Optimal. Leaf size=328 \[ -\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{c^{3/4} \sqrt {e} \sqrt {c-d x^2} (b c-a d)}-\frac {d \sqrt {e x}}{c e \sqrt {c-d x^2} (b c-a d)}+\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)}+\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.49, antiderivative size = 328, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.267, Rules used = {466, 414, 523, 224, 221, 409, 1219, 1218} \[ -\frac {d^{3/4} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{c^{3/4} \sqrt {e} \sqrt {c-d x^2} (b c-a d)}-\frac {d \sqrt {e x}}{c e \sqrt {c-d x^2} (b c-a d)}+\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)}+\frac {b \sqrt [4]{c} \sqrt {1-\frac {d x^2}{c}} \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{a \sqrt [4]{d} \sqrt {e} \sqrt {c-d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 224
Rule 409
Rule 414
Rule 466
Rule 523
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {1}{\sqrt {e x} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \left (c-\frac {d x^4}{e^2}\right )^{3/2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=-\frac {d \sqrt {e x}}{c (b c-a d) e \sqrt {c-d x^2}}-\frac {e \operatorname {Subst}\left (\int \frac {-\frac {2 b c-a d}{e^2}-\frac {b d x^4}{e^4}}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c (b c-a d)}\\ &=-\frac {d \sqrt {e x}}{c (b c-a d) e \sqrt {c-d x^2}}+\frac {(2 b) \operatorname {Subst}\left (\int \frac {1}{\left (a-\frac {b x^4}{e^2}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{(b c-a d) e}-\frac {d \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{c (b c-a d) e}\\ &=-\frac {d \sqrt {e x}}{c (b c-a d) e \sqrt {c-d x^2}}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a (b c-a d) e}+\frac {b \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{a (b c-a d) e}-\frac {\left (d \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{c (b c-a d) e \sqrt {c-d x^2}}\\ &=-\frac {d \sqrt {e x}}{c (b c-a d) e \sqrt {c-d x^2}}-\frac {d^{3/4} \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 )}{c^{3/4} (b c-a d) \sqrt {e} \sqrt {c-d x^2}}+\frac {\left (b \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1-\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{a (b c-a d) e \sqrt {c-d x^2}}+\frac {\left (b \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {Subst}\left (\int \frac {1}{\left (1+\frac {\sqrt {b} x^2}{\sqrt {a} e}\right ) \sqrt {1-\frac {d x^4}{c e^2}}} \, dx,x,\sqrt {e x}\right )}{a (b c-a d) e \sqrt {c-d x^2}}\\ &=-\frac {d \sqrt {e x}}{c (b c-a d) e \sqrt {c-d x^2}}-\frac {d^{3/4} \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 )}{c^{3/4} (b c-a d) \sqrt {e} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} \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 )}{a \sqrt [4]{d} (b c-a d) \sqrt {e} \sqrt {c-d x^2}}+\frac {b \sqrt [4]{c} \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 )}{a \sqrt [4]{d} (b c-a d) \sqrt {e} \sqrt {c-d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.13, size = 147, normalized size = 0.45 \[ \frac {5 x \sqrt {1-\frac {d x^2}{c}} (2 b c-a d) F_1\left (\frac {1}{4};\frac {1}{2},1;\frac {5}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )+b d x^3 \sqrt {1-\frac {d x^2}{c}} F_1\left (\frac {5}{4};\frac {1}{2},1;\frac {9}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )-5 a d x}{5 a c \sqrt {e x} \sqrt {c-d x^2} (b c-a d)} \]
Warning: Unable to verify antiderivative.
[In]
[Out]
________________________________________________________________________________________
fricas [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int -\frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.04, size = 708, normalized size = 2.16 \[ \frac {\left (-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, b^{2} c^{2} \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )+\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, b^{2} c^{2} \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )+2 \sqrt {a b}\, a \,d^{2} x -2 \sqrt {a b}\, b c d x +\sqrt {2}\, \sqrt {a b}\, \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, a d \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {2}\, \sqrt {a b}\, \sqrt {c d}\, \sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, b c \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}\, b c \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b -\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )-\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {2}\, \sqrt {\frac {-d x +\sqrt {c d}}{\sqrt {c d}}}\, \sqrt {-\frac {d x}{\sqrt {c d}}}\, \sqrt {a b}\, \sqrt {c d}\, b c \EllipticPi \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {c d}\, b}{\sqrt {c d}\, b +\sqrt {a b}\, d}, \frac {\sqrt {2}}{2}\right )\right ) \sqrt {-d \,x^{2}+c}\, b d}{2 \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \sqrt {a b}\, \left (a d -b c \right ) \left (d \,x^{2}-c \right ) \sqrt {e x}\, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {1}{{\left (b x^{2} - a\right )} {\left (-d x^{2} + c\right )}^{\frac {3}{2}} \sqrt {e x}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {1}{\sqrt {e\,x}\,\left (a-b\,x^2\right )\,{\left (c-d\,x^2\right )}^{3/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {1}{- a c \sqrt {e x} \sqrt {c - d x^{2}} + a d x^{2} \sqrt {e x} \sqrt {c - d x^{2}} + b c x^{2} \sqrt {e x} \sqrt {c - d x^{2}} - b d x^{4} \sqrt {e x} \sqrt {c - d x^{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________