Optimal. Leaf size=111 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]
[Out]
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Rubi [A] time = 0.145275, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 19, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.21 \[ \frac{c^3 \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{3/2}}-\frac{c^2 \sqrt{b x+c x^2}}{8 b x^{3/2}}-\frac{c \sqrt{b x+c x^2}}{4 x^{5/2}}-\frac{\left (b x+c x^2\right )^{3/2}}{3 x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[(b*x + c*x^2)^(3/2)/x^(11/2),x]
[Out]
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Rubi in Sympy [A] time = 17.1847, size = 95, normalized size = 0.86 \[ - \frac{c \sqrt{b x + c x^{2}}}{4 x^{\frac{5}{2}}} - \frac{\left (b x + c x^{2}\right )^{\frac{3}{2}}}{3 x^{\frac{9}{2}}} - \frac{c^{2} \sqrt{b x + c x^{2}}}{8 b x^{\frac{3}{2}}} + \frac{c^{3} \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2+b*x)**(3/2)/x**(11/2),x)
[Out]
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Mathematica [A] time = 0.0945137, size = 94, normalized size = 0.85 \[ \frac{\sqrt{x (b+c x)} \left (3 c^3 x^3 \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} \sqrt{b+c x} \left (8 b^2+14 b c x+3 c^2 x^2\right )\right )}{24 b^{3/2} x^{7/2} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
[In] Integrate[(b*x + c*x^2)^(3/2)/x^(11/2),x]
[Out]
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Maple [A] time = 0.015, size = 90, normalized size = 0.8 \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 3\,{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-3\,{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}-14\,x{b}^{3/2}c\sqrt{cx+b}-8\,{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{3}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2+b*x)^(3/2)/x^(11/2),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^(11/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.237189, size = 1, normalized size = 0.01 \[ \left [\frac{3 \, c^{3} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) - 2 \,{\left (3 \, c^{2} x^{2} + 14 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{3}{2}} x^{4}}, \frac{3 \, c^{3} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (3 \, c^{2} x^{2} + 14 \, b c x + 8 \, b^{2}\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/x^(11/2),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**(11/2),x)
[Out]
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GIAC/XCAS [A] time = 0.244208, size = 97, normalized size = 0.87 \[ -\frac{1}{24} \, c^{3}{\left (\frac{3 \, \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b} + \frac{3 \,{\left (c x + b\right )}^{\frac{5}{2}} + 8 \,{\left (c x + b\right )}^{\frac{3}{2}} b - 3 \, \sqrt{c x + b} b^{2}}{b c^{3} x^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)/x^(11/2),x, algorithm="giac")
[Out]