Optimal. Leaf size=282 \[ \frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{60 b^{15/4} \sqrt{a+b x^4}}-\frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{a+b x^4}}+\frac{77 a^2 x \sqrt{a+b x^4}}{30 b^{7/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{77 a x^3 \sqrt{a+b x^4}}{90 b^3}+\frac{11 x^7 \sqrt{a+b x^4}}{18 b^2}-\frac{x^{11}}{2 b \sqrt{a+b x^4}} \]
[Out]
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Rubi [A] time = 0.303914, antiderivative size = 282, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333 \[ \frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{60 b^{15/4} \sqrt{a+b x^4}}-\frac{77 a^{9/4} \left (\sqrt{a}+\sqrt{b} x^2\right ) \sqrt{\frac{a+b x^4}{\left (\sqrt{a}+\sqrt{b} x^2\right )^2}} E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac{1}{2}\right )}{30 b^{15/4} \sqrt{a+b x^4}}+\frac{77 a^2 x \sqrt{a+b x^4}}{30 b^{7/2} \left (\sqrt{a}+\sqrt{b} x^2\right )}-\frac{77 a x^3 \sqrt{a+b x^4}}{90 b^3}+\frac{11 x^7 \sqrt{a+b x^4}}{18 b^2}-\frac{x^{11}}{2 b \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Int[x^14/(a + b*x^4)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 35.9243, size = 258, normalized size = 0.91 \[ - \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{30 b^{\frac{15}{4}} \sqrt{a + b x^{4}}} + \frac{77 a^{\frac{9}{4}} \sqrt{\frac{a + b x^{4}}{\left (\sqrt{a} + \sqrt{b} x^{2}\right )^{2}}} \left (\sqrt{a} + \sqrt{b} x^{2}\right ) F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{b} x}{\sqrt [4]{a}} \right )}\middle | \frac{1}{2}\right )}{60 b^{\frac{15}{4}} \sqrt{a + b x^{4}}} + \frac{77 a^{2} x \sqrt{a + b x^{4}}}{30 b^{\frac{7}{2}} \left (\sqrt{a} + \sqrt{b} x^{2}\right )} - \frac{77 a x^{3} \sqrt{a + b x^{4}}}{90 b^{3}} - \frac{x^{11}}{2 b \sqrt{a + b x^{4}}} + \frac{11 x^{7} \sqrt{a + b x^{4}}}{18 b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**14/(b*x**4+a)**(3/2),x)
[Out]
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Mathematica [C] time = 0.344506, size = 183, normalized size = 0.65 \[ \frac{-231 a^{5/2} \sqrt{\frac{b x^4}{a}+1} F\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+231 a^{5/2} \sqrt{\frac{b x^4}{a}+1} E\left (\left .i \sinh ^{-1}\left (\sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} x\right )\right |-1\right )+\sqrt{b} x^3 \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \left (-77 a^2-22 a b x^4+10 b^2 x^8\right )}{90 b^{7/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{a}}} \sqrt{a+b x^4}} \]
Antiderivative was successfully verified.
[In] Integrate[x^14/(a + b*x^4)^(3/2),x]
[Out]
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Maple [C] time = 0.018, size = 157, normalized size = 0.6 \[ -{\frac{{x}^{3}{a}^{2}}{2\,{b}^{3}}{\frac{1}{\sqrt{ \left ({x}^{4}+{\frac{a}{b}} \right ) b}}}}+{\frac{{x}^{7}}{9\,{b}^{2}}\sqrt{b{x}^{4}+a}}-{\frac{16\,a{x}^{3}}{45\,{b}^{3}}\sqrt{b{x}^{4}+a}}+{{\frac{77\,i}{30}}{a}^{{\frac{5}{2}}}\sqrt{1-{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}}\sqrt{1+{i{x}^{2}\sqrt{b}{\frac{1}{\sqrt{a}}}}} \left ({\it EllipticF} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) -{\it EllipticE} \left ( x\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}},i \right ) \right ){b}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{{i\sqrt{b}{\frac{1}{\sqrt{a}}}}}}}{\frac{1}{\sqrt{b{x}^{4}+a}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^14/(b*x^4+a)^(3/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 10.2945, size = 37, normalized size = 0.13 \[ \frac{x^{15} \Gamma \left (\frac{15}{4}\right ){{}_{2}F_{1}\left (\begin{matrix} \frac{3}{2}, \frac{15}{4} \\ \frac{19}{4} \end{matrix}\middle |{\frac{b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac{3}{2}} \Gamma \left (\frac{19}{4}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**14/(b*x**4+a)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{14}}{{\left (b x^{4} + a\right )}^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x^14/(b*x^4 + a)^(3/2),x, algorithm="giac")
[Out]