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

Optimal. Leaf size=179 \[ \frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]

[Out]

1/8*x*(e*x^2+d*x+c)/a/(-b*x^4+a)^2+1/32*x*(5*e*x^2+6*d*x+7*c)/a^2/(-b*x^4+a)+3/16*d*arctanh(x^2*b^(1/2)/a^(1/2
))/a^(5/2)/b^(1/2)+1/64*arctan(b^(1/4)*x/a^(1/4))*(-5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)+1/64*arctanh(b^
(1/4)*x/a^(1/4))*(5*e*a^(1/2)+21*c*b^(1/2))/a^(11/4)/b^(3/4)

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Rubi [A]  time = 0.17, antiderivative size = 179, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 6, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {1855, 1876, 275, 208, 1167, 205} \[ \frac {\left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {\left (5 \sqrt {a} e+21 \sqrt {b} c\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}+\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*(c + d*x + e*x^2))/(8*a*(a - b*x^4)^2) + (x*(7*c + 6*d*x + 5*e*x^2))/(32*a^2*(a - b*x^4)) + ((21*Sqrt[b]*c
- 5*Sqrt[a]*e)*ArcTan[(b^(1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + ((21*Sqrt[b]*c + 5*Sqrt[a]*e)*ArcTanh[(b^(
1/4)*x)/a^(1/4)])/(64*a^(11/4)*b^(3/4)) + (3*d*ArcTanh[(Sqrt[b]*x^2)/Sqrt[a]])/(16*a^(5/2)*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 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}{\left (a-b x^4\right )^3} \, dx &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}-\frac {\int \frac {-7 c-6 d x-5 e x^2}{\left (a-b x^4\right )^2} \, dx}{8 a}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {\int \frac {21 c+12 d x+5 e x^2}{a-b x^4} \, dx}{32 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {\int \left (\frac {12 d x}{a-b x^4}+\frac {21 c+5 e x^2}{a-b x^4}\right ) \, dx}{32 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {\int \frac {21 c+5 e x^2}{a-b x^4} \, dx}{32 a^2}+\frac {(3 d) \int \frac {x}{a-b x^4} \, dx}{8 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {(3 d) \operatorname {Subst}\left (\int \frac {1}{a-b x^2} \, dx,x,x^2\right )}{16 a^2}-\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \int \frac {1}{-\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}+5 e\right ) \int \frac {1}{\sqrt {a} \sqrt {b}-b x^2} \, dx}{64 a^2}\\ &=\frac {x \left (c+d x+e x^2\right )}{8 a \left (a-b x^4\right )^2}+\frac {x \left (7 c+6 d x+5 e x^2\right )}{32 a^2 \left (a-b x^4\right )}+\frac {\left (\frac {21 \sqrt {b} c}{\sqrt {a}}-5 e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{9/4} b^{3/4}}+\frac {\left (21 \sqrt {b} c+5 \sqrt {a} e\right ) \tanh ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 a^{11/4} b^{3/4}}+\frac {3 d \tanh ^{-1}\left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{5/2} \sqrt {b}}\\ \end {align*}

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Mathematica [A]  time = 0.29, size = 244, normalized size = 1.36 \[ \frac {-\frac {\log \left (\sqrt [4]{a}-\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt {b} c+12 \sqrt {a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac {\log \left (\sqrt [4]{a}+\sqrt [4]{b} x\right ) \left (5 a^{3/4} e+21 \sqrt [4]{a} \sqrt {b} c-12 \sqrt {a} \sqrt [4]{b} d\right )}{b^{3/4}}+\frac {16 a^2 x (c+x (d+e x))}{\left (a-b x^4\right )^2}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {b} c-5 \sqrt {a} e\right ) \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{b^{3/4}}+\frac {4 a x (7 c+x (6 d+5 e x))}{a-b x^4}+\frac {12 \sqrt {a} d \log \left (\sqrt {a}+\sqrt {b} x^2\right )}{\sqrt {b}}}{128 a^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

[Out]

