Integrand size = 28, antiderivative size = 308 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=-\frac {c+e x^2}{8 b \left (a+b x^4\right )^2}-\frac {x \left (d+f x^2\right )}{8 b \left (a+b x^4\right )^2}+\frac {e x^2}{16 a b \left (a+b x^4\right )}+\frac {x \left (d+3 f x^2\right )}{32 a b \left (a+b x^4\right )}+\frac {e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{16 a^{3/2} b^{3/2}}-\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d+\sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{64 \sqrt {2} a^{7/4} b^{7/4}} \] Output:
-1/8*(e*x^2+c)/b/(b*x^4+a)^2-1/8*x*(f*x^2+d)/b/(b*x^4+a)^2+1/16*e*x^2/a/b/ (b*x^4+a)+1/32*x*(3*f*x^2+d)/a/b/(b*x^4+a)+1/16*e*arctan(b^(1/2)*x^2/a^(1/ 2))/a^(3/2)/b^(3/2)+3/128*(b^(1/2)*d+a^(1/2)*f)*arctan(-1+2^(1/2)*b^(1/4)* x/a^(1/4))*2^(1/2)/a^(7/4)/b^(7/4)+3/128*(b^(1/2)*d+a^(1/2)*f)*arctan(1+2^ (1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(7/4)/b^(7/4)+3/128*(b^(1/2)*d-a^(1/2)* f)*arctanh(2^(1/2)*a^(1/4)*b^(1/4)*x/(a^(1/2)+b^(1/2)*x^2))*2^(1/2)/a^(7/4 )/b^(7/4)
Time = 0.25 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.07 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {\frac {8 b^{3/4} x (d+x (2 e+3 f x))}{a \left (a+b x^4\right )}-\frac {32 b^{3/4} (c+x (d+x (e+f x)))}{\left (a+b x^4\right )^2}-\frac {2 \left (3 \sqrt {2} \sqrt {b} d+8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {2 \left (3 \sqrt {2} \sqrt {b} d-8 \sqrt [4]{a} \sqrt [4]{b} e+3 \sqrt {2} \sqrt {a} f\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (-\sqrt {b} d+\sqrt {a} f\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {3 \sqrt {2} \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{256 b^{7/4}} \] Input:
Integrate[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^3,x]
Output:
((8*b^(3/4)*x*(d + x*(2*e + 3*f*x)))/(a*(a + b*x^4)) - (32*b^(3/4)*(c + x* (d + x*(e + f*x))))/(a + b*x^4)^2 - (2*(3*Sqrt[2]*Sqrt[b]*d + 8*a^(1/4)*b^ (1/4)*e + 3*Sqrt[2]*Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^ (7/4) + (2*(3*Sqrt[2]*Sqrt[b]*d - 8*a^(1/4)*b^(1/4)*e + 3*Sqrt[2]*Sqrt[a]* f)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (3*Sqrt[2]*(-(Sqrt[b ]*d) + Sqrt[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/ a^(7/4) + (3*Sqrt[2]*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/4) *b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4))/(256*b^(7/4))
Time = 0.96 (sec) , antiderivative size = 351, normalized size of antiderivative = 1.14, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.179, Rules used = {2363, 2394, 25, 2415, 2009}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
| \(\displaystyle \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2363 |
| \(\displaystyle \frac {\int \frac {3 f x^2+2 e x+d}{\left (b x^4+a\right )^2}dx}{8 b}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (d+2 e x+3 f x^2\right )}{4 a \left (a+b x^4\right )}-\frac {\int -\frac {3 f x^2+4 e x+3 d}{b x^4+a}dx}{4 a}}{8 b}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {3 f x^2+4 e x+3 d}{b x^4+a}dx}{4 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 b}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {4 e x}{b x^4+a}+\frac {3 f x^2+3 d}{b x^4+a}\right )dx}{4 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 b}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {3 \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}+\frac {3 \arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) \left (\sqrt {a} f+\sqrt {b} d\right )}{2 \sqrt {2} a^{3/4} b^{3/4}}-\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {3 \left (\sqrt {b} d-\sqrt {a} f\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{3/4} b^{3/4}}+\frac {2 e \arctan \left (\frac {\sqrt {b} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {b}}}{4 a}+\frac {x \left (d+2 e x+3 f x^2\right )}{4 a \left (a+b x^4\right )}}{8 b}-\frac {c+d x+e x^2+f x^3}{8 b \left (a+b x^4\right )^2}\) |
