Optimal. Leaf size=365 \[ -\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{c-d x^2}} \]
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Rubi [A] time = 1.52213, antiderivative size = 365, normalized size of antiderivative = 1., number of steps used = 13, number of rules used = 11, integrand size = 30, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.367 \[ -\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (-\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}+\frac{\sqrt [4]{c} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} (b c-a d) \Pi \left (\frac{\sqrt{b} \sqrt{c}}{\sqrt{a} \sqrt{d}};\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{\sqrt{a} b^{3/2} \sqrt [4]{d} \sqrt{c-d x^2}}-\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} F\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{c-d x^2}}+\frac{2 c^{3/4} \sqrt [4]{d} \sqrt{e} \sqrt{1-\frac{d x^2}{c}} E\left (\left .\sin ^{-1}\left (\frac{\sqrt [4]{d} \sqrt{e x}}{\sqrt [4]{c} \sqrt{e}}\right )\right |-1\right )}{b \sqrt{c-d x^2}} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2),x]
[Out]
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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)**(1/2)*(-d*x**2+c)**(1/2)/(-b*x**2+a),x)
[Out]
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Mathematica [C] time = 0.345066, size = 164, normalized size = 0.45 \[ -\frac{14 a c x \sqrt{e x} \sqrt{c-d x^2} F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )}{3 \left (a-b x^2\right ) \left (2 x^2 \left (a d F_1\left (\frac{7}{4};\frac{1}{2},1;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )-2 b c F_1\left (\frac{7}{4};-\frac{1}{2},2;\frac{11}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )-7 a c F_1\left (\frac{3}{4};-\frac{1}{2},1;\frac{7}{4};\frac{d x^2}{c},\frac{b x^2}{a}\right )\right )} \]
Warning: Unable to verify antiderivative.
[In] Integrate[(Sqrt[e*x]*Sqrt[c - d*x^2])/(a - b*x^2),x]
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Maple [B] time = 0.024, size = 701, normalized size = 1.9 \[ \text{result too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((e*x)^(1/2)*(-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. \[ -\int \frac{\sqrt{-d x^{2} + c} \sqrt{e x}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*sqrt(e*x)/(b*x^2 - a),x, algorithm="maxima")
[Out]
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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)*sqrt(e*x)/(b*x^2 - a),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \int \frac{\sqrt{e x} \sqrt{c - d x^{2}}}{- a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((e*x)**(1/2)*(-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. \[ \int -\frac{\sqrt{-d x^{2} + c} \sqrt{e x}}{b x^{2} - a}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(-sqrt(-d*x^2 + c)*sqrt(e*x)/(b*x^2 - a),x, algorithm="giac")
[Out]