\(\int \frac {(c+d x^4)^3}{x^2 (a+b x^4)^2} \, dx\) [216]

Optimal result

Integrand size = 22, antiderivative size = 247 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=-\frac {c^3}{a^2 x}+\frac {d^3 x^3}{3 b^2}-\frac {(b c-a d)^3 x^3}{4 a^2 b^2 \left (a+b x^4\right )}+\frac {(b c-a d)^2 (5 b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}-\frac {(b c-a d)^2 (5 b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}}+\frac {(b c-a d)^2 (5 b c+7 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{8 \sqrt {2} a^{9/4} b^{11/4}} \] Output:

-c^3/a^2/x+1/3*d^3*x^3/b^2-1/4*(-a*d+b*c)^3*x^3/a^2/b^2/(b*x^4+a)-1/16*(-a 
*d+b*c)^2*(7*a*d+5*b*c)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(9/ 
4)/b^(11/4)-1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctan(1+2^(1/2)*b^(1/4)*x/a^( 
1/4))*2^(1/2)/a^(9/4)/b^(11/4)+1/16*(-a*d+b*c)^2*(7*a*d+5*b*c)*arctanh(2^( 
1/2)*a^(1/4)*b^(1/4)*x/(a^(1/2)+b^(1/2)*x^2))*2^(1/2)/a^(9/4)/b^(11/4)
 

Mathematica [A] (verified)

Time = 0.18 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.25 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\frac {1}{96} \left (-\frac {96 c^3}{a^2 x}+\frac {32 d^3 x^3}{b^2}+\frac {24 (-b c+a d)^3 x^3}{a^2 b^2 \left (a+b x^4\right )}+\frac {6 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{9/4} b^{11/4}}-\frac {6 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{9/4} b^{11/4}}-\frac {3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{9/4} b^{11/4}}+\frac {3 \sqrt {2} (b c-a d)^2 (5 b c+7 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{9/4} b^{11/4}}\right ) \] Input:

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

Output:

((-96*c^3)/(a^2*x) + (32*d^3*x^3)/b^2 + (24*(-(b*c) + a*d)^3*x^3)/(a^2*b^2 
*(a + b*x^4)) + (6*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 - (Sqrt[ 
2]*b^(1/4)*x)/a^(1/4)])/(a^(9/4)*b^(11/4)) - (6*Sqrt[2]*(b*c - a*d)^2*(5*b 
*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(a^(9/4)*b^(11/4)) - 
(3*Sqrt[2]*(b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^( 
1/4)*x + Sqrt[b]*x^2])/(a^(9/4)*b^(11/4)) + (3*Sqrt[2]*(b*c - a*d)^2*(5*b* 
c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(a^(9/4 
)*b^(11/4)))/96
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 353, normalized size of antiderivative = 1.43, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {968, 25, 1040, 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.

\(\displaystyle \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx\)

\(\Big \downarrow \) 968

\(\displaystyle \frac {\left (c+d x^4\right )^2 (b c-a d)}{4 a b x \left (a+b x^4\right )}-\frac {\int -\frac {\left (d x^4+c\right ) \left (c (5 b c-a d)-d (3 b c-7 a d) x^4\right )}{x^2 \left (b x^4+a\right )}dx}{4 a b}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {\left (d x^4+c\right ) \left (c (5 b c-a d)-d (3 b c-7 a d) x^4\right )}{x^2 \left (b x^4+a\right )}dx}{4 a b}+\frac {\left (c+d x^4\right )^2 (b c-a d)}{4 a b x \left (a+b x^4\right )}\)

