Optimal. Leaf size=165 \[ \frac {b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}}-\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} (7 b B-10 A c)}{128 c^4}+\frac {b \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{48 c^3}-\frac {x \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \]
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Rubi [A] time = 0.16, antiderivative size = 165, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {794, 670, 640, 612, 620, 206} \begin {gather*} -\frac {b^2 (b+2 c x) \sqrt {b x+c x^2} (7 b B-10 A c)}{128 c^4}+\frac {b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}}+\frac {b \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{48 c^3}-\frac {x \left (b x+c x^2\right )^{3/2} (7 b B-10 A c)}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 206
Rule 612
Rule 620
Rule 640
Rule 670
Rule 794
Rubi steps
\begin {align*} \int x^2 (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (2 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right ) \int x^2 \sqrt {b x+c x^2} \, dx}{5 c}\\ &=-\frac {(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {(b (7 b B-10 A c)) \int x \sqrt {b x+c x^2} \, dx}{16 c^2}\\ &=\frac {b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}-\frac {\left (b^2 (7 b B-10 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{32 c^3}\\ &=-\frac {b^2 (7 b B-10 A c) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (b^4 (7 b B-10 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{256 c^4}\\ &=-\frac {b^2 (7 b B-10 A c) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {\left (b^4 (7 b B-10 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{128 c^4}\\ &=-\frac {b^2 (7 b B-10 A c) (b+2 c x) \sqrt {b x+c x^2}}{128 c^4}+\frac {b (7 b B-10 A c) \left (b x+c x^2\right )^{3/2}}{48 c^3}-\frac {(7 b B-10 A c) x \left (b x+c x^2\right )^{3/2}}{40 c^2}+\frac {B x^2 \left (b x+c x^2\right )^{3/2}}{5 c}+\frac {b^4 (7 b B-10 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{128 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.25, size = 148, normalized size = 0.90 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\frac {15 b^{7/2} (7 b B-10 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (10 b^3 c (15 A+7 B x)-4 b^2 c^2 x (25 A+14 B x)+16 b c^3 x^2 (5 A+3 B x)+96 c^4 x^3 (5 A+4 B x)-105 b^4 B\right )\right )}{1920 c^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.54, size = 153, normalized size = 0.93 \begin {gather*} \frac {\left (10 A b^4 c-7 b^5 B\right ) \log \left (-2 \sqrt {c} \sqrt {b x+c x^2}+b+2 c x\right )}{256 c^{9/2}}+\frac {\sqrt {b x+c x^2} \left (150 A b^3 c-100 A b^2 c^2 x+80 A b c^3 x^2+480 A c^4 x^3-105 b^4 B+70 b^3 B c x-56 b^2 B c^2 x^2+48 b B c^3 x^3+384 B c^4 x^4\right )}{1920 c^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 302, normalized size = 1.83 \begin {gather*} \left [-\frac {15 \, {\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{3840 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (384 \, B c^{5} x^{4} - 105 \, B b^{4} c + 150 \, A b^{3} c^{2} + 48 \, {\left (B b c^{4} + 10 \, A c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{2} c^{3} - 10 \, A b c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{3} c^{2} - 10 \, A b^{2} