Optimal. Leaf size=551 \[ \frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) \text{EllipticF}\left (\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right ),-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e (e x)^{3/2}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]
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Rubi [A] time = 1.29642, antiderivative size = 551, normalized size of antiderivative = 1., number of steps used = 17, number of rules used = 13, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.433, Rules used = {466, 471, 579, 584, 307, 224, 221, 1200, 1199, 424, 490, 1219, 1218} \[ \frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt [4]{d} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{2 \sqrt [4]{c} \sqrt{c-d x^2} (b c-a d)^3}+\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}-\frac{\sqrt{b} \sqrt [4]{c} e^{5/2} \sqrt{1-\frac{d x^2}{c}} (7 a d+3 b c) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{4 \sqrt{a} \sqrt [4]{d} \sqrt{c-d x^2} (b c-a d)^3}+\frac{d e (e x)^{3/2} (a d+4 b c)}{2 c \sqrt{c-d x^2} (b c-a d)^3}+\frac{5 d e (e x)^{3/2}}{6 \left (c-d x^2\right )^{3/2} (b c-a d)^2}+\frac{e (e x)^{3/2}}{2 \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2} (b c-a d)} \]
Antiderivative was successfully verified.
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Rule 466
Rule 471
Rule 579
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{(e x)^{5/2}}{\left (a-b x^2\right )^2 \left (c-d x^2\right )^{5/2}} \, dx &=\frac{2 \operatorname{Subst}\left (\int \frac{x^6}{\left (a-\frac{b x^4}{e^2}\right )^2 \left (c-\frac{d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt{e x}\right )}{e}\\ &=\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}-\frac{e \operatorname{Subst}\left (\int \frac{x^2 \left (3 c+\frac{7 d x^4}{e^2}\right )}{\left (a-\frac{b x^4}{e^2}\right ) \left (c-\frac{d x^4}{e^2}\right )^{5/2}} \, dx,x,\sqrt{e x}\right )}{2 (b c-a d)}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{e^3 \operatorname{Subst}\left (\int \frac{x^2 \left (-\frac{6 c (3 b c+2 a d)}{e^2}-\frac{30 b c d x^4}{e^4}\right )}{\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 )}{12 c (b c-a d)^2}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}-\frac{e^5 \operatorname{Subst}\left (\int \frac{x^2 \left (\frac{12 c \left (3 b^2 c^2+11 a b c d+a^2 d^2\right )}{e^4}-\frac{12 b c d (4 b c+a d) x^4}{e^6}\right )}{\left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{24 c^2 (b c-a d)^3}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}-\frac{e^5 \operatorname{Subst}\left (\int \left (\frac{12 c d (4 b c+a d) x^2}{e^4 \sqrt{c-\frac{d x^4}{e^2}}}+\frac{12 \left (3 b^2 c^3+7 a b c^2 d\right ) x^2}{e^4 \left (a-\frac{b x^4}{e^2}\right ) \sqrt{c-\frac{d x^4}{e^2}}}\right ) \, dx,x,\sqrt{e x}\right )}{24 c^2 (b c-a d)^3}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}-\frac{(d (4 b c+a d) e) \operatorname{Subst}\left (\int \frac{x^2}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 c (b c-a d)^3}-\frac{(b (3 b c+7 a d) e) \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 )}{2 (b c-a d)^3}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\left (\sqrt{d} (4 b c+a d) e^2\right ) \operatorname{Subst}\left (\int \frac{1}{\sqrt{c-\frac{d x^4}{e^2}}} \, dx,x,\sqrt{e x}\right )}{2 \sqrt{c} (b c-a d)^3}-\frac{\left (\sqrt{d} (4 b c+a d) e^2\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 )}{2 \sqrt{c} (b c-a d)^3}-\frac{\left (\sqrt{b} (3 b c+7 a d) e^3\right ) \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 )}{4 (b c-a d)^3}+\frac{\left (\sqrt{b} (3 b c+7 a d) e^3\right ) \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 )}{4 (b c-a d)^3}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\left (\sqrt{d} (4 b c+a d) e^2 \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 )}{2 \sqrt{c} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} (4 b c+a d) e^2 \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 )}{2 \sqrt{c} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{b} (3 b c+7 a d) e^3 \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 )}{4 (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\left (\sqrt{b} (3 b c+7 a d) e^3 \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 )}{4 (b c-a d)^3 \sqrt{c-d x^2}}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 )}{4 \sqrt{a} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 )}{4 \sqrt{a} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\left (\sqrt{d} (4 b c+a d) e^2 \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 )}{2 \sqrt{c} (b c-a d)^3 \sqrt{c-d x^2}}\\ &=\frac{5 d e (e x)^{3/2}}{6 (b c-a d)^2 \left (c-d x^2\right )^{3/2}}+\frac{e (e x)^{3/2}}{2 (b c-a d) \left (a-b x^2\right ) \left (c-d x^2\right )^{3/2}}+\frac{d (4 b c+a d) e (e x)^{3/2}}{2 c (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\sqrt [4]{d} (4 b c+a d) e^{5/2} \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 )}{2 \sqrt [4]{c} (b c-a d)^3 \sqrt{c-d x^2}}+\frac{\sqrt{b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 )}{4 \sqrt{a} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}-\frac{\sqrt{b} \sqrt [4]{c} (3 b c+7 a d) e^{5/2} \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 )}{4 \sqrt{a} \sqrt [4]{d} (b c-a d)^3 \sqrt{c-d x^2}}\\ \end{align*}
Mathematica [C] time = 0.455667, size = 278, normalized size = 0.5 \[ -\frac{e (e x)^{3/2} \left (7 \left (b x^2-a\right ) \left (c-d x^2\right ) \sqrt{1-\frac{d x^2}{c}} \left (a^2 d^2+11 a b c d+3 b^2 c^2\right ) F_1\left (\frac{3}{4};\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+7 a \left (a^2 d^2 \left (c-3 d x^2\right )+a b d \left (11 c^2-10 c d x^2+3 d^2 x^4\right )+b^2 c \left (3 c^2-17 c d x^2+12 d^2 x^4\right )\right )+3 b d x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \sqrt{1-\frac{d x^2}{c}} (a d+4 b c) F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}{42 a c \left (b x^2-a\right ) \left (c-d x^2\right )^{3/2} (b c-a d)^3} \]
Warning: Unable to verify antiderivative.
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Maple [B] time = 0.046, size = 5078, normalized size = 9.2 \begin{align*} \text{output 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{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{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(-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{\left (e x\right )^{\frac{5}{2}}}{{\left (b x^{2} - a\right )}^{2}{\left (-d x^{2} + c\right )}^{\frac{5}{2}}}\,{d x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
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