Optimal. Leaf size=105 \[ \frac {c (4 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}-\frac {\sqrt {b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}} \]
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Rubi [A] time = 0.09, antiderivative size = 105, 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, 672, 660, 207} \begin {gather*} -\frac {\sqrt {b x+c x^2} (4 b B-3 A c)}{4 b^2 x^{3/2}}+\frac {c (4 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 672
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{5/2} \sqrt {b x+c x^2}} \, dx &=-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {\left (-\frac {5}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{2 b}\\ &=-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}}-\frac {(4 b B-3 A c) \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {(c (4 b B-3 A c)) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}}-\frac {(4 b B-3 A c) \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {(c (4 b B-3 A c)) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^2}\\ &=-\frac {A \sqrt {b x+c x^2}}{2 b x^{5/2}}-\frac {(4 b B-3 A c) \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}+\frac {c (4 b B-3 A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 92, normalized size = 0.88 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (c x^2 (4 b B-3 A c) \tanh ^{-1}\left (\sqrt {\frac {c x}{b}+1}\right )+b \sqrt {\frac {c x}{b}+1} (-2 A b+3 A c x-4 b B x)\right )}{4 b^3 x^{5/2} \sqrt {\frac {c x}{b}+1}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.23, size = 87, normalized size = 0.83 \begin {gather*} \frac {\left (4 b B c-3 A c^2\right ) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{4 b^{5/2}}+\frac {\sqrt {b x+c x^2} (-2 A b+3 A c x-4 b B x)}{4 b^2 x^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 188, normalized size = 1.79 \begin {gather*} \left [-\frac {{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt {b} x^{3} \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 \, A b^{2} + {\left (4 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b^{3} x^{3}}, -\frac {{\left (4 \, B b c - 3 \, A c^{2}\right )} \sqrt {-b} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (2 \, A b^{2} + {\left (4 \, B b^{2} - 3 \, A b c\right )} x\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b^{3} x^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.26, size = 111, normalized size = 1.06 \begin {gather*} -\frac {\frac {{\left (4 \, B b c^{2} - 3 \, A c^{3}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {4 \, {\left (c x + b\right )}^{\frac {3}{2}} B b c^{2} - 4 \, \sqrt {c x + b} B b^{2} c^{2} - 3 \, {\left (c x + b\right )}^{\frac {3}{2}} A c^{3} + 5 \, \sqrt {c x + b} A b c^{3}}{b^{2} c^{2} x^{2}}}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 109, normalized size = 1.04 \begin {gather*} -\frac {\sqrt {\left (c x +b \right ) x}\, \left (3 A \,c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-4 B b c \,x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-3 \sqrt {c x +b}\, A \sqrt {b}\, c x +4 \sqrt {c x +b}\, B \,b^{\frac {3}{2}} x +2 \sqrt {c x +b}\, A \,b^{\frac {3}{2}}\right )}{4 \sqrt {c x +b}\, b^{\frac {5}{2}} x^{\frac {5}{2}}} \end {gather*}
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 \begin {gather*} \int \frac {B x + A}{\sqrt {c x^{2} + b x} x^{\frac {5}{2}}}\,{d x} \end {gather*}
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 \begin {gather*} \int \frac {A+B\,x}{x^{5/2}\,\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
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 \begin {gather*} \int \frac {A + B x}{x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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