Integrand size = 19, antiderivative size = 337 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=\frac {d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x}{b^5}+\frac {d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5}{5 b^4}+\frac {d^4 (5 b c-2 a d) x^9}{9 b^3}+\frac {d^5 x^{13}}{13 b^2}+\frac {(b c-a d)^5 x}{4 a b^5 \left (a+b x^4\right )}-\frac {(b c-a d)^4 (3 b c+17 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{21/4}}+\frac {(b c-a d)^4 (3 b c+17 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{8 \sqrt {2} a^{7/4} b^{21/4}}+\frac {(b c-a d)^4 (3 b c+17 a d) \text {arctanh}\left (\frac {\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x}{\sqrt {a}+\sqrt {b} x^2}\right )}{8 \sqrt {2} a^{7/4} b^{21/4}} \] Output:
d^2*(-4*a^3*d^3+15*a^2*b*c*d^2-20*a*b^2*c^2*d+10*b^3*c^3)*x/b^5+1/5*d^3*(3 *a^2*d^2-10*a*b*c*d+10*b^2*c^2)*x^5/b^4+1/9*d^4*(-2*a*d+5*b*c)*x^9/b^3+1/1 3*d^5*x^13/b^2+1/4*(-a*d+b*c)^5*x/a/b^5/(b*x^4+a)+1/16*(-a*d+b*c)^4*(17*a* d+3*b*c)*arctan(-1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/a^(7/4)/b^(21/4)+1/1 6*(-a*d+b*c)^4*(17*a*d+3*b*c)*arctan(1+2^(1/2)*b^(1/4)*x/a^(1/4))*2^(1/2)/ a^(7/4)/b^(21/4)+1/16*(-a*d+b*c)^4*(17*a*d+3*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^(7/4)/b^(21/4)
Time = 0.42 (sec) , antiderivative size = 391, normalized size of antiderivative = 1.16 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=\frac {18720 \sqrt [4]{b} d^2 \left (10 b^3 c^3-20 a b^2 c^2 d+15 a^2 b c d^2-4 a^3 d^3\right ) x+3744 b^{5/4} d^3 \left (10 b^2 c^2-10 a b c d+3 a^2 d^2\right ) x^5+2080 b^{9/4} d^4 (5 b c-2 a d) x^9+1440 b^{13/4} d^5 x^{13}+\frac {4680 \sqrt [4]{b} (b c-a d)^5 x}{a \left (a+b x^4\right )}-\frac {1170 \sqrt {2} (b c-a d)^4 (3 b c+17 a d) \arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}+\frac {1170 \sqrt {2} (b c-a d)^4 (3 b c+17 a d) \arctan \left (1+\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right )}{a^{7/4}}-\frac {585 \sqrt {2} (b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt {a}-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}+\frac {585 \sqrt {2} (b c-a d)^4 (3 b c+17 a d) \log \left (\sqrt {a}+\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {b} x^2\right )}{a^{7/4}}}{18720 b^{21/4}} \] Input:
Integrate[(c + d*x^4)^5/(a + b*x^4)^2,x]
Output:
(18720*b^(1/4)*d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d ^3)*x + 3744*b^(5/4)*d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^5 + 2080* b^(9/4)*d^4*(5*b*c - 2*a*d)*x^9 + 1440*b^(13/4)*d^5*x^13 + (4680*b^(1/4)*( b*c - a*d)^5*x)/(a*(a + b*x^4)) - (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17* a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4) + (1170*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/a^(7/4 ) - (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1 /4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4) + (585*Sqrt[2]*(b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/a^(7/4)) /(18720*b^(21/4))
Time = 1.01 (sec) , antiderivative size = 407, normalized size of antiderivative = 1.21, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {915, 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.