\(\Big \downarrow \) 1040

\(\displaystyle \frac {\int \left (-\frac {(a d-5 b c) c^2}{a x^2}-\frac {d^2 (3 b c-7 a d) x^2}{b}-\frac {(a d-b c)^2 (5 b c+7 a d) x^2}{a b \left (b x^4+a\right )}\right )dx}{4 a b}+\frac {\left (c+d x^4\right )^2 (b c-a d)}{4 a b x \left (a+b x^4\right )}\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^2 (7 a d+5 b c)}{2 \sqrt {2} a^{5/4} b^{7/4}}-\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^2 (7 a d+5 b c)}{2 \sqrt {2} a^{5/4} b^{7/4}}-\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}+\frac {(b c-a d)^2 (7 a d+5 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{4 \sqrt {2} a^{5/4} b^{7/4}}-\frac {c^2 (5 b c-a d)}{a x}-\frac {d^2 x^3 (3 b c-7 a d)}{3 b}}{4 a b}+\frac {\left (c+d x^4\right )^2 (b c-a d)}{4 a b x \left (a+b x^4\right )}\)

Input:

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

Output:

((b*c - a*d)*(c + d*x^4)^2)/(4*a*b*x*(a + b*x^4)) + (-((c^2*(5*b*c - a*d)) 
/(a*x)) - (d^2*(3*b*c - 7*a*d)*x^3)/(3*b) + ((b*c - a*d)^2*(5*b*c + 7*a*d) 
*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*Sqrt[2]*a^(5/4)*b^(7/4)) - (( 
b*c - a*d)^2*(5*b*c + 7*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(2*S 
qrt[2]*a^(5/4)*b^(7/4)) - ((b*c - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] - Sqr 
t[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(4*Sqrt[2]*a^(5/4)*b^(7/4)) + ((b*c 
 - a*d)^2*(5*b*c + 7*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b 
]*x^2])/(4*Sqrt[2]*a^(5/4)*b^(7/4)))/(4*a*b)
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 968

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_ 
))^(q_), x_Symbol] :> Simp[(-(c*b - a*d))*(e*x)^(m + 1)*(a + b*x^n)^(p + 1) 
*((c + d*x^n)^(q - 1)/(a*b*e*n*(p + 1))), x] + Simp[1/(a*b*n*(p + 1))   Int 
[(e*x)^m*(a + b*x^n)^(p + 1)*(c + d*x^n)^(q - 2)*Simp[c*(c*b*n*(p + 1) + (c 
*b - a*d)*(m + 1)) + d*(c*b*n*(p + 1) + (c*b - a*d)*(m + n*(q - 1) + 1))*x^ 
n, x], x], x] /; FreeQ[{a, b, c, d, e, m}, x] && NeQ[b*c - a*d, 0] && IGtQ[ 
n, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, e, m, n, p, q, 
 x]
 

rule 1040

Int[((g_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n 
_))^(q_.)*((e_) + (f_.)*(x_)^(n_))^(r_.), x_Symbol] :> Int[ExpandIntegrand[ 
(g*x)^m*(a + b*x^n)^p*(c + d*x^n)^q*(e + f*x^n)^r, x], x] /; FreeQ[{a, b, c 
, d, e, f, g, m, n}, x] && IGtQ[p, -2] && IGtQ[q, 0] && IGtQ[r, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.12 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.89

