Integrand size = 19, antiderivative size = 167 \[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a+b x}}-\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} d \sqrt {a+b x}} \]
[Out]
Time = 0.15 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {65, 313, 230, 227, 1214, 1213, 435} \[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a+b x}}-\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right ),-1\right )}{b^{3/4} d \sqrt {a+b x}} \]
[In]
[Out]
Rule 65
Rule 227
Rule 230
Rule 313
Rule 435
Rule 1213
Rule 1214
Rubi steps \begin{align*} \text {integral}& = \frac {4 \text {Subst}\left (\int \frac {x^2}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{d} \\ & = -\frac {\left (4 \sqrt {b c-a d}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt {b} d}+\frac {\left (4 \sqrt {b c-a d}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {a-\frac {b c}{d}+\frac {b x^4}{d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt {b} d} \\ & = -\frac {\left (4 \sqrt {b c-a d} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt {b} d \sqrt {a+b x}}+\frac {\left (4 \sqrt {b c-a d} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}{\sqrt {1+\frac {b x^4}{\left (a-\frac {b c}{d}\right ) d}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt {b} d \sqrt {a+b x}} \\ & = -\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a+b x}}+\frac {\left (4 \sqrt {b c-a d} \sqrt {\frac {d (a+b x)}{-b c+a d}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}}{\sqrt {1-\frac {\sqrt {b} x^2}{\sqrt {b c-a d}}}} \, dx,x,\sqrt [4]{c+d x}\right )}{\sqrt {b} d \sqrt {a+b x}} \\ & = \frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a+b x}}-\frac {4 (b c-a d)^{3/4} \sqrt {-\frac {d (a+b x)}{b c-a d}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} \sqrt [4]{c+d x}}{\sqrt [4]{b c-a d}}\right )\right |-1\right )}{b^{3/4} d \sqrt {a+b x}} \\ \end{align*}
Result contains higher order function than in optimal. Order 5 vs. order 4 in optimal.
Time = 0.02 (sec) , antiderivative size = 71, normalized size of antiderivative = 0.43 \[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\frac {2 \sqrt {a+b x} \sqrt [4]{\frac {b (c+d x)}{b c-a d}} \operatorname {Hypergeometric2F1}\left (\frac {1}{4},\frac {1}{2},\frac {3}{2},\frac {d (a+b x)}{-b c+a d}\right )}{b \sqrt [4]{c+d x}} \]
[In]
[Out]
\[\int \frac {1}{\sqrt {b x +a}\, \left (d x +c \right )^{\frac {1}{4}}}d x\]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\sqrt {a + b x} \sqrt [4]{c + d x}}\, dx \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
\[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\int { \frac {1}{\sqrt {b x + a} {\left (d x + c\right )}^{\frac {1}{4}}} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {1}{\sqrt {a+b x} \sqrt [4]{c+d x}} \, dx=\int \frac {1}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{1/4}} \,d x \]
[In]
[Out]