Input:
Int[(x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^3,x]
Output:
-1/8*(c + d*x + e*x^2 + f*x^3)/(b*(a + b*x^4)^2) + ((x*(d + 2*e*x + 3*f*x^ 2))/(4*a*(a + b*x^4)) + ((2*e*ArcTan[(Sqrt[b]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt [b]) - (3*(Sqrt[b]*d + Sqrt[a]*f)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)]) /(2*Sqrt[2]*a^(3/4)*b^(3/4)) + (3*(Sqrt[b]*d + Sqrt[a]*f)*ArcTan[1 + (Sqrt [2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*b^(3/4)) - (3*(Sqrt[b]*d - Sqr t[a]*f)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2] *a^(3/4)*b^(3/4)) + (3*(Sqrt[b]*d - Sqrt[a]*f)*Log[Sqrt[a] + Sqrt[2]*a^(1/ 4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(3/4)*b^(3/4)))/(4*a))/(8*b)
Int[(Pq_)*(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[Pq*(( a + b*x^n)^(p + 1)/(b*n*(p + 1))), x] - Simp[1/(b*n*(p + 1)) Int[D[Pq, x] *(a + b*x^n)^(p + 1), x], x] /; FreeQ[{a, b, m, n}, x] && PolyQ[Pq, x] && E qQ[m - n + 1, 0] && LtQ[p, -1]
Int[(Pq_)*((a_) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(-x)*Pq*((a + b *x^n)^(p + 1)/(a*n*(p + 1))), x] + Simp[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]
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
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.15 (sec) , antiderivative size = 114, normalized size of antiderivative = 0.37
| method | result | size |
| risch | \(\frac {\frac {3 f \,x^{7}}{32 a}+\frac {e \,x^{6}}{16 a}+\frac {d \,x^{5}}{32 a}-\frac {f \,x^{3}}{32 b}-\frac {e \,x^{2}}{16 b}-\frac {3 d x}{32 b}-\frac {c}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (3 f \,\textit {\_R}^{2}+4 e \textit {\_R} +3 d \right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 b^{2} a}\) | \(114\) |
| default | \(\frac {\frac {3 f \,x^{7}}{32 a}+\frac {e \,x^{6}}{16 a}+\frac {d \,x^{5}}{32 a}-\frac {f \,x^{3}}{32 b}-\frac {e \,x^{2}}{16 b}-\frac {3 d x}{32 b}-\frac {c}{8 b}}{\left (b \,x^{4}+a \right )^{2}}+\frac {\frac {3 d \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 a}+\frac {2 e \arctan \left (x^{2} \sqrt {\frac {b}{a}}\right )}{\sqrt {a b}}+\frac {3 f \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{32 b a}\) | \(304\) |
Input:
int(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(3/32*f/a*x^7+1/16/a*e*x^6+1/32*d/a*x^5-1/32*f*x^3/b-1/16*e*x^2/b-3/32*d*x /b-1/8/b*c)/(b*x^4+a)^2+1/128/b^2/a*sum((3*_R^2*f+4*_R*e+3*d)/_R^3*ln(x-_R ),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 7.58 (sec) , antiderivative size = 124542, normalized size of antiderivative = 404.36 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Timed out. \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\text {Timed out} \] Input:
integrate(x**3*(f*x**3+e*x**2+d*x+c)/(b*x**4+a)**3,x)
Output:
Timed out