| \(\displaystyle \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx\) |
| \(\Big \downarrow \) 915 |
| \(\displaystyle \int \left (\frac {d^3 x^4 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{b^4}+\frac {d^2 \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac {5 b d x^4 (b c-a d)^4+(4 a d+b c) (b c-a d)^4}{b^5 \left (a+b x^4\right )^2}+\frac {d^4 x^8 (5 b c-2 a d)}{b^3}+\frac {d^5 x^{12}}{b^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\arctan \left (1-\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}\right ) (b c-a d)^4 (17 a d+3 b c)}{8 \sqrt {2} a^{7/4} b^{21/4}}+\frac {\arctan \left (\frac {\sqrt {2} \sqrt [4]{b} x}{\sqrt [4]{a}}+1\right ) (b c-a d)^4 (17 a d+3 b c)}{8 \sqrt {2} a^{7/4} b^{21/4}}-\frac {(b c-a d)^4 (17 a d+3 b c) \log \left (-\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{21/4}}+\frac {(b c-a d)^4 (17 a d+3 b c) \log \left (\sqrt {2} \sqrt [4]{a} \sqrt [4]{b} x+\sqrt {a}+\sqrt {b} x^2\right )}{16 \sqrt {2} a^{7/4} b^{21/4}}+\frac {d^3 x^5 \left (3 a^2 d^2-10 a b c d+10 b^2 c^2\right )}{5 b^4}+\frac {d^2 x \left (-4 a^3 d^3+15 a^2 b c d^2-20 a b^2 c^2 d+10 b^3 c^3\right )}{b^5}+\frac {x (b c-a d)^5}{4 a b^5 \left (a+b x^4\right )}+\frac {d^4 x^9 (5 b c-2 a d)}{9 b^3}+\frac {d^5 x^{13}}{13 b^2}\) |
Input:
Int[(c + d*x^4)^5/(a + b*x^4)^2,x]
Output:
(d^2*(10*b^3*c^3 - 20*a*b^2*c^2*d + 15*a^2*b*c*d^2 - 4*a^3*d^3)*x)/b^5 + ( d^3*(10*b^2*c^2 - 10*a*b*c*d + 3*a^2*d^2)*x^5)/(5*b^4) + (d^4*(5*b*c - 2*a *d)*x^9)/(9*b^3) + (d^5*x^13)/(13*b^2) + ((b*c - a*d)^5*x)/(4*a*b^5*(a + b *x^4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*ArcTan[1 - (Sqrt[2]*b^(1/4)*x)/a^ (1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*Arc Tan[1 + (Sqrt[2]*b^(1/4)*x)/a^(1/4)])/(8*Sqrt[2]*a^(7/4)*b^(21/4)) - ((b*c - a*d)^4*(3*b*c + 17*a*d)*Log[Sqrt[a] - Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[ b]*x^2])/(16*Sqrt[2]*a^(7/4)*b^(21/4)) + ((b*c - a*d)^4*(3*b*c + 17*a*d)*L og[Sqrt[a] + Sqrt[2]*a^(1/4)*b^(1/4)*x + Sqrt[b]*x^2])/(16*Sqrt[2]*a^(7/4) *b^(21/4))
Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Int[PolynomialDivide[(a + b*x^n)^p, (c + d*x^n)^(-q), x], x] /; FreeQ[{a , b, c, d}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IGtQ[p, 0] && ILtQ[q, 0] && GeQ[p, -q]
Result contains higher order function than in optimal. Order 9 vs. order 3.
Time = 0.92 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.90
| method | result | size |
| risch | \(\frac {d^{5} x^{13}}{13 b^{2}}-\frac {2 d^{5} a \,x^{9}}{9 b^{3}}+\frac {5 d^{4} c \,x^{9}}{9 b^{2}}+\frac {3 d^{5} a^{2} x^{5}}{5 b^{4}}-\frac {2 d^{4} a c \,x^{5}}{b^{3}}+\frac {2 d^{3} c^{2} x^{5}}{b^{2}}-\frac {4 d^{5} a^{3} x}{b^{5}}+\frac {15 d^{4} a^{2} c x}{b^{4}}-\frac {20 d^{3} a \,c^{2} x}{b^{3}}+\frac {10 d^{2} c^{3} x}{b^{2}}-\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right ) x}{4 a \,b^{5} \left (b \,x^{4}+a \right )}+\frac {\munderset {\textit {\_R} =\operatorname {RootOf}\left (b \,\textit {\_Z}^{4}+a \right )}{\sum }\frac {\left (17 a^{5} d^{5}-65 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-50 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 c^{5} b^{5}\right ) \ln \left (x -\textit {\_R} \right )}{\textit {\_R}^{3}}}{16 b^{6} a}\) | \(304\) |
