Integrand size = 37, antiderivative size = 366 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\frac {1}{8} \left (-8 a^2+9 b+4 a x\right ) \sqrt [4]{b x^3+a x^4}+\frac {\left (-32 a^4+8 a^2 b-5 b^2\right ) \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{3/4}}+\frac {\left (32 a^4-8 a^2 b+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{16 a^{3/4}}+\text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {-2 a^5 \log (x)+a^3 b \log (x)+2 a^5 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )-a^3 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right )+a^4 \log (x) \text {$\#$1}^4-a^2 b \log (x) \text {$\#$1}^4+b^2 \log (x) \text {$\#$1}^4-a^4 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4+a^2 b \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4-b^2 \log \left (\sqrt [4]{b x^3+a x^4}-x \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ] \]
Time = 1.17 (sec) , antiderivative size = 454, normalized size of antiderivative = 1.24 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (-16 a^{11/4} x^{3/4} \sqrt [4]{b+a x}+18 a^{3/4} b x^{3/4} \sqrt [4]{b+a x}+8 a^{7/4} x^{7/4} \sqrt [4]{b+a x}-32 a^4 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+8 a^2 b \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-5 b^2 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )+\left (32 a^4-8 a^2 b+5 b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-4 a^{3/4} \text {RootSum}\left [2 a^2-b-3 a \text {$\#$1}^4+\text {$\#$1}^8\&,\frac {2 a^5 \log (x)-a^3 b \log (x)-8 a^5 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )+4 a^3 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right )-a^4 \log (x) \text {$\#$1}^4+a^2 b \log (x) \text {$\#$1}^4-b^2 \log (x) \text {$\#$1}^4+4 a^4 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4-4 a^2 b \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4+4 b^2 \log \left (\sqrt [4]{b+a x}-\sqrt [4]{x} \text {$\#$1}\right ) \text {$\#$1}^4}{3 a \text {$\#$1}^3-2 \text {$\#$1}^7}\&\right ]\right )}{16 a^{3/4} \left (x^3 (b+a x)\right )^{3/4}} \]
(x^(9/4)*(b + a*x)^(3/4)*(-16*a^(11/4)*x^(3/4)*(b + a*x)^(1/4) + 18*a^(3/4 )*b*x^(3/4)*(b + a*x)^(1/4) + 8*a^(7/4)*x^(7/4)*(b + a*x)^(1/4) - 32*a^4*A rcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + 8*a^2*b*ArcTan[(a^(1/4)*x^(1/4) )/(b + a*x)^(1/4)] - 5*b^2*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] + (32 *a^4 - 8*a^2*b + 5*b^2)*ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)] - 4*a^( 3/4)*RootSum[2*a^2 - b - 3*a*#1^4 + #1^8 & , (2*a^5*Log[x] - a^3*b*Log[x] - 8*a^5*Log[(b + a*x)^(1/4) - x^(1/4)*#1] + 4*a^3*b*Log[(b + a*x)^(1/4) - x^(1/4)*#1] - a^4*Log[x]*#1^4 + a^2*b*Log[x]*#1^4 - b^2*Log[x]*#1^4 + 4*a^ 4*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^4 - 4*a^2*b*Log[(b + a*x)^(1/4) - x ^(1/4)*#1]*#1^4 + 4*b^2*Log[(b + a*x)^(1/4) - x^(1/4)*#1]*#1^4)/(3*a*#1^3 - 2*#1^7) & ]))/(16*a^(3/4)*(x^3*(b + a*x))^(3/4))
Leaf count is larger than twice the leaf count of optimal. \(1255\) vs. \(2(366)=732\).
Time = 4.81 (sec) , antiderivative size = 1255, normalized size of antiderivative = 3.43, number of steps used = 16, number of rules used = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.405, Rules used = {2027, 2467, 25, 1201, 25, 90, 60, 73, 854, 827, 216, 219, 2035, 7279, 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.
