Optimal. Leaf size=328 \[ \frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+5 b B)}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (A c+5 b B)}{5 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{5 b x^{11/2}} \]
[Out]
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Rubi [A] time = 0.701611, antiderivative size = 328, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.25 \[ \frac{2 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}-\frac{4 \sqrt [4]{c} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (A c+5 b B) E\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{5 b^{3/4} \sqrt{b x^2+c x^4}}+\frac{4 \sqrt{c} x^{3/2} \left (b+c x^2\right ) (A c+5 b B)}{5 b \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{b x^2+c x^4}}-\frac{2 \sqrt{b x^2+c x^4} (A c+5 b B)}{5 b x^{3/2}}-\frac{2 A \left (b x^2+c x^4\right )^{3/2}}{5 b x^{11/2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 59.9583, size = 309, normalized size = 0.94 \[ - \frac{2 A \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{5 b x^{\frac{11}{2}}} + \frac{4 \sqrt{c} \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b \sqrt{x} \left (\sqrt{b} + \sqrt{c} x\right )} - \frac{2 \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{5 b x^{\frac{3}{2}}} - \frac{4 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}} E\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} x \left (b + c x^{2}\right )} + \frac{2 \sqrt [4]{c} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (A c + 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{5 b^{\frac{3}{4}} x \left (b + c x^{2}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*(c*x**4+b*x**2)**(1/2)/x**(9/2),x)
[Out]
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Mathematica [C] time = 1.03788, size = 219, normalized size = 0.67 \[ \frac{2 \left (\sqrt{b} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (5 B x^2-A\right ) \left (b+c x^2\right )+2 \sqrt{c} x^{7/2} \sqrt{\frac{b}{c x^2}+1} (A c+5 b B) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )-2 \sqrt{c} x^{7/2} \sqrt{\frac{b}{c x^2}+1} (A c+5 b B) E\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )\right )}{5 \sqrt{b} x^{3/2} \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x^2)*Sqrt[b*x^2 + c*x^4])/x^(9/2),x]
[Out]
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Maple [A] time = 0.05, size = 422, normalized size = 1.3 \[{\frac{2}{ \left ( 5\,c{x}^{2}+5\,b \right ) b}\sqrt{c{x}^{4}+b{x}^{2}} \left ( 2\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}bc-A\sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{2}\sqrt{{1 \left ( -cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}}\sqrt{-{cx{\frac{1}{\sqrt{-bc}}}}}{\it EllipticF} \left ( \sqrt{{1 \left ( cx+\sqrt{-bc} \right ){\frac{1}{\sqrt{-bc}}}}},{\frac{\sqrt{2}}{2}} \right ){x}^{2}bc+10\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticE} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}-5\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ){x}^{2}{b}^{2}-2\,A{c}^{2}{x}^{4}-5\,B{x}^{4}bc-3\,Abc{x}^{2}-5\,B{b}^{2}{x}^{2}-{b}^{2}A \right ){x}^{-{\frac{7}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*(c*x^4+b*x^2)^(1/2)/x^(9/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(9/2),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(9/2),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)*(c*x**4+b*x**2)**(1/2)/x**(9/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )}}{x^{\frac{9}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)/x^(9/2),x, algorithm="giac")
[Out]