Integrand size = 30, antiderivative size = 414 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \]
[Out]
Time = 0.56 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.00, number of steps used = 15, number of rules used = 12, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.400, Rules used = {477, 489, 598, 313, 230, 227, 1214, 1213, 435, 504, 1233, 1232} \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=-\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \operatorname {EllipticPi}\left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}},\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) \operatorname {EllipticF}\left (\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right ),-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {2 c^{3/4} e^{5/2} \sqrt {1-\frac {d x^2}{c}} (2 b c-5 a d) E\left (\left .\arcsin \left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b} \]
[In]
[Out]
Rule 227
Rule 230
Rule 313
Rule 435
Rule 477
Rule 489
Rule 504
Rule 598
Rule 1213
Rule 1214
Rule 1232
Rule 1233
Rubi steps \begin{align*} \text {integral}& = \frac {2 \text {Subst}\left (\int \frac {x^6 \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {(2 e) \text {Subst}\left (\int \frac {x^2 \left (3 a c+\frac {(2 b c-5 a d) x^4}{e^2}\right )}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {(2 e) \text {Subst}\left (\int \left (-\frac {(2 b c-5 a d) x^2}{b \sqrt {c-\frac {d x^4}{e^2}}}-\frac {5 \left (-a b c+a^2 d\right ) x^2}{b \left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}}\right ) \, dx,x,\sqrt {e x}\right )}{5 b} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {(2 (2 b c-5 a d) e) \text {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2}+\frac {(2 a (b c-a d) e) \text {Subst}\left (\int \frac {x^2}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^2} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d}}+\frac {\left (a (b c-a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2}}-\frac {\left (a (b c-a d) e^3\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2}} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d} \sqrt {c-d x^2}}+\frac {\left (a (b c-a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e-\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2} \sqrt {c-d x^2}}-\frac {\left (a (b c-a d) e^3 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {1}{\left (\sqrt {a} e+\sqrt {b} x^2\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{b^{5/2} \sqrt {c-d x^2}} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {\left (2 \sqrt {c} (2 b c-5 a d) e^2 \sqrt {1-\frac {d x^2}{c}}\right ) \text {Subst}\left (\int \frac {\sqrt {1+\frac {\sqrt {d} x^2}{\sqrt {c} e}}}{\sqrt {1-\frac {\sqrt {d} x^2}{\sqrt {c} e}}} \, dx,x,\sqrt {e x}\right )}{5 b^2 \sqrt {d} \sqrt {c-d x^2}} \\ & = -\frac {2 e (e x)^{3/2} \sqrt {c-d x^2}}{5 b}-\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}+\frac {2 c^{3/4} (2 b c-5 a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{5 b^2 d^{3/4} \sqrt {c-d x^2}}-\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt {a} \sqrt [4]{c} (b c-a d) e^{5/2} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b^{5/2} \sqrt [4]{d} \sqrt {c-d x^2}} \\ \end{align*}
Result contains higher order function than in optimal. Order 6 vs. order 4 in optimal.
Time = 11.13 (sec) , antiderivative size = 143, normalized size of antiderivative = 0.35 \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\frac {2 e (e x)^{3/2} \left (-7 a \left (c-d x^2\right )+7 a c \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {3}{4},\frac {1}{2},1,\frac {7}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )+(2 b c-5 a d) x^2 \sqrt {1-\frac {d x^2}{c}} \operatorname {AppellF1}\left (\frac {7}{4},\frac {1}{2},1,\frac {11}{4},\frac {d x^2}{c},\frac {b x^2}{a}\right )\right )}{35 a b \sqrt {c-d x^2}} \]
[In]
[Out]
Time = 4.12 (sec) , antiderivative size = 528, normalized size of antiderivative = 1.28
method | result | size |
risch | \(-\frac {2 x^{2} \sqrt {-d \,x^{2}+c}\, e^{3}}{5 b \sqrt {e x}}+\frac {\left (\frac {\left (5 a d -2 b c \right ) \sqrt {c d}\, \sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {2 \left (x -\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \left (-\frac {2 \sqrt {c d}\, E\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}+\frac {\sqrt {c d}\, F\left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )}{d}\right )}{b d \sqrt {-d e \,x^{3}+c e x}}+\frac {5 \left (a d -b c \right ) a \left (\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}-\frac {\sqrt {a b}}{b}\right )}+\frac {\sqrt {c d}\, \sqrt {\frac {d x}{\sqrt {c d}}+1}\, \sqrt {-\frac {2 d x}{\sqrt {c d}}+2}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \Pi \left (\sqrt {\frac {\left (x +\frac {\sqrt {c d}}{d}\right ) d}{\sqrt {c d}}}, -\frac {\sqrt {c d}}{d \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}, \frac {\sqrt {2}}{2}\right )}{2 b d \sqrt {-d e \,x^{3}+c e x}\, \left (-\frac {\sqrt {c d}}{d}+\frac {\sqrt {a b}}{b}\right )}\right )}{b}\right ) e^{3} \sqrt {\left (-d \,x^{2}+c \right ) e x}}{5 b \sqrt {e x}\, \sqrt {-d \,x^{2}+c}}\) | \(528\) |
elliptic | \(\text {Expression too large to display}\) | \(1003\) |
default | \(\text {Expression too large to display}\) | \(1480\) |
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\text {Timed out} \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=- \int \frac {\left (e x\right )^{\frac {5}{2}} \sqrt {c - d x^{2}}}{- a + b x^{2}}\, dx \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\int { -\frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a} \,d x } \]
[In]
[Out]
\[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\int { -\frac {\sqrt {-d x^{2} + c} \left (e x\right )^{\frac {5}{2}}}{b x^{2} - a} \,d x } \]
[In]
[Out]
Timed out. \[ \int \frac {(e x)^{5/2} \sqrt {c-d x^2}}{a-b x^2} \, dx=\int \frac {{\left (e\,x\right )}^{5/2}\,\sqrt {c-d\,x^2}}{a-b\,x^2} \,d x \]
[In]
[Out]