method	result	size
default	\(\frac {d^{3} x^{3}}{3 b^{2}}-\frac {\frac {\left (-\frac {1}{4} a^{3} d^{3}+\frac {3}{4} a^{2} b c \,d^{2}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {1}{4} b^{3} c^{3}\right ) x^{3}}{b \,x^{4}+a}+\frac {\left (\frac {7}{4} a^{3} d^{3}-\frac {3}{4} a \,b^{2} c^{2} d +\frac {5}{4} b^{3} c^{3}-\frac {9}{4} a^{2} b c \,d^{2}\right ) \sqrt {2}\, \left (\ln \left (\frac {x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{8 b \left (\frac {a}{b}\right )^{\frac {1}{4}}}}{a^{2} b^{2}}-\frac {c^{3}}{a^{2} x}\)	\(221\)
risch	\(\frac {d^{3} x^{3}}{3 b^{2}}+\frac {\frac {\left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right ) x^{4}}{4 a^{2}}-\frac {c^{3} b^{2}}{a}}{b^{2} x \left (b \,x^{4}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (2401 a^{12} d^{12}-12348 a^{11} b c \,d^{11}+19698 a^{10} b^{2} c^{2} d^{10}+2324 a^{9} b^{3} c^{3} d^{9}-37665 a^{8} b^{4} c^{4} d^{8}+27144 a^{7} b^{5} c^{5} d^{7}+19068 a^{6} b^{6} c^{6} d^{6}-28728 a^{5} b^{7} c^{7} d^{5}+1071 a^{4} b^{8} c^{8} d^{4}+11060 a^{3} b^{9} c^{9} d^{3}-3150 a^{2} b^{10} c^{10} d^{2}-1500 a \,b^{11} c^{11} d +625 b^{12} c^{12}+a^{9} b^{3} \textit {\_Z}^{4}\right )}{\sum }\textit {\_R} \ln \left (\left (9604 a^{12} d^{12}-49392 a^{11} b c \,d^{11}+78792 a^{10} b^{2} c^{2} d^{10}+9296 a^{9} b^{3} c^{3} d^{9}-150660 a^{8} b^{4} c^{4} d^{8}+108576 a^{7} b^{5} c^{5} d^{7}+76272 a^{6} b^{6} c^{6} d^{6}-114912 a^{5} b^{7} c^{7} d^{5}+4284 a^{4} b^{8} c^{8} d^{4}+44240 a^{3} b^{9} c^{9} d^{3}-12600 a^{2} b^{10} c^{10} d^{2}-6000 a \,b^{11} c^{11} d +2500 b^{12} c^{12}+5 \textit {\_R}^{4} a^{9} b^{3}\right ) x +\left (7 a^{10} b^{2} d^{3}-9 a^{9} b^{3} c \,d^{2}-3 a^{8} b^{4} c^{2} d +5 a^{7} b^{5} c^{3}\right ) \textit {\_R}^{3}\right )}{16 b^{2}}\)	\(498\)

Input:

int((d*x^4+c)^3/x^2/(b*x^4+a)^2,x,method=_RETURNVERBOSE)
 

Output:

1/3*d^3*x^3/b^2-1/a^2/b^2*((-1/4*a^3*d^3+3/4*a^2*b*c*d^2-3/4*a*b^2*c^2*d+1 
/4*b^3*c^3)*x^3/(b*x^4+a)+1/8*(7/4*a^3*d^3-3/4*a*b^2*c^2*d+5/4*b^3*c^3-9/4 
*a^2*b*c*d^2)/b/(a/b)^(1/4)*2^(1/2)*(ln((x^2-(a/b)^(1/4)*x*2^(1/2)+(a/b)^( 
1/2))/(x^2+(a/b)^(1/4)*x*2^(1/2)+(a/b)^(1/2)))+2*arctan(2^(1/2)/(a/b)^(1/4 
)*x+1)+2*arctan(2^(1/2)/(a/b)^(1/4)*x-1)))-c^3/a^2/x
 

