Optimal. Leaf size=73 \[ -\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}-\frac {2 B \sqrt {b x+c x^2}}{x}+2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \]
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Rubi [A] time = 0.07, antiderivative size = 73, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {792, 662, 620, 206} \[ -\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}-\frac {2 B \sqrt {b x+c x^2}}{x}+2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \]
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 662
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \sqrt {b x+c x^2}}{x^3} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}+B \int \frac {\sqrt {b x+c x^2}}{x^2} \, dx\\ &=-\frac {2 B \sqrt {b x+c x^2}}{x}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}+(B c) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=-\frac {2 B \sqrt {b x+c x^2}}{x}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}+(2 B c) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=-\frac {2 B \sqrt {b x+c x^2}}{x}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{3 b x^3}+2 B \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}
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Mathematica [C] time = 0.06, size = 86, normalized size = 1.18 \[ \frac {2 \sqrt {x (b+c x)} \left ((b+c x) \sqrt {\frac {c x}{b}+1} (b B-A c)-b^2 B \, _2F_1\left (-\frac {3}{2},-\frac {3}{2};-\frac {1}{2};-\frac {c x}{b}\right )\right )}{3 b c x^2 \sqrt {\frac {c x}{b}+1}} \]
Antiderivative was successfully verified.
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fricas [A] time = 1.22, size = 141, normalized size = 1.93 \[ \left [\frac {3 \, B b \sqrt {c} x^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x} {\left (A b + {\left (3 \, B b + A c\right )} x\right )}}{3 \, b x^{2}}, -\frac {2 \, {\left (3 \, B b \sqrt {-c} x^{2} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x} {\left (A b + {\left (3 \, B b + A c\right )} x\right )}\right )}}{3 \, b x^{2}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.26, size = 151, normalized size = 2.07 \[ -B \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \frac {2 \, {\left (3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b \sqrt {c} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A c^{\frac {3}{2}} + 3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b c + A b^{2} \sqrt {c}\right )}}{3 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} \sqrt {c}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 89, normalized size = 1.22 \[ B \sqrt {c}\, \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )+\frac {2 \sqrt {c \,x^{2}+b x}\, B c}{b}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B}{b \,x^{2}}-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A}{3 b \,x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.95, size = 85, normalized size = 1.16 \[ {\left (\sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {2 \, \sqrt {c x^{2} + b x}}{x}\right )} B - \frac {2}{3} \, A {\left (\frac {\sqrt {c x^{2} + b x} c}{b x} + \frac {\sqrt {c x^{2} + b x}}{x^{2}}\right )} \]
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 \[ \int \frac {\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right )}{x^3} \,d x \]
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 \[ \int \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{3}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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