Timed out

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giac [B]  time = 0.23, size = 340, normalized size = 1.90 \[ -\frac {\sqrt {2} {\left (21 \, b^{2} c - 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d + 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c + 12 \, \sqrt {2} \left (-a b^{3}\right )^{\frac {1}{4}} b d - 5 \, \sqrt {-a b} b e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (-\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (-\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} + \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} + \frac {\sqrt {2} {\left (21 \, b^{2} c - 5 \, \sqrt {-a b} b e\right )} \log \left (x^{2} - \sqrt {2} x \left (-\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {-\frac {a}{b}}\right )}{256 \, \left (-a b^{3}\right )^{\frac {3}{4}} a^{2}} - \frac {5 \, b x^{7} e + 6 \, b d x^{6} + 7 \, b c x^{5} - 9 \, a x^{3} e - 10 \, a d x^{2} - 11 \, a c x}{32 \, {\left (b x^{4} - a\right )}^{2} a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/128*sqrt(2)*(21*b^2*c - 12*sqrt(2)*(-a*b^3)^(1/4)*b*d + 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 + 12*sqrt(2)*(-a*b^3)^(1/4)*b*d -
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*s
qrt(2)*(21*b^2*c - 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) + 1/2
56*sqrt(2)*(21*b^2*c - 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*x^7*e + 6*b*d*x^6 + 7*b*c*x^5 - 9*a*x^3*e - 10*a*d*x^2 - 11*a*c*x)/((b*x^4 - a)^2*a^2)

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maple [B]  time = 0.05, size = 286, normalized size = 1.60 \[ \frac {e \,x^{3}}{8 \left (b \,x^{4}-a \right )^{2} a}+\frac {d \,x^{2}}{8 \left (b \,x^{4}-a \right )^{2} a}-\frac {5 e \,x^{3}}{32 \left (b \,x^{4}-a \right ) a^{2}}+\frac {c x}{8 \left (b \,x^{4}-a \right )^{2} a}-\frac {3 d \,x^{2}}{16 \left (b \,x^{4}-a \right ) a^{2}}-\frac {7 c x}{32 \left (b \,x^{4}-a \right ) a^{2}}-\frac {3 d \ln \left (\frac {\sqrt {a b}\, x^{2}-a}{-\sqrt {a b}\, x^{2}-a}\right )}{32 \sqrt {a b}\, a^{2}}-\frac {5 e \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {5 e \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \left (\frac {a}{b}\right )^{\frac {1}{4}} a^{2} b}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \arctan \left (\frac {x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{64 a^{3}}+\frac {21 \left (\frac {a}{b}\right )^{\frac {1}{4}} c \ln \left (\frac {x +\left (\frac {a}{b}\right )^{\frac {1}{4}}}{x -\left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 a^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [A]  time = 3.12, size = 230, normalized size = 1.28 \[ -\frac {5 \, b e x^{7} + 6 \, b d x^{6} + 7 \, b c x^{5} - 9 \, a e x^{3} - 10 \, a d x^{2} - 11 \, a c x}{32 \, {\left (a^{2} b^{2} x^{8} - 2 \, a^{3} b x^{4} + a^{4}\right )}} + \frac {\frac {12 \, d \log \left (\sqrt {b} x^{2} + \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} - \frac {12 \, d \log \left (\sqrt {b} x^{2} - \sqrt {a}\right )}{\sqrt {a} \sqrt {b}} + \frac {2 \, {\left (21 \, \sqrt {b} c - 5 \, \sqrt {a} e\right )} \arctan \left (\frac {\sqrt {b} x}{\sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {{\left (21 \, \sqrt {b} c + 5 \, \sqrt {a} e\right )} \log \left (\frac {\sqrt {b} x - \sqrt {\sqrt {a} \sqrt {b}}}{\sqrt {b} x + \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}}}{128 \, a^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/32*(5*b*e*x^7 + 6*b*d*x^6 + 7*b*c*x^5 - 9*a*e*x^3 - 10*a*d*x^2 - 11*a*c*x)/(a^2*b^2*x^8 - 2*a^3*b*x^4 + a^4
) + 1/128*(12*d*log(sqrt(b)*x^2 + sqrt(a))/(sqrt(a)*sqrt(b)) - 12*d*log(sqrt(b)*x^2 - sqrt(a))/(sqrt(a)*sqrt(b
)) + 2*(21*sqrt(b)*c - 5*sqrt(a)*e)*arctan(sqrt(b)*x/sqrt(sqrt(a)*sqrt(b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))*sqr
t(b)) - (21*sqrt(b)*c + 5*sqrt(a)*e)*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^2

