Optimal. Leaf size=372 \[ -\frac{2 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2} (2 b c-7 a d)}{21 b^2 d}-\frac{2 e (e x)^{5/2} \sqrt{c-d x^2}}{7 b} \]
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Rubi [A] time = 2.05913, antiderivative size = 372, 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 \[ -\frac{2 \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} \left (-21 a^2 d^2+14 a b c d+2 b^2 c^2\right ) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{21 b^3 d^{5/4} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{a \sqrt [4]{c} e^{7/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^3 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{2 e^3 \sqrt{e x} \sqrt{c-d x^2} (2 b c-7 a d)}{21 b^2 d}-\frac{2 e (e x)^{5/2} \sqrt{c-d x^2}}{7 b} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]
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Rubi in Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((e*x)**(7/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
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Mathematica [C] time = 1.74449, size = 382, normalized size = 1.03 \[ \frac{2 (e x)^{7/2} \left (-\frac{9 a c x^2 \left (21 a^2 d^2-14 a b c d-2 b^2 c^2\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+\frac{25 a^2 c^2 (7 a d-2 b c) F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{\left (a-b x^2\right ) \left (2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )}+5 \left (c-d x^2\right ) \left (-7 a d+2 b c-3 b d x^2\right )\right )}{105 b^2 d x^3 \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((e*x)^(7/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]
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Maple [B] time = 0.097, size = 1479, normalized size = 4. \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(7/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{7}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*(e*x)^(7/2)/(b*x^2 - a),x, algorithm="maxima")
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Fricas [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*(e*x)^(7/2)/(b*x^2 - a),x, algorithm="fricas")
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(7/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{7}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*(e*x)^(7/2)/(b*x^2 - a),x, algorithm="giac")
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