Optimal. Leaf size=127 \[ -\frac {b^2 (5 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {b \sqrt {b x+c x^2} (5 b B-6 A c)}{8 c^3}-\frac {x \sqrt {b x+c x^2} (5 b B-6 A c)}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c} \]
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Rubi [A] time = 0.12, antiderivative size = 127, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {794, 670, 640, 620, 206} \begin {gather*} -\frac {b^2 (5 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}+\frac {b \sqrt {b x+c x^2} (5 b B-6 A c)}{8 c^3}-\frac {x \sqrt {b x+c x^2} (5 b B-6 A c)}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 620
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\sqrt {b x+c x^2}} \, dx &=\frac {B x^2 \sqrt {b x+c x^2}}{3 c}+\frac {\left (2 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right ) \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx}{3 c}\\ &=-\frac {(5 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c}+\frac {(b (5 b B-6 A c)) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{8 c^2}\\ &=\frac {b (5 b B-6 A c) \sqrt {b x+c x^2}}{8 c^3}-\frac {(5 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (b^2 (5 b B-6 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c^3}\\ &=\frac {b (5 b B-6 A c) \sqrt {b x+c x^2}}{8 c^3}-\frac {(5 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c}-\frac {\left (b^2 (5 b B-6 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^3}\\ &=\frac {b (5 b B-6 A c) \sqrt {b x+c x^2}}{8 c^3}-\frac {(5 b B-6 A c) x \sqrt {b x+c x^2}}{12 c^2}+\frac {B x^2 \sqrt {b x+c x^2}}{3 c}-\frac {b^2 (5 b B-6 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{7/2}}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 116, normalized size = 0.91 \begin {gather*} \frac {\sqrt {c} x (b+c x) \left (-2 b c (9 A+5 B x)+4 c^2 x (3 A+2 B x)+15 b^2 B\right )-3 b^{5/2} \sqrt {x} \sqrt {\frac {c x}{b}+1} (5 b B-6 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{24 c^{7/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 105, normalized size = 0.83 \begin {gather*} \frac {\sqrt {b x+c x^2} \left (-18 A b c+12 A c^2 x+15 b^2 B-10 b B c x+8 B c^2 x^2\right )}{24 c^3}+\frac {\left (5 b^3 B-6 A b^2 c\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{16 c^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.45, size = 207, normalized size = 1.63 \begin {gather*} \left [-\frac {3 \, {\left (5 \, B b^{3} - 6 \, 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} + 15 \, B b^{2} c - 18 \, A b c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{48 \, c^{4}}, \frac {3 \, {\left (5 \, B b^{3} - 6 \, 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} + 15 \, B b^{2} c - 18 \, A b c^{2} - 2 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{24 \, c^{4}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.23, size = 109, normalized size = 0.86 \begin {gather*} \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (\frac {4 \, B x}{c} - \frac {5 \, B b c - 6 \, A c^{2}}{c^{3}}\right )} x + \frac {3 \, {\left (5 \, B b^{2} - 6 \, A b c\right )}}{c^{3}}\right )} + \frac {{\left (5 \, B b^{3} - 6 \, 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 {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 163, normalized size = 1.28 \begin {gather*} \frac {\sqrt {c \,x^{2}+b x}\, B \,x^{2}}{3 c}+\frac {3 A \,b^{2} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}-\frac {5 B \,b^{3} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {7}{2}}}+\frac {\sqrt {c \,x^{2}+b x}\, A x}{2 c}-\frac {5 \sqrt {c \,x^{2}+b x}\, B b x}{12 c^{2}}-\frac {3 \sqrt {c \,x^{2}+b x}\, A b}{4 c^{2}}+\frac {5 \sqrt {c \,x^{2}+b x}\, B \,b^{2}}{8 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 160, normalized size = 1.26 \begin {gather*} \frac {\sqrt {c x^{2} + b x} B x^{2}}{3 \, c} - \frac {5 \, \sqrt {c x^{2} + b x} B b x}{12 \, c^{2}} + \frac {\sqrt {c x^{2} + b x} A x}{2 \, c} - \frac {5 \, B b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {7}{2}}} + \frac {3 \, A b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} + \frac {5 \, \sqrt {c x^{2} + b x} B b^{2}}{8 \, c^{3}} - \frac {3 \, \sqrt {c x^{2} + b x} A b}{4 \, c^{2}} \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 {x^2\,\left (A+B\,x\right )}{\sqrt {c\,x^2+b\,x}} \,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 {x^{2} \left (A + B x\right )}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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