Integrand size = 28, antiderivative size = 193 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (445536 a b^6 d+21216 a^2 b^5 d x-24960 a^3 b^4 d x^2+30720 a^4 b^3 d x^3-40960 a^5 b^2 d x^4+65536 a^6 b d x^5-262144 a^7 d x^6+348075 b^7 c x^7+1392300 a b^6 c x^8\right )}{2784600 a b^6 x^7}+\frac {3 b^2 c \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{7/4}}-\frac {3 b^2 c \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{7/4}} \]
1/2784600*(a*x^4+b*x^3)^(1/4)*(1392300*a*b^6*c*x^8+348075*b^7*c*x^7-262144 *a^7*d*x^6+65536*a^6*b*d*x^5-40960*a^5*b^2*d*x^4+30720*a^4*b^3*d*x^3-24960 *a^3*b^4*d*x^2+21216*a^2*b^5*d*x+445536*a*b^6*d)/a/b^6/x^7+3/16*b^2*c*arct an(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(7/4)-3/16*b^2*c*arctanh(a^(1/4)*x/(a* x^4+b*x^3)^(1/4))/a^(7/4)
Time = 0.41 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.08 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\frac {\sqrt [4]{x^3 (b+a x)} \left (2 a^{3/4} \sqrt [4]{b+a x} \left (21216 a^2 b^5 d x-24960 a^3 b^4 d x^2+30720 a^4 b^3 d x^3-40960 a^5 b^2 d x^4+65536 a^6 b d x^5-262144 a^7 d x^6+348075 b^7 c x^7+55692 a b^6 \left (8 d+25 c x^8\right )\right )-1044225 b^8 c x^{25/4} \arctan \left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )-1044225 b^8 c x^{25/4} \text {arctanh}\left (\frac {\sqrt [4]{b+a x}}{\sqrt [4]{a} \sqrt [4]{x}}\right )\right )}{5569200 a^{7/4} b^6 x^7 \sqrt [4]{b+a x}} \]
((x^3*(b + a*x))^(1/4)*(2*a^(3/4)*(b + a*x)^(1/4)*(21216*a^2*b^5*d*x - 249 60*a^3*b^4*d*x^2 + 30720*a^4*b^3*d*x^3 - 40960*a^5*b^2*d*x^4 + 65536*a^6*b *d*x^5 - 262144*a^7*d*x^6 + 348075*b^7*c*x^7 + 55692*a*b^6*(8*d + 25*c*x^8 )) - 1044225*b^8*c*x^(25/4)*ArcTan[(b + a*x)^(1/4)/(a^(1/4)*x^(1/4))] - 10 44225*b^8*c*x^(25/4)*ArcTanh[(b + a*x)^(1/4)/(a^(1/4)*x^(1/4))]))/(5569200 *a^(7/4)*b^6*x^7*(b + a*x)^(1/4))
Time = 0.60 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.77, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.071, Rules used = {2449, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4+b x^3} \left (c x^8-d\right )}{x^8} \, dx\) |
| \(\Big \downarrow \) 2449 |
| \(\displaystyle \int \left (c \sqrt [4]{a x^4+b x^3}-\frac {d \sqrt [4]{a x^4+b x^3}}{x^8}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {3 b^2 c x^{9/4} (a x+b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {3 b^2 c x^{9/4} (a x+b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{16 a^{7/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {32768 a^5 d \left (a x^4+b x^3\right )^{5/4}}{348075 b^6 x^5}+\frac {8192 a^4 d \left (a x^4+b x^3\right )^{5/4}}{69615 b^5 x^6}-\frac {1024 a^3 d \left (a x^4+b x^3\right )^{5/4}}{7735 b^4 x^7}+\frac {256 a^2 d \left (a x^4+b x^3\right )^{5/4}}{1785 b^3 x^8}-\frac {16 a d \left (a x^4+b x^3\right )^{5/4}}{105 b^2 x^9}+\frac {1}{2} c x \sqrt [4]{a x^4+b x^3}+\frac {b c \sqrt [4]{a x^4+b x^3}}{8 a}+\frac {4 d \left (a x^4+b x^3\right )^{5/4}}{25 b x^{10}}\) |
(b*c*(b*x^3 + a*x^4)^(1/4))/(8*a) + (c*x*(b*x^3 + a*x^4)^(1/4))/2 + (4*d*( b*x^3 + a*x^4)^(5/4))/(25*b*x^10) - (16*a*d*(b*x^3 + a*x^4)^(5/4))/(105*b^ 2*x^9) + (256*a^2*d*(b*x^3 + a*x^4)^(5/4))/(1785*b^3*x^8) - (1024*a^3*d*(b *x^3 + a*x^4)^(5/4))/(7735*b^4*x^7) + (8192*a^4*d*(b*x^3 + a*x^4)^(5/4))/( 69615*b^5*x^6) - (32768*a^5*d*(b*x^3 + a*x^4)^(5/4))/(348075*b^6*x^5) + (3 *b^2*c*x^(9/4)*(b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/ (16*a^(7/4)*(b*x^3 + a*x^4)^(3/4)) - (3*b^2*c*x^(9/4)*(b + a*x)^(3/4)*ArcT anh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(16*a^(7/4)*(b*x^3 + a*x^4)^(3/4))
3.25.5.3.1 Defintions of rubi rules used
Int[(Pq_)*((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_))^(p_), x_S ymbol] :> Int[ExpandIntegrand[(c*x)^m*Pq*(a*x^j + b*x^n)^p, x], x] /; FreeQ [{a, b, c, j, m, n, p}, x] && (PolyQ[Pq, x] || PolyQ[Pq, x^n]) && !Integer Q[p] && NeQ[n, j]
