Integrand size = 17, antiderivative size = 288 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{32 a^2 \left (a+c x^4\right )}+\frac {3 d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{8 a^{5/2} \sqrt {c}}-\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2+5 \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x}{\sqrt {a}+\sqrt {c} x^2}\right )}{64 \sqrt {2} a^{11/4} c^{3/4}} \] Output:
1/8*x*(e*x+d)^2/a/(c*x^4+a)^2+1/32*x*(5*e^2*x^2+12*d*e*x+7*d^2)/a^2/(c*x^4 +a)+3/8*d*e*arctan(c^(1/2)*x^2/a^(1/2))/a^(5/2)/c^(1/2)+1/128*(21*c^(1/2)* d^2+5*a^(1/2)*e^2)*arctan(-1+2^(1/2)*c^(1/4)*x/a^(1/4))*2^(1/2)/a^(11/4)/c ^(3/4)+1/128*(21*c^(1/2)*d^2+5*a^(1/2)*e^2)*arctan(1+2^(1/2)*c^(1/4)*x/a^( 1/4))*2^(1/2)/a^(11/4)/c^(3/4)+1/128*(21*c^(1/2)*d^2-5*a^(1/2)*e^2)*arctan h(2^(1/2)*a^(1/4)*c^(1/4)*x/(a^(1/2)+c^(1/2)*x^2))*2^(1/2)/a^(11/4)/c^(3/4 )
Time = 0.34 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {32 a^2 x (d+e x)^2}{\left (a+c x^4\right )^2}+\frac {8 a x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{a+c x^4}-\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {c} d^2+48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {2 \sqrt [4]{a} \left (21 \sqrt {2} \sqrt {c} d^2-48 \sqrt [4]{a} \sqrt [4]{c} d e+5 \sqrt {2} \sqrt {a} e^2\right ) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right )}{c^{3/4}}+\frac {\sqrt {2} \left (-21 \sqrt [4]{a} \sqrt {c} d^2+5 a^{3/4} e^2\right ) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}+\frac {\sqrt {2} \left (21 \sqrt [4]{a} \sqrt {c} d^2-5 a^{3/4} e^2\right ) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {c} x^2\right )}{c^{3/4}}}{256 a^3} \] Input:
Integrate[(d + e*x)^2/(a + c*x^4)^3,x]
Output:
((32*a^2*x*(d + e*x)^2)/(a + c*x^4)^2 + (8*a*x*(7*d^2 + 12*d*e*x + 5*e^2*x ^2))/(a + c*x^4) - (2*a^(1/4)*(21*Sqrt[2]*Sqrt[c]*d^2 + 48*a^(1/4)*c^(1/4) *d*e + 5*Sqrt[2]*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^( 3/4) + (2*a^(1/4)*(21*Sqrt[2]*Sqrt[c]*d^2 - 48*a^(1/4)*c^(1/4)*d*e + 5*Sqr t[2]*Sqrt[a]*e^2)*ArcTan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/c^(3/4) + (Sqrt [2]*(-21*a^(1/4)*Sqrt[c]*d^2 + 5*a^(3/4)*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4 )*c^(1/4)*x + Sqrt[c]*x^2])/c^(3/4) + (Sqrt[2]*(21*a^(1/4)*Sqrt[c]*d^2 - 5 *a^(3/4)*e^2)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/c^(3 /4))/(256*a^3)
Time = 0.99 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.30, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.353, Rules used = {2394, 25, 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.
