Integrand size = 19, antiderivative size = 123 \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\frac {\left (77 b^3-44 a b^2 x+32 a^2 b x^2+384 a^3 x^3\right ) \sqrt [4]{b x^3+a x^4}}{1536 a^3}+\frac {77 b^4 \arctan \left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{1024 a^{15/4}}-\frac {77 b^4 \text {arctanh}\left (\frac {\sqrt [4]{a} x}{\sqrt [4]{b x^3+a x^4}}\right )}{1024 a^{15/4}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 224, normalized size of antiderivative = 1.82, number of steps used = 10, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.421, Rules used = {2046, 2049, 2057, 65, 338, 304, 209, 212} \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\frac {77 b^4 x^{9/4} (a x+b)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{1024 a^{15/4} \left (a x^4+b x^3\right )^{3/4}}-\frac {77 b^4 x^{9/4} (a x+b)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{a x+b}}\right )}{1024 a^{15/4} \left (a x^4+b x^3\right )^{3/4}}+\frac {77 b^3 \sqrt [4]{a x^4+b x^3}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{a x^4+b x^3}}{384 a^2}+\frac {1}{4} x^3 \sqrt [4]{a x^4+b x^3}+\frac {b x^2 \sqrt [4]{a x^4+b x^3}}{48 a} \]
[In]
[Out]
Rule 65
Rule 209
Rule 212
Rule 304
Rule 338
Rule 2046
Rule 2049
Rule 2057
Rubi steps \begin{align*} \text {integral}& = \frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}+\frac {1}{16} b \int \frac {x^5}{\left (b x^3+a x^4\right )^{3/4}} \, dx \\ & = \frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (11 b^2\right ) \int \frac {x^4}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{192 a} \\ & = -\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}+\frac {\left (77 b^3\right ) \int \frac {x^3}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{1536 a^2} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (77 b^4\right ) \int \frac {x^2}{\left (b x^3+a x^4\right )^{3/4}} \, dx}{2048 a^3} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (77 b^4 x^{9/4} (b+a x)^{3/4}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x)^{3/4}} \, dx}{2048 a^3 \left (b x^3+a x^4\right )^{3/4}} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (77 b^4 x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b+a x^4\right )^{3/4}} \, dx,x,\sqrt [4]{x}\right )}{512 a^3 \left (b x^3+a x^4\right )^{3/4}} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (77 b^4 x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {x^2}{1-a x^4} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{512 a^3 \left (b x^3+a x^4\right )^{3/4}} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}-\frac {\left (77 b^4 x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1-\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{7/2} \left (b x^3+a x^4\right )^{3/4}}+\frac {\left (77 b^4 x^{9/4} (b+a x)^{3/4}\right ) \text {Subst}\left (\int \frac {1}{1+\sqrt {a} x^2} \, dx,x,\frac {\sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{7/2} \left (b x^3+a x^4\right )^{3/4}} \\ & = \frac {77 b^3 \sqrt [4]{b x^3+a x^4}}{1536 a^3}-\frac {11 b^2 x \sqrt [4]{b x^3+a x^4}}{384 a^2}+\frac {b x^2 \sqrt [4]{b x^3+a x^4}}{48 a}+\frac {1}{4} x^3 \sqrt [4]{b x^3+a x^4}+\frac {77 b^4 x^{9/4} (b+a x)^{3/4} \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{15/4} \left (b x^3+a x^4\right )^{3/4}}-\frac {77 b^4 x^{9/4} (b+a x)^{3/4} \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )}{1024 a^{15/4} \left (b x^3+a x^4\right )^{3/4}} \\ \end{align*}
Time = 0.49 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.14 \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\frac {x^{9/4} (b+a x)^{3/4} \left (2 a^{3/4} x^{3/4} \sqrt [4]{b+a x} \left (77 b^3-44 a b^2 x+32 a^2 b x^2+384 a^3 x^3\right )+231 b^4 \arctan \left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )-231 b^4 \text {arctanh}\left (\frac {\sqrt [4]{a} \sqrt [4]{x}}{\sqrt [4]{b+a x}}\right )\right )}{3072 a^{15/4} \left (x^3 (b+a x)\right )^{3/4}} \]
[In]
[Out]
Time = 0.32 (sec) , antiderivative size = 153, normalized size of antiderivative = 1.24
method | result | size |
pseudoelliptic | \(\frac {1536 \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {15}{4}} x^{3}+128 a^{\frac {11}{4}} b \,x^{2} \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}-176 a^{\frac {7}{4}} b^{2} x \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}+308 b^{3} \left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}} a^{\frac {3}{4}}-231 \ln \left (\frac {a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{-a^{\frac {1}{4}} x +\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}\right ) b^{4}-462 \arctan \left (\frac {\left (x^{3} \left (a x +b \right )\right )^{\frac {1}{4}}}{a^{\frac {1}{4}} x}\right ) b^{4}}{6144 a^{\frac {15}{4}}}\) | \(153\) |
[In]
[Out]
Result contains complex when optimal does not.
