Optimal. Leaf size=204 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4} (11 b B-9 A c)}{231 b^3 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-9 A c)}{77 b^2 x^{9/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{11 b x^{13/2}} \]
[Out]
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Rubi [A] time = 0.572306, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.179 \[ \frac{5 c^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (11 b B-9 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 b^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 c \sqrt{b x^2+c x^4} (11 b B-9 A c)}{231 b^3 x^{5/2}}-\frac{2 \sqrt{b x^2+c x^4} (11 b B-9 A c)}{77 b^2 x^{9/2}}-\frac{2 A \sqrt{b x^2+c x^4}}{11 b x^{13/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x^2)/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Rubi in Sympy [A] time = 44.6411, size = 201, normalized size = 0.99 \[ - \frac{2 A \sqrt{b x^{2} + c x^{4}}}{11 b x^{\frac{13}{2}}} + \frac{2 \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 b^{2} x^{\frac{9}{2}}} - \frac{10 c \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 b^{3} x^{\frac{5}{2}}} - \frac{5 c^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (9 A c - 11 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 b^{\frac{13}{4}} x \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)/x**(11/2)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.350868, size = 181, normalized size = 0.89 \[ \frac{-2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (3 A \left (7 b^2-9 b c x^2+15 c^2 x^4\right )+11 b B x^2 \left (3 b-5 c x^2\right )\right )-10 i c^2 x^{13/2} \sqrt{\frac{b}{c x^2}+1} (9 A c-11 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{231 b^3 x^{9/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x^2)/(x^(11/2)*Sqrt[b*x^2 + c*x^4]),x]
[Out]
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Maple [A] time = 0.026, size = 274, normalized size = 1.3 \[ -{\frac{1}{231\,{b}^{3}} \left ( 45\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{5}{c}^{2}-55\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{x}^{5}bc+90\,A{c}^{3}{x}^{6}-110\,B{x}^{6}b{c}^{2}+36\,Ab{c}^{2}{x}^{4}-44\,B{x}^{4}{b}^{2}c-12\,A{b}^{2}c{x}^{2}+66\,B{x}^{2}{b}^{3}+42\,A{b}^{3} \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}{x}^{-{\frac{9}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)/x^(11/2)/(c*x^4+b*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),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((B*x**2+A)/x**(11/2)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{B x^{2} + A}{\sqrt{c x^{4} + b x^{2}} x^{\frac{11}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)/(sqrt(c*x^4 + b*x^2)*x^(11/2)),x, algorithm="giac")
[Out]