Integrand size = 28, antiderivative size = 195 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\frac {\sqrt [4]{b x^3+a x^4} \left (327680 a^5 b^2 c+65536 a^6 b c x-262144 a^7 c x^2+21945 b^7 d x^3-12540 a b^6 d x^4+9120 a^2 b^5 d x^5-7296 a^3 b^4 d x^6+6144 a^4 b^3 d x^7+122880 a^5 b^2 d x^8\right )}{737280 a^5 b^2 x^3}+\frac {1463 b^6 d \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{32768 a^{23/4}}-\frac {1463 b^6 d \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{32768 a^{23/4}} \]
1/737280*(a*x^4+b*x^3)^(1/4)*(122880*a^5*b^2*d*x^8+6144*a^4*b^3*d*x^7-7296 *a^3*b^4*d*x^6+9120*a^2*b^5*d*x^5-12540*a*b^6*d*x^4+21945*b^7*d*x^3-262144 *a^7*c*x^2+65536*a^6*b*c*x+327680*a^5*b^2*c)/a^5/b^2/x^3+1463/32768*b^6*d* arctan(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)-1463/32768*b^6*d*arctanh(a^ (1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(23/4)
Time = 0.98 (sec) , antiderivative size = 206, normalized size of antiderivative = 1.06 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\frac {(b+a x)^{3/4} \left (2 a^{3/4} \sqrt [4]{b+a x} \left (65536 a^6 b c x-262144 a^7 c x^2+21945 b^7 d x^3-12540 a b^6 d x^4+9120 a^2 b^5 d x^5-7296 a^3 b^4 d x^6+6144 a^4 b^3 d x^7+40960 a^5 b^2 \left (8 c+3 d x^8\right )\right )+65835 b^8 d x^{9/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-65835 b^8 d x^{9/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{1474560 a^{23/4} b^2 \left (x^3 (b+a x)\right )^{3/4}} \]
((b + a*x)^(3/4)*(2*a^(3/4)*(b + a*x)^(1/4)*(65536*a^6*b*c*x - 262144*a^7* c*x^2 + 21945*b^7*d*x^3 - 12540*a*b^6*d*x^4 + 9120*a^2*b^5*d*x^5 - 7296*a^ 3*b^4*d*x^6 + 6144*a^4*b^3*d*x^7 + 40960*a^5*b^2*(8*c + 3*d*x^8)) + 65835* b^8*d*x^(9/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] - 65835*b^8*d*x^(9 /4)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]))/(1474560*a^(23/4)*b^2*(x^ 3*(b + a*x))^(3/4))
Time = 0.61 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.75, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4+b x^3} \left (d x^8-c\right )}{x^4} \, dx\) |
| \(\Big \downarrow \) 2449 |
| \(\displaystyle \int \left (d x^4 \sqrt [4]{a x^4+b x^3}-\frac {c \sqrt [4]{a x^4+b x^3}}{x^4}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {1463 b^6 d x^{9/4} (a x+b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{32768 a^{23/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {1463 b^6 d x^{9/4} (a x+b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{32768 a^{23/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {1463 b^5 d \sqrt [4]{a x^4+b x^3}}{49152 a^5}-\frac {209 b^4 d x \sqrt [4]{a x^4+b x^3}}{12288 a^4}+\frac {19 b^3 d x^2 \sqrt [4]{a x^4+b x^3}}{1536 a^3}-\frac {19 b^2 d x^3 \sqrt [4]{a x^4+b x^3}}{1920 a^2}-\frac {16 a c \left (a x^4+b x^3\right )^{5/4}}{45 b^2 x^5}+\frac {4 c \left (a x^4+b x^3\right )^{5/4}}{9 b x^6}+\frac {b d x^4 \sqrt [4]{a x^4+b x^3}}{120 a}+\frac {1}{6} d x^5 \sqrt [4]{a x^4+b x^3}\) |
(1463*b^5*d*(b*x^3 + a*x^4)^(1/4))/(49152*a^5) - (209*b^4*d*x*(b*x^3 + a*x ^4)^(1/4))/(12288*a^4) + (19*b^3*d*x^2*(b*x^3 + a*x^4)^(1/4))/(1536*a^3) - (19*b^2*d*x^3*(b*x^3 + a*x^4)^(1/4))/(1920*a^2) + (b*d*x^4*(b*x^3 + a*x^4 )^(1/4))/(120*a) + (d*x^5*(b*x^3 + a*x^4)^(1/4))/6 + (4*c*(b*x^3 + a*x^4)^ (5/4))/(9*b*x^6) - (16*a*c*(b*x^3 + a*x^4)^(5/4))/(45*b^2*x^5) + (1463*b^6 *d*x^(9/4)*(b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(327 68*a^(23/4)*(b*x^3 + a*x^4)^(3/4)) - (1463*b^6*d*x^(9/4)*(b + a*x)^(3/4)*A rcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(32768*a^(23/4)*(b*x^3 + a*x^4) ^(3/4))
3.25.17.3.1 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 = 0.49 (sec) , antiderivative size = 191, normalized size of antiderivative = 0.98
| method | result | size |
| pseudoelliptic | \(\frac {-\frac {1463 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{8} d \,x^{3}}{65536}-\frac {1463 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{8} d \,x^{3}}{32768}+\frac {19 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} \left (\frac {2048 b^{2} \left (\frac {3 d \,x^{8}}{8}+c \right ) a^{\frac {23}{4}}}{57}+x \left (a^{\frac {11}{4}} b^{5} d \,x^{4}-\frac {4 a^{\frac {15}{4}} b^{4} d \,x^{5}}{5}+\frac {64 a^{\frac {19}{4}} b^{3} d \,x^{6}}{95}+\frac {2048 a^{\frac {27}{4}} b c}{285}-\frac {8192 a^{\frac {31}{4}} c x}{285}+\frac {77 b^{6} x^{2} d \left (a^{\frac {3}{4}} b -\frac {4 a^{\frac {7}{4}} x}{7}\right )}{32}\right )\right )}{1536}}{x^{3} a^{\frac {23}{4}} b^{2}}\) | \(191\) |
