Optimal. Leaf size=96 \[ \frac{14 a^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a-b x^2}}+\frac{14}{45} a x \left (a-b x^2\right )^{3/4}+\frac{2}{9} x \left (a-b x^2\right )^{7/4} \]
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Rubi [A] time = 0.0746524, antiderivative size = 96, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 12, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{14 a^{5/2} \sqrt [4]{1-\frac{b x^2}{a}} E\left (\left .\frac{1}{2} \sin ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a-b x^2}}+\frac{14}{45} a x \left (a-b x^2\right )^{3/4}+\frac{2}{9} x \left (a-b x^2\right )^{7/4} \]
Antiderivative was successfully verified.
[In] Int[(a - b*x^2)^(7/4),x]
[Out]
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Rubi in Sympy [A] time = 9.24125, size = 83, normalized size = 0.86 \[ \frac{14 a^{\frac{5}{2}} \sqrt [4]{1 - \frac{b x^{2}}{a}} E\left (\frac{\operatorname{asin}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{15 \sqrt{b} \sqrt [4]{a - b x^{2}}} + \frac{14 a x \left (a - b x^{2}\right )^{\frac{3}{4}}}{45} + \frac{2 x \left (a - b x^{2}\right )^{\frac{7}{4}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((-b*x**2+a)**(7/4),x)
[Out]
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Mathematica [C] time = 0.0501919, size = 77, normalized size = 0.8 \[ \frac{21 a^2 x \sqrt [4]{1-\frac{b x^2}{a}} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};\frac{b x^2}{a}\right )+24 a^2 x-34 a b x^3+10 b^2 x^5}{45 \sqrt [4]{a-b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a - b*x^2)^(7/4),x]
[Out]
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Maple [F] time = 0.045, size = 0, normalized size = 0. \[ \int \left ( -b{x}^{2}+a \right ) ^{{\frac{7}{4}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((-b*x^2+a)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int{\left (-b x^{2} + a\right )}^{\frac{7}{4}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(7/4),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left ({\left (-b x^{2} + a\right )}^{\frac{7}{4}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(7/4),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.2884, size = 27, normalized size = 0.28 \[ a^{\frac{7}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{2 i \pi }}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x**2+a)**(7/4),x)
[Out]
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GIAC/XCAS [F(-2)] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((-b*x^2 + a)^(7/4),x, algorithm="giac")
[Out]