Time = 0.57 (sec) , antiderivative size = 195, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {3 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{8} c \,x^{7}}{16}-\frac {3 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{8} c \,x^{7}}{8}+\left (b^{6} \left (c \,x^{8}+\frac {8 d}{25}\right ) a^{\frac {7}{4}}+\frac {8 \left (a^{\frac {11}{4}} b^{5} d -\frac {20 x \left (a^{\frac {15}{4}} b^{4} d -\frac {16 x \left (a^{\frac {19}{4}} b^{3} d -\frac {4 x \left (a^{\frac {23}{4}} b^{2} d -\frac {8 x \left (\frac {348075 b^{7} c \,x^{2} a^{\frac {3}{4}}}{65536}+a^{\frac {27}{4}} b d -4 a^{\frac {31}{4}} d x \right )}{5}\right )}{3}\right )}{13}\right )}{17}\right ) x}{525}\right ) \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{2 a^{\frac {7}{4}} x^{7} b^{6}}\) | \(195\) |
1/2*(-3/16*ln((a^(1/4)*x+(x^3*(a*x+b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x+b))^(1 /4)))*b^8*c*x^7-3/8*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))*b^8*c*x^7+(b^6 *(c*x^8+8/25*d)*a^(7/4)+8/525*(a^(11/4)*b^5*d-20/17*x*(a^(15/4)*b^4*d-16/1 3*x*(a^(19/4)*b^3*d-4/3*x*(a^(23/4)*b^2*d-8/5*x*(348075/65536*b^7*c*x^2*a^ (3/4)+a^(27/4)*b*d-4*a^(31/4)*d*x)))))*x)*(x^3*(a*x+b))^(1/4))/a^(7/4)/x^7 /b^6
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 374, normalized size of antiderivative = 1.94 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=-\frac {1044225 \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a b^{6} x^{7} \log \left (\frac {3 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2} c + \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a^{2} x\right )}}{x}\right ) + 1044225 i \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a b^{6} x^{7} \log \left (\frac {3 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2} c + i \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a^{2} x\right )}}{x}\right ) - 1044225 i \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a b^{6} x^{7} \log \left (\frac {3 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2} c - i \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a^{2} x\right )}}{x}\right ) - 1044225 \, \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a b^{6} x^{7} \log \left (\frac {3 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{2} c - \left (\frac {b^{8} c^{4}}{a^{7}}\right )^{\frac {1}{4}} a^{2} x\right )}}{x}\right ) - 4 \, {\left (1392300 \, a b^{6} c x^{8} + 348075 \, b^{7} c x^{7} - 262144 \, a^{7} d x^{6} + 65536 \, a^{6} b d x^{5} - 40960 \, a^{5} b^{2} d x^{4} + 30720 \, a^{4} b^{3} d x^{3} - 24960 \, a^{3} b^{4} d x^{2} + 21216 \, a^{2} b^{5} d x + 445536 \, a b^{6} d\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{11138400 \, a b^{6} x^{7}} \]
-1/11138400*(1044225*(b^8*c^4/a^7)^(1/4)*a*b^6*x^7*log(3*((a*x^4 + b*x^3)^ (1/4)*b^2*c + (b^8*c^4/a^7)^(1/4)*a^2*x)/x) + 1044225*I*(b^8*c^4/a^7)^(1/4 )*a*b^6*x^7*log(3*((a*x^4 + b*x^3)^(1/4)*b^2*c + I*(b^8*c^4/a^7)^(1/4)*a^2 *x)/x) - 1044225*I*(b^8*c^4/a^7)^(1/4)*a*b^6*x^7*log(3*((a*x^4 + b*x^3)^(1 /4)*b^2*c - I*(b^8*c^4/a^7)^(1/4)*a^2*x)/x) - 1044225*(b^8*c^4/a^7)^(1/4)* a*b^6*x^7*log(3*((a*x^4 + b*x^3)^(1/4)*b^2*c - (b^8*c^4/a^7)^(1/4)*a^2*x)/ x) - 4*(1392300*a*b^6*c*x^8 + 348075*b^7*c*x^7 - 262144*a^7*d*x^6 + 65536* a^6*b*d*x^5 - 40960*a^5*b^2*d*x^4 + 30720*a^4*b^3*d*x^3 - 24960*a^3*b^4*d* x^2 + 21216*a^2*b^5*d*x + 445536*a*b^6*d)*(a*x^4 + b*x^3)^(1/4))/(a*b^6*x^ 7)
\[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (c x^{8} - d\right )}{x^{8}}\, dx \]
\[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\int { \frac {{\left (c x^{8} - d\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x^{8}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 358 vs. \(2 (173) = 346\).
