Integrand size = 28, antiderivative size = 198 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {\left (101 b^2+52 a b x+32 a^2 x^2\right ) \sqrt [4]{b x^3+a x^4}}{96 a^3}-\frac {155 b^3 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{15/4}}+\frac {2 \sqrt [4]{2} b^3 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{15/4}}+\frac {155 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{64 a^{15/4}}-\frac {2 \sqrt [4]{2} b^3 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{a^{15/4}} \]
1/96*(32*a^2*x^2+52*a*b*x+101*b^2)*(a*x^4+b*x^3)^(1/4)/a^3-155/64*b^3*arct an(a^(1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(15/4)+2*2^(1/4)*b^3*arctan(2^(1/4)*a^ (1/4)*x/(a*x^4+b*x^3)^(1/4))/a^(15/4)+155/64*b^3*arctanh(a^(1/4)*x/(a*x^4+ b*x^3)^(1/4))/a^(15/4)-2*2^(1/4)*b^3*arctanh(2^(1/4)*a^(1/4)*x/(a*x^4+b*x^ 3)^(1/4))/a^(15/4)
Time = 0.81 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.15 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (202 a^{3/4} b^2 x^{3/4} \sqrt [4]{b+a x}+104 a^{7/4} b x^{7/4} \sqrt [4]{b+a x}+64 a^{11/4} x^{11/4} \sqrt [4]{b+a x}-465 b^3 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+384 \sqrt [4]{2} b^3 \arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+465 b^3 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-384 \sqrt [4]{2} b^3 \text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{192 a^{15/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(b + a*x)^(3/4)*(202*a^(3/4)*b^2*x^(3/4)*(b + a*x)^(1/4) + 104*a^ (7/4)*b*x^(7/4)*(b + a*x)^(1/4) + 64*a^(11/4)*x^(11/4)*(b + a*x)^(1/4) - 4 65*b^3*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + 384*2^(1/4)*b^3*ArcTan[ (2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + 465*b^3*ArcTanh[(a^(1/4)*x^(1 /4))/(b + a*x)^(1/4)] - 384*2^(1/4)*b^3*ArcTanh[(2^(1/4)*a^(1/4)*x^(1/4))/ (b + a*x)^(1/4)]))/(192*a^(15/4)*(x^3*(b + a*x))^(3/4))
Time = 0.57 (sec) , antiderivative size = 386, normalized size of antiderivative = 1.95, number of steps used = 23, number of rules used = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.786, Rules used = {1948, 25, 112, 27, 173, 105, 105, 104, 27, 827, 216, 219, 1194, 27, 87, 57, 60, 73, 854, 827, 216, 219}
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.
| \(\displaystyle \int \frac {x^2 \sqrt [4]{a x^4+b x^3}}{a x-b} \, dx\) |
| \(\Big \downarrow \) 1948 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {x^{11/4} \sqrt [4]{b+a x}}{b-a x}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{11/4} \sqrt [4]{b+a x}}{b-a x}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 112 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {\int \frac {b x^{7/4} (11 b+13 a x)}{4 (b-a x) (b+a x)^{3/4}}dx}{3 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \int \frac {x^{7/4} (11 b+13 a x)}{(b-a x) (b+a x)^{3/4}}dx}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 173 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \int \frac {x^{7/4}}{(b-a x) (b+a x)^{11/4}}dx-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\int \frac {x^{3/4}}{(b-a x) (b+a x)^{7/4}}dx}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 105 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {\int \frac {1}{\sqrt [4]{x} (b-a x) (b+a x)^{3/4}}dx}{2 a}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 104 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \int \frac {\sqrt {x}}{b \sqrt {b+a x} \left (1-\frac {2 a x}{b+a x}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \int \frac {\sqrt {x}}{\sqrt {b+a x} \left (1-\frac {2 a x}{b+a