Time = 0.12 (sec) , antiderivative size = 343, normalized size of antiderivative = 1.11 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {3 \, b f x^{7} + 2 \, b e x^{6} + b d x^{5} - a f x^{3} - 2 \, a e x^{2} - 3 \, a d x - 4 \, a c}{32 \, {\left (a b^{3} x^{8} + 2 \, a^{2} b^{2} x^{4} + a^{3} b\right )}} + \frac {\frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} - \frac {3 \, \sqrt {2} {\left (\sqrt {b} d - \sqrt {a} f\right )} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f - 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}} + \frac {2 \, {\left (3 \, \sqrt {2} a^{\frac {1}{4}} b^{\frac {3}{4}} d + 3 \, \sqrt {2} a^{\frac {3}{4}} b^{\frac {1}{4}} f + 8 \, \sqrt {a} \sqrt {b} e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {b}} b^{\frac {3}{4}}}}{256 \, a b} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="maxima")
Output:
1/32*(3*b*f*x^7 + 2*b*e*x^6 + b*d*x^5 - a*f*x^3 - 2*a*e*x^2 - 3*a*d*x - 4* a*c)/(a*b^3*x^8 + 2*a^2*b^2*x^4 + a^3*b) + 1/256*(3*sqrt(2)*(sqrt(b)*d - s qrt(a)*f)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)* b^(3/4)) - 3*sqrt(2)*(sqrt(b)*d - sqrt(a)*f)*log(sqrt(b)*x^2 - sqrt(2)*a^( 1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(3/4)) + 2*(3*sqrt(2)*a^(1/4)*b^(3/4) *d + 3*sqrt(2)*a^(3/4)*b^(1/4)*f - 8*sqrt(a)*sqrt(b)*e)*arctan(1/2*sqrt(2) *(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*s qrt(sqrt(a)*sqrt(b))*b^(3/4)) + 2*(3*sqrt(2)*a^(1/4)*b^(3/4)*d + 3*sqrt(2) *a^(3/4)*b^(1/4)*f + 8*sqrt(a)*sqrt(b)*e)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(a^(3/4)*sqrt(sqrt(a)*sq rt(b))*b^(3/4)))/(a*b)
Time = 0.13 (sec) , antiderivative size = 334, normalized size of antiderivative = 1.08 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\frac {3 \, b f x^{7} + 2 \, b e x^{6} + b d x^{5} - a f x^{3} - 2 \, a e x^{2} - 3 \, a d x - 4 \, a c}{32 \, {\left (b x^{4} + a\right )}^{2} a b} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {\sqrt {2} {\left (4 \, \sqrt {2} \sqrt {a b} b^{2} e + 3 \, \left (a b^{3}\right )^{\frac {1}{4}} b^{2} d + 3 \, \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{b}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{b}\right )^{\frac {1}{4}}}\right )}{128 \, a^{2} b^{4}} + \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} - \frac {3 \, \sqrt {2} {\left (\left (a b^{3}\right )^{\frac {1}{4}} b^{2} d - \left (a b^{3}\right )^{\frac {3}{4}} f\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{256 \, a^{2} b^{4}} \] Input:
integrate(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x, algorithm="giac")
Output:
1/32*(3*b*f*x^7 + 2*b*e*x^6 + b*d*x^5 - a*f*x^3 - 2*a*e*x^2 - 3*a*d*x - 4* a*c)/((b*x^4 + a)^2*a*b) + 1/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b^2*e + 3*(a *b^3)^(1/4)*b^2*d + 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*( a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 1/128*sqrt(2)*(4*sqrt(2)*sqrt(a*b)*b^ 2*e + 3*(a*b^3)^(1/4)*b^2*d + 3*(a*b^3)^(3/4)*f)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^4) + 3/256*sqrt(2)*((a*b^3)^(1/4 )*b^2*d - (a*b^3)^(3/4)*f)*log(x^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a ^2*b^4) - 3/256*sqrt(2)*((a*b^3)^(1/4)*b^2*d - (a*b^3)^(3/4)*f)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^4)
