Optimal. Leaf size=204 \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]
[Out]
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Rubi [A] time = 0.526323, antiderivative size = 204, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 6, integrand size = 28, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.214 \[ \frac{2 b^{7/4} x \left (\sqrt{b}+\sqrt{c} x\right ) \sqrt{\frac{b+c x^2}{\left (\sqrt{b}+\sqrt{c} x\right )^2}} (5 b B-11 A c) F\left (2 \tan ^{-1}\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}}\right )|\frac{1}{2}\right )}{231 c^{9/4} \sqrt{b x^2+c x^4}}-\frac{4 b \sqrt{b x^2+c x^4} (5 b B-11 A c)}{231 c^2 \sqrt{x}}-\frac{2 x^{3/2} \sqrt{b x^2+c x^4} (5 b B-11 A c)}{77 c}+\frac{2 B \left (b x^2+c x^4\right )^{3/2}}{11 c \sqrt{x}} \]
Antiderivative was successfully verified.
[In] Int[Sqrt[x]*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]
[Out]
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Rubi in Sympy [A] time = 44.3876, size = 199, normalized size = 0.98 \[ \frac{2 B \left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{11 c \sqrt{x}} - \frac{2 b^{\frac{7}{4}} \sqrt{\frac{b + c x^{2}}{\left (\sqrt{b} + \sqrt{c} x\right )^{2}}} \left (\sqrt{b} + \sqrt{c} x\right ) \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}} F\left (2 \operatorname{atan}{\left (\frac{\sqrt [4]{c} \sqrt{x}}{\sqrt [4]{b}} \right )}\middle | \frac{1}{2}\right )}{231 c^{\frac{9}{4}} x \left (b + c x^{2}\right )} + \frac{4 b \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{231 c^{2} \sqrt{x}} + \frac{2 x^{\frac{3}{2}} \left (11 A c - 5 B b\right ) \sqrt{b x^{2} + c x^{4}}}{77 c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x**2+A)*x**(1/2)*(c*x**4+b*x**2)**(1/2),x)
[Out]
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Mathematica [C] time = 0.720126, size = 159, normalized size = 0.78 \[ \frac{1}{231} \sqrt{x^2 \left (b+c x^2\right )} \left (\frac{4 b c \left (11 A+3 B x^2\right )+6 c^2 x^2 \left (11 A+7 B x^2\right )-20 b^2 B}{c^2 \sqrt{x}}+\frac{4 i b^2 \sqrt{\frac{b}{c x^2}+1} (5 b B-11 A c) F\left (\left .i \sinh ^{-1}\left (\frac{\sqrt{\frac{i \sqrt{b}}{\sqrt{c}}}}{\sqrt{x}}\right )\right |-1\right )}{c^2 \sqrt{\frac{i \sqrt{b}}{\sqrt{c}}} \left (b+c x^2\right )}\right ) \]
Antiderivative was successfully verified.
[In] Integrate[Sqrt[x]*(A + B*x^2)*Sqrt[b*x^2 + c*x^4],x]
[Out]
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Maple [A] time = 0.038, size = 283, normalized size = 1.4 \[ -{\frac{2}{ \left ( 231\,c{x}^{2}+231\,b \right ){c}^{3}}\sqrt{c{x}^{4}+b{x}^{2}} \left ( -21\,B{x}^{7}{c}^{4}+11\,A\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{2}c-33\,A{x}^{5}{c}^{4}-5\,B\sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{2}\sqrt{{\frac{-cx+\sqrt{-bc}}{\sqrt{-bc}}}}\sqrt{-{\frac{cx}{\sqrt{-bc}}}}{\it EllipticF} \left ( \sqrt{{\frac{cx+\sqrt{-bc}}{\sqrt{-bc}}}},1/2\,\sqrt{2} \right ) \sqrt{-bc}{b}^{3}-27\,B{x}^{5}b{c}^{3}-55\,A{x}^{3}b{c}^{3}+4\,B{x}^{3}{b}^{2}{c}^{2}-22\,Ax{b}^{2}{c}^{2}+10\,Bx{b}^{3}c \right ){x}^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x^2+A)*x^(1/2)*(c*x^4+b*x^2)^(1/2),x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)*sqrt(x),x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)*sqrt(x),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{x} \sqrt{x^{2} \left (b + c x^{2}\right )} \left (A + B x^{2}\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x**2+A)*x**(1/2)*(c*x**4+b*x**2)**(1/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \sqrt{c x^{4} + b x^{2}}{\left (B x^{2} + A\right )} \sqrt{x}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^4 + b*x^2)*(B*x^2 + A)*sqrt(x),x, algorithm="giac")
[Out]