x}\right )}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+a x}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt {a}}-\frac {\int \frac {1}{\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+a x}}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt {a}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\int \frac {1}{1-\frac {\sqrt {2} \sqrt {a} \sqrt {x}}{\sqrt {b+a x}}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {2} \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\int \frac {x^{7/4} \left (13 x^2 a^5+50 b x a^4+85 b^2 a^3\right )}{(b+a x)^{11/4}}dx}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 1194 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {\int \frac {5 a^4 b x^{7/4} (136 b+41 a x)}{4 (b+a x)^{11/4}}dx}{2 a}+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 27 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \int \frac {x^{7/4} (136 b+41 a x)}{(b+a x)^{11/4}}dx+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 87 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \int \frac {x^{7/4}}{(b+a x)^{7/4}}dx\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 57 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \int \frac {x^{3/4}}{(b+a x)^{3/4}}dx}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{4 a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{(b+a x)^{3/4}}d\sqrt [4]{x}}{a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}\right )}{a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )+\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (\frac {b \left (96 b^3 \left (\frac {\frac {2 \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{2} \sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2\ 2^{3/4} a^{3/4}}\right )}{a b}-\frac {2 x^{3/4}}{3 a b (a x+b)^{3/4}}}{2 a}-\frac {2 x^{7/4}}{7 a b (a x+b)^{7/4}}\right )-\frac {\frac {13 a^4 x^{15/4}}{2 (a x+b)^{7/4}}+\frac {5}{8} a^3 b \left (\frac {380 x^{11/4}}{7 (a x+b)^{7/4}}-\frac {93}{7} \left (\frac {7 \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )}{3 a}-\frac {4 x^{7/4}}{3 a (a x+b)^{3/4}}\right )\right )}{a^3}\right )}{12 a}-\frac {x^{11/4} \sqrt [4]{a x+b}}{3 a}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
-(((b*x^3 + a*x^4)^(1/4)*(-1/3*(x^(11/4)*(b + a*x)^(1/4))/a + (b*(-(((13*a ^4*x^(15/4))/(2*(b + a*x)^(7/4)) + (5*a^3*b*((380*x^(11/4))/(7*(b + a*x)^( 7/4)) - (93*((-4*x^(7/4))/(3*a*(b + a*x)^(3/4)) + (7*((x^(3/4)*(b + a*x)^( 1/4))/a - (3*b*(-1/2*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/a^(3/4) + A rcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(2*a^(3/4))))/a))/(3*a)))/7))/8) /a^3) + 96*b^3*((-2*x^(7/4))/(7*a*b*(b + a*x)^(7/4)) + ((-2*x^(3/4))/(3*a* b*(b + a*x)^(3/4)) + (2*(-1/2*ArcTan[(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^( 1/4)]/(2^(3/4)*a^(3/4)) + ArcTanh[(2^(1/4)*a^(1/4)*x^(1/4))/(b + a*x)^(1/4 )]/(2*2^(3/4)*a^(3/4))))/(a*b))/(2*a))))/(12*a)))/(x^(3/4)*(b + a*x)^(1/4) ))
3.25.39.3.1 Defintions of rubi rules used
Int[(a_)*(Fx_), x_Symbol] :> Simp[a Int[Fx, x], x] /; FreeQ[a, x] && !Ma tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] & & GtQ[n, 0] && LtQ[m, -1] && !(IntegerQ[n] && !IntegerQ[m]) && !(ILeQ[m + n + 2, 0] && (FractionQ[m] || GeQ[2*n + m + 1, 0])) && IntLinearQ[a, b, c , d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ (a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( b*(m + n + 1))) Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] && !(IGtQ[m, 0] && ( !Integer Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) && !ILtQ[m + n + 2, 0] && IntLinear Q[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ {p = Denominator[m]}, Simp[p/b Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL inearQ[a, b, c, d, m, n, x]
Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p _.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p + 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(f*(p + 1)*(c*f - d*e)) Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] /; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege rQ[p] || !(IntegerQ[n] || !(EqQ[e, 0] || !(EqQ[c, 0] || LtQ[p, n]))))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x _)), x_] :> With[{q = Denominator[m]}, Simp[q Subst[Int[x^(q*(m + 1) - 1) /(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] ] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^(m + 1)*(c + d*x)^n*((e + f*x)^(p + 1)/((m + 1)*(b*e - a*f))), x] - Simp[n*((d*e - c*f)/((m + 1)*(b*e - a*f))) Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1)*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, m, p}, x] && EqQ[m + n + p + 2, 0] && GtQ[n, 0] && (SumSimplerQ[m, 1] || !SumSimplerQ[p, 1]) && NeQ[m, -1]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[(a + b*x)^m*(c + d*x)^n*((e + f*x)^(p + 1)/(f*(m + n + p + 1))), x] - Simp[1/(f*(m + n + p + 1)) Int[(a + b*x)^(m - 1)*(c + d*x) ^(n - 1)*(e + f*x)^p*Simp[c*m*(b*e - a*f) + a*n*(d*e - c*f) + (d*m*(b*e - a *f) + b*n*(d*e - c*f))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, p}, x] && GtQ[m, 0] && GtQ[n, 0] && NeQ[m + n + p + 1, 0] && (IntegersQ[2*m, 2*n, 2*p ] || (IntegersQ[m, n + p] || IntegersQ[p, m + n]))
Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((g_.) + (h_.)*(x_ )))/((e_.) + (f_.)*(x_)), x_] :> Simp[(f*g - e*h)*((c*f - d*e)^(m + n + 1)/ f^(m + n + 2)) Int[(a + b*x)^m/((c + d*x)^(m + 1)*(e + f*x)), x], x] + Si mp[1/f^(m + n + 2) Int[((a + b*x)^m/(c + d*x)^(m + 1))*ExpandToSum[(f^(m + n + 2)*(c + d*x)^(m + n + 1)*(g + h*x) - (f*g - e*h)*(c*f - d*e)^(m + n + 1))/(e + f*x), x], x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x] && IGtQ[m + n + 1, 0] && (LtQ[m, 0] || SumSimplerQ[m, 1] || !SumSimplerQ[n, 1])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[b, 2]))*A rcTan[Rt[b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (GtQ[a , 0] || GtQ[b, 0])
Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt Q[a, 0] || LtQ[b, 0])
Int[(x_)^2/((a_) + (b_.)*(x_)^4), x_Symbol] :> With[{r = Numerator[Rt[-a/b, 2]], s = Denominator[Rt[-a/b, 2]]}, Simp[s/(2*b) Int[1/(r + s*x^2), x], x] - Simp[s/(2*b) Int[1/(r - s*x^2), x], x]] /; FreeQ[{a, b}, x] && !GtQ [a/b, 0]
Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[a^(p + (m + 1)/n) Subst[Int[x^m/(1 - b*x^n)^(p + (m + 1)/n + 1), x], x, x/(a + b*x^n )^(1/n)], x] /; FreeQ[{a, b}, x] && IGtQ[n, 0] && LtQ[-1, p, 0] && NeQ[p, - 2^(-1)] && IntegersQ[m, p + (m + 1)/n]
Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Simp[c^p*(d + e*x)^(m + 2*p)*((f + g*x )^(n + 1)/(g*e^(2*p)*(m + n + 2*p + 1))), x] + Simp[1/(g*e^(2*p)*(m + n + 2 *p + 1)) Int[(d + e*x)^m*(f + g*x)^n*ExpandToSum[g*(m + n + 2*p + 1)*(e^( 2*p)*(a + b*x + c*x^2)^p - c^p*(d + e*x)^(2*p)) - c^p*(e*f - d*g)*(m + 2*p) *(d + e*x)^(2*p - 1), x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && IGtQ [p, 0] && !IntegerQ[m] && !IntegerQ[n] && NeQ[m + n + 2*p + 1, 0]
Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + (d_.)*(x_)^(n_.))^(q_.), x_Symbol] :> Simp[e^IntPart[m]*(e*x)^FracPart[m]*( (a*x^j + b*x^(j + n))^FracPart[p]/(x^(FracPart[m] + j*FracPart[p])*(a + b*x ^n)^FracPart[p])) Int[x^(m + j*p)*(a + b*x^n)^p*(c + d*x^n)^q, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p, q}, x] && EqQ[jn, j + n] && !IntegerQ[p] && NeQ[b*c - a*d, 0] && !(EqQ[n, 1] && EqQ[j, 1])
