g^2*h*z - 51200*a^5*b^3*e^2*h*z - 903168*a^4*b^4*c^
2*h*z + 55296*a^5*b^3*d*g^2*z + 153600*a^4*b^4*d*e^2*z + 2709504*a^3*b^5*c^2*d*z - 5760*a^3*b^2*d*e*g*h + 4032
0*a^2*b^3*c*d*e*h + 8640*a^2*b^3*d^2*e*g - 6720*a^3*b^2*c*e*h^2 - 6300*a^2*b^3*c*e^2*g + 960*a^4*b*e*g*h^2 - 6
0480*a*b^4*c*d^2*e - 3072*a^4*b*d*h^3 + 111132*a*b^4*c^3*g + 13824*a^3*b^2*d^2*h^2 + 450*a^3*b^2*e^2*g^2 - 238
14*a^2*b^3*c^2*g^2 - 27648*a^2*b^3*d^3*h + 2268*a^3*b^2*c*g^3 + 22050*a*b^4*c^2*e^2 - 625*a^2*b^3*e^4 - 81*a^4
*b*g^4 + 20736*a*b^4*d^4 + 256*a^5*h^4 - 194481*b^5*c^4, z, k), k, 1, 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((h*x**5+g*x**4+f*x**3+e*x**2+d*x+c)/(-b*x**4+a)**3,x)

[Out]

Timed out

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