Optimal. Leaf size=66 \[ -\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}} \]
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Rubi [A] time = 0.05, antiderivative size = 66, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {792, 660, 207} \begin {gather*} -\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^{3/2} \sqrt {b x+c x^2}} \, dx &=-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}}+\frac {\left (-\frac {3}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{b}\\ &=-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}}+\frac {\left (2 \left (-\frac {3}{2} (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b}\\ &=-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}}-\frac {(2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 73, normalized size = 1.11 \begin {gather*} \frac {-x \sqrt {b+c x} (2 b B-A c) \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )-A \sqrt {b} (b+c x)}{b^{3/2} \sqrt {x} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.19, size = 64, normalized size = 0.97 \begin {gather*} \frac {(A c-2 b B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{b^{3/2}}-\frac {A \sqrt {b x+c x^2}}{b x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 144, normalized size = 2.18 \begin {gather*} \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 \, \sqrt {c x^{2} + b x} A b \sqrt {x}}{2 \, b^{2} 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 ) - \sqrt {c x^{2} + b x} A b \sqrt {x}}{b^{2} x^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 58, normalized size = 0.88 \begin {gather*} -\frac {\frac {\sqrt {c x + b} A c}{b x} - \frac {{\left (2 \, B b c - A c^{2}\right )} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b}}{c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 71, normalized size = 1.08 \begin {gather*} \frac {\sqrt {\left (c x +b \right ) x}\, \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 )-\sqrt {c x +b}\, A \sqrt {b}\right )}{\sqrt {c x +b}\, b^{\frac {3}{2}} x^{\frac {3}{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 {3}{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 = -0.02 \begin {gather*} \int \frac {A+B\,x}{x^{3/2}\,\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 {3}{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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