Time = 0.78 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.11
| method | result | size |
| pseudoelliptic | \(\frac {128 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {11}{4}} x^{2}+208 a^{\frac {7}{4}} b x \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}+404 b^{2} \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}-384 \,2^{\frac {1}{4}} \ln \left (\frac {-x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x 2^{\frac {1}{4}} a^{\frac {1}{4}}-\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{3}-768 \,2^{\frac {1}{4}} \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} 2^{\frac {3}{4}}}{2 x \,a^{\frac {1}{4}}}\right ) b^{3}+465 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{3}+930 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{3}}{384 a^{\frac {15}{4}}}\) | \(219\) |
1/384*(128*(x^3*(a*x+b))^(1/4)*a^(11/4)*x^2+208*a^(7/4)*b*x*(x^3*(a*x+b))^ (1/4)+404*b^2*(x^3*(a*x+b))^(1/4)*a^(3/4)-384*2^(1/4)*ln((-x*2^(1/4)*a^(1/ 4)-(x^3*(a*x+b))^(1/4))/(x*2^(1/4)*a^(1/4)-(x^3*(a*x+b))^(1/4)))*b^3-768*2 ^(1/4)*arctan(1/2*(x^3*(a*x+b))^(1/4)/x*2^(3/4)/a^(1/4))*b^3+465*ln((a^(1/ 4)*x+(x^3*(a*x+b))^(1/4))/(-a^(1/4)*x+(x^3*(a*x+b))^(1/4)))*b^3+930*arctan (1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))*b^3)/a^(15/4)
Result contains complex when optimal does not.
Time = 0.26 (sec) , antiderivative size = 485, normalized size of antiderivative = 2.45 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=-\frac {384 \cdot 2^{\frac {1}{4}} a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {2^{\frac {1}{4}} a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}}{x}\right ) - 384 \cdot 2^{\frac {1}{4}} a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {2^{\frac {1}{4}} a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}}{x}\right ) + 384 i \cdot 2^{\frac {1}{4}} a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {i \cdot 2^{\frac {1}{4}} a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}}{x}\right ) - 384 i \cdot 2^{\frac {1}{4}} a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {-i \cdot 2^{\frac {1}{4}} a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}}{x}\right ) - 465 \, a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {155 \, {\left (a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) + 465 \, a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {155 \, {\left (a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) + 465 i \, a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {155 \, {\left (i \, a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) - 465 i \, a^{3} \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {155 \, {\left (-i \, a^{4} x \left (\frac {b^{12}}{a^{15}}\right )^{\frac {1}{4}} - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{3}\right )}}{x}\right ) - 4 \, {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (32 \, a^{2} x^{2} + 52 \, a b x + 101 \, b^{2}\right )}}{384 \, a^{3}} \]