| \(\displaystyle \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}-\frac {\int -\frac {7 d^2+12 e x d+5 e^2 x^2}{\left (c x^4+a\right )^2}dx}{8 a}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\int \frac {7 d^2+12 e x d+5 e^2 x^2}{\left (c x^4+a\right )^2}dx}{8 a}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2394 |
| \(\displaystyle \frac {\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{4 a \left (a+c x^4\right )}-\frac {\int -\frac {21 d^2+24 e x d+5 e^2 x^2}{c x^4+a}dx}{4 a}}{8 a}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle \frac {\frac {\int \frac {21 d^2+24 e x d+5 e^2 x^2}{c x^4+a}dx}{4 a}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2415 |
| \(\displaystyle \frac {\frac {\int \left (\frac {24 d e x}{c x^4+a}+\frac {21 d^2+5 e^2 x^2}{c x^4+a}\right )dx}{4 a}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {\frac {-\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}\right ) \left (5 \sqrt {a} e^2+21 \sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{c} x}{\sqrt [4]{a}}+1\right ) \left (5 \sqrt {a} e^2+21 \sqrt {c} d^2\right )}{2 \sqrt {2} a^{3/4} c^{3/4}}-\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {\left (21 \sqrt {c} d^2-5 \sqrt {a} e^2\right ) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{c} x+\sqrt {a}+\sqrt {c} x^2\right )}{4 \sqrt {2} a^{3/4} c^{3/4}}+\frac {12 d e \arctan \left (\frac {\sqrt {c} x^2}{\sqrt {a}}\right )}{\sqrt {a} \sqrt {c}}}{4 a}+\frac {x \left (7 d^2+12 d e x+5 e^2 x^2\right )}{4 a \left (a+c x^4\right )}}{8 a}+\frac {x (d+e x)^2}{8 a \left (a+c x^4\right )^2}\) |
Input:
Int[(d + e*x)^2/(a + c*x^4)^3,x]
Output:
(x*(d + e*x)^2)/(8*a*(a + c*x^4)^2) + ((x*(7*d^2 + 12*d*e*x + 5*e^2*x^2))/ (4*a*(a + c*x^4)) + ((12*d*e*ArcTan[(Sqrt[c]*x^2)/Sqrt[a]])/(Sqrt[a]*Sqrt[ c]) - ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*ArcTan[1 - (Sqrt[2]*c^(1/4)*x)/a^( 1/4)])/(2*Sqrt[2]*a^(3/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 + 5*Sqrt[a]*e^2)*Arc Tan[1 + (Sqrt[2]*c^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(3/4)*c^(3/4)) - ((21*S qrt[c]*d^2 - 5*Sqrt[a]*e^2)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt [c]*x^2])/(4*Sqrt[2]*a^(3/4)*c^(3/4)) + ((21*Sqrt[c]*d^2 - 5*Sqrt[a]*e^2)* Log[Sqrt[a] + Sqrt[2]*a^(1/4)*c^(1/4)*x + Sqrt[c]*x^2])/(4*Sqrt[2]*a^(3/4) *c^(3/4)))/(4*a))/(8*a)
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 = 0.27 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.44
| method | result | size |
| risch | \(\frac {\frac {5 c \,e^{2} x^{7}}{32 a^{2}}+\frac {3 c d e \,x^{6}}{8 a^{2}}+\frac {7 c \,d^{2} x^{5}}{32 a^{2}}+\frac {9 e^{2} x^{3}}{32 a}+\frac {5 d e \,x^{2}}{8 a}+\frac {11 d^{2} x}{32 a}}{\left (c \,x^{4}+a \right )^{2}}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (c \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (5 e^{2} \textit {\_R}^{2}+24 d e \textit {\_R} +21 d^{2}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{128 a^{2} c}\) | \(126\) |
| default | \(d^{2} \left (\frac {x}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {7 x}{32 a \left (c \,x^{4}+a \right )}+\frac {21 \left (\frac {a}{c}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a^{2}}}{a}\right )+2 d e \left (\frac {x^{2}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {3 x^{2}}{16 a \left (c \,x^{4}+a \right )}+\frac {3 \arctan \left (\sqrt {\frac {c}{a}}\, x^{2}\right )}{16 a \sqrt {a c}}}{a}\right )+e^{2} \left (\frac {x^{3}}{8 a \left (c \,x^{4}+a \right )^{2}}+\frac {\frac {5 x^{3}}{32 a \left (c \,x^{4}+a \right )}+\frac {5 \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}{x^{2}+\left (\frac {a}{c}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{c}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{c}\right )^{\frac {1}{4}}}-1\right )\right )}{256 a c \left (\frac {a}{c}\right )^{\frac {1}{4}}}}{a}\right )\) | \(360\) |
Input:
int((e*x+d)^2/(c*x^4+a)^3,x,method=_RETURNVERBOSE)
Output:
(5/32*c*e^2/a^2*x^7+3/8*c*d*e/a^2*x^6+7/32*c*d^2/a^2*x^5+9/32*e^2/a*x^3+5/ 8*d*e/a*x^2+11/32*d^2/a*x)/(c*x^4+a)^2+1/128/a^2/c*sum((5*_R^2*e^2+24*_R*d *e+21*d^2)/_R^3*ln(x-_R),_R=RootOf(_Z^4*c+a))
Result contains complex when optimal does not.