Time = 0.32 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.85 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\frac {\frac {2088450 \, \sqrt {2} b^{3} c \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {2088450 \, \sqrt {2} b^{3} c \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {1044225 \, \sqrt {2} b^{3} c \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a} + \frac {1044225 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} c \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{2}} + \frac {2784600 \, {\left ({\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{3} c + 3 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{3} c\right )} x^{2}}{a b^{2}} + \frac {256 \, {\left (13923 \, {\left (a + \frac {b}{x}\right )}^{\frac {25}{4}} b^{120} d - 82875 \, {\left (a + \frac {b}{x}\right )}^{\frac {21}{4}} a b^{120} d + 204750 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} a^{2} b^{120} d - 267750 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a^{3} b^{120} d + 193375 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{4} b^{120} d - 69615 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{5} b^{120} d\right )}}{b^{125}}}{22276800 \, b} \]
1/22276800*(2088450*sqrt(2)*b^3*c*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a) + 2088450*sqrt(2)*b^3*c*arc tan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/((-a )^(3/4)*a) + 1044225*sqrt(2)*b^3*c*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a) + 1044225*sqrt(2)*(-a)^(1/4)*b^ 3*c*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/a^ 2 + 2784600*((a + b/x)^(5/4)*b^3*c + 3*(a + b/x)^(1/4)*a*b^3*c)*x^2/(a*b^2 ) + 256*(13923*(a + b/x)^(25/4)*b^120*d - 82875*(a + b/x)^(21/4)*a*b^120*d + 204750*(a + b/x)^(17/4)*a^2*b^120*d - 267750*(a + b/x)^(13/4)*a^3*b^120 *d + 193375*(a + b/x)^(9/4)*a^4*b^120*d - 69615*(a + b/x)^(5/4)*a^5*b^120* d)/b^125)/b
Time = 9.34 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.07 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-d+c x^8\right )}{x^8} \, dx=\frac {4\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{25\,x^7}-\frac {16\,a^2\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{1785\,b^2\,x^5}+\frac {256\,a^3\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{23205\,b^3\,x^4}-\frac {1024\,a^4\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{69615\,b^4\,x^3}+\frac {8192\,a^5\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{348075\,b^5\,x^2}-\frac {32768\,a^6\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{348075\,b^6\,x}+\frac {4\,c\,x\,{\left (a\,x^4+b\,x^3\right )}^{1/4}\,{{}}_2{\mathrm {F}}_1\left (-\frac {1}{4},\frac {7}{4};\ \frac {11}{4};\ -\frac {a\,x}{b}\right )}{7\,{\left (\frac {a\,x}{b}+1\right )}^{1/4}}+\frac {4\,a\,d\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{525\,b\,x^6} \]
(4*d*(a*x^4 + b*x^3)^(1/4))/(25*x^7) - (16*a^2*d*(a*x^4 + b*x^3)^(1/4))/(1 785*b^2*x^5) + (256*a^3*d*(a*x^4 + b*x^3)^(1/4))/(23205*b^3*x^4) - (1024*a ^4*d*(a*x^4 + b*x^3)^(1/4))/(69615*b^4*x^3) + (8192*a^5*d*(a*x^4 + b*x^3)^ (1/4))/(348075*b^5*x^2) - (32768*a^6*d*(a*x^4 + b*x^3)^(1/4))/(348075*b^6* x) + (4*c*x*(a*x^4 + b*x^3)^(1/4)*hypergeom([-1/4, 7/4], 11/4, -(a*x)/b))/ (7*((a*x)/b + 1)^(1/4)) + (4*a*d*(a*x^4 + b*x^3)^(1/4))/(525*b*x^6)