Integrand size = 29, antiderivative size = 162 \[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=\frac {\sqrt [4]{-b x^3+a x^4} \left (6144 a^3 d-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 c x^4\right )}{1536 a^3 x}+\frac {\left (77 b^4 c+2048 a^4 d\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{1024 a^{15/4}}+\frac {\left (-77 b^4 c-2048 a^4 d\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{-b x^3+a x^4}}\right )}{1024 a^{15/4}} \]
1/1536*(a*x^4-b*x^3)^(1/4)*(384*a^3*c*x^4-32*a^2*b*c*x^3-44*a*b^2*c*x^2-77 *b^3*c*x+6144*a^3*d)/a^3/x+1/1024*(2048*a^4*d+77*b^4*c)*arctan(a^(1/4)*x/( a*x^4-b*x^3)^(1/4))/a^(15/4)+1/1024*(-2048*a^4*d-77*b^4*c)*arctanh(a^(1/4) *x/(a*x^4-b*x^3)^(1/4))/a^(15/4)
Time = 0.72 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.14 \[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=\frac {x^2 (-b+a x)^{3/4} \left (2 a^{3/4} \sqrt [4]{-b+a x} \left (-77 b^3 c x-44 a b^2 c x^2-32 a^2 b c x^3+384 a^3 \left (16 d+c x^4\right )\right )+3 \left (77 b^4 c+2048 a^4 d\right ) \sqrt [4]{x} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )-3 \left (77 b^4 c+2048 a^4 d\right ) \sqrt [4]{x} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{-b+a x}}\right )\right )}{3072 a^{15/4} \left (x^3 (-b+a x)\right )^{3/4}} \]
(x^2*(-b + a*x)^(3/4)*(2*a^(3/4)*(-b + a*x)^(1/4)*(-77*b^3*c*x - 44*a*b^2* c*x^2 - 32*a^2*b*c*x^3 + 384*a^3*(16*d + c*x^4)) + 3*(77*b^4*c + 2048*a^4* d)*x^(1/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)] - 3*(77*b^4*c + 2048 *a^4*d)*x^(1/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)]))/(3072*a^(15/ 4)*(x^3*(-b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(392\) vs. \(2(162)=324\).
Time = 0.64 (sec) , antiderivative size = 392, normalized size of antiderivative = 2.42, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, 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.
| \(\displaystyle \int \frac {\sqrt [4]{a x^4-b x^3} \left (c x^4-d\right )}{x^2} \, dx\) |
| \(\Big \downarrow \) 2449 |
| \(\displaystyle \int \left (c x^2 \sqrt [4]{a x^4-b x^3}-\frac {d \sqrt [4]{a x^4-b x^3}}{x^2}\right )dx\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle \frac {77 b^4 c x^{9/4} (a x-b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{1024 a^{15/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {77 b^4 c x^{9/4} (a x-b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{1024 a^{15/4} \left (a x^4-b x^3\right )^{3/4}}-\frac {77 b^3 c \sqrt [4]{a x^4-b x^3}}{1536 a^3}-\frac {11 b^2 c x \sqrt [4]{a x^4-b x^3}}{384 a^2}+\frac {2 \sqrt [4]{a} d x^{9/4} (a x-b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{\left (a x^4-b x^3\right )^{3/4}}-\frac {2 \sqrt [4]{a} d x^{9/4} (a x-b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x-b}}\right )}{\left (a x^4-b x^3\right )^{3/4}}+\frac {1}{4} c x^3 \sqrt [4]{a x^4-b x^3}-\frac {b c x^2 \sqrt [4]{a x^4-b x^3}}{48 a}+\frac {4 d \sqrt [4]{a x^4-b x^3}}{x}\) |
(-77*b^3*c*(-(b*x^3) + a*x^4)^(1/4))/(1536*a^3) + (4*d*(-(b*x^3) + a*x^4)^ (1/4))/x - (11*b^2*c*x*(-(b*x^3) + a*x^4)^(1/4))/(384*a^2) - (b*c*x^2*(-(b *x^3) + a*x^4)^(1/4))/(48*a) + (c*x^3*(-(b*x^3) + a*x^4)^(1/4))/4 + (77*b^ 4*c*x^(9/4)*(-b + a*x)^(3/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/( 1024*a^(15/4)*(-(b*x^3) + a*x^4)^(3/4)) + (2*a^(1/4)*d*x^(9/4)*(-b + a*x)^ (3/4)*ArcTan[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(-(b*x^3) + a*x^4)^(3/4) - (77*b^4*c*x^(9/4)*(-b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x) ^(1/4)])/(1024*a^(15/4)*(-(b*x^3) + a*x^4)^(3/4)) - (2*a^(1/4)*d*x^(9/4)*( -b + a*x)^(3/4)*ArcTanh[(a^(1/4)*x^(1/4))/(-b + a*x)^(1/4)])/(-(b*x^3) + a *x^4)^(3/4)
