Optimal. Leaf size=397 \[ \frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{3 a c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-5 a d)}{3 a c^2 e (e x)^{3/2} (b c-a d)}-\frac{d}{c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)} \]
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Rubi [A] time = 0.789251, antiderivative size = 397, normalized size of antiderivative = 1., number of steps used = 11, number of rules used = 9, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.3, Rules used = {466, 472, 583, 523, 224, 221, 409, 1219, 1218} \[ \frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{b^2 \sqrt [4]{c} \sqrt{1-\frac{d x^2}{c}} \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{a^2 \sqrt [4]{d} e^{5/2} \sqrt{c-d x^2} (b c-a d)}+\frac{d^{3/4} \sqrt{1-\frac{d x^2}{c}} (2 b c-5 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 a c^{7/4} e^{5/2} \sqrt{c-d x^2} (b c-a d)}-\frac{\sqrt{c-d x^2} (2 b c-5 a d)}{3 a c^2 e (e x)^{3/2} (b c-a d)}-\frac{d}{c e (e x)^{3/2} \sqrt{c-d x^2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 472
Rule 583
Rule 523
Rule 224
Rule 221
Rule 409
Rule 1219
Rule 1218
Rubi steps
\begin{align*} \int \frac{1}{(e x)^{5/2} \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{1}{x^4 \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}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{e \operatorname{Subst}\left (\int \frac{-\frac{2 b c-5 a d}{e^2}-\frac{5 b d x^4}{e^4}}{x^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}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{(2 b c-5 a d) \sqrt{c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac{e \operatorname{Subst}\left (\int \frac{\frac{6 b^2 c^2+2 a b c d-5 a^2 d^2}{e^4}-\frac{b d (2 b c-5 a d) x^4}{e^6}}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 a c^2 (b c-a d)}\\ &=-\frac{d}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{(2 b c-5 a d) \sqrt{c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac{\left (2 b^2\right ) \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 )}{a (b c-a d) e^3}+\frac{(d (2 b c-5 a d)) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{3 a c^2 (b c-a d) e^3}\\ &=-\frac{d}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{(2 b c-5 a d) \sqrt{c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac{b^2 \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^2 (b c-a d) e^3}+\frac{b^2 \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^2 (b c-a d) e^3}+\frac{\left (d (2 b c-5 a 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 )}{3 a c^2 (b c-a d) e^3 \sqrt{c-d x^2}}\\ &=-\frac{d}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{(2 b c-5 a d) \sqrt{c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac{d^{3/4} (2 b c-5 a 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 )}{3 a c^{7/4} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{\left (b^2 \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^2 (b c-a d) e^3 \sqrt{c-d x^2}}+\frac{\left (b^2 \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^2 (b c-a d) e^3 \sqrt{c-d x^2}}\\ &=-\frac{d}{c (b c-a d) e (e x)^{3/2} \sqrt{c-d x^2}}-\frac{(2 b c-5 a d) \sqrt{c-d x^2}}{3 a c^2 (b c-a d) e (e x)^{3/2}}+\frac{d^{3/4} (2 b c-5 a 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 )}{3 a c^{7/4} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{b^2 \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^2 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt{c-d x^2}}+\frac{b^2 \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^2 \sqrt [4]{d} (b c-a d) e^{5/2} \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.247626, size = 197, normalized size = 0.5 \[ \frac{x \left (5 x^2 \sqrt{1-\frac{d x^2}{c}} \left (-5 a^2 d^2+2 a b c d+6 b^2 c^2\right ) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+b d x^4 \sqrt{1-\frac{d x^2}{c}} (5 a d-2 b c) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+5 a \left (a d \left (2 c-5 d x^2\right )-2 b c \left (c-d x^2\right )\right )\right )}{15 a^2 c^2 (e x)^{5/2} \sqrt{c-d x^2} (b c-a d)} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.033, size = 896, normalized size = 2.3 \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 )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [F(-2)] time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: UnboundLocalError} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [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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Giac [F] time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{1}{{\left (b x^{2} - a\right )}{\left (-d x^{2} + c\right )}^{\frac{3}{2}} \left (e x\right )^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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