Time = 8.11 (sec) , antiderivative size = 91420, normalized size of antiderivative = 317.43 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\text {Too large to display} \] Input:
integrate((e*x+d)^2/(c*x^4+a)^3,x, algorithm="fricas")
Output:
Too large to include
Time = 3.72 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.30 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\operatorname {RootSum} {\left (268435456 t^{4} a^{11} c^{3} + 25755648 t^{2} a^{6} c^{2} d^{2} e^{2} + t \left (307200 a^{4} c d e^{5} - 5419008 a^{3} c^{2} d^{5} e\right ) + 625 a^{2} e^{8} + 111906 a c d^{4} e^{4} + 194481 c^{2} d^{8}, \left ( t \mapsto t \log {\left (x + \frac {262144000 t^{3} a^{10} c^{2} e^{6} + 46110081024 t^{3} a^{9} c^{3} d^{4} e^{2} - 1645608960 t^{2} a^{7} c^{2} d^{3} e^{5} + 3641573376 t^{2} a^{6} c^{3} d^{7} e + 32688000 t a^{5} c d^{2} e^{8} + 3128219136 t a^{4} c^{2} d^{6} e^{4} + 522764928 t a^{3} c^{3} d^{10} + 225000 a^{3} d e^{11} - 43338240 a^{2} c d^{5} e^{7} - 523431720 a c^{2} d^{9} e^{3}}{15625 a^{3} e^{12} - 21357225 a^{2} c d^{4} e^{8} - 376741449 a c^{2} d^{8} e^{4} + 85766121 c^{3} d^{12}} \right )} \right )\right )} + \frac {11 a d^{2} x + 20 a d e x^{2} + 9 a e^{2} x^{3} + 7 c d^{2} x^{5} + 12 c d e x^{6} + 5 c e^{2} x^{7}}{32 a^{4} + 64 a^{3} c x^{4} + 32 a^{2} c^{2} x^{8}} \] Input:
integrate((e*x+d)**2/(c*x**4+a)**3,x)
Output:
RootSum(268435456*_t**4*a**11*c**3 + 25755648*_t**2*a**6*c**2*d**2*e**2 + _t*(307200*a**4*c*d*e**5 - 5419008*a**3*c**2*d**5*e) + 625*a**2*e**8 + 111 906*a*c*d**4*e**4 + 194481*c**2*d**8, Lambda(_t, _t*log(x + (262144000*_t* *3*a**10*c**2*e**6 + 46110081024*_t**3*a**9*c**3*d**4*e**2 - 1645608960*_t **2*a**7*c**2*d**3*e**5 + 3641573376*_t**2*a**6*c**3*d**7*e + 32688000*_t* a**5*c*d**2*e**8 + 3128219136*_t*a**4*c**2*d**6*e**4 + 522764928*_t*a**3*c **3*d**10 + 225000*a**3*d*e**11 - 43338240*a**2*c*d**5*e**7 - 523431720*a* c**2*d**9*e**3)/(15625*a**3*e**12 - 21357225*a**2*c*d**4*e**8 - 376741449* a*c**2*d**8*e**4 + 85766121*c**3*d**12)))) + (11*a*d**2*x + 20*a*d*e*x**2 + 9*a*e**2*x**3 + 7*c*d**2*x**5 + 12*c*d*e*x**6 + 5*c*e**2*x**7)/(32*a**4 + 64*a**3*c*x**4 + 32*a**2*c**2*x**8)
Time = 0.12 (sec) , antiderivative size = 364, normalized size of antiderivative = 1.26 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {5 \, c e^{2} x^{7} + 12 \, c d e x^{6} + 7 \, c d^{2} x^{5} + 9 \, a e^{2} x^{3} + 20 \, a d e x^{2} + 11 \, a d^{2} x}{32 \, {\left (a^{2} c^{2} x^{8} + 2 \, a^{3} c x^{4} + a^{4}\right )}} + \frac {\frac {\sqrt {2} {\left (21 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (21 \, \sqrt {c} d^{2} - 5 \, \sqrt {a} e^{2}\right )} \log \left (\sqrt {c} x^{2} - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {3}{4}} c^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} - 48 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x + \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}} + \frac {2 \, {\left (21 \, \sqrt {2} a^{\frac {1}{4}} c^{\frac {3}{4}} d^{2} + 5 \, \sqrt {2} a^{\frac {3}{4}} c^{\frac {1}{4}} e^{2} + 48 \, \sqrt {a} \sqrt {c} d e\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {c} x - \sqrt {2} a^{\frac {1}{4}} c^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {c}}}\right )}{a^{\frac {3}{4}} \sqrt {\sqrt {a} \sqrt {c}} c^{\frac {3}{4}}}}{256 \, a^{2}} \] Input:
integrate((e*x+d)^2/(c*x^4+a)^3,x, algorithm="maxima")
Output:
1/32*(5*c*e^2*x^7 + 12*c*d*e*x^6 + 7*c*d^2*x^5 + 9*a*e^2*x^3 + 20*a*d*e*x^ 2 + 11*a*d^2*x)/(a^2*c^2*x^8 + 2*a^3*c*x^4 + a^4) + 1/256*(sqrt(2)*(21*sqr t(c)*d^2 - 5*sqrt(a)*e^2)*log(sqrt(c)*x^2 + sqrt(2)*a^(1/4)*c^(1/4)*x + sq rt(a))/(a^(3/4)*c^(3/4)) - sqrt(2)*(21*sqrt(c)*d^2 - 5*sqrt(a)*e^2)*log(sq rt(c)*x^2 - sqrt(2)*a^(1/4)*c^(1/4)*x + sqrt(a))/(a^(3/4)*c^(3/4)) + 2*(21 *sqrt(2)*a^(1/4)*c^(3/4)*d^2 + 5*sqrt(2)*a^(3/4)*c^(1/4)*e^2 - 48*sqrt(a)* sqrt(c)*d*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x + sqrt(2)*a^(1/4)*c^(1/4))/sq rt(sqrt(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(3/4)) + 2*(21*sqrt( 2)*a^(1/4)*c^(3/4)*d^2 + 5*sqrt(2)*a^(3/4)*c^(1/4)*e^2 + 48*sqrt(a)*sqrt(c )*d*e)*arctan(1/2*sqrt(2)*(2*sqrt(c)*x - sqrt(2)*a^(1/4)*c^(1/4))/sqrt(sqr t(a)*sqrt(c)))/(a^(3/4)*sqrt(sqrt(a)*sqrt(c))*c^(3/4)))/a^2
Time = 0.12 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.24 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {5 \, c e^{2} x^{7} + 12 \, c d e x^{6} + 7 \, c d^{2} x^{5} + 9 \, a e^{2} x^{3} + 20 \, a d e x^{2} + 11 \, a d^{2} x}{32 \, {\left (c x^{4} + a\right )}^{2} a^{2}} + \frac {\sqrt {2} {\left (24 \, \sqrt {2} \sqrt {a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x + \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (24 \, \sqrt {2} \sqrt {a c} c^{2} d e + 21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} + 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (2 \, x - \sqrt {2} \left (\frac {a}{c}\right )^{\frac {1}{4}}\right )}}{2 \, \left (\frac {a}{c}\right )^{\frac {1}{4}}}\right )}{128 \, a^{3} c^{3}} + \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} - \frac {\sqrt {2} {\left (21 \, \left (a c^{3}\right )^{\frac {1}{4}} c^{2} d^{2} - 5 \, \left (a c^{3}\right )^{\frac {3}{4}} e^{2}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{c}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{c}}\right )}{256 \, a^{3} c^{3}} \] Input:
integrate((e*x+d)^2/(c*x^4+a)^3,x, algorithm="giac")
Output:
1/32*(5*c*e^2*x^7 + 12*c*d*e*x^6 + 7*c*d^2*x^5 + 9*a*e^2*x^3 + 20*a*d*e*x^ 2 + 11*a*d^2*x)/((c*x^4 + a)^2*a^2) + 1/128*sqrt(2)*(24*sqrt(2)*sqrt(a*c)* c^2*d*e + 21*(a*c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2)*arctan(1/2*sqrt( 2)*(2*x + sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 1/128*sqrt(2)*(24* sqrt(2)*sqrt(a*c)*c^2*d*e + 21*(a*c^3)^(1/4)*c^2*d^2 + 5*(a*c^3)^(3/4)*e^2 )*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/c)^(1/4))/(a/c)^(1/4))/(a^3*c^3) + 1/256*sqrt(2)*(21*(a*c^3)^(1/4)*c^2*d^2 - 5*(a*c^3)^(3/4)*e^2)*log(x^2 + s qrt(2)*x*(a/c)^(1/4) + sqrt(a/c))/(a^3*c^3) - 1/256*sqrt(2)*(21*(a*c^3)^(1 /4)*c^2*d^2 - 5*(a*c^3)^(3/4)*e^2)*log(x^2 - sqrt(2)*x*(a/c)^(1/4) + sqrt( a/c))/(a^3*c^3)
Time = 21.86 (sec) , antiderivative size = 676, normalized size of antiderivative = 2.35 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx=\frac {\frac {11\,d^2\,x}{32\,a}+\frac {9\,e^2\,x^3}{32\,a}+\frac {7\,c\,d^2\,x^5}{32\,a^2}+\frac {5\,c\,e^2\,x^7}{32\,a^2}+\frac {5\,d\,e\,x^2}{8\,a}+\frac {3\,c\,d\,e\,x^6}{8\,a^2}}{a^2+2\,a\,c\,x^4+c^2\,x^8}+\left (\sum _{k=1}^4\ln \left (-\frac {c\,\left (125\,a\,e^6-9891\,c\,d^4\,e^2+{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )}^2\,a^5\,c^2\,d^2\,344064-8784\,c\,d^3\,e^3\,x-\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^3\,c\,e^4\,x\,3200+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^2\,c^2\,d^4\,x\,56448+\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\,a^3\,c\,d\,e^3\,30720-{\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )}^2\,a^5\,c^2\,d\,e\,x\,393216\right )}{a^6\,32768}\right )\,\mathrm {root}\left (268435456\,a^{11}\,c^3\,z^4+25755648\,a^6\,c^2\,d^2\,e^2\,z^2-5419008\,a^3\,c^2\,d^5\,e\,z+307200\,a^4\,c\,d\,e^5\,z+111906\,a\,c\,d^4\,e^4+194481\,c^2\,d^8+625\,a^2\,e^8,z,k\right )\right ) \] Input:
