Integrand size = 23, antiderivative size = 277 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}} \]
[Out]
Time = 0.20 (sec) , antiderivative size = 277, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.304, Rules used = {427, 537, 230, 227, 418, 1233, 1232} \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=-\frac {\sqrt [4]{a} b^{3/4} \sqrt {1-\frac {b x^4}{a}} (3 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \operatorname {EllipticPi}\left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} \sqrt {1-\frac {b x^4}{a}} (b c-a d)^2 \operatorname {EllipticPi}\left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}},\arcsin \left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right ),-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {b x \sqrt {a-b x^4}}{3 d} \]
[In]
[Out]
Rule 227
Rule 230
Rule 418
Rule 427
Rule 537
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\int \frac {a (b c-3 a d)-b (3 b c-5 a d) x^4}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{3 d} \\ & = \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {(b (3 b c-5 a d)) \int \frac {1}{\sqrt {a-b x^4}} \, dx}{3 d^2}+\frac {(b c-a d)^2 \int \frac {1}{\sqrt {a-b x^4} \left (c-d x^4\right )} \, dx}{d^2} \\ & = \frac {b x \sqrt {a-b x^4}}{3 d}+\frac {(b c-a d)^2 \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^2}+\frac {(b c-a d)^2 \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {a-b x^4}} \, dx}{2 c d^2}-\frac {\left (b (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\sqrt {1-\frac {b x^4}{a}}} \, dx}{3 d^2 \sqrt {a-b x^4}} \\ & = \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1-\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{2 c d^2 \sqrt {a-b x^4}}+\frac {\left ((b c-a d)^2 \sqrt {1-\frac {b x^4}{a}}\right ) \int \frac {1}{\left (1+\frac {\sqrt {d} x^2}{\sqrt {c}}\right ) \sqrt {1-\frac {b x^4}{a}}} \, dx}{2 c d^2 \sqrt {a-b x^4}} \\ & = \frac {b x \sqrt {a-b x^4}}{3 d}-\frac {\sqrt [4]{a} b^{3/4} (3 b c-5 a d) \sqrt {1-\frac {b x^4}{a}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{3 d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \Pi \left (-\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}}+\frac {\sqrt [4]{a} (b c-a d)^2 \sqrt {1-\frac {b x^4}{a}} \Pi \left (\frac {\sqrt {a} \sqrt {d}}{\sqrt {b} \sqrt {c}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )\right |-1\right )}{2 \sqrt [4]{b} c d^2 \sqrt {a-b x^4}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 10.35 (sec) , antiderivative size = 341, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=-\frac {x \left (\frac {b (-3 b c+5 a d) x^4 \sqrt {1-\frac {b x^4}{a}} \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )}{c}+\frac {5 \left (5 a c \left (3 a^2 d-a b d x^4+b^2 x^4 \left (-c+d x^4\right )\right ) \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 b x^4 \left (a-b x^4\right ) \left (c-d x^4\right ) \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}{\left (-c+d x^4\right ) \left (5 a c \operatorname {AppellF1}\left (\frac {1}{4},\frac {1}{2},1,\frac {5}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+2 x^4 \left (2 a d \operatorname {AppellF1}\left (\frac {5}{4},\frac {1}{2},2,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )+b c \operatorname {AppellF1}\left (\frac {5}{4},\frac {3}{2},1,\frac {9}{4},\frac {b x^4}{a},\frac {d x^4}{c}\right )\right )\right )}\right )}{15 d \sqrt {a-b x^4}} \]
[In]
[Out]
Result contains higher order function than in optimal. Order 9 vs. order 4.
Time = 6.26 (sec) , antiderivative size = 306, normalized size of antiderivative = 1.10
method | result | size |
risch | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\frac {b \left (5 a d -3 b c \right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{d \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}+\frac {\left (-3 a^{2} d^{2}+6 a b c d -3 b^{2} c^{2}\right ) \left (\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}}{\underline {\hspace {1.25 ex}}\alpha ^{3}}\right )}{8 d^{2}}}{3 d}\) | \(306\) |
default | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(307\) |
elliptic | \(\frac {b x \sqrt {-b \,x^{4}+a}}{3 d}+\frac {\left (\frac {b \left (2 a d -b c \right )}{d^{2}}-\frac {b a}{3 d}\right ) \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, F\left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, i\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, \sqrt {-b \,x^{4}+a}}-\frac {\munderset {\underline {\hspace {1.25 ex}}\alpha =\operatorname {RootOf}\left (d \,\textit {\_Z}^{4}-c \right )}{\sum }\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\operatorname {arctanh}\left (\frac {-2 b \,x^{2} \underline {\hspace {1.25 ex}}\alpha ^{2}+2 a}{2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {-b \,x^{4}+a}}\right )}{\sqrt {\frac {a d -b c}{d}}}-\frac {2 \underline {\hspace {1.25 ex}}\alpha ^{3} d \sqrt {1-\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \sqrt {1+\frac {x^{2} \sqrt {b}}{\sqrt {a}}}\, \Pi \left (x \sqrt {\frac {\sqrt {b}}{\sqrt {a}}}, \frac {\sqrt {a}\, \underline {\hspace {1.25 ex}}\alpha ^{2} d}{\sqrt {b}\, c}, \frac {\sqrt {-\frac {\sqrt {b}}{\sqrt {a}}}}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}}\right )}{\sqrt {\frac {\sqrt {b}}{\sqrt {a}}}\, c \sqrt {-b \,x^{4}+a}}\right )}{\underline {\hspace {1.25 ex}}\alpha ^{3}}}{8 d^{3}}\) | \(307\) |
[In]
[Out]
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=- \int \frac {a \sqrt {a - b x^{4}}}{- c + d x^{4}}\, dx - \int \left (- \frac {b x^{4} \sqrt {a - b x^{4}}}{- c + d x^{4}}\right )\, dx \]
[In]
[Out]
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \]
[In]
[Out]
\[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int { -\frac {{\left (-b x^{4} + a\right )}^{\frac {3}{2}}}{d x^{4} - c} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {\left (a-b x^4\right )^{3/2}}{c-d x^4} \, dx=\int \frac {{\left (a-b\,x^4\right )}^{3/2}}{c-d\,x^4} \,d x \]
[In]
[Out]