Optimal. Leaf size=204 \[ -\frac{5 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 b \sqrt{b x^2+c x^4} (9 b B-11 A c)}{231 c^3 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-11 A c)}{77 c^2}+\frac{2 B x^{7/2} \sqrt{b x^2+c x^4}}{11 c} \]
[Out]
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Rubi [A] time = 0.556507, 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 \[ -\frac{5 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (9 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{13/4} \sqrt{b x^2+c x^4}}+\frac{10 b \sqrt{b x^2+c x^4} (9 b B-11 A c)}{231 c^3 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (9 b B-11 A c)}{77 c^2}+\frac{2 B x^{7/2} \sqrt{b x^2+c x^4}}{11 c} \]
Antiderivative was successfully verified.
[In] Int[(x^(9/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 44.1369, size = 201, normalized size = 0.99 \[ \frac{2 B x^{\frac{7}{2}} \sqrt{b x^{2} + c x^{4}}}{11 c} + \frac{5 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{13}{4}} x \left (b + c x^{2}\right )} - \frac{10 b \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 c^{3} \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \left (11 A c - 9 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**(9/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.305409, size = 176, normalized size = 0.86 \[ \frac{2 x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right ) \left (-b c \left (55 A+27 B x^2\right )+3 c^2 x^2 \left (11 A+7 B x^2\right )+45 b^2 B\right )+10 i b^2 x^2 \sqrt{\frac{b}{c x^2}+1} (11 A c-9 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{231 c^3 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^(9/2)*(A + B*x^2))/Sqrt[b*x^2 + c*x^4],x]
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Maple [A] time = 0.025, size = 274, normalized size = 1.3 \[{\frac{1}{231\,{c}^{4}}\sqrt{x} \left ( 42\,B{x}^{7}{c}^{4}+55\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{2}c+66\,A{x}^{5}{c}^{4}-45\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{3}-12\,B{x}^{5}b{c}^{3}-44\,A{x}^{3}b{c}^{3}+36\,B{x}^{3}{b}^{2}{c}^{2}-110\,Ax{b}^{2}{c}^{2}+90\,Bx{b}^{3}c \right ){\frac{1}{\sqrt{c{x}^{4}+b{x}^{2}}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^(9/2)*(B*x^2+A)/(c*x^4+b*x^2)^(1/2),x)
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{{\left (B x^{6} + A x^{4}\right )} \sqrt{x}}{\sqrt{c x^{4} + b x^{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^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**(9/2)*(B*x**2+A)/(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (B x^{2} + A\right )} x^{\frac{9}{2}}}{\sqrt{c x^{4} + b x^{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x^2 + A)*x^(9/2)/sqrt(c*x^4 + b*x^2),x, algorithm="giac")
[Out]