Optimal. Leaf size=111 \[ -\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]
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Rubi [A] time = 0.0861093, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.364 \[ -\frac{14 a^{5/2} \sqrt [4]{\frac{b x^2}{a}+1} E\left (\left .\frac{1}{2} \tan ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a}}\right )\right |2\right )}{15 \sqrt{b} \sqrt [4]{a+b x^2}}+\frac{14 a^2 x}{15 \sqrt [4]{a+b x^2}}+\frac{14}{45} a x \left (a+b x^2\right )^{3/4}+\frac{2}{9} x \left (a+b x^2\right )^{7/4} \]
Antiderivative was successfully verified.
[In] Int[(a + b*x^2)^(7/4),x]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{7 a^{3} \int \frac{1}{\left (a + b x^{2}\right )^{\frac{5}{4}}}\, dx}{15} + \frac{14 a^{2} x}{15 \sqrt [4]{a + b x^{2}}} + \frac{14 a x \left (a + b x^{2}\right )^{\frac{3}{4}}}{45} + \frac{2 x \left (a + b x^{2}\right )^{\frac{7}{4}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x**2+a)**(7/4),x)
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Mathematica [C] time = 0.0478442, size = 76, normalized size = 0.68 \[ \frac{21 a^2 x \sqrt [4]{\frac{b x^2}{a}+1} \, _2F_1\left (\frac{1}{4},\frac{1}{2};\frac{3}{2};-\frac{b x^2}{a}\right )+24 a^2 x+34 a b x^3+10 b^2 x^5}{45 \sqrt [4]{a+b x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(a + b*x^2)^(7/4),x]
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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((b*x^2+a)^(7/4),x)
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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")
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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")
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Sympy [A] time = 10.2539, size = 26, normalized size = 0.23 \[ a^{\frac{7}{4}} x{{}_{2}F_{1}\left (\begin{matrix} - \frac{7}{4}, \frac{1}{2} \\ \frac{3}{2} \end{matrix}\middle |{\frac{b x^{2} e^{i \pi }}{a}} \right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x**2+a)**(7/4),x)
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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")
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