Optimal. Leaf size=379 \[ \frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}} \]
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Rubi [A] time = 0.683064, antiderivative size = 379, normalized size of antiderivative = 1., number of steps used = 15, number of rules used = 12, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.4, Rules used = {466, 480, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ -\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt [4]{d} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}} \]
Antiderivative was successfully verified.
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Rule 466
Rule 480
Rule 584
Rule 307
Rule 224
Rule 221
Rule 1200
Rule 1199
Rule 424
Rule 490
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{(e x)^{3/2} \left (a-b x^2\right ) \sqrt{c-d x^2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{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 )}{e}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{2 \operatorname{Subst}\left (\int \frac{x^2 \left (\frac{b c-a d}{e^2}+\frac{b d x^4}{e^4}\right )}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{a c e}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{2 \operatorname{Subst}\left (\int \left (-\frac{d x^2}{e^2 \sqrt{c-\frac{d x^4}{e^2}}}+\frac{b c x^2}{e^2 \left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}}\right ) \, dx,x,\sqrt{e x}\right )}{a c e}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{(2 b) \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 )}{a e^3}-\frac{(2 d) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{a c e^3}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{\left (2 \sqrt{d}\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{a \sqrt{c} e^2}-\frac{\left (2 \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 )}{a \sqrt{c} e^2}+\frac{\sqrt{b} \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 )}{a e}-\frac{\sqrt{b} \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 )}{a e}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{\left (2 \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 )}{a \sqrt{c} e^2 \sqrt{c-d x^2}}-\frac{\left (2 \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 )}{a \sqrt{c} e^2 \sqrt{c-d x^2}}+\frac{\left (\sqrt{b} \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 )}{a e \sqrt{c-d x^2}}-\frac{\left (\sqrt{b} \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 )}{a e \sqrt{c-d x^2}}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}+\frac{2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{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^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{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^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}-\frac{\left (2 \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 )}{a \sqrt{c} e^2 \sqrt{c-d x^2}}\\ &=-\frac{2 \sqrt{c-d x^2}}{a c e \sqrt{e x}}-\frac{2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}+\frac{2 \sqrt [4]{d} \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 )}{a \sqrt [4]{c} e^{3/2} \sqrt{c-d x^2}}-\frac{\sqrt{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^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}+\frac{\sqrt{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^{3/2} \sqrt [4]{d} e^{3/2} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.116515, size = 146, normalized size = 0.39 \[ \frac{x \left (6 b d x^4 \sqrt{1-\frac{d x^2}{c}} F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+14 x^2 \sqrt{1-\frac{d x^2}{c}} (b c-a d) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-42 a \left (c-d x^2\right )\right )}{21 a^2 c (e x)^{3/2} \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.03, size = 835, normalized size = 2.2 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F] time = 0., size = 0, normalized size = 0. \begin{align*} -\int \frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F] time = 0., size = 0, normalized size = 0. \begin{align*} - \int \frac{1}{- a \left (e x\right )^{\frac{3}{2}} \sqrt{c - d x^{2}} + b x^{2} \left (e x\right )^{\frac{3}{2}} \sqrt{c - d x^{2}}}\, dx \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} - a\right )} \sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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