Optimal. Leaf size=97 \[ \frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}-\frac {(b+2 c x) \sqrt {b x+c x^2} (b B-2 A c)}{8 c^2}+\frac {B \left (b x+c x^2\right )^{3/2}}{3 c} \]
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Rubi [A] time = 0.04, antiderivative size = 97, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {640, 612, 620, 206} \begin {gather*} \frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}-\frac {(b+2 c x) \sqrt {b x+c x^2} (b B-2 A c)}{8 c^2}+\frac {B \left (b x+c x^2\right )^{3/2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rubi steps
\begin {align*} \int (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {(-b B+2 A c) \int \sqrt {b x+c x^2} \, dx}{2 c}\\ &=-\frac {(b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {\left (b^2 (b B-2 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^2}\\ &=-\frac {(b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {\left (b^2 (b B-2 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c^2}\\ &=-\frac {(b B-2 A c) (b+2 c x) \sqrt {b x+c x^2}}{8 c^2}+\frac {B \left (b x+c x^2\right )^{3/2}}{3 c}+\frac {b^2 (b B-2 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{5/2}}\\ \end {align*}
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Mathematica [A] time = 0.18, size = 108, normalized size = 1.11 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {3 b^{3/2} (b B-2 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (2 b c (3 A+B x)+4 c^2 x (3 A+2 B x)-3 b^2 B\right )\right )}{24 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.00, size = 105, normalized size = 1.08 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (6 A b c+12 A c^2 x-3 b^2 B+2 b B c x+8 B c^2 x^2\right )}{24 c^2}+\frac {\left (2 A b^2 c-b^3 B\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{16 c^{5/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 204, normalized size = 2.10 \begin {gather*} \left [-\frac {3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (8 \, B c^{3} x^{2} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{3}}, -\frac {3 \, {\left (B b^{3} - 2 \, A b^{2} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (8 \, B c^{3} x^{2} - 3 \, B b^{2} c + 6 \, A b c^{2} + 2 \, {\left (B b c^{2} + 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{3}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.27, size = 102, normalized size = 1.05 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, B x + \frac {B b c + 6 \, A c^{2}}{c^{2}}\right )} x - \frac {3 \, {\left (B b^{2} - 2 \, A b c\right )}}{c^{2}}\right )} - \frac {{\left (B b^{3} - 2 \, A b^{2} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 157, normalized size = 1.62 \begin {gather*} -\frac {A \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}+\frac {B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {5}{2}}}+\frac {\sqrt {c \,x^{2}+b x}\, A x}{2}-\frac {\sqrt {c \,x^{2}+b x}\, B b x}{4 c}+\frac {\sqrt {c \,x^{2}+b x}\, A b}{4 c}-\frac {\sqrt {c \,x^{2}+b x}\, B \,b^{2}}{8 c^{2}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B}{3 c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 154, normalized size = 1.59 \begin {gather*} \frac {1}{2} \, \sqrt {c x^{2} + b x} A x - \frac {\sqrt {c x^{2} + b x} B b x}{4 \, c} + \frac {B b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} - \frac {A b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {3}{2}}} - \frac {\sqrt {c x^{2} + b x} B b^{2}}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B}{3 \, c} + \frac {\sqrt {c x^{2} + b x} A b}{4 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.69, size = 127, normalized size = 1.31 \begin {gather*} A\,\sqrt {c\,x^2+b\,x}\,\left (\frac {x}{2}+\frac {b}{4\,c}\right )+\frac {B\,b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {B\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}-\frac {A\,b^2\,\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{8\,c^{3/2}} \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 \sqrt {x \left (b + c x\right )} \left (A + B x\right )\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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