Fricas [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 0.11 (sec) , antiderivative size = 2090, normalized size of antiderivative = 8.46 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/48*(16*a^2*b*d^3*x^8 - 48*a*b^2*c^3 - 4*(15*b^3*c^3 - 9*a*b^2*c^2*d + 9* 
a^2*b*c*d^2 - 7*a^3*d^3)*x^4 - 3*(a^2*b^3*x^5 + a^3*b^2*x)*(-(625*b^12*c^1 
2 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 
1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 271 
44*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698* 
a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4 
)*log(a^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d 
^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 
+ 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 
2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2*c^2*d^10 - 12348*a^11*b*c*d^11 + 240 
1*a^12*d^12)/(a^9*b^11))^(3/4) + (125*b^9*c^9 - 225*a*b^8*c^8*d - 540*a^2* 
b^7*c^7*d^2 + 1308*a^3*b^6*c^6*d^3 + 342*a^4*b^5*c^5*d^4 - 2430*a^5*b^4*c^ 
4*d^5 + 1140*a^6*b^3*c^3*d^6 + 1260*a^7*b^2*c^2*d^7 - 1323*a^8*b*c*d^8 + 3 
43*a^9*d^9)*x) - 3*(-I*a^2*b^3*x^5 - I*a^3*b^2*x)*(-(625*b^12*c^12 - 1500* 
a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11060*a^3*b^9*c^9*d^3 + 1071*a^4* 
b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19068*a^6*b^6*c^6*d^6 + 27144*a^7*b^ 
5*c^5*d^7 - 37665*a^8*b^4*c^4*d^8 + 2324*a^9*b^3*c^3*d^9 + 19698*a^10*b^2* 
c^2*d^10 - 12348*a^11*b*c*d^11 + 2401*a^12*d^12)/(a^9*b^11))^(1/4)*log(I*a 
^7*b^8*(-(625*b^12*c^12 - 1500*a*b^11*c^11*d - 3150*a^2*b^10*c^10*d^2 + 11 
060*a^3*b^9*c^9*d^3 + 1071*a^4*b^8*c^8*d^4 - 28728*a^5*b^7*c^7*d^5 + 19...
 

Sympy [A] (verification not implemented)

Time = 21.43 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.72 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\frac {- 4 a b^{2} c^{3} + x^{4} \left (a^{3} d^{3} - 3 a^{2} b c d^{2} + 3 a b^{2} c^{2} d - 5 b^{3} c^{3}\right )}{4 a^{3} b^{2} x + 4 a^{2} b^{3} x^{5}} + \operatorname {RootSum} {\left (65536 t^{4} a^{9} b^{11} + 2401 a^{12} d^{12} - 12348 a^{11} b c d^{11} + 19698 a^{10} b^{2} c^{2} d^{10} + 2324 a^{9} b^{3} c^{3} d^{9} - 37665 a^{8} b^{4} c^{4} d^{8} + 27144 a^{7} b^{5} c^{5} d^{7} + 19068 a^{6} b^{6} c^{6} d^{6} - 28728 a^{5} b^{7} c^{7} d^{5} + 1071 a^{4} b^{8} c^{8} d^{4} + 11060 a^{3} b^{9} c^{9} d^{3} - 3150 a^{2} b^{10} c^{10} d^{2} - 1500 a b^{11} c^{11} d + 625 b^{12} c^{12}, \left ( t \mapsto t \log {\left (- \frac {4096 t^{3} a^{7} b^{8}}{343 a^{9} d^{9} - 1323 a^{8} b c d^{8} + 1260 a^{7} b^{2} c^{2} d^{7} + 1140 a^{6} b^{3} c^{3} d^{6} - 2430 a^{5} b^{4} c^{4} d^{5} + 342 a^{4} b^{5} c^{5} d^{4} + 1308 a^{3} b^{6} c^{6} d^{3} - 540 a^{2} b^{7} c^{7} d^{2} - 225 a b^{8} c^{8} d + 125 b^{9} c^{9}} + x \right )} \right )\right )} + \frac {d^{3} x^{3}}{3 b^{2}} \] Input:

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

Output:

