Optimal. Leaf size=68 \[ -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {2 \sqrt {x} (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.05, antiderivative size = 68, 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 = {788, 660, 207} \begin {gather*} -\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{b^{3/2}}-\frac {2 \sqrt {x} (b B-A c)}{b c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 788
Rubi steps
\begin {align*} \int \frac {\sqrt {x} (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) \sqrt {x}}{b c \sqrt {b x+c x^2}}+\frac {A \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{b}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{b c \sqrt {b x+c x^2}}+\frac {(2 A) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{b}\\ &=-\frac {2 (b B-A c) \sqrt {x}}{b c \sqrt {b x+c x^2}}-\frac {2 A \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 = 69, normalized size = 1.01 \begin {gather*} -\frac {2 \sqrt {x} \left (\sqrt {b} (b B-A c)+A c \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )\right )}{b^{3/2} c \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.75, size = 75, normalized size = 1.10 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (A c-b B)}{b c \sqrt {x} (b+c x)}-\frac {2 A \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {b x+c x^2}}\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 192, normalized size = 2.82 \begin {gather*} \left [\frac {{\left (A c^{2} x^{2} + A b c x\right )} \sqrt {b} \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 (B b^{2} - A b c\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{b^{2} c^{2} x^{2} + b^{3} c x}, \frac {2 \, {\left ({\left (A c^{2} x^{2} + A b c x\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) - {\left (B b^{2} - A b c\right )} \sqrt {c x^{2} + b x} \sqrt {x}\right )}}{b^{2} c^{2} x^{2} + b^{3} c x}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 96, normalized size = 1.41 \begin {gather*} \frac {2 \, A \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b} - \frac {2 \, {\left (B b - A c\right )}}{\sqrt {c x + b} b c} - \frac {2 \, {\left (A \sqrt {b} c \arctan \left (\frac {\sqrt {b}}{\sqrt {-b}}\right ) - B \sqrt {-b} b + A \sqrt {-b} c\right )}}{\sqrt {-b} b^{\frac {3}{2}} c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 63, normalized size = 0.93 \begin {gather*} -\frac {2 \sqrt {\left (c x +b \right ) x}\, \left (\sqrt {c x +b}\, A c \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-A \sqrt {b}\, c +B \,b^{\frac {3}{2}}\right )}{\left (c x +b \right ) b^{\frac {3}{2}} c \sqrt {x}} \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 {{\left (B x + A\right )} \sqrt {x}}{{\left (c x^{2} + b x\right )}^{\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 = -0.01 \begin {gather*} \int \frac {\sqrt {x}\,\left (A+B\,x\right )}{{\left (c\,x^2+b\,x\right )}^{3/2}} \,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 {\sqrt {x} \left (A + B x\right )}{\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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