int((d + e*x)^2/(a + c*x^4)^3,x)
Output:
((11*d^2*x)/(32*a) + (9*e^2*x^3)/(32*a) + (7*c*d^2*x^5)/(32*a^2) + (5*c*e^ 2*x^7)/(32*a^2) + (5*d*e*x^2)/(8*a) + (3*c*d*e*x^6)/(8*a^2))/(a^2 + c^2*x^ 8 + 2*a*c*x^4) + symsum(log(-(c*(125*a*e^6 - 9891*c*d^4*e^2 + 344064*root( 268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^ 5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a ^2*e^8, z, k)^2*a^5*c^2*d^2 - 8784*c*d^3*e^3*x - 3200*root(268435456*a^11* c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200* a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)*a ^3*c*e^4*x + 56448*root(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2* z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)*a^2*c^2*d^4*x + 30720*root(268435456 *a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 3 07200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z , k)*a^3*c*d*e^3 - 393216*root(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d ^2*e^2*z^2 - 5419008*a^3*c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d ^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k)^2*a^5*c^2*d*e*x))/(32768*a^6) )*root(268435456*a^11*c^3*z^4 + 25755648*a^6*c^2*d^2*e^2*z^2 - 5419008*a^3 *c^2*d^5*e*z + 307200*a^4*c*d*e^5*z + 111906*a*c*d^4*e^4 + 194481*c^2*d^8 + 625*a^2*e^8, z, k), k, 1, 4)
Time = 0.16 (sec) , antiderivative size = 1265, normalized size of antiderivative = 4.39 \[ \int \frac {(d+e x)^2}{\left (a+c x^4\right )^3} \, dx =\text {Too large to display} \] Input:
int((e*x+d)^2/(c*x^4+a)^3,x)
Output:
( - 10*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt( c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*e**2 - 20*c**(1/4)*a**(3/4)*sqrt(2 )*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2 )))*a*c*e**2*x**4 - 10*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*s qrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*e**2*x**8 - 42*c** (3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c** (1/4)*a**(1/4)*sqrt(2)))*a**2*d**2 - 84*c**(3/4)*a**(1/4)*sqrt(2)*atan((c* *(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*d* *2*x**4 - 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2 *sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*d**2*x**8 - 96*sqrt(c)*sqrt( a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt( 2)))*a**2*d*e - 192*sqrt(c)*sqrt(a)*atan((c**(1/4)*a**(1/4)*sqrt(2) - 2*sq rt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a*c*d*e*x**4 - 96*sqrt(c)*sqrt(a)*at an((c**(1/4)*a**(1/4)*sqrt(2) - 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))* c**2*d*e*x**8 + 10*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt( 2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*a**2*e**2 + 20*c**(1/4)*a** (3/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a** (1/4)*sqrt(2)))*a*c*e**2*x**4 + 10*c**(1/4)*a**(3/4)*sqrt(2)*atan((c**(1/4 )*a**(1/4)*sqrt(2) + 2*sqrt(c)*x)/(c**(1/4)*a**(1/4)*sqrt(2)))*c**2*e**2*x **8 + 42*c**(3/4)*a**(1/4)*sqrt(2)*atan((c**(1/4)*a**(1/4)*sqrt(2) + 2*...