(-4*a*b**2*c**3 + x**4*(a**3*d**3 - 3*a**2*b*c*d**2 + 3*a*b**2*c**2*d - 5* 
b**3*c**3))/(4*a**3*b**2*x + 4*a**2*b**3*x**5) + RootSum(65536*_t**4*a**9* 
b**11 + 2401*a**12*d**12 - 12348*a**11*b*c*d**11 + 19698*a**10*b**2*c**2*d 
**10 + 2324*a**9*b**3*c**3*d**9 - 37665*a**8*b**4*c**4*d**8 + 27144*a**7*b 
**5*c**5*d**7 + 19068*a**6*b**6*c**6*d**6 - 28728*a**5*b**7*c**7*d**5 + 10 
71*a**4*b**8*c**8*d**4 + 11060*a**3*b**9*c**9*d**3 - 3150*a**2*b**10*c**10 
*d**2 - 1500*a*b**11*c**11*d + 625*b**12*c**12, Lambda(_t, _t*log(-4096*_t 
**3*a**7*b**8/(343*a**9*d**9 - 1323*a**8*b*c*d**8 + 1260*a**7*b**2*c**2*d* 
*7 + 1140*a**6*b**3*c**3*d**6 - 2430*a**5*b**4*c**4*d**5 + 342*a**4*b**5*c 
**5*d**4 + 1308*a**3*b**6*c**6*d**3 - 540*a**2*b**7*c**7*d**2 - 225*a*b**8 
*c**8*d + 125*b**9*c**9) + x))) + d**3*x**3/(3*b**2)
 

Maxima [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 299, normalized size of antiderivative = 1.21 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\frac {d^{3} x^{3}}{3 \, b^{2}} - \frac {4 \, a b^{2} c^{3} + {\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d + 3 \, a^{2} b c d^{2} - a^{3} d^{3}\right )} x^{4}}{4 \, {\left (a^{2} b^{3} x^{5} + a^{3} b^{2} x\right )}} - \frac {{\left (5 \, b^{3} c^{3} - 3 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + 7 \, a^{3} d^{3}\right )} {\left (\frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} + \frac {2 \, \sqrt {2} \arctan \left (\frac {\sqrt {2} {\left (2 \, \sqrt {b} x - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}}\right )}}{2 \, \sqrt {\sqrt {a} \sqrt {b}}}\right )}{\sqrt {\sqrt {a} \sqrt {b}} \sqrt {b}} - \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} + \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}} + \frac {\sqrt {2} \log \left (\sqrt {b} x^{2} - \sqrt {2} a^{\frac {1}{4}} b^{\frac {1}{4}} x + \sqrt {a}\right )}{a^{\frac {1}{4}} b^{\frac {3}{4}}}\right )}}{32 \, a^{2} b^{2}} \] Input:

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

Output:

1/3*d^3*x^3/b^2 - 1/4*(4*a*b^2*c^3 + (5*b^3*c^3 - 3*a*b^2*c^2*d + 3*a^2*b* 
c*d^2 - a^3*d^3)*x^4)/(a^2*b^3*x^5 + a^3*b^2*x) - 1/32*(5*b^3*c^3 - 3*a*b^ 
2*c^2*d - 9*a^2*b*c*d^2 + 7*a^3*d^3)*(2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt 
(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt 
(b))*sqrt(b)) + 2*sqrt(2)*arctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4 
)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)))/(sqrt(sqrt(a)*sqrt(b))*sqrt(b)) - sqrt(2 
)*log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(1/4)*b^(3/4)) 
 + sqrt(2)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(1/4) 
*b^(3/4)))/(a^2*b^2)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 499 vs. \(2 (196) = 392\).

Time = 0.13 (sec) , antiderivative size = 499, normalized size of antiderivative = 2.02 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\frac {d^{3} x^{3}}{3 \, b^{2}} - \frac {5 \, b^{3} c^{3} x^{4} - 3 \, a b^{2} c^{2} d x^{4} + 3 \, a^{2} b c d^{2} x^{4} - a^{3} d^{3} x^{4} + 4 \, a b^{2} c^{3}}{4 \, {\left (b x^{5} + a x\right )} a^{2} b^{2}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\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 )}{16 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\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 )}{16 \, a^{3} b^{5}} + \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (x^{2} + \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{3} b^{5}} - \frac {\sqrt {2} {\left (5 \, \left (a b^{3}\right )^{\frac {3}{4}} b^{3} c^{3} - 3 \, \left (a b^{3}\right )^{\frac {3}{4}} a b^{2} c^{2} d - 9 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{2} b c d^{2} + 7 \, \left (a b^{3}\right )^{\frac {3}{4}} a^{3} d^{3}\right )} \log \left (x^{2} - \sqrt {2} x \left (\frac {a}{b}\right )^{\frac {1}{4}} + \sqrt {\frac {a}{b}}\right )}{32 \, a^{3} b^{5}} \] Input:

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

Output:

1/3*d^3*x^3/b^2 - 1/4*(5*b^3*c^3*x^4 - 3*a*b^2*c^2*d*x^4 + 3*a^2*b*c*d^2*x 
^4 - a^3*d^3*x^4 + 4*a*b^2*c^3)/((b*x^5 + a*x)*a^2*b^2) - 1/16*sqrt(2)*(5* 
(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2* 
b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b) 
^(1/4))/(a/b)^(1/4))/(a^3*b^5) - 1/16*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3 
*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4) 
*a^3*d^3)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^3 
*b^5) + 1/32*sqrt(2)*(5*(a*b^3)^(3/4)*b^3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2* 
d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b^3)^(3/4)*a^3*d^3)*log(x^2 + sqrt( 
2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^5) - 1/32*sqrt(2)*(5*(a*b^3)^(3/4)*b^ 
3*c^3 - 3*(a*b^3)^(3/4)*a*b^2*c^2*d - 9*(a*b^3)^(3/4)*a^2*b*c*d^2 + 7*(a*b 
^3)^(3/4)*a^3*d^3)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^3*b^5)
 

Mupad [B] (verification not implemented)

Time = 3.50 (sec) , antiderivative size = 651, normalized size of antiderivative = 2.64 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx=\frac {\frac {x^4\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-5\,b^3\,c^3\right )}{4\,a^2}-\frac {b^2\,c^3}{a}}{b^3\,x^5+a\,b^2\,x}+\frac {d^3\,x^3}{3\,b^2}-\frac {\mathrm {atan}\left (\frac {x\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (12544\,a^{13}\,b^8\,d^6-32256\,a^{12}\,b^9\,c\,d^5+9984\,a^{11}\,b^{10}\,c^2\,d^4+31744\,a^{10}\,b^{11}\,c^3\,d^3-20736\,a^9\,b^{12}\,c^4\,d^2-7680\,a^8\,b^{13}\,c^5\,d+6400\,a^7\,b^{14}\,c^6\right )}{8\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (10976\,a^{14}\,b^5\,d^9-42336\,a^{13}\,b^6\,c\,d^8+40320\,a^{12}\,b^7\,c^2\,d^7+36480\,a^{11}\,b^8\,c^3\,d^6-77760\,a^{10}\,b^9\,c^4\,d^5+10944\,a^9\,b^{10}\,c^5\,d^4+41856\,a^8\,b^{11}\,c^6\,d^3-17280\,a^7\,b^{12}\,c^7\,d^2-7200\,a^6\,b^{13}\,c^8\,d+4000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )}{8\,{\left (-a\right )}^{9/4}\,b^{11/4}}-\frac {\mathrm {atan}\left (\frac {x\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,\left (12544\,a^{13}\,b^8\,d^6-32256\,a^{12}\,b^9\,c\,d^5+9984\,a^{11}\,b^{10}\,c^2\,d^4+31744\,a^{10}\,b^{11}\,c^3\,d^3-20736\,a^9\,b^{12}\,c^4\,d^2-7680\,a^8\,b^{13}\,c^5\,d+6400\,a^7\,b^{14}\,c^6\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{9/4}\,b^{11/4}\,\left (10976\,a^{14}\,b^5\,d^9-42336\,a^{13}\,b^6\,c\,d^8+40320\,a^{12}\,b^7\,c^2\,d^7+36480\,a^{11}\,b^8\,c^3\,d^6-77760\,a^{10}\,b^9\,c^4\,d^5+10944\,a^9\,b^{10}\,c^5\,d^4+41856\,a^8\,b^{11}\,c^6\,d^3-17280\,a^7\,b^{12}\,c^7\,d^2-7200\,a^6\,b^{13}\,c^8\,d+4000\,a^5\,b^{14}\,c^9\right )}\right )\,{\left (a\,d-b\,c\right )}^2\,\left (7\,a\,d+5\,b\,c\right )\,1{}\mathrm {i}}{8\,{\left (-a\right )}^{9/4}\,b^{11/4}} \] Input:

