Optimal. Leaf size=83 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
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Rubi [A] time = 0.10441, antiderivative size = 83, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.158 \[ -\frac{3 c^2 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{3 c \sqrt{b x+c x^2}}{4 x^{3/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{2 x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(3/2)/x^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 12.7864, size = 76, normalized size = 0.92 \[ - \frac{3 c \sqrt{b x + c x^{2}}}{4 x^{\frac{3}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{2 x^{\frac{7}{2}}} - \frac{3 c^{2} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(3/2)/x**(9/2),x)
[Out]
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Mathematica [A] time = 0.0693698, size = 82, normalized size = 0.99 \[ -\frac{\sqrt{x (b+c x)} \left (3 c^2 x^2 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )+\sqrt{b} \sqrt{b+c x} (2 b+5 c x)\right )}{4 \sqrt{b} x^{5/2} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(3/2)/x^(9/2),x]
[Out]
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Maple [A] time = 0.015, size = 72, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){c}^{2}{x}^{2}+5\,xc\sqrt{cx+b}\sqrt{b}+2\,{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(3/2)/x^(9/2),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^2 + b*x)^(3/2)/x^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232388, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{2} x^{3} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \, \sqrt{c x^{2} + b x}{\left (5 \, c x + 2 \, b\right )} \sqrt{b} \sqrt{x}}{8 \, \sqrt{b} x^{3}}, -\frac{3 \, c^{2} x^{3} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) + \sqrt{c x^{2} + b x}{\left (5 \, c x + 2 \, b\right )} \sqrt{-b} \sqrt{x}}{4 \, \sqrt{-b} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/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((c*x**2+b*x)**(3/2)/x**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.236076, size = 74, normalized size = 0.89 \[ \frac{1}{4} \, c^{2}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{5 \,{\left (c x + b\right )}^{\frac{3}{2}} - 3 \, \sqrt{c x + b} b}{c^{2} x^{2}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/x^(9/2),x, algorithm="giac")
[Out]