| default | \(-\frac {d^{2} \left (-\frac {1}{13} b^{3} d^{3} x^{13}+\frac {2}{9} a \,b^{2} d^{3} x^{9}-\frac {5}{9} b^{3} c \,d^{2} x^{9}-\frac {3}{5} a^{2} b \,d^{3} x^{5}+2 a \,b^{2} c \,d^{2} x^{5}-2 b^{3} c^{2} d \,x^{5}+4 a^{3} d^{3} x -15 a^{2} b c \,d^{2} x +20 a \,b^{2} c^{2} d x -10 b^{3} c^{3} x \right )}{b^{5}}+\frac {-\frac {\left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -c^{5} b^{5}\right ) x}{4 a \left (b \,x^{4}+a \right )}+\frac {\left (17 a^{5} d^{5}-65 a^{4} b c \,d^{4}+90 a^{3} b^{2} c^{2} d^{3}-50 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d +3 c^{5} b^{5}\right ) \left (\frac {a}{b}\right )^{\frac {1}{4}} \sqrt {2}\, \left (\ln \left (\frac {x^{2}+\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}{x^{2}-\left (\frac {a}{b}\right )^{\frac {1}{4}} x \sqrt {2}+\sqrt {\frac {a}{b}}}\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}+1\right )+2 \arctan \left (\frac {\sqrt {2}\, x}{\left (\frac {a}{b}\right )^{\frac {1}{4}}}-1\right )\right )}{32 a^{2}}}{b^{5}}\) | \(373\) |
Input:
int((d*x^4+c)^5/(b*x^4+a)^2,x,method=_RETURNVERBOSE)
Output:
1/13*d^5*x^13/b^2-2/9*d^5/b^3*a*x^9+5/9*d^4/b^2*c*x^9+3/5*d^5/b^4*a^2*x^5- 2*d^4/b^3*a*c*x^5+2*d^3/b^2*c^2*x^5-4*d^5/b^5*a^3*x+15*d^4/b^4*a^2*c*x-20* d^3/b^3*a*c^2*x+10*d^2/b^2*c^3*x-1/4*(a^5*d^5-5*a^4*b*c*d^4+10*a^3*b^2*c^2 *d^3-10*a^2*b^3*c^3*d^2+5*a*b^4*c^4*d-b^5*c^5)/a*x/b^5/(b*x^4+a)+1/16/b^6/ a*sum((17*a^5*d^5-65*a^4*b*c*d^4+90*a^3*b^2*c^2*d^3-50*a^2*b^3*c^3*d^2+5*a *b^4*c^4*d+3*b^5*c^5)/_R^3*ln(x-_R),_R=RootOf(_Z^4*b+a))
Result contains complex when optimal does not.
Time = 0.12 (sec) , antiderivative size = 2884, normalized size of antiderivative = 8.56 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \] Input:
integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="fricas")
Output:
1/9360*(720*a*b^4*d^5*x^17 + 80*(65*a*b^4*c*d^4 - 17*a^2*b^3*d^5)*x^13 + 2 08*(90*a*b^4*c^2*d^3 - 65*a^2*b^3*c*d^4 + 17*a^3*b^2*d^5)*x^9 + 1872*(50*a *b^4*c^3*d^2 - 90*a^2*b^3*c^2*d^3 + 65*a^3*b^2*c*d^4 - 17*a^4*b*d^5)*x^5 + 585*(a*b^6*x^4 + a^2*b^5)*(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^2* b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 13264 *a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^10* b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144*a ^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d^1 7 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20)/( a^7*b^21))^(1/4)*log(a^2*b^5*(-(81*b^20*c^20 + 540*a*b^19*c^19*d - 4050*a^ 2*b^18*c^18*d^2 - 15780*a^3*b^17*c^17*d^3 + 132205*a^4*b^16*c^16*d^4 - 132 64*a^5*b^15*c^15*d^5 - 1960920*a^6*b^14*c^14*d^6 + 6137200*a^7*b^13*c^13*d ^7 - 500110*a^8*b^12*c^12*d^8 - 48530040*a^9*b^11*c^11*d^9 + 174873556*a^1 0*b^10*c^10*d^10 - 360900280*a^11*b^9*c^9*d^11 + 517559250*a^12*b^8*c^8*d^ 12 - 548231440*a^13*b^7*c^7*d^13 + 438700840*a^14*b^6*c^6*d^14 - 266040144 *a^15*b^5*c^5*d^15 + 120836285*a^16*b^4*c^4*d^16 - 39944900*a^17*b^3*c^3*d ^17 + 9094830*a^18*b^2*c^2*d^18 - 1277380*a^19*b*c*d^19 + 83521*a^20*d^20) /(a^7*b^21))^(1/4) + (3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + ...
