Optimal. Leaf size=315 \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b} \]
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Rubi [A] time = 1.37871, 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 \[ -\frac{2 \sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (2 b c-3 a d) F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{3 b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} e^{3/2} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b^2 \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 e \sqrt{e x} \sqrt{c-d x^2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[((e*x)^(3/2)*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)**(3/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
[Out]
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Mathematica [C] time = 0.653605, size = 418, normalized size = 1.33 \[ \frac{2 e \sqrt{e x} \left (\frac{10 x^2 \left (a-b x^2\right ) \left (c-d x^2\right ) \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c \left (5 a c-2 a d x^2-7 b c x^2+5 b d x^4\right ) F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{9}{4};\frac{1}{2},2;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{9}{4};\frac{3}{2},1;\frac{13}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+9 a c F_1\left (\frac{5}{4};\frac{1}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}-\frac{25 a^2 c^2 F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{2 x^2 \left (2 b c F_1\left (\frac{5}{4};\frac{1}{2},2;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )+a d F_1\left (\frac{5}{4};\frac{3}{2},1;\frac{9}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )+5 a c F_1\left (\frac{1}{4};\frac{1}{2},1;\frac{5}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}\right )}{15 b \left (b x^2-a\right ) \sqrt{c-d x^2}} \]
Warning: Unable to verify antiderivative.
[In] Integrate[((e*x)^(3/2)*Sqrt[c - d*x^2])/(a - b*x^2),x]
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Maple [B] time = 0.026, size = 1286, normalized size = 4.1 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(3/2)*(-d*x^2+c)^(1/2)/(-b*x^2+a),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ -\int \frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*(e*x)^(3/2)/(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)^(3/2)/(b*x^2 - a),x, algorithm="fricas")
[Out]
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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)**(3/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int -\frac{\sqrt{-d x^{2} + c} \left (e x\right )^{\frac{3}{2}}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*(e*x)^(3/2)/(b*x^2 - a),x, algorithm="giac")
[Out]