| \(\displaystyle \int \frac {\left (a x^2+b x\right ) \sqrt [4]{a x^4+b x^3}}{a x-b+x^2} \, dx\) |
| \(\Big \downarrow \) 2027 |
| \(\displaystyle \int \frac {x (a x+b) \sqrt [4]{a x^4+b x^3}}{a x-b+x^2}dx\) |
| \(\Big \downarrow \) 2467 |
| \(\displaystyle \frac {\sqrt [4]{a x^4+b x^3} \int -\frac {x^{7/4} (b+a x)^{5/4}}{-x^2-a x+b}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \int \frac {x^{7/4} (b+a x)^{5/4}}{-x^2-a x+b}dx}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 1201 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \int \frac {x^{3/4} \left (a^2-x a-2 b\right )}{(b+a x)^{3/4}}dx+\int -\frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 25 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \int \frac {x^{3/4} \left (a^2-x a-2 b\right )}{(b+a x)^{3/4}}dx-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 90 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \int \frac {x^{3/4}}{(b+a x)^{3/4}}dx-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 60 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}}dx}{4 a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 73 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{(b+a x)^{3/4}}d\sqrt [4]{x}}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 854 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \int \frac {\sqrt {x}}{1-a x}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 827 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\int \frac {1}{\sqrt {a} \sqrt {x}+1}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 216 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\int \frac {1}{1-\sqrt {a} \sqrt {x}}d\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}}{2 \sqrt {a}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 219 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-\int \frac {x^{3/4} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}dx\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2035 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-4 \int \frac {x^{3/2} \left (a \left (a^2-2 b\right ) b-\left (a^4-b a^2+b^2\right ) x\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}d\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 7279 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{a x+b}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{2 a^{3/4}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{a x+b}\right )-4 \int \left (\frac {\sqrt {x} \left (a^4-b a^2+b^2\right )}{(b+a x)^{3/4}}+\frac {\sqrt {x} \left (a \left (a^4-b^2\right ) x-b \left (a^4-b a^2+b^2\right )\right )}{(b+a x)^{3/4} \left (-x^2-a x+b\right )}\right )d\sqrt [4]{x}\right )}{x^{3/4} \sqrt [4]{a x+b}}\) |
| \(\Big \downarrow \) 2009 |
| \(\displaystyle -\frac {\sqrt [4]{a x^4+b x^3} \left (a \left (\frac {1}{8} \left (8 a^2-9 b\right ) \left (\frac {x^{3/4} \sqrt [4]{b+a x}}{a}-\frac {3 b \left (\frac {\text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}-\frac {\arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}\right )}{a}\right )-\frac {1}{2} x^{7/4} \sqrt [4]{b+a x}\right )-4 \left (-\frac {\left (a^4-b a^2+b^2\right ) \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}-\frac {a \left (a^4-b^2\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {b \left (a^4-b a^2+b^2\right ) \arctan \left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {a \left (a^4-b^2\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {b \left (a^4-b a^2+b^2\right ) \arctan \left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {\left (a^4-b a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{2 a^{3/4}}+\frac {a \left (a^4-b^2\right ) \left (a-\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}+\frac {b \left (a^4-b a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2-\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a-\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} \sqrt [4]{a-\sqrt {a^2+4 b}} \left (a^2-\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {a \left (a^4-b^2\right ) \left (a+\sqrt {a^2+4 b}\right )^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{2 \sqrt {a^2+4 b} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}-\frac {b \left (a^4-b a^2+b^2\right ) \text {arctanh}\left (\frac {\sqrt [4]{a^2+\sqrt {a^2+4 b} a-2 b} \sqrt [4]{x}}{\sqrt [4]{a+\sqrt {a^2+4 b}} \sqrt [4]{b+a x}}\right )}{\sqrt {a^2+4 b} \sqrt [4]{a+\sqrt {a^2+4 b}} \left (a^2+\sqrt {a^2+4 b} a-2 b\right )^{3/4}}\right )\right )}{x^{3/4} \sqrt [4]{b+a x}}\) |
-(((b*x^3 + a*x^4)^(1/4)*(a*(-1/2*(x^(7/4)*(b + a*x)^(1/4)) + ((8*a^2 - 9* b)*((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) + ArcTanh[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)]/(2*a^(3/ 4))))/a))/8) - 4*(-1/2*((a^4 - a^2*b + b^2)*ArcTan[(a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/a^(3/4) - (b*(a^4 - a^2*b + b^2)*ArcTan[((a^2 - 2*b - a*Sqrt[ a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))]) /(Sqrt[a^2 + 4*b]*(a - Sqrt[a^2 + 4*b])^(1/4)*(a^2 - 2*b - a*Sqrt[a^2 + 4* b])^(3/4)) - (a*(a^4 - b^2)*(a - Sqrt[a^2 + 4*b])^(3/4)*ArcTan[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a^2 + 4*b])^(1/4)*(b + a*x )^(1/4))])/(2*Sqrt[a^2 + 4*b]*(a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) + (b* (a^4 - a^2*b + b^2)*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4)) /((a + Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(Sqrt[a^2 + 4*b]*(a + Sqr t[a^2 + 4*b])^(1/4)*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(3/4)) + (a*(a^4 - b^2 )*(a + Sqrt[a^2 + 4*b])^(3/4)*ArcTan[((a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(1/4 )*x^(1/4))/((a + Sqrt[a^2 + 4*b])^(1/4)*(b + a*x)^(1/4))])/(2*Sqrt[a^2 + 4 *b]*(a^2 - 2*b + a*Sqrt[a^2 + 4*b])^(3/4)) + ((a^4 - a^2*b + b^2)*ArcTanh[ (a^(1/4)*x^(1/4))/(b + a*x)^(1/4)])/(2*a^(3/4)) + (b*(a^4 - a^2*b + b^2)*A rcTanh[((a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(1/4)*x^(1/4))/((a - Sqrt[a^2 + 4* b])^(1/4)*(b + a*x)^(1/4))])/(Sqrt[a^2 + 4*b]*(a - Sqrt[a^2 + 4*b])^(1/4)* (a^2 - 2*b - a*Sqrt[a^2 + 4*b])^(3/4)) + (a*(a^4 - b^2)*(a - Sqrt[a^2 +...
