Optimal. Leaf size=230 \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]
[Out]
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Rubi [A] time = 0.425478, antiderivative size = 230, normalized size of antiderivative = 1., number of steps used = 12, number of rules used = 9, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.6 \[ \frac{5 \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )}{8 \sqrt{2} b^{9/4}}+\frac{5 \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )}{4 \sqrt{2} b^{9/4}}-\frac{5 \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )}{4 \sqrt{2} b^{9/4}}-\frac{x^{5/2}}{2 b \left (a+b x^2\right )}+\frac{5 \sqrt{x}}{2 b^2} \]
Antiderivative was successfully verified.
[In] Int[x^(7/2)/(a + b*x^2)^2,x]
[Out]
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Rubi in Sympy [A] time = 70.1972, size = 216, normalized size = 0.94 \[ \frac{5 \sqrt{2} \sqrt [4]{a} \log{\left (- \sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{a} \log{\left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x} + \sqrt{a} + \sqrt{b} x \right )}}{16 b^{\frac{9}{4}}} + \frac{5 \sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 - \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{9}{4}}} - \frac{5 \sqrt{2} \sqrt [4]{a} \operatorname{atan}{\left (1 + \frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}} \right )}}{8 b^{\frac{9}{4}}} - \frac{x^{\frac{5}{2}}}{2 b \left (a + b x^{2}\right )} + \frac{5 \sqrt{x}}{2 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(7/2)/(b*x**2+a)**2,x)
[Out]
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Mathematica [A] time = 0.238733, size = 212, normalized size = 0.92 \[ \frac{\frac{8 a \sqrt [4]{b} \sqrt{x}}{a+b x^2}+5 \sqrt{2} \sqrt [4]{a} \log \left (-\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )-5 \sqrt{2} \sqrt [4]{a} \log \left (\sqrt{2} \sqrt [4]{a} \sqrt [4]{b} \sqrt{x}+\sqrt{a}+\sqrt{b} x\right )+10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (1-\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}\right )-10 \sqrt{2} \sqrt [4]{a} \tan ^{-1}\left (\frac{\sqrt{2} \sqrt [4]{b} \sqrt{x}}{\sqrt [4]{a}}+1\right )+32 \sqrt [4]{b} \sqrt{x}}{16 b^{9/4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^(7/2)/(a + b*x^2)^2,x]
[Out]
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Maple [A] time = 0.018, size = 158, normalized size = 0.7 \[ 2\,{\frac{\sqrt{x}}{{b}^{2}}}+{\frac{a}{2\,{b}^{2} \left ( b{x}^{2}+a \right ) }\sqrt{x}}-{\frac{5\,\sqrt{2}}{16\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\ln \left ({1 \left ( x+\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) \left ( x-\sqrt [4]{{\frac{a}{b}}}\sqrt{x}\sqrt{2}+\sqrt{{\frac{a}{b}}} \right ) ^{-1}} \right ) }-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}+1 \right ) }-{\frac{5\,\sqrt{2}}{8\,{b}^{2}}\sqrt [4]{{\frac{a}{b}}}\arctan \left ({\sqrt{2}\sqrt{x}{\frac{1}{\sqrt [4]{{\frac{a}{b}}}}}}-1 \right ) } \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(7/2)/(b*x^2+a)^2,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(b*x^2 + a)^2,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.288531, size = 239, normalized size = 1.04 \[ \frac{20 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \arctan \left (\frac{b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}}}{\sqrt{b^{4} \sqrt{-\frac{a}{b^{9}}} + x} + \sqrt{x}}\right ) - 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) + 5 \,{\left (b^{3} x^{2} + a b^{2}\right )} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} \log \left (-5 \, b^{2} \left (-\frac{a}{b^{9}}\right )^{\frac{1}{4}} + 5 \, \sqrt{x}\right ) + 4 \,{\left (4 \, b x^{2} + 5 \, a\right )} \sqrt{x}}{8 \,{\left (b^{3} x^{2} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(b*x^2 + a)^2,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(x**(7/2)/(b*x**2+a)**2,x)
[Out]
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GIAC/XCAS [A] time = 0.218843, size = 265, normalized size = 1.15 \[ -\frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} + 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}} \arctan \left (-\frac{\sqrt{2}{\left (\sqrt{2} \left (\frac{a}{b}\right )^{\frac{1}{4}} - 2 \, \sqrt{x}\right )}}{2 \, \left (\frac{a}{b}\right )^{\frac{1}{4}}}\right )}{8 \, b^{3}} - \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{3}} + \frac{5 \, \sqrt{2} \left (a b^{3}\right )^{\frac{1}{4}}{\rm ln}\left (-\sqrt{2} \sqrt{x} \left (\frac{a}{b}\right )^{\frac{1}{4}} + x + \sqrt{\frac{a}{b}}\right )}{16 \, b^{3}} + \frac{a \sqrt{x}}{2 \,{\left (b x^{2} + a\right )} b^{2}} + \frac{2 \, \sqrt{x}}{b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^(7/2)/(b*x^2 + a)^2,x, algorithm="giac")
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