3.22.81.3.1 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 = 0.34 (sec) , antiderivative size = 164, normalized size of antiderivative = 1.01
| method | result | size |
| pseudoelliptic | \(-\frac {2 \left (\frac {x \left (a^{4} d +\frac {77 b^{4} c}{2048}\right ) \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}\right )}{2}+x \left (a^{4} d +\frac {77 b^{4} c}{2048}\right ) \arctan \left (\frac {\left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right )+\frac {\left (\left (-12 c \,x^{4}-192 d \right ) a^{\frac {15}{4}}+c x b \left (a^{\frac {11}{4}} x^{2}+\frac {77 b^{2} a^{\frac {3}{4}}}{32}+\frac {11 a^{\frac {7}{4}} b x}{8}\right )\right ) \left (x^{3} \left (a x -b \right )\right )^{\frac {1}{4}}}{96}\right )}{a^{\frac {15}{4}} x}\) | \(164\) |
-2*(1/2*x*(a^4*d+77/2048*b^4*c)*ln((-a^(1/4)*x-(x^3*(a*x-b))^(1/4))/(a^(1/ 4)*x-(x^3*(a*x-b))^(1/4)))+x*(a^4*d+77/2048*b^4*c)*arctan(1/a^(1/4)/x*(x^3 *(a*x-b))^(1/4))+1/96*((-12*c*x^4-192*d)*a^(15/4)+c*x*b*(a^(11/4)*x^2+77/3 2*b^2*a^(3/4)+11/8*a^(7/4)*b*x))*(x^3*(a*x-b))^(1/4))/a^(15/4)/x
Result contains complex when optimal does not.
Time = 0.25 (sec) , antiderivative size = 740, normalized size of antiderivative = 4.57 \[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=-\frac {3 \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} + {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 3 \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} - {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) + 3 i \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {i \, a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} + {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 3 i \, a^{3} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {-i \, a^{4} x \left (\frac {35153041 \, b^{16} c^{4} + 3739918336 \, a^{4} b^{12} c^{3} d + 149208170496 \, a^{8} b^{8} c^{2} d^{2} + 2645699854336 \, a^{12} b^{4} c d^{3} + 17592186044416 \, a^{16} d^{4}}{a^{15}}\right )^{\frac {1}{4}} + {\left (77 \, b^{4} c + 2048 \, a^{4} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{x}\right ) - 4 \, {\left (384 \, a^{3} c x^{4} - 32 \, a^{2} b c x^{3} - 44 \, a b^{2} c x^{2} - 77 \, b^{3} c x + 6144 \, a^{3} d\right )} {\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}}}{6144 \, a^{3} x} \]
-1/6144*(3*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 1492081 70496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16 *d^4)/a^15)^(1/4)*log((a^4*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3 *d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 1759218 6044416*a^16*d^4)/a^15)^(1/4) + (77*b^4*c + 2048*a^4*d)*(a*x^4 - b*x^3)^(1 /4))/x) - 3*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208 170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^1 6*d^4)/a^15)^(1/4)*log(-(a^4*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c ^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592 186044416*a^16*d^4)/a^15)^(1/4) - (77*b^4*c + 2048*a^4*d)*(a*x^4 - b*x^3)^ (1/4))/x) + 3*I*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 14 9208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416 *a^16*d^4)/a^15)^(1/4)*log((I*a^4*x*((35153041*b^16*c^4 + 3739918336*a^4*b ^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 17592186044416*a^16*d^4)/a^15)^(1/4) + (77*b^4*c + 2048*a^4*d)*(a*x^4 - b* x^3)^(1/4))/x) - 3*I*a^3*x*((35153041*b^16*c^4 + 3739918336*a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c*d^3 + 175921860 44416*a^16*d^4)/a^15)^(1/4)*log((-I*a^4*x*((35153041*b^16*c^4 + 3739918336 *a^4*b^12*c^3*d + 149208170496*a^8*b^8*c^2*d^2 + 2645699854336*a^12*b^4*c* d^3 + 17592186044416*a^16*d^4)/a^15)^(1/4) + (77*b^4*c + 2048*a^4*d)*(a...
