Optimal. Leaf size=165 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac{3 c^4 \sqrt{b x^2+c x^4}}{256 b^3 x^3}+\frac{c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{160 b x^7}-\frac{\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac{3 c \sqrt{b x^2+c x^4}}{80 x^9} \]
[Out]
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Rubi [A] time = 0.43497, antiderivative size = 165, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{3 c^5 \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{b x^2+c x^4}}\right )}{256 b^{7/2}}-\frac{3 c^4 \sqrt{b x^2+c x^4}}{256 b^3 x^3}+\frac{c^3 \sqrt{b x^2+c x^4}}{128 b^2 x^5}-\frac{c^2 \sqrt{b x^2+c x^4}}{160 b x^7}-\frac{\left (b x^2+c x^4\right )^{3/2}}{10 x^{13}}-\frac{3 c \sqrt{b x^2+c x^4}}{80 x^9} \]
Antiderivative was successfully verified.
[In] Int[(b*x^2 + c*x^4)^(3/2)/x^14,x]
[Out]
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Rubi in Sympy [A] time = 47.7787, size = 150, normalized size = 0.91 \[ - \frac{3 c \sqrt{b x^{2} + c x^{4}}}{80 x^{9}} - \frac{\left (b x^{2} + c x^{4}\right )^{\frac{3}{2}}}{10 x^{13}} - \frac{c^{2} \sqrt{b x^{2} + c x^{4}}}{160 b x^{7}} + \frac{c^{3} \sqrt{b x^{2} + c x^{4}}}{128 b^{2} x^{5}} - \frac{3 c^{4} \sqrt{b x^{2} + c x^{4}}}{256 b^{3} x^{3}} + \frac{3 c^{5} \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{b x^{2} + c x^{4}}} \right )}}{256 b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**4+b*x**2)**(3/2)/x**14,x)
[Out]
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Mathematica [A] time = 0.270946, size = 137, normalized size = 0.83 \[ -\frac{\sqrt{b+c x^2} \left (\sqrt{b} \sqrt{b+c x^2} \left (128 b^4+176 b^3 c x^2+8 b^2 c^2 x^4-10 b c^3 x^6+15 c^4 x^8\right )-15 c^5 x^{10} \log \left (\sqrt{b} \sqrt{b+c x^2}+b\right )+15 c^5 x^{10} \log (x)\right )}{1280 b^{7/2} x^9 \sqrt{x^2 \left (b+c x^2\right )}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x^2 + c*x^4)^(3/2)/x^14,x]
[Out]
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Maple [A] time = 0.043, size = 186, normalized size = 1.1 \[{\frac{1}{1280\,{x}^{13}{b}^{5}} \left ( c{x}^{4}+b{x}^{2} \right ) ^{{\frac{3}{2}}} \left ( 15\,\ln \left ( 2\,{\frac{\sqrt{b}\sqrt{c{x}^{2}+b}+b}{x}} \right ){b}^{3/2}{x}^{10}{c}^{5}-5\, \left ( c{x}^{2}+b \right ) ^{3/2}{x}^{10}{c}^{5}+5\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{8}{c}^{4}-15\,\sqrt{c{x}^{2}+b}{x}^{10}b{c}^{5}+10\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{6}b{c}^{3}-40\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{4}{b}^{2}{c}^{2}+80\, \left ( c{x}^{2}+b \right ) ^{5/2}{x}^{2}{b}^{3}c-128\, \left ( c{x}^{2}+b \right ) ^{5/2}{b}^{4} \right ) \left ( c{x}^{2}+b \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^4+b*x^2)^(3/2)/x^14,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)/x^14,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.33097, size = 1, normalized size = 0.01 \[ \left [\frac{15 \, \sqrt{b} c^{5} x^{11} \log \left (-\frac{{\left (c x^{3} + 2 \, b x\right )} \sqrt{b} + 2 \, \sqrt{c x^{4} + b x^{2}} b}{x^{3}}\right ) - 2 \,{\left (15 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\right )} \sqrt{c x^{4} + b x^{2}}}{2560 \, b^{4} x^{11}}, -\frac{15 \, \sqrt{-b} c^{5} x^{11} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{c x^{4} + b x^{2}}}\right ) +{\left (15 \, b c^{4} x^{8} - 10 \, b^{2} c^{3} x^{6} + 8 \, b^{3} c^{2} x^{4} + 176 \, b^{4} c x^{2} + 128 \, b^{5}\right )} \sqrt{c x^{4} + b x^{2}}}{1280 \, b^{4} x^{11}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)/x^14,x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (x^{2} \left (b + c x^{2}\right )\right )^{\frac{3}{2}}}{x^{14}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**4+b*x**2)**(3/2)/x**14,x)
[Out]
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GIAC/XCAS [A] time = 0.34379, size = 149, normalized size = 0.9 \[ -\frac{1}{1280} \, c^{5}{\left (\frac{15 \, \arctan \left (\frac{\sqrt{c x^{2} + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{15 \,{\left (c x^{2} + b\right )}^{\frac{9}{2}} - 70 \,{\left (c x^{2} + b\right )}^{\frac{7}{2}} b + 128 \,{\left (c x^{2} + b\right )}^{\frac{5}{2}} b^{2} + 70 \,{\left (c x^{2} + b\right )}^{\frac{3}{2}} b^{3} - 15 \, \sqrt{c x^{2} + b} b^{4}}{b^{3} c^{5} x^{10}}\right )}{\rm sign}\left (x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^4 + b*x^2)^(3/2)/x^14,x, algorithm="giac")
[Out]