Time = 168.42 (sec) , antiderivative size = 619, normalized size of antiderivative = 1.84 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=x^{9} \left (- \frac {2 a d^{5}}{9 b^{3}} + \frac {5 c d^{4}}{9 b^{2}}\right ) + x^{5} \cdot \left (\frac {3 a^{2} d^{5}}{5 b^{4}} - \frac {2 a c d^{4}}{b^{3}} + \frac {2 c^{2} d^{3}}{b^{2}}\right ) + x \left (- \frac {4 a^{3} d^{5}}{b^{5}} + \frac {15 a^{2} c d^{4}}{b^{4}} - \frac {20 a c^{2} d^{3}}{b^{3}} + \frac {10 c^{3} d^{2}}{b^{2}}\right ) + \frac {x \left (- a^{5} d^{5} + 5 a^{4} b c d^{4} - 10 a^{3} b^{2} c^{2} d^{3} + 10 a^{2} b^{3} c^{3} d^{2} - 5 a b^{4} c^{4} d + b^{5} c^{5}\right )}{4 a^{2} b^{5} + 4 a b^{6} x^{4}} + \operatorname {RootSum} {\left (65536 t^{4} a^{7} b^{21} + 83521 a^{20} d^{20} - 1277380 a^{19} b c d^{19} + 9094830 a^{18} b^{2} c^{2} d^{18} - 39944900 a^{17} b^{3} c^{3} d^{17} + 120836285 a^{16} b^{4} c^{4} d^{16} - 266040144 a^{15} b^{5} c^{5} d^{15} + 438700840 a^{14} b^{6} c^{6} d^{14} - 548231440 a^{13} b^{7} c^{7} d^{13} + 517559250 a^{12} b^{8} c^{8} d^{12} - 360900280 a^{11} b^{9} c^{9} d^{11} + 174873556 a^{10} b^{10} c^{10} d^{10} - 48530040 a^{9} b^{11} c^{11} d^{9} - 500110 a^{8} b^{12} c^{12} d^{8} + 6137200 a^{7} b^{13} c^{13} d^{7} - 1960920 a^{6} b^{14} c^{14} d^{6} - 13264 a^{5} b^{15} c^{15} d^{5} + 132205 a^{4} b^{16} c^{16} d^{4} - 15780 a^{3} b^{17} c^{17} d^{3} - 4050 a^{2} b^{18} c^{18} d^{2} + 540 a b^{19} c^{19} d + 81 b^{20} c^{20}, \left ( t \mapsto t \log {\left (\frac {16 t a^{2} b^{5}}{17 a^{5} d^{5} - 65 a^{4} b c d^{4} + 90 a^{3} b^{2} c^{2} d^{3} - 50 a^{2} b^{3} c^{3} d^{2} + 5 a b^{4} c^{4} d + 3 b^{5} c^{5}} + x \right )} \right )\right )} + \frac {d^{5} x^{13}}{13 b^{2}} \] Input:
integrate((d*x**4+c)**5/(b*x**4+a)**2,x)
Output:
x**9*(-2*a*d**5/(9*b**3) + 5*c*d**4/(9*b**2)) + x**5*(3*a**2*d**5/(5*b**4) - 2*a*c*d**4/b**3 + 2*c**2*d**3/b**2) + x*(-4*a**3*d**5/b**5 + 15*a**2*c* d**4/b**4 - 20*a*c**2*d**3/b**3 + 10*c**3*d**2/b**2) + x*(-a**5*d**5 + 5*a **4*b*c*d**4 - 10*a**3*b**2*c**2*d**3 + 10*a**2*b**3*c**3*d**2 - 5*a*b**4* c**4*d + b**5*c**5)/(4*a**2*b**5 + 4*a*b**6*x**4) + RootSum(65536*_t**4*a* *7*b**21 + 83521*a**20*d**20 - 1277380*a**19*b*c*d**19 + 9094830*a**18*b** 2*c**2*d**18 - 39944900*a**17*b**3*c**3*d**17 + 120836285*a**16*b**4*c**4* d**16 - 266040144*a**15*b**5*c**5*d**15 + 438700840*a**14*b**6*c**6*d**14 - 548231440*a**13*b**7*c**7*d**13 + 517559250*a**12*b**8*c**8*d**12 - 3609 00280*a**11*b**9*c**9*d**11 + 174873556*a**10*b**10*c**10*d**10 - 48530040 *a**9*b**11*c**11*d**9 - 500110*a**8*b**12*c**12*d**8 + 6137200*a**7*b**13 *c**13*d**7 - 1960920*a**6*b**14*c**14*d**6 - 13264*a**5*b**15*c**15*d**5 + 132205*a**4*b**16*c**16*d**4 - 15780*a**3*b**17*c**17*d**3 - 4050*a**2*b **18*c**18*d**2 + 540*a*b**19*c**19*d + 81*b**20*c**20, Lambda(_t, _t*log( 16*_t*a**2*b**5/(17*a**5*d**5 - 65*a**4*b*c*d**4 + 90*a**3*b**2*c**2*d**3 - 50*a**2*b**3*c**3*d**2 + 5*a*b**4*c**4*d + 3*b**5*c**5) + x))) + d**5*x* *13/(13*b**2)
