Optimal. Leaf size=78 \[ \frac{2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac{2 \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a+b x^2\right )^{3/4}} \]
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Rubi [A] time = 0.0533191, antiderivative size = 78, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.273 \[ \frac{2 x}{3 a \left (a+b x^2\right )^{3/4}}+\frac{2 \left (\frac{b x^2}{a}+1\right )^{3/4} F\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{3 \sqrt{a} \sqrt{b} \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(-7/4),x]
[Out]
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Rubi in Sympy [A] time = 5.97665, size = 68, normalized size = 0.87 \[ \frac{2 x}{3 a \left (a + b x^{2}\right )^{\frac{3}{4}}} + \frac{2 \left (1 + \frac{b x^{2}}{a}\right )^{\frac{3}{4}} F\left (\frac{\operatorname{atan}{\left (\frac{\sqrt{b} x}{\sqrt{a}} \right )}}{2}\middle | 2\right )}{3 \sqrt{a} \sqrt{b} \left (a + b x^{2}\right )^{\frac{3}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(b*x**2+a)**(7/4),x)
[Out]
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Mathematica [C] time = 0.0478071, size = 55, normalized size = 0.71 \[ \frac{x \left (\left (\frac{b x^2}{a}+1\right )^{3/4} \, _2F_1\left (\frac{1}{2},\frac{3}{4};\frac{3}{2};-\frac{b x^2}{a}\right )+2\right )}{3 a \left (a+b x^2\right )^{3/4}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(-7/4),x]
[Out]
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Maple [F] time = 0.044, 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(1/(b*x^2+a)^(7/4),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
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. \[{\rm integral}\left (\frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}, x\right ) \]
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 = 4.04126, size = 24, normalized size = 0.31 \[ \frac{x{{}_{2}F_{1}\left (\begin{matrix} \frac{1}{2}, \frac{7}{4} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )}}{a^{\frac{7}{4}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(b*x**2+a)**(7/4),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{{\left (b x^{2} + a\right )}^{\frac{7}{4}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x^2 + a)^(-7/4),x, algorithm="giac")
[Out]