3.30.62.3.1 Defintions of rubi rules used
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*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p + 2)) Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0]
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), x_Symbol] :> Simp[g/c^2 Int[Simp[2*c*e*f + c*d*g - b* e*g + c*e*g*x, x]*(d + e*x)^(m - 1)*(f + g*x)^(n - 2), x], x] + Simp[1/c^2 Int[Simp[c^2*d*f^2 - 2*a*c*e*f*g - a*c*d*g^2 + a*b*e*g^2 + (c^2*e*f^2 + 2 *c^2*d*f*g - 2*b*c*e*f*g - b*c*d*g^2 + b^2*e*g^2 - a*c*e*g^2)*x, x]*(d + e* x)^(m - 1)*((f + g*x)^(n - 2)/(a + b*x + c*x^2)), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && !IntegerQ[m] && !IntegerQ[n] && GtQ[m, 0] && GtQ[n, 1 ]
Int[(Fx_.)*((a_.)*(x_)^(r_.) + (b_.)*(x_)^(s_.))^(p_.), x_Symbol] :> Int[x^ (p*r)*(a + b*x^(s - r))^p*Fx, x] /; FreeQ[{a, b, r, s}, x] && IntegerQ[p] & & PosQ[s - r] && !(EqQ[p, 1] && EqQ[u, 1])
Int[(Fx_)*(x_)^(m_), x_Symbol] :> With[{k = Denominator[m]}, Simp[k Subst [Int[x^(k*(m + 1) - 1)*SubstPower[Fx, x, k], x], x, x^(1/k)], x]] /; Fracti onQ[m] && AlgebraicFunctionQ[Fx, x]
Int[(Fx_.)*(Px_)^(p_), x_Symbol] :> With[{r = Expon[Px, x, Min]}, Simp[Px^F racPart[p]/(x^(r*FracPart[p])*ExpandToSum[Px/x^r, x]^FracPart[p]) Int[x^( p*r)*ExpandToSum[Px/x^r, x]^p*Fx, x], x] /; IGtQ[r, 0]] /; FreeQ[p, x] && P olyQ[Px, x] && !IntegerQ[p] && !MonomialQ[Px, x] && !PolyQ[Fx, x]
Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[ {v = RationalFunctionExpand[u/(a + b*x^n + c*x^(2*n)), x]}, Int[v, x] /; Su mQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]
Time = 0.41 (sec) , antiderivative size = 372, normalized size of antiderivative = 1.02
| method | result | size |
| pseudoelliptic | \(\frac {-32 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {11}{4}}+16 a^{\frac {7}{4}} x \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}+36 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} b \,a^{\frac {3}{4}}+64 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{4}+32 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) a^{4}+32 \left (\munderset {\textit {\_R} =\operatorname {RootOf}\left (\textit {\_Z}^{8}-3 a \,\textit {\_Z}^{4}+2 a^{2}-b \right )}{\sum }\frac {\left (\textit {\_R}^{4} a^{4}-\textit {\_R}^{4} a^{2} b +\textit {\_R}^{4} b^{2}-2 a^{5}+a^{3} b \right ) \ln \left (\frac {-\textit {\_R} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{x}\right )}{\textit {\_R}^{3} \left (2 \textit {\_R}^{4}-3 a \right )}\right ) a^{\frac {3}{4}}-16 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) a^{2} b -8 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) a^{2} b +10 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{2}+5 \ln \left (\frac {-a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x -\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{2}}{32 a^{\frac {3}{4}}}\) | \(372\) |