-1/384*(384*2^(1/4)*a^3*(b^12/a^15)^(1/4)*log((2^(1/4)*a^4*x*(b^12/a^15)^( 1/4) + (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 384*2^(1/4)*a^3*(b^12/a^15)^(1/4)*l og(-(2^(1/4)*a^4*x*(b^12/a^15)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^3)/x) + 384 *I*2^(1/4)*a^3*(b^12/a^15)^(1/4)*log((I*2^(1/4)*a^4*x*(b^12/a^15)^(1/4) + (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 384*I*2^(1/4)*a^3*(b^12/a^15)^(1/4)*log((- I*2^(1/4)*a^4*x*(b^12/a^15)^(1/4) + (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 465*a^ 3*(b^12/a^15)^(1/4)*log(155*(a^4*x*(b^12/a^15)^(1/4) + (a*x^4 + b*x^3)^(1/ 4)*b^3)/x) + 465*a^3*(b^12/a^15)^(1/4)*log(-155*(a^4*x*(b^12/a^15)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^3)/x) + 465*I*a^3*(b^12/a^15)^(1/4)*log(-155*(I*a ^4*x*(b^12/a^15)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^3)/x) - 465*I*a^3*(b^12/a ^15)^(1/4)*log(-155*(-I*a^4*x*(b^12/a^15)^(1/4) - (a*x^4 + b*x^3)^(1/4)*b^ 3)/x) - 4*(a*x^4 + b*x^3)^(1/4)*(32*a^2*x^2 + 52*a*b*x + 101*b^2))/a^3
\[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int \frac {x^{2} \sqrt [4]{x^{3} \left (a x + b\right )}}{a x - b}\, dx \]
\[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x^{2}}{a x - b} \,d x } \]
Leaf count of result is larger than twice the leaf count of optimal. 461 vs. \(2 (158) = 316\).
Time = 0.37 (sec) , antiderivative size = 461, normalized size of antiderivative = 2.33 \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=-\frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 )}{a^{4}} - \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b^{3} \arctan \left (-\frac {2^{\frac {1}{4}} {\left (2^{\frac {3}{4}} \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 )}{a^{4}} - \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{2 \, a^{4}} + \frac {2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} b^{3} \log \left (-2^{\frac {3}{4}} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {2} \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{2 \, a^{4}} + \frac {155 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \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 )}{128 \, a^{4}} + \frac {155 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \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 )}{128 \, a^{4}} + \frac {155 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{3} \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 )}{256 \, a^{4}} + \frac {155 \, \sqrt {2} b^{3} \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 )}{256 \, \left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {{\left (101 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} b^{3} - 150 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a b^{3} + 81 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{2} b^{3}\right )} x^{3}}{96 \, a^{3} b^{3}} \]
-2^(3/4)*(-a)^(1/4)*b^3*arctan(1/2*2^(1/4)*(2^(3/4)*(-a)^(1/4) + 2*(a + b/ x)^(1/4))/(-a)^(1/4))/a^4 - 2^(3/4)*(-a)^(1/4)*b^3*arctan(-1/2*2^(1/4)*(2^ (3/4)*(-a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^4 - 1/2*2^(3/4)*(-a)^( 1/4)*b^3*log(2^(3/4)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt( a + b/x))/a^4 + 1/2*2^(3/4)*(-a)^(1/4)*b^3*log(-2^(3/4)*(-a)^(1/4)*(a + b/ x)^(1/4) + sqrt(2)*sqrt(-a) + sqrt(a + b/x))/a^4 + 155/128*sqrt(2)*(-a)^(1 /4)*b^3*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) + 2*(a + b/x)^(1/4))/(-a)^( 1/4))/a^4 + 155/128*sqrt(2)*(-a)^(1/4)*b^3*arctan(-1/2*sqrt(2)*(sqrt(2)*(- a)^(1/4) - 2*(a + b/x)^(1/4))/(-a)^(1/4))/a^4 + 155/256*sqrt(2)*(-a)^(1/4) *b^3*log(sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/a^ 4 + 155/256*sqrt(2)*b^3*log(-sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/((-a)^(3/4)*a^3) + 1/96*(101*(a + b/x)^(9/4)*b^3 - 150*( a + b/x)^(5/4)*a*b^3 + 81*(a + b/x)^(1/4)*a^2*b^3)*x^3/(a^3*b^3)
Timed out. \[ \int \frac {x^2 \sqrt [4]{b x^3+a x^4}}{-b+a x} \, dx=-\int \frac {x^2\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{b-a\,x} \,d x \]