Optimal. Leaf size=365 \[ -\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}} \]
[Out]
________________________________________________________________________________________
Rubi [A] time = 0.54, antiderivative size = 365, normalized size of antiderivative = 1.00, number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.367, Rules used = {466, 491, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (-\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} (b c-a d) \Pi \left (\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}};\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}}+\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \sqrt {1-\frac {d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac {\sqrt [4]{d} \sqrt {e x}}{\sqrt [4]{c} \sqrt {e}}\right )\right |-1\right )}{b \sqrt {c-d x^2}} \]
Antiderivative was successfully verified.
[In]
[Out]
Rule 221
Rule 224
Rule 307
Rule 424
Rule 466
Rule 490
Rule 491
Rule 1199
Rule 1200
Rule 1218
Rule 1219
Rubi steps
\begin {align*} \int \frac {\sqrt {e x} \sqrt {c-d x^2}}{a-b x^2} \, dx &=\frac {2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c-\frac {d x^4}{e^2}}}{a-\frac {b x^4}{e^2}} \, dx,x,\sqrt {e x}\right )}{e}\\ &=\frac {(2 d) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b e}+\frac {(2 (b c-a d)) \operatorname {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 e}\\ &=-\frac {\left (2 \sqrt {c} \sqrt {d}\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {c-\frac {d x^4}{e^2}}} \, dx,x,\sqrt {e x}\right )}{b}+\frac {\left (2 \sqrt {c} \sqrt {d}\right ) \operatorname {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 )}{b}+\frac {((b c-a d) e) \operatorname {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^{3/2}}-\frac {((b c-a d) e) \operatorname {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^{3/2}}\\ &=-\frac {\left (2 \sqrt {c} \sqrt {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 )}{b \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 )}{b \sqrt {c-d x^2}}+\frac {\left ((b c-a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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^{3/2} \sqrt {c-d x^2}}-\frac {\left ((b c-a d) e \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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^{3/2} \sqrt {c-d x^2}}\\ &=-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \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 )}{b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) \sqrt {e} \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 )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {e} \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 )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\left (2 \sqrt {c} \sqrt {d} \sqrt {1-\frac {d x^2}{c}}\right ) \operatorname {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 )}{b \sqrt {c-d x^2}}\\ &=\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \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 )}{b \sqrt {c-d x^2}}-\frac {2 c^{3/4} \sqrt [4]{d} \sqrt {e} \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 )}{b \sqrt {c-d x^2}}-\frac {\sqrt [4]{c} (b c-a d) \sqrt {e} \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 )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}+\frac {\sqrt [4]{c} (b c-a d) \sqrt {e} \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 )}{\sqrt {a} b^{3/2} \sqrt [4]{d} \sqrt {c-d x^2}}\\ \end {align*}
________________________________________________________________________________________
Mathematica [C] time = 0.04, size = 69, normalized size = 0.19 \[ \frac {2 x \sqrt {e x} \sqrt {c-d x^2} F_1\left (\frac {3}{4};-\frac {1}{2},1;\frac {7}{4};\frac {d x^2}{c},\frac {b x^2}{a}\right )}{3 a \sqrt {1-\frac {d x^2}{c}}} \]
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 \[ \int -\frac {\sqrt {-d x^{2} + c} \sqrt {e x}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maple [B] time = 0.02, size = 701, normalized size = 1.92 \[ -\frac {\sqrt {e x}\, \sqrt {-d \,x^{2}+c}\, \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}}}\, \left (4 a b c d \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-2 a b c d \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )-a b c d \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 )-a b c d \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 )-4 b^{2} c^{2} \EllipticE \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )+2 b^{2} c^{2} \EllipticF \left (\sqrt {\frac {d x +\sqrt {c d}}{\sqrt {c d}}}, \frac {\sqrt {2}}{2}\right )+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 )+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 {a b}\, \sqrt {c d}\, a d \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 {a b}\, \sqrt {c d}\, a d \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 {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 {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 ) d}{2 \left (d \,x^{2}-c \right ) \left (\sqrt {c d}\, b +\sqrt {a b}\, d \right ) \left (\sqrt {c d}\, b -\sqrt {a b}\, d \right ) b x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ -\int \frac {\sqrt {-d x^{2} + c} \sqrt {e x}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
mupad [F] time = 0.00, size = -1, normalized size = -0.00 \[ \int \frac {\sqrt {e\,x}\,\sqrt {c-d\,x^2}}{a-b\,x^2} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________
sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ - \int \frac {\sqrt {e x} \sqrt {c - d x^{2}}}{- a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In]
[Out]
________________________________________________________________________________________