Integrand size = 25, antiderivative size = 345 \[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\frac {\arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{2^{3/4} b-2^{3/4} a x+2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\arctan \left (\frac {2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}{-2^{3/4} b+2^{3/4} a x+2 \sqrt [4]{a} \sqrt [4]{b} \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}}-\frac {\text {arctanh}\left (\frac {-\frac {b^2}{\sqrt [4]{2}}+2^{3/4} a b x-\frac {a^2 x^2}{\sqrt [4]{2}}-2 \sqrt [4]{2} \sqrt {a} \sqrt {b} \sqrt {b^2 x+a^2 x^3}}{-2 \sqrt [4]{a} b^{5/4} \sqrt [4]{b^2 x+a^2 x^3}+2 a^{5/4} \sqrt [4]{b} x \sqrt [4]{b^2 x+a^2 x^3}}\right )}{2\ 2^{3/4} a^{3/4} b^{3/4}} \]
[Out]
Result contains higher order function than in optimal. Order 6 vs. order 3 in optimal.
Time = 0.18 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.43, number of steps used = 8, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {2081, 973, 477, 525, 524} \[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\frac {4 x \sqrt [4]{\frac {a^2 x^2}{b^2}+1} \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {1}{4},\frac {11}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{a^2 x^3+b^2 x}}-\frac {4 a x^2 \sqrt [4]{\frac {a^2 x^2}{b^2}+1} \operatorname {AppellF1}\left (\frac {7}{8},1,\frac {1}{4},\frac {15}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{a^2 x^3+b^2 x}} \]
[In]
[Out]
Rule 477
Rule 524
Rule 525
Rule 973
Rule 2081
Rubi steps \begin{align*} \text {integral}& = \frac {\left (\sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} (b+a x) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}} \\ & = -\frac {\left (a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {x^{3/4}}{\left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \int \frac {1}{\sqrt [4]{x} \left (b^2-a^2 x^2\right ) \sqrt [4]{b^2+a^2 x^2}} \, dx}{\sqrt [4]{b^2 x+a^2 x^3}} \\ & = -\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{b^2+a^2 x^2}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{b^2+a^2 x^8}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}} \\ & = -\frac {\left (4 a \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {x^6}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}}+\frac {\left (4 b \sqrt [4]{x} \sqrt [4]{1+\frac {a^2 x^2}{b^2}}\right ) \text {Subst}\left (\int \frac {x^2}{\left (b^2-a^2 x^8\right ) \sqrt [4]{1+\frac {a^2 x^8}{b^2}}} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{b^2 x+a^2 x^3}} \\ & = \frac {4 x \sqrt [4]{1+\frac {a^2 x^2}{b^2}} \operatorname {AppellF1}\left (\frac {3}{8},1,\frac {1}{4},\frac {11}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{3 b \sqrt [4]{b^2 x+a^2 x^3}}-\frac {4 a x^2 \sqrt [4]{1+\frac {a^2 x^2}{b^2}} \operatorname {AppellF1}\left (\frac {7}{8},1,\frac {1}{4},\frac {15}{8},\frac {a^2 x^2}{b^2},-\frac {a^2 x^2}{b^2}\right )}{7 b^2 \sqrt [4]{b^2 x+a^2 x^3}} \\ \end{align*}
\[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx \]
[In]
[Out]
\[\int \frac {1}{\left (a x +b \right ) \left (a^{2} x^{3}+b^{2} x \right )^{\frac {1}{4}}}d x\]
[In]
[Out]
Timed out. \[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\int \frac {1}{\sqrt [4]{x \left (a^{2} x^{2} + b^{2}\right )} \left (a x + b\right )}\, dx \]
[In]
[Out]
\[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\int { \frac {1}{{\left (a^{2} x^{3} + b^{2} x\right )}^{\frac {1}{4}} {\left (a x + b\right )}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\int { \frac {1}{{\left (a^{2} x^{3} + b^{2} x\right )}^{\frac {1}{4}} {\left (a x + b\right )}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{(b+a x) \sqrt [4]{b^2 x+a^2 x^3}} \, dx=\int \frac {1}{{\left (a^2\,x^3+b^2\,x\right )}^{1/4}\,\left (b+a\,x\right )} \,d x \]
[In]
[Out]