Optimal. Leaf size=113 \[ -\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}}+\frac {b (b+2 c x) \sqrt {b x+c x^2} (5 b B-8 A c)}{64 c^3}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \]
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Rubi [A] time = 0.05, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {779, 612, 620, 206} \begin {gather*} -\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}}+\frac {b (b+2 c x) \sqrt {b x+c x^2} (5 b B-8 A c)}{64 c^3}-\frac {\left (b x+c x^2\right )^{3/2} (-8 A c+5 b B-6 B c x)}{24 c^2} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 779
Rubi steps
\begin {align*} \int x (A+B x) \sqrt {b x+c x^2} \, dx &=-\frac {(5 b B-8 A c-6 B c x) \left (b x+c x^2\right )^{3/2}}{24 c^2}+\frac {(b (5 b B-8 A c)) \int \sqrt {b x+c x^2} \, dx}{16 c^2}\\ &=\frac {b (5 b B-8 A c) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (b x+c x^2\right )^{3/2}}{24 c^2}-\frac {\left (b^3 (5 b B-8 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{128 c^3}\\ &=\frac {b (5 b B-8 A c) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (b x+c x^2\right )^{3/2}}{24 c^2}-\frac {\left (b^3 (5 b B-8 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{64 c^3}\\ &=\frac {b (5 b B-8 A c) (b+2 c x) \sqrt {b x+c x^2}}{64 c^3}-\frac {(5 b B-8 A c-6 B c x) \left (b x+c x^2\right )^{3/2}}{24 c^2}-\frac {b^3 (5 b B-8 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{64 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.21, size = 128, normalized size = 1.13 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (-2 b^2 c (12 A+5 B x)+8 b c^2 x (2 A+B x)+16 c^3 x^2 (4 A+3 B x)+15 b^3 B\right )-\frac {3 b^{5/2} (5 b B-8 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}\right )}{192 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.42, size = 129, normalized size = 1.14 \begin {gather*} \frac {\left (5 b^4 B-8 A b^3 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{128 c^{7/2}}+\frac {\sqrt {b x+c x^2} \left (-24 A b^2 c+16 A b c^2 x+64 A c^3 x^2+15 b^3 B-10 b^2 B c x+8 b B c^2 x^2+48 B c^3 x^3\right )}{192 c^3} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 253, normalized size = 2.24 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (48 \, B c^{4} x^{3} + 15 \, B b^{3} c - 24 \, A b^{2} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{384 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (48 \, B c^{4} x^{3} + 15 \, B b^{3} c - 24 \, A b^{2} c^{2} + 8 \, {\left (B b c^{3} + 8 \, A c^{4}\right )} x^{2} - 2 \, {\left (5 \, B b^{2} c^{2} - 8 \, A b c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{192 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.43, size = 132, normalized size = 1.17 \begin {gather*} \frac {1}{192} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, B x + \frac {B b c^{2} + 8 \, A c^{3}}{c^{3}}\right )} x - \frac {5 \, B b^{2} c - 8 \, A b c^{2}}{c^{3}}\right )} x + \frac {3 \, {\left (5 \, B b^{3} - 8 \, A b^{2} c\right )}}{c^{3}}\right )} + \frac {{\left (5 \, B b^{4} - 8 \, A b^{3} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{128 \, c^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 201, normalized size = 1.78 \begin {gather*} \frac {A \,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 {5 B \,b^{4} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{128 c^{\frac {7}{2}}}-\frac {\sqrt {c \,x^{2}+b x}\, A b x}{4 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{2} x}{32 c^{2}}-\frac {\sqrt {c \,x^{2}+b x}\, A \,b^{2}}{8 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{3}}{64 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B x}{4 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A}{3 c}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b}{24 c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.90, size = 198, normalized size = 1.75 \begin {gather*} \frac {5 \, \sqrt {c x^{2} + b x} B b^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B x}{4 \, c} - \frac {\sqrt {c x^{2} + b x} A b x}{4 \, c} - \frac {5 \, B b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} + \frac {A b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {5}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} B b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{24 \, c^{2}} - \frac {\sqrt {c x^{2} + b x} A b^{2}}{8 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A}{3 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.52, size = 165, normalized size = 1.46 \begin {gather*} \frac {A\,\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}+\frac {B\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,B\,b\,\left (\frac {b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}}+\frac {\sqrt {c\,x^2+b\,x}\,\left (-3\,b^2+2\,b\,c\,x+8\,c^2\,x^2\right )}{24\,c^2}\right )}{8\,c}+\frac {A\,b^3\,\ln \left (\frac {b+2\,c\,x}{\sqrt {c}}+2\,\sqrt {c\,x^2+b\,x}\right )}{16\,c^{5/2}} \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 x \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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