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mupad [B]  time = 5.11, size = 826, normalized size = 4.61 \[ \frac {\frac {5\,d\,x^2}{16\,a}+\frac {9\,e\,x^3}{32\,a}+\frac {11\,c\,x}{32\,a}-\frac {7\,b\,c\,x^5}{32\,a^2}-\frac {3\,b\,d\,x^6}{16\,a^2}-\frac {5\,b\,e\,x^7}{32\,a^2}}{a^2-2\,a\,b\,x^4+b^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\frac {b\,\left (125\,a\,e^3+3024\,b\,c\,d^2-2205\,b\,c^2\,e+1728\,b\,d^3\,x+{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,c\,344064+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,e^2\,x\,3200-2520\,b\,c\,d\,e\,x+\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^2\,b^2\,c^2\,x\,56448-{\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )}^2\,a^5\,b^2\,d\,x\,196608-\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\,a^3\,b\,d\,e\,15360\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,b^3\,z^4-6881280\,a^6\,b^2\,c\,e\,z^2-4718592\,a^6\,b^2\,d^2\,z^2+2709504\,a^3\,b^2\,c^2\,d\,z+153600\,a^4\,b\,d\,e^2\,z-60480\,a\,b\,c\,d^2\,e+22050\,a\,b\,c^2\,e^2+20736\,a\,b\,d^4-625\,a^2\,e^4-194481\,b^2\,c^4,z,k\right )\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

((5*d*x^2)/(16*a) + (9*e*x^3)/(32*a) + (11*c*x)/(32*a) - (7*b*c*x^5)/(32*a^2) - (3*b*d*x^6)/(16*a^2) - (5*b*e*
x^7)/(32*a^2))/(a^2 + b^2*x^8 - 2*a*b*x^4) + symsum(log(-(b*(125*a*e^3 + 3024*b*c*d^2 - 2205*b*c^2*e + 1728*b*
d^3*x + 344064*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b
^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 1944
81*b^2*c^4, z, k)^2*a^5*b^2*c + 3200*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d
^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^
4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^3*b*e^2*x - 2520*b*c*d*e*x + 56448*root(268435456*a^11*b^3*z^4 - 688
1280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*
d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)*a^2*b^2*c^2*x - 196608*root(26
8435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^
4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k)^2*a^
5*b^2*d*x - 15360*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*a^6*b^2*d^2*z^2 + 2709504*a^
3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 20736*a*b*d^4 - 625*a^2*e^4 - 1
94481*b^2*c^4, z, k)*a^3*b*d*e))/(32768*a^6))*root(268435456*a^11*b^3*z^4 - 6881280*a^6*b^2*c*e*z^2 - 4718592*
a^6*b^2*d^2*z^2 + 2709504*a^3*b^2*c^2*d*z + 153600*a^4*b*d*e^2*z - 60480*a*b*c*d^2*e + 22050*a*b*c^2*e^2 + 207
36*a*b*d^4 - 625*a^2*e^4 - 194481*b^2*c^4, z, k), k, 1, 4)