19/1536/a^(23/4)*(-231/128*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*d*x^3-231/64*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1 /4))*b^8*d*x^3+(x^3*(a*x+b))^(1/4)*(2048/57*b^2*(3/8*d*x^8+c)*a^(23/4)+x*( a^(11/4)*b^5*d*x^4-4/5*a^(15/4)*b^4*d*x^5+64/95*a^(19/4)*b^3*d*x^6+2048/28 5*a^(27/4)*b*c-8192/285*a^(31/4)*c*x+77/32*b^6*x^2*d*(a^(3/4)*b-4/7*a^(7/4 )*x))))/x^3/b^2
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 384, normalized size of antiderivative = 1.97 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=-\frac {65835 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d + \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) + 65835 i \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d + i \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) - 65835 i \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d - i \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) - 65835 \, \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{5} b^{2} x^{3} \log \left (\frac {1463 \, {\left ({\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{6} d - \left (\frac {b^{24} d^{4}}{a^{23}}\right )^{\frac {1}{4}} a^{6} x\right )}}{x}\right ) - 4 \, {\left (122880 \, a^{5} b^{2} d x^{8} + 6144 \, a^{4} b^{3} d x^{7} - 7296 \, a^{3} b^{4} d x^{6} + 9120 \, a^{2} b^{5} d x^{5} - 12540 \, a b^{6} d x^{4} + 21945 \, b^{7} d x^{3} - 262144 \, a^{7} c x^{2} + 65536 \, a^{6} b c x + 327680 \, a^{5} b^{2} c\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{2949120 \, a^{5} b^{2} x^{3}} \]
-1/2949120*(65835*(b^24*d^4/a^23)^(1/4)*a^5*b^2*x^3*log(1463*((a*x^4 + b*x ^3)^(1/4)*b^6*d + (b^24*d^4/a^23)^(1/4)*a^6*x)/x) + 65835*I*(b^24*d^4/a^23 )^(1/4)*a^5*b^2*x^3*log(1463*((a*x^4 + b*x^3)^(1/4)*b^6*d + I*(b^24*d^4/a^ 23)^(1/4)*a^6*x)/x) - 65835*I*(b^24*d^4/a^23)^(1/4)*a^5*b^2*x^3*log(1463*( (a*x^4 + b*x^3)^(1/4)*b^6*d - I*(b^24*d^4/a^23)^(1/4)*a^6*x)/x) - 65835*(b ^24*d^4/a^23)^(1/4)*a^5*b^2*x^3*log(1463*((a*x^4 + b*x^3)^(1/4)*b^6*d - (b ^24*d^4/a^23)^(1/4)*a^6*x)/x) - 4*(122880*a^5*b^2*d*x^8 + 6144*a^4*b^3*d*x ^7 - 7296*a^3*b^4*d*x^6 + 9120*a^2*b^5*d*x^5 - 12540*a*b^6*d*x^4 + 21945*b ^7*d*x^3 - 262144*a^7*c*x^2 + 65536*a^6*b*c*x + 327680*a^5*b^2*c)*(a*x^4 + b*x^3)^(1/4))/(a^5*b^2*x^3)
\[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x + b\right )} \left (- c + d x^{8}\right )}{x^{4}}\, dx \]
\[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\int { \frac {{\left (d x^{8} - c\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{x^{4}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 359 vs. \(2 (175) = 350\).
Time = 0.36 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.84 \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\frac {\frac {131670 \, \sqrt {2} b^{7} d \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {131670 \, \sqrt {2} b^{7} d \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {65835 \, \sqrt {2} b^{7} d \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{5}} + \frac {65835 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{7} d \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{6}} + \frac {24 \, {\left (7315 \, {\left (a + \frac {b}{x}\right )}^{\frac {21}{4}} b^{7} d - 40755 \, {\left (a + \frac {b}{x}\right )}^{\frac {17}{4}} a b^{7} d + 92910 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} a^{2} b^{7} d - 109782 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a^{3} b^{7} d + 69327 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{4} b^{7} d + 21945 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{5} b^{7} d\right )} x^{6}}{a^{5} b^{6}} + \frac {524288 \, {\left (5 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{8} c - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{8} c\right )}}{b^{9}}}{5898240 \, b} \]
1/5898240*(131670*sqrt(2)*b^7*d*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2 *(a + b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^5) + 131670*sqrt(2)*b^7*d*arct an(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/((-a) ^(3/4)*a^5) + 65835*sqrt(2)*b^7*d*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^5) + 65835*sqrt(2)*(-a)^(1/4)*b^7 *d*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/a^6 + 24*(7315*(a + b/x)^(21/4)*b^7*d - 40755*(a + b/x)^(17/4)*a*b^7*d + 9291 0*(a + b/x)^(13/4)*a^2*b^7*d - 109782*(a + b/x)^(9/4)*a^3*b^7*d + 69327*(a + b/x)^(5/4)*a^4*b^7*d + 21945*(a + b/x)^(1/4)*a^5*b^7*d)*x^6/(a^5*b^6) + 524288*(5*(a + b/x)^(9/4)*b^8*c - 9*(a + b/x)^(5/4)*a*b^8*c)/b^9)/b
Timed out. \[ \int \frac {\sqrt [4]{b x^3+a x^4} \left (-c+d x^8\right )}{x^4} \, dx=\int -\frac {\left (c-d\,x^8\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^4} \,d x \]