\[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=\int \frac {\sqrt [4]{x^{3} \left (a x - b\right )} \left (c x^{4} - d\right )}{x^{2}}\, dx \]
\[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=\int { \frac {{\left (a x^{4} - b x^{3}\right )}^{\frac {1}{4}} {\left (c x^{4} - d\right )}}{x^{2}} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 350 vs. \(2 (142) = 284\).
Time = 0.31 (sec) , antiderivative size = 350, normalized size of antiderivative = 2.16 \[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=\frac {49152 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} b d + \frac {6 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {6 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {3 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} - \frac {3 \, \sqrt {2} {\left (77 \, b^{5} c + 2048 \, a^{4} b d\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a - \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {8 \, {\left (77 \, {\left (a - \frac {b}{x}\right )}^{\frac {13}{4}} b^{5} c - 275 \, {\left (a - \frac {b}{x}\right )}^{\frac {9}{4}} a b^{5} c + 351 \, {\left (a - \frac {b}{x}\right )}^{\frac {5}{4}} a^{2} b^{5} c + 231 \, {\left (a - \frac {b}{x}\right )}^{\frac {1}{4}} a^{3} b^{5} c\right )} x^{4}}{a^{3} b^{4}}}{12288 \, b} \]
1/12288*(49152*(a - b/x)^(1/4)*b*d + 6*sqrt(2)*(77*b^5*c + 2048*a^4*b*d)*a rctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a - b/x)^(1/4))/(-a)^(1/4))/((- a)^(3/4)*a^3) + 6*sqrt(2)*(77*b^5*c + 2048*a^4*b*d)*arctan(-1/2*sqrt(2)*(s qrt(2)*(-a)^(1/4) - 2*(a - b/x)^(1/4))/(-a)^(1/4))/((-a)^(3/4)*a^3) + 3*sq rt(2)*(77*b^5*c + 2048*a^4*b*d)*log(sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + s qrt(-a) + sqrt(a - b/x))/((-a)^(3/4)*a^3) - 3*sqrt(2)*(77*b^5*c + 2048*a^4 *b*d)*log(-sqrt(2)*(-a)^(1/4)*(a - b/x)^(1/4) + sqrt(-a) + sqrt(a - b/x))/ ((-a)^(3/4)*a^3) + 8*(77*(a - b/x)^(13/4)*b^5*c - 275*(a - b/x)^(9/4)*a*b^ 5*c + 351*(a - b/x)^(5/4)*a^2*b^5*c + 231*(a - b/x)^(1/4)*a^3*b^5*c)*x^4/( a^3*b^4))/b
Timed out. \[ \int \frac {\sqrt [4]{-b x^3+a x^4} \left (-d+c x^4\right )}{x^2} \, dx=-\int \frac {\left (d-c\,x^4\right )\,{\left (a\,x^4-b\,x^3\right )}^{1/4}}{x^2} \,d x \]