Optimal. Leaf size=142 \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]
[Out]
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Rubi [A] time = 0.280801, antiderivative size = 142, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.167 \[ -\frac{c^2 (6 b B-5 A c) \tanh ^{-1}\left (\frac{\sqrt{b x+c x^2}}{\sqrt{b} \sqrt{x}}\right )}{8 b^{7/2}}+\frac{c \sqrt{b x+c x^2} (6 b B-5 A c)}{8 b^3 x^{3/2}}-\frac{\sqrt{b x+c x^2} (6 b B-5 A c)}{12 b^2 x^{5/2}}-\frac{A \sqrt{b x+c x^2}}{3 b x^{7/2}} \]
Antiderivative was successfully verified.
[In] Int[(A + B*x)/(x^(7/2)*Sqrt[b*x + c*x^2]),x]
[Out]
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Rubi in Sympy [A] time = 18.6606, size = 131, normalized size = 0.92 \[ - \frac{A \sqrt{b x + c x^{2}}}{3 b x^{\frac{7}{2}}} + \frac{\left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{12 b^{2} x^{\frac{5}{2}}} - \frac{c \left (5 A c - 6 B b\right ) \sqrt{b x + c x^{2}}}{8 b^{3} x^{\frac{3}{2}}} + \frac{c^{2} \left (5 A c - 6 B b\right ) \operatorname{atanh}{\left (\frac{\sqrt{b x + c x^{2}}}{\sqrt{b} \sqrt{x}} \right )}}{8 b^{\frac{7}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((B*x+A)/x**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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Mathematica [A] time = 0.14635, size = 115, normalized size = 0.81 \[ \frac{3 c^2 x^3 \sqrt{b+c x} (5 A c-6 b B) \tanh ^{-1}\left (\frac{\sqrt{b+c x}}{\sqrt{b}}\right )-\sqrt{b} (b+c x) \left (A \left (8 b^2-10 b c x+15 c^2 x^2\right )+6 b B x (2 b-3 c x)\right )}{24 b^{7/2} x^{5/2} \sqrt{x (b+c x)}} \]
Antiderivative was successfully verified.
[In] Integrate[(A + B*x)/(x^(7/2)*Sqrt[b*x + c*x^2]),x]
[Out]
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Maple [A] time = 0.027, size = 147, normalized size = 1. \[{\frac{1}{24}\sqrt{x \left ( cx+b \right ) } \left ( 15\,A{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}{c}^{3}-18\,B{\it Artanh} \left ({\frac{\sqrt{cx+b}}{\sqrt{b}}} \right ){x}^{3}b{c}^{2}-15\,A{x}^{2}{c}^{2}\sqrt{b}\sqrt{cx+b}+18\,B{x}^{2}{b}^{3/2}c\sqrt{cx+b}+10\,Ax{b}^{3/2}c\sqrt{cx+b}-12\,Bx{b}^{5/2}\sqrt{cx+b}-8\,A{b}^{5/2}\sqrt{cx+b} \right ){b}^{-{\frac{7}{2}}}{x}^{-{\frac{7}{2}}}{\frac{1}{\sqrt{cx+b}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((B*x+A)/x^(7/2)/(c*x^2+b*x)^(1/2),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((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.289036, size = 1, normalized size = 0.01 \[ \left [-\frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} \log \left (-\frac{2 \, \sqrt{c x^{2} + b x} b \sqrt{x} +{\left (c x^{2} + 2 \, b x\right )} \sqrt{b}}{x^{2}}\right ) + 2 \,{\left (8 \, A b^{2} - 3 \,{\left (6 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{b} \sqrt{x}}{48 \, b^{\frac{7}{2}} x^{4}}, -\frac{3 \,{\left (6 \, B b c^{2} - 5 \, A c^{3}\right )} x^{4} \arctan \left (\frac{\sqrt{-b} \sqrt{x}}{\sqrt{c x^{2} + b x}}\right ) +{\left (8 \, A b^{2} - 3 \,{\left (6 \, B b c - 5 \, A c^{2}\right )} x^{2} + 2 \,{\left (6 \, B b^{2} - 5 \, A b c\right )} x\right )} \sqrt{c x^{2} + b x} \sqrt{-b} \sqrt{x}}{24 \, \sqrt{-b} b^{3} x^{4}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),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)/x**(7/2)/(c*x**2+b*x)**(1/2),x)
[Out]
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GIAC/XCAS [A] time = 0.345696, size = 194, normalized size = 1.37 \[ \frac{\frac{3 \,{\left (6 \, B b c^{3} - 5 \, A c^{4}\right )} \arctan \left (\frac{\sqrt{c x + b}}{\sqrt{-b}}\right )}{\sqrt{-b} b^{3}} + \frac{18 \,{\left (c x + b\right )}^{\frac{5}{2}} B b c^{3} - 48 \,{\left (c x + b\right )}^{\frac{3}{2}} B b^{2} c^{3} + 30 \, \sqrt{c x + b} B b^{3} c^{3} - 15 \,{\left (c x + b\right )}^{\frac{5}{2}} A c^{4} + 40 \,{\left (c x + b\right )}^{\frac{3}{2}} A b c^{4} - 33 \, \sqrt{c x + b} A b^{2} c^{4}}{b^{3} c^{3} x^{3}}}{24 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((B*x + A)/(sqrt(c*x^2 + b*x)*x^(7/2)),x, algorithm="giac")
[Out]