Time = 0.31 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.15 \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=-\frac {231 \, a^{3} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} \log \left (\frac {77 \, {\left (a^{4} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} x + {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{4}\right )}}{x}\right ) - 231 \, a^{3} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {77 \, {\left (a^{4} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{4}\right )}}{x}\right ) - 231 i \, a^{3} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {77 \, {\left (i \, a^{4} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{4}\right )}}{x}\right ) + 231 i \, a^{3} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} \log \left (-\frac {77 \, {\left (-i \, a^{4} \left (\frac {b^{16}}{a^{15}}\right )^{\frac {1}{4}} x - {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} b^{4}\right )}}{x}\right ) - 4 \, {\left (384 \, a^{3} x^{3} + 32 \, a^{2} b x^{2} - 44 \, a b^{2} x + 77 \, b^{3}\right )} {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}}}{6144 \, a^{3}} \]
[In]
[Out]
\[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\int x^{2} \sqrt [4]{x^{3} \left (a x + b\right )}\, dx \]
[In]
[Out]
\[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\int { {\left (a x^{4} + b x^{3}\right )}^{\frac {1}{4}} x^{2} \,d x } \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 278 vs. \(2 (103) = 206\).
Time = 0.30 (sec) , antiderivative size = 278, normalized size of antiderivative = 2.26 \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\frac {\frac {462 \, \sqrt {2} b^{5} \arctan \left (\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} + 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {462 \, \sqrt {2} b^{5} \arctan \left (-\frac {\sqrt {2} {\left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} - 2 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}}\right )}}{2 \, \left (-a\right )^{\frac {1}{4}}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {231 \, \sqrt {2} b^{5} \log \left (\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{\left (-a\right )^{\frac {3}{4}} a^{3}} + \frac {231 \, \sqrt {2} \left (-a\right )^{\frac {1}{4}} b^{5} \log \left (-\sqrt {2} \left (-a\right )^{\frac {1}{4}} {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} + \sqrt {-a} + \sqrt {a + \frac {b}{x}}\right )}{a^{4}} + \frac {8 \, {\left (77 \, {\left (a + \frac {b}{x}\right )}^{\frac {13}{4}} b^{5} - 275 \, {\left (a + \frac {b}{x}\right )}^{\frac {9}{4}} a b^{5} + 351 \, {\left (a + \frac {b}{x}\right )}^{\frac {5}{4}} a^{2} b^{5} + 231 \, {\left (a + \frac {b}{x}\right )}^{\frac {1}{4}} a^{3} b^{5}\right )} x^{4}}{a^{3} b^{4}}}{12288 \, b} \]
[In]
[Out]
Timed out. \[ \int x^2 \sqrt [4]{b x^3+a x^4} \, dx=\int x^2\,{\left (a\,x^4+b\,x^3\right )}^{1/4} \,d x \]
[In]
[Out]