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

Output:

((x^4*(a^3*d^3 - 5*b^3*c^3 + 3*a*b^2*c^2*d - 3*a^2*b*c*d^2))/(4*a^2) - (b^ 
2*c^3)/a)/(b^3*x^5 + a*b^2*x) + (d^3*x^3)/(3*b^2) - (atan((x*(a*d - b*c)^2 
*(7*a*d + 5*b*c)*(6400*a^7*b^14*c^6 + 12544*a^13*b^8*d^6 - 7680*a^8*b^13*c 
^5*d - 32256*a^12*b^9*c*d^5 - 20736*a^9*b^12*c^4*d^2 + 31744*a^10*b^11*c^3 
*d^3 + 9984*a^11*b^10*c^2*d^4))/(8*(-a)^(9/4)*b^(11/4)*(4000*a^5*b^14*c^9 
+ 10976*a^14*b^5*d^9 - 7200*a^6*b^13*c^8*d - 42336*a^13*b^6*c*d^8 - 17280* 
a^7*b^12*c^7*d^2 + 41856*a^8*b^11*c^6*d^3 + 10944*a^9*b^10*c^5*d^4 - 77760 
*a^10*b^9*c^4*d^5 + 36480*a^11*b^8*c^3*d^6 + 40320*a^12*b^7*c^2*d^7)))*(a* 
d - b*c)^2*(7*a*d + 5*b*c))/(8*(-a)^(9/4)*b^(11/4)) - (atan((x*(a*d - b*c) 
^2*(7*a*d + 5*b*c)*(6400*a^7*b^14*c^6 + 12544*a^13*b^8*d^6 - 7680*a^8*b^13 
*c^5*d - 32256*a^12*b^9*c*d^5 - 20736*a^9*b^12*c^4*d^2 + 31744*a^10*b^11*c 
^3*d^3 + 9984*a^11*b^10*c^2*d^4)*1i)/(8*(-a)^(9/4)*b^(11/4)*(4000*a^5*b^14 
*c^9 + 10976*a^14*b^5*d^9 - 7200*a^6*b^13*c^8*d - 42336*a^13*b^6*c*d^8 - 1 
7280*a^7*b^12*c^7*d^2 + 41856*a^8*b^11*c^6*d^3 + 10944*a^9*b^10*c^5*d^4 - 
77760*a^10*b^9*c^4*d^5 + 36480*a^11*b^8*c^3*d^6 + 40320*a^12*b^7*c^2*d^7)) 
)*(a*d - b*c)^2*(7*a*d + 5*b*c)*1i)/(8*(-a)^(9/4)*b^(11/4))
 

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 1356, normalized size of antiderivative = 5.49 \[ \int \frac {\left (c+d x^4\right )^3}{x^2 \left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:

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

Output:

(42*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**4*d**3*x - 54*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**3*b*c*d**2*x + 42*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**3*b*d**3*x**5 - 18* 
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*b**2*c**2*d*x - 54*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*b**2*c*d**2*x**5 + 30*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**3*c**3*x 
 - 18*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**3*c**2*d*x**5 + 30*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**4*c**3*x**5 - 42*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**4*d**3*x + 
54*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**3*b*c*d**2*x - 42*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**3*b*d**3*x**5 + 18*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*b**2*c**2*...