Optimal. Leaf size=60 \[ \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}-\frac {2 x (b B-A c)}{b c \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.03, antiderivative size = 60, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.150, Rules used = {777, 620, 206} \begin {gather*} \frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}-\frac {2 x (b B-A c)}{b c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 777
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (b B-A c) x}{b c \sqrt {b x+c x^2}}+\frac {B \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c}\\ &=-\frac {2 (b B-A c) x}{b c \sqrt {b x+c x^2}}+\frac {(2 B) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=-\frac {2 (b B-A c) x}{b c \sqrt {b x+c x^2}}+\frac {2 B \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.07, size = 79, normalized size = 1.32 \begin {gather*} \frac {2 \sqrt {c} x (A c-b B)+2 b^{3/2} B \sqrt {x} \sqrt {\frac {c x}{b}+1} \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{b c^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.37, size = 76, normalized size = 1.27 \begin {gather*} \frac {2 \sqrt {b x+c x^2} (A c-b B)}{b c (b+c x)}-\frac {B \log \left (-2 c^{3/2} \sqrt {b x+c x^2}+b c+2 c^2 x\right )}{c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 164, normalized size = 2.73 \begin {gather*} \left [\frac {{\left (B b c x + B b^{2}\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (B b c - A c^{2}\right )} \sqrt {c x^{2} + b x}}{b c^{3} x + b^{2} c^{2}}, -\frac {2 \, {\left ({\left (B b c x + B b^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (B b c - A c^{2}\right )} \sqrt {c x^{2} + b x}\right )}}{b c^{3} x + b^{2} c^{2}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 67, normalized size = 1.12 \begin {gather*} \frac {2 A x}{\sqrt {c \,x^{2}+b x}\, b}-\frac {2 B x}{\sqrt {c \,x^{2}+b x}\, c}+\frac {B \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.87, size = 65, normalized size = 1.08 \begin {gather*} \frac {2 \, A x}{\sqrt {c x^{2} + b x} b} - \frac {2 \, B x}{\sqrt {c x^{2} + b x} c} + \frac {B \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.33, size = 64, normalized size = 1.07 \begin {gather*} \frac {B\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{c^{3/2}}+\frac {2\,A\,x}{b\,\sqrt {x\,\left (b+c\,x\right )}}-\frac {2\,B\,x}{c\,\sqrt {c\,x^2+b\,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 {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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