1/32*(-32*(x^3*(a*x+b))^(1/4)*a^(11/4)+16*a^(7/4)*x*(x^3*(a*x+b))^(1/4)+36 *(x^3*(a*x+b))^(1/4)*b*a^(3/4)+64*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))* a^4+32*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a*x+b))^(1/4)) )*a^4+32*sum((_R^4*a^4-_R^4*a^2*b+_R^4*b^2-2*a^5+a^3*b)*ln((-_R*x+(x^3*(a* x+b))^(1/4))/x)/_R^3/(2*_R^4-3*a),_R=RootOf(_Z^8-3*_Z^4*a+2*a^2-b))*a^(3/4 )-16*arctan(1/a^(1/4)/x*(x^3*(a*x+b))^(1/4))*a^2*b-8*ln((-a^(1/4)*x-(x^3*( a*x+b))^(1/4))/(a^(1/4)*x-(x^3*(a*x+b))^(1/4)))*a^2*b+10*arctan(1/a^(1/4)/ x*(x^3*(a*x+b))^(1/4))*b^2+5*ln((-a^(1/4)*x-(x^3*(a*x+b))^(1/4))/(a^(1/4)* x-(x^3*(a*x+b))^(1/4)))*b^2)/a^(3/4)
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 10.59 (sec) , antiderivative size = 7889, normalized size of antiderivative = 21.55 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\text {Too large to display} \]
Not integrable
Time = 5.68 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.07 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int \frac {x \sqrt [4]{x^{3} \left (a x + b\right )} \left (a x + b\right )}{a x - b + x^{2}}\, dx \]
Not integrable
Time = 0.23 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.10 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int { \frac {{\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} {\left (a x^{2} + b x\right )}}{a x + x^{2} - b} \,d x } \]
Result contains higher order function than in optimal. Order 3 vs. order 1.
Time = 5.83 (sec) , antiderivative size = 310, normalized size of antiderivative = 0.85 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=-\frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{32 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{32 \, \left (-a\right )^{\frac {3}{4}}} - \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{64 \, \left (-a\right )^{\frac {3}{4}}} + \frac {\sqrt {2} {\left (32 \, a^{4} - 8 \, a^{2} b + 5 \, b^{2}\right )} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{64 \, \left (-a\right )^{\frac {3}{4}}} - \frac {{\left (8 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{2} b - 8 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{3} b - 9 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} b^{2} + 5 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a b^{2}\right )} x^{2}}{8 \, b^{2}} \]
-1/32*sqrt(2)*(32*a^4 - 8*a^2*b + 5*b^2)*arctan(1/2*sqrt(2)*(sqrt(2)*(-a)^ (1/4) + 2*(a + b/x)^(1/4))/(-a)^(1/4))/(-a)^(3/4) - 1/32*sqrt(2)*(32*a^4 - 8*a^2*b + 5*b^2)*arctan(-1/2*sqrt(2)*(sqrt(2)*(-a)^(1/4) - 2*(a + b/x)^(1 /4))/(-a)^(1/4))/(-a)^(3/4) - 1/64*sqrt(2)*(32*a^4 - 8*a^2*b + 5*b^2)*log( sqrt(2)*(-a)^(1/4)*(a + b/x)^(1/4) + sqrt(-a) + sqrt(a + b/x))/(-a)^(3/4) + 1/64*sqrt(2)*(32*a^4 - 8*a^2*b + 5*b^2)*log(-sqrt(2)*(-a)^(1/4)*(a + b/x )^(1/4) + sqrt(-a) + sqrt(a + b/x))/(-a)^(3/4) - 1/8*(8*(a + b/x)^(5/4)*a^ 2*b - 8*(a + b/x)^(1/4)*a^3*b - 9*(a + b/x)^(5/4)*b^2 + 5*(a + b/x)^(1/4)* a*b^2)*x^2/b^2
Not integrable
Time = 7.22 (sec) , antiderivative size = 37, normalized size of antiderivative = 0.10 \[ \int \frac {\left (b x+a x^2\right ) \sqrt [4]{b x^3+a x^4}}{-b+a x+x^2} \, dx=\int \frac {\left (a\,x^2+b\,x\right )\,{\left (a\,x^4+b\,x^3\right )}^{1/4}}{x^2+a\,x-b} \,d x \]