Optimal. Leaf size=137 \[ \frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]
[Out]
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Rubi [A] time = 0.288395, antiderivative size = 137, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208 \[ \frac{3 c \sqrt{b x+c x^2} (A c+4 b B)}{4 b \sqrt{x}}-\frac{3 c (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{4 \sqrt{b}}-\frac{\left (b x+c x^2\right )^{3/2} (A c+4 b B)}{4 b x^{5/2}}-\frac{A \left (b x+c x^2\right )^{5/2}}{2 b x^{9/2}} \]
Antiderivative was successfully verified.
[In] Int[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(9/2),x]
[Out]
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Rubi in Sympy [A] time = 17.7308, size = 124, normalized size = 0.91 \[ - \frac{A \left (b x + c x^{2}\right )^{\frac{5}{2}}}{2 b x^{\frac{9}{2}}} + \frac{3 c \left (A c + 4 B b\right ) \sqrt{b x + c x^{2}}}{4 b \sqrt{x}} - \frac{\left (A c + 4 B b\right ) \left (b x + c x^{2}\right )^{\frac{3}{2}}}{4 b x^{\frac{5}{2}}} - \frac{3 c \left (A c + 4 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{4 \sqrt{b}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)*(c*x**2+b*x)**(3/2)/x**(9/2),x)
[Out]
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Mathematica [A] time = 0.143693, size = 101, normalized size = 0.74 \[ -\frac{\sqrt{x (b+c x)} \left (3 c x^2 (A c+4 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )+\sqrt{b} \sqrt{b+c x} (A (2 b+5 c x)+4 B x (b-2 c x))\right )}{4 \sqrt{b} x^{5/2} \sqrt{b+c x}} \]
Antiderivative was successfully verified.
[In] Integrate[((A + B*x)*(b*x + c*x^2)^(3/2))/x^(9/2),x]
[Out]
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Maple [A] time = 0.026, size = 126, normalized size = 0.9 \[ -{\frac{1}{4}\sqrt{x \left ( cx+b \right ) } \left ( 3\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}{c}^{2}+12\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{2}bc-8\,B{x}^{2}c\sqrt{b}\sqrt{cx+b}+5\,Axc\sqrt{cx+b}\sqrt{b}+4\,Bx{b}^{3/2}\sqrt{cx+b}+2\,A{b}^{3/2}\sqrt{cx+b} \right ){x}^{-{\frac{5}{2}}}{\frac{1}{\sqrt{cx+b}}}{\frac{1}{\sqrt{b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)*(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)*(B*x + A)/x^(9/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.299076, size = 1, normalized size = 0.01 \[ \left [\frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{c x^{2} + b x} x^{\frac{3}{2}} \log \left (\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} -{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (8 \, B c^{2} x^{3} - 2 \, A b^{2} +{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} -{\left (4 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{b}}{8 \, \sqrt{c x^{2} + b x} \sqrt{b} x^{\frac{3}{2}}}, -\frac{3 \,{\left (4 \, B b c + A c^{2}\right )} \sqrt{c x^{2} + b x} x^{\frac{3}{2}} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) -{\left (8 \, B c^{2} x^{3} - 2 \, A b^{2} +{\left (4 \, B b c - 5 \, A c^{2}\right )} x^{2} -{\left (4 \, B b^{2} + 7 \, A b c\right )} x\right )} \sqrt{-b}}{4 \, \sqrt{c x^{2} + b x} \sqrt{-b} x^{\frac{3}{2}}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(B*x + 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+A)*(c*x**2+b*x)**(3/2)/x**(9/2),x)
[Out]
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GIAC/XCAS [A] time = 0.313692, size = 161, normalized size = 1.18 \[ \frac{8 \, \sqrt{c x + b} B c^{2} + \frac{3 \,{\left (4 \, B b c^{2} + A c^{3}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b}} - \frac{4 \,{\left (c x + b\right )}^{\frac{3}{2}} B b c^{2} - 4 \, \sqrt{c x + b} B b^{2} c^{2} + 5 \,{\left (c x + b\right )}^{\frac{3}{2}} A c^{3} - 3 \, \sqrt{c x + b} A b c^{3}}{c^{2} x^{2}}}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2 + b*x)^(3/2)*(B*x + A)/x^(9/2),x, algorithm="giac")
[Out]