Optimal. Leaf size=95 \[ \frac {\sqrt {b x+c x^2} (A c+2 b B)}{b \sqrt {x}}-\frac {(A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 95, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.167, Rules used = {792, 664, 660, 207} \[ \frac {\sqrt {b x+c x^2} (A c+2 b B)}{b \sqrt {x}}-\frac {(A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 664
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^{5/2}} \, dx &=-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}}+\frac {\left (-\frac {5}{2} (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^{3/2}} \, dx}{b}\\ &=\frac {(2 b B+A c) \sqrt {b x+c x^2}}{b \sqrt {x}}-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}}+\frac {1}{2} (2 b B+A c) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx\\ &=\frac {(2 b B+A c) \sqrt {b x+c x^2}}{b \sqrt {x}}-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}}+(2 b B+A c) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )\\ &=\frac {(2 b B+A c) \sqrt {b x+c x^2}}{b \sqrt {x}}-\frac {A \left (b x+c x^2\right )^{3/2}}{b x^{5/2}}-\frac {(2 b B+A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{\sqrt {b}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 80, normalized size = 0.84 \[ -\frac {\sqrt {x (b+c x)} \left (\sqrt {b} (A-2 B x) \sqrt {b+c x}+x (A c+2 b B) \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{\sqrt {b} x^{3/2} \sqrt {b+c x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.99, size = 157, normalized size = 1.65 \[ \left [\frac {{\left (2 \, B b + A c\right )} \sqrt {b} x^{2} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (2 \, B b x - A b\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{2 \, b x^{2}}, \frac {{\left (2 \, B b + A c\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (2 \, B b x - A b\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{b x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.28, size = 61, normalized size = 0.64 \[ \frac {2 \, \sqrt {c x + b} B c - \frac {\sqrt {c x + b} A c}{x} + \frac {{\left (2 \, B b c + A c^{2}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b}}}{c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 86, normalized size = 0.91 \[ \frac {\left (-A c x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-2 B b x \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )+2 \sqrt {c x +b}\, B \sqrt {b}\, x -\sqrt {c x +b}\, A \sqrt {b}\right ) \sqrt {\left (c x +b \right ) x}}{\sqrt {c x +b}\, \sqrt {b}\, x^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {c x^{2} + b x} {\left (B x + A\right )}}{x^{\frac {5}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{x^{5/2}} \,d x \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{\frac {5}{2}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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