Leaf count of result is larger than twice the leaf count of optimal. 644 vs. \(2 (282) = 564\).
Time = 0.12 (sec) , antiderivative size = 644, normalized size of antiderivative = 1.91 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=\frac {{\left (b^{5} c^{5} - 5 \, a b^{4} c^{4} d + 10 \, a^{2} b^{3} c^{3} d^{2} - 10 \, a^{3} b^{2} c^{2} d^{3} + 5 \, a^{4} b c d^{4} - a^{5} d^{5}\right )} x}{4 \, {\left (a b^{6} x^{4} + a^{2} b^{5}\right )}} + \frac {45 \, b^{3} d^{5} x^{13} + 65 \, {\left (5 \, b^{3} c d^{4} - 2 \, a b^{2} d^{5}\right )} x^{9} + 117 \, {\left (10 \, b^{3} c^{2} d^{3} - 10 \, a b^{2} c d^{4} + 3 \, a^{2} b d^{5}\right )} x^{5} + 585 \, {\left (10 \, b^{3} c^{3} d^{2} - 20 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - 4 \, a^{3} d^{5}\right )} x}{585 \, b^{5}} + \frac {\frac {2 \, \sqrt {2} {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 50 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 65 \, a^{4} b c d^{4} + 17 \, a^{5} d^{5}\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 )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {2 \, \sqrt {2} {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 50 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 65 \, a^{4} b c d^{4} + 17 \, a^{5} d^{5}\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 )}{\sqrt {a} \sqrt {\sqrt {a} \sqrt {b}}} + \frac {\sqrt {2} {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 50 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 65 \, a^{4} b c d^{4} + 17 \, a^{5} d^{5}\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 {1}{4}}} - \frac {\sqrt {2} {\left (3 \, b^{5} c^{5} + 5 \, a b^{4} c^{4} d - 50 \, a^{2} b^{3} c^{3} d^{2} + 90 \, a^{3} b^{2} c^{2} d^{3} - 65 \, a^{4} b c d^{4} + 17 \, a^{5} d^{5}\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 {1}{4}}}}{32 \, a b^{5}} \] Input:
integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="maxima")
Output:
1/4*(b^5*c^5 - 5*a*b^4*c^4*d + 10*a^2*b^3*c^3*d^2 - 10*a^3*b^2*c^2*d^3 + 5 *a^4*b*c*d^4 - a^5*d^5)*x/(a*b^6*x^4 + a^2*b^5) + 1/585*(45*b^3*d^5*x^13 + 65*(5*b^3*c*d^4 - 2*a*b^2*d^5)*x^9 + 117*(10*b^3*c^2*d^3 - 10*a*b^2*c*d^4 + 3*a^2*b*d^5)*x^5 + 585*(10*b^3*c^3*d^2 - 20*a*b^2*c^2*d^3 + 15*a^2*b*c* d^4 - 4*a^3*d^5)*x)/b^5 + 1/32*(2*sqrt(2)*(3*b^5*c^5 + 5*a*b^4*c^4*d - 50* a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)*arctan (1/2*sqrt(2)*(2*sqrt(b)*x + sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt(b)) )/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + 2*sqrt(2)*(3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)*ar ctan(1/2*sqrt(2)*(2*sqrt(b)*x - sqrt(2)*a^(1/4)*b^(1/4))/sqrt(sqrt(a)*sqrt (b)))/(sqrt(a)*sqrt(sqrt(a)*sqrt(b))) + sqrt(2)*(3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)* log(sqrt(b)*x^2 + sqrt(2)*a^(1/4)*b^(1/4)*x + sqrt(a))/(a^(3/4)*b^(1/4)) - sqrt(2)*(3*b^5*c^5 + 5*a*b^4*c^4*d - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2* d^3 - 65*a^4*b*c*d^4 + 17*a^5*d^5)*log(sqrt(b)*x^2 - sqrt(2)*a^(1/4)*b^(1/ 4)*x + sqrt(a))/(a^(3/4)*b^(1/4)))/(a*b^5)
Leaf count of result is larger than twice the leaf count of optimal. 798 vs. \(2 (282) = 564\).
Time = 0.14 (sec) , antiderivative size = 798, normalized size of antiderivative = 2.37 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:
integrate((d*x^4+c)^5/(b*x^4+a)^2,x, algorithm="giac")
Output:
1/16*sqrt(2)*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*( a*b^3)^(1/4)*a^2*b^3*c^3*d^2 + 90*(a*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^ 3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*arctan(1/2*sqrt(2)*(2*x + sqrt(2)*(a/b)^(1/4))/(a/b)^(1/4))/(a^2*b^6) + 1/16*sqrt(2)*(3*(a*b^3)^(1/ 4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c^3*d^ 2 + 90*(a*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*( a*b^3)^(1/4)*a^5*d^5)*arctan(1/2*sqrt(2)*(2*x - sqrt(2)*(a/b)^(1/4))/(a/b) ^(1/4))/(a^2*b^6) + 1/32*sqrt(2)*(3*(a*b^3)^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4 )*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c^3*d^2 + 90*(a*b^3)^(1/4)*a^3*b^ 2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*log(x ^2 + sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a^2*b^6) - 1/32*sqrt(2)*(3*(a*b^3 )^(1/4)*b^5*c^5 + 5*(a*b^3)^(1/4)*a*b^4*c^4*d - 50*(a*b^3)^(1/4)*a^2*b^3*c ^3*d^2 + 90*(a*b^3)^(1/4)*a^3*b^2*c^2*d^3 - 65*(a*b^3)^(1/4)*a^4*b*c*d^4 + 17*(a*b^3)^(1/4)*a^5*d^5)*log(x^2 - sqrt(2)*x*(a/b)^(1/4) + sqrt(a/b))/(a ^2*b^6) + 1/4*(b^5*c^5*x - 5*a*b^4*c^4*d*x + 10*a^2*b^3*c^3*d^2*x - 10*a^3 *b^2*c^2*d^3*x + 5*a^4*b*c*d^4*x - a^5*d^5*x)/((b*x^4 + a)*a*b^5) + 1/585* (45*b^24*d^5*x^13 + 325*b^24*c*d^4*x^9 - 130*a*b^23*d^5*x^9 + 1170*b^24*c^ 2*d^3*x^5 - 1170*a*b^23*c*d^4*x^5 + 351*a^2*b^22*d^5*x^5 + 5850*b^24*c^3*d ^2*x - 11700*a*b^23*c^2*d^3*x + 8775*a^2*b^22*c*d^4*x - 2340*a^3*b^21*d^5* x)/b^26
Time = 1.00 (sec) , antiderivative size = 2490, normalized size of antiderivative = 7.39 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx=\text {Too large to display} \] Input:
int((c + d*x^4)^5/(a + b*x^4)^2,x)
Output:
x*((10*c^3*d^2)/b^2 - (2*a*((2*a*((2*a*d^5)/b^3 - (5*c*d^4)/b^2))/b - (a^2 *d^5)/b^4 + (10*c^2*d^3)/b^2))/b + (a^2*((2*a*d^5)/b^3 - (5*c*d^4)/b^2))/b ^2) - x^9*((2*a*d^5)/(9*b^3) - (5*c*d^4)/(9*b^2)) + x^5*((2*a*((2*a*d^5)/b ^3 - (5*c*d^4)/b^2))/(5*b) - (a^2*d^5)/(5*b^4) + (2*c^2*d^3)/b^2) + (d^5*x ^13)/(13*b^2) - (x*(a^5*d^5 - b^5*c^5 - 10*a^2*b^3*c^3*d^2 + 10*a^3*b^2*c^ 2*d^3 + 5*a*b^4*c^4*d - 5*a^4*b*c*d^4))/(4*a*(a*b^5 + b^6*x^4)) + (atan((( ((x*(289*a^10*d^10 + 9*b^10*c^10 - 275*a^2*b^8*c^8*d^2 + 40*a^3*b^7*c^7*d^ 3 + 3010*a^4*b^6*c^6*d^4 - 9548*a^5*b^5*c^5*d^5 + 14770*a^6*b^4*c^4*d^6 - 13400*a^7*b^3*c^3*d^7 + 7285*a^8*b^2*c^2*d^8 + 30*a*b^9*c^9*d - 2210*a^9*b *c*d^9))/(4*a^2*b^7) - ((a*d - b*c)^4*(17*a*d + 3*b*c)*(17*a^5*d^5 + 3*b^5 *c^5 - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 + 5*a*b^4*c^4*d - 65*a^4*b* c*d^4))/(4*(-a)^(7/4)*b^(29/4)))*(a*d - b*c)^4*(17*a*d + 3*b*c)*1i)/(16*(- a)^(7/4)*b^(21/4)) + (((x*(289*a^10*d^10 + 9*b^10*c^10 - 275*a^2*b^8*c^8*d ^2 + 40*a^3*b^7*c^7*d^3 + 3010*a^4*b^6*c^6*d^4 - 9548*a^5*b^5*c^5*d^5 + 14 770*a^6*b^4*c^4*d^6 - 13400*a^7*b^3*c^3*d^7 + 7285*a^8*b^2*c^2*d^8 + 30*a* b^9*c^9*d - 2210*a^9*b*c*d^9))/(4*a^2*b^7) + ((a*d - b*c)^4*(17*a*d + 3*b* c)*(17*a^5*d^5 + 3*b^5*c^5 - 50*a^2*b^3*c^3*d^2 + 90*a^3*b^2*c^2*d^3 + 5*a *b^4*c^4*d - 65*a^4*b*c*d^4))/(4*(-a)^(7/4)*b^(29/4)))*(a*d - b*c)^4*(17*a *d + 3*b*c)*1i)/(16*(-a)^(7/4)*b^(21/4)))/((((x*(289*a^10*d^10 + 9*b^10*c^ 10 - 275*a^2*b^8*c^8*d^2 + 40*a^3*b^7*c^7*d^3 + 3010*a^4*b^6*c^6*d^4 - ...
Time = 0.26 (sec) , antiderivative size = 2146, normalized size of antiderivative = 6.37 \[ \int \frac {\left (c+d x^4\right )^5}{\left (a+b x^4\right )^2} \, dx =\text {Too large to display} \] Input:
int((d*x^4+c)^5/(b*x^4+a)^2,x)
Output:
( - 19890*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) - 2*sq rt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**6*d**5 + 76050*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**5*b*c*d**4 - 19890*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**5*b*d**5*x* *4 - 105300*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**4*b**2*c**2*d**3 + 76050*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**4*b**2*c*d**4*x**4 + 58500*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**3*b**3*c**3*d**2 - 105300*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**3*b**3*c **2*d**3*x**4 - 5850*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqr t(2) - 2*sqrt(b)*x)/(b**(1/4)*a**(1/4)*sqrt(2)))*a**2*b**4*c**4*d + 58500* 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*b**4*c**3*d**2*x**4 - 3510*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**5*c**5 - 5850*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**5*c**4*d *x**4 - 3510*b**(3/4)*a**(1/4)*sqrt(2)*atan((b**(1/4)*a**(1/4)*sqrt(2) ...