Optimal. Leaf size=315 \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{3 b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b} \]
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Rubi [A] time = 0.508994, antiderivative size = 315, normalized size of antiderivative = 1., 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, 478, 523, 224, 221, 409, 1219, 1218} \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b} \]
Antiderivative was successfully verified.
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Rule 466
Rule 478
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{(e x)^{3/2} \sqrt{c-d x^2}}{a-b x^2} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^4 \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 \sqrt{e x} \sqrt{c-d x^2}}{3 b}+\frac{(2 e) \operatorname{Subst}\left (\int \frac{a c+\frac{(2 b c-3 a d) x^4}{e^2}}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 b}\\ &=-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b}-\frac{(2 (2 b c-3 a d) e) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 b^2}+\frac{(2 a (b c-a d) e) \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^2}\\ &=-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b}+\frac{((b c-a d) e) \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 )}{b^2}+\frac{((b c-a d) e) \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 )}{b^2}-\frac{\left (2 (2 b c-3 a d) e \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 )}{3 b^2 \sqrt{c-d x^2}}\\ &=-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b}-\frac{2 \sqrt [4]{c} (2 b c-3 a d) e^{3/2} \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 )}{3 b^2 \sqrt [4]{d} \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 (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 )}{b^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 (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 )}{b^2 \sqrt{c-d x^2}}\\ &=-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b}-\frac{2 \sqrt [4]{c} (2 b c-3 a d) e^{3/2} \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 )}{3 b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} (b c-a d) e^{3/2} \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^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} (b c-a d) e^{3/2} \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^2 \sqrt [4]{d} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.117163, size = 143, normalized size = 0.45 \[ \frac{2 e \sqrt{e x} \left (x^2 \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+5 a c \sqrt{1-\frac{d x^2}{c}} F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-5 a \left (c-d x^2\right )\right )}{15 a b \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.017, size = 1286, normalized size = 4.1 \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{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{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{\left (e x\right )^{\frac{3}{2}} \sqrt{c - d x^{2}}}{- a + b 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{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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