Time = 7.08 (sec) , antiderivative size = 521, normalized size of antiderivative = 1.69 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx=\left (\sum _{k=1}^4\ln \left (-\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\,\left (\frac {3\,b^2\,d}{2}-2\,b^2\,e\,x\right )+\frac {3\,e\,f}{32\,a}+\frac {x\,\left (144\,a\,b^2\,d^2-144\,a^2\,b\,f^2\right )}{4096\,a^3\,b}\right )-\frac {3\,\left (9\,b\,d^2\,f-16\,b\,d\,e^2+9\,a\,f^3\right )}{32768\,a^3\,b^2}+\frac {x\,\left (8\,e^3-9\,d\,e\,f\right )}{4096\,a^3\,b}\right )\,\mathrm {root}\left (268435456\,a^7\,b^7\,z^4+589824\,a^4\,b^4\,d\,f\,z^2+524288\,a^4\,b^4\,e^2\,z^2+18432\,a^3\,b^2\,e\,f^2\,z-18432\,a^2\,b^3\,d^2\,e\,z-576\,a\,b\,d\,e^2\,f+162\,a\,b\,d^2\,f^2+256\,a\,b\,e^4+81\,a^2\,f^4+81\,b^2\,d^4,z,k\right )\right )-\frac {\frac {c}{8\,b}-\frac {d\,x^5}{32\,a}-\frac {e\,x^6}{16\,a}+\frac {e\,x^2}{16\,b}-\frac {3\,f\,x^7}{32\,a}+\frac {f\,x^3}{32\,b}+\frac {3\,d\,x}{32\,b}}{a^2+2\,a\,b\,x^4+b^2\,x^8} \] Input:
int((x^3*(c + d*x + e*x^2 + f*x^3))/(a + b*x^4)^3,x)
Output:
symsum(log((x*(8*e^3 - 9*d*e*f))/(4096*a^3*b) - (3*(9*a*f^3 - 16*b*d*e^2 + 9*b*d^2*f))/(32768*a^3*b^2) - root(268435456*a^7*b^7*z^4 + 589824*a^4*b^4 *d*f*z^2 + 524288*a^4*b^4*e^2*z^2 + 18432*a^3*b^2*e*f^2*z - 18432*a^2*b^3* d^2*e*z - 576*a*b*d*e^2*f + 162*a*b*d^2*f^2 + 256*a*b*e^4 + 81*a^2*f^4 + 8 1*b^2*d^4, z, k)*(root(268435456*a^7*b^7*z^4 + 589824*a^4*b^4*d*f*z^2 + 52 4288*a^4*b^4*e^2*z^2 + 18432*a^3*b^2*e*f^2*z - 18432*a^2*b^3*d^2*e*z - 576 *a*b*d*e^2*f + 162*a*b*d^2*f^2 + 256*a*b*e^4 + 81*a^2*f^4 + 81*b^2*d^4, z, k)*((3*b^2*d)/2 - 2*b^2*e*x) + (3*e*f)/(32*a) + (x*(144*a*b^2*d^2 - 144*a ^2*b*f^2))/(4096*a^3*b)))*root(268435456*a^7*b^7*z^4 + 589824*a^4*b^4*d*f* z^2 + 524288*a^4*b^4*e^2*z^2 + 18432*a^3*b^2*e*f^2*z - 18432*a^2*b^3*d^2*e *z - 576*a*b*d*e^2*f + 162*a*b*d^2*f^2 + 256*a*b*e^4 + 81*a^2*f^4 + 81*b^2 *d^4, z, k), k, 1, 4) - (c/(8*b) - (d*x^5)/(32*a) - (e*x^6)/(16*a) + (e*x^ 2)/(16*b) - (3*f*x^7)/(32*a) + (f*x^3)/(32*b) + (3*d*x)/(32*b))/(a^2 + b^2 *x^8 + 2*a*b*x^4)
Time = 0.19 (sec) , antiderivative size = 1208, normalized size of antiderivative = 3.92 \[ \int \frac {x^3 \left (c+d x+e x^2+f x^3\right )}{\left (a+b x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int(x^3*(f*x^3+e*x^2+d*x+c)/(b*x^4+a)^3,x)
Output:
( - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b )*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*f - 12*b**(1/4)*a**(3/4)*sqrt(2)*at an((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))* a*b*f*x**4 - 6*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*f*x**8 - 6*b**(3/4)*a**(1/ 4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/ 4)*sqrt(2)))*a**2*d - 12*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4) *sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*d*x**4 - 6*b**(3/ 4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/ 4)*a**(1/4)*sqrt(2)))*b**2*d*x**8 - 16*sqrt(b)*sqrt(a)*atan((b**(1/4)*a**( 1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*e - 32*sqrt( b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/ 4)*sqrt(2)))*a*b*e*x**4 - 16*sqrt(b)*sqrt(a)*atan((b**(1/4)*a**(1/4)*sqrt( 2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*b**2*e*x**8 + 6*b**(1/4)*a* *(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a* *(1/4)*sqrt(2)))*a**2*f + 12*b**(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**( 1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a*b*f*x**4 + 6*b* *(1/4)*a**(3/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b* *(1/4)*a**(1/4)*sqrt(2)))*b**2*f*x**8 + 6*b**(3/4)*a**(1/4)*sqrt(2)*atan(( b**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a...