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sympy [B]  time = 45.34, size = 563, normalized size = 3.15 \[ - \operatorname {RootSum} {\left (268435456 t^{4} a^{11} b^{3} + t^{2} \left (- 6881280 a^{6} b^{2} c e - 4718592 a^{6} b^{2} d^{2}\right ) + t \left (- 153600 a^{4} b d e^{2} - 2709504 a^{3} b^{2} c^{2} d\right ) - 625 a^{2} e^{4} + 22050 a b c^{2} e^{2} - 60480 a b c d^{2} e + 20736 a b d^{4} - 194481 b^{2} c^{4}, \left (t \mapsto t \log {\left (x + \frac {- 262144000 t^{3} a^{10} b^{2} e^{3} - 4624220160 t^{3} a^{9} b^{3} c^{2} e + 12683575296 t^{3} a^{9} b^{3} c d^{2} + 309657600 t^{2} a^{7} b^{2} c d e^{2} - 283115520 t^{2} a^{7} b^{2} d^{3} e - 1820786688 t^{2} a^{6} b^{3} c^{3} d + 5040000 t a^{5} b c e^{4} + 6912000 t a^{5} b d^{2} e^{3} + 118540800 t a^{4} b^{2} c^{3} e^{2} - 365783040 t a^{4} b^{2} c^{2} d^{2} e - 111476736 t a^{4} b^{2} c d^{4} + 522764928 t a^{3} b^{3} c^{5} + 112500 a^{3} d e^{5} - 4536000 a^{2} b c d^{3} e^{2} + 2488320 a^{2} b d^{5} e + 58344300 a b^{2} c^{4} d e - 80015040 a b^{2} c^{3} d^{3}}{15625 a^{3} e^{6} + 275625 a^{2} b c^{2} e^{4} - 3024000 a^{2} b c d^{2} e^{3} + 2073600 a^{2} b d^{4} e^{2} - 4862025 a b^{2} c^{4} e^{2} + 53343360 a b^{2} c^{3} d^{2} e - 36578304 a b^{2} c^{2} d^{4} - 85766121 b^{3} c^{6}} \right )} \right )\right )} - \frac {- 11 a c x - 10 a d x^{2} - 9 a e x^{3} + 7 b c x^{5} + 6 b d x^{6} + 5 b e x^{7}}{32 a^{4} - 64 a^{3} b x^{4} + 32 a^{2} b^{2} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-RootSum(268435456*_t**4*a**11*b**3 + _t**2*(-6881280*a**6*b**2*c*e - 4718592*a**6*b**2*d**2) + _t*(-153600*a*
*4*b*d*e**2 - 2709504*a**3*b**2*c**2*d) - 625*a**2*e**4 + 22050*a*b*c**2*e**2 - 60480*a*b*c*d**2*e + 20736*a*b
*d**4 - 194481*b**2*c**4, Lambda(_t, _t*log(x + (-262144000*_t**3*a**10*b**2*e**3 - 4624220160*_t**3*a**9*b**3
*c**2*e + 12683575296*_t**3*a**9*b**3*c*d**2 + 309657600*_t**2*a**7*b**2*c*d*e**2 - 283115520*_t**2*a**7*b**2*
d**3*e - 1820786688*_t**2*a**6*b**3*c**3*d + 5040000*_t*a**5*b*c*e**4 + 6912000*_t*a**5*b*d**2*e**3 + 11854080
0*_t*a**4*b**2*c**3*e**2 - 365783040*_t*a**4*b**2*c**2*d**2*e - 111476736*_t*a**4*b**2*c*d**4 + 522764928*_t*a
**3*b**3*c**5 + 112500*a**3*d*e**5 - 4536000*a**2*b*c*d**3*e**2 + 2488320*a**2*b*d**5*e + 58344300*a*b**2*c**4
*d*e - 80015040*a*b**2*c**3*d**3)/(15625*a**3*e**6 + 275625*a**2*b*c**2*e**4 - 3024000*a**2*b*c*d**2*e**3 + 20
73600*a**2*b*d**4*e**2 - 4862025*a*b**2*c**4*e**2 + 53343360*a*b**2*c**3*d**2*e - 36578304*a*b**2*c**2*d**4 -
85766121*b**3*c**6)))) - (-11*a*c*x - 10*a*d*x**2 - 9*a*e*x**3 + 7*b*c*x**5 + 6*b*d*x**6 + 5*b*e*x**7)/(32*a**
4 - 64*a**3*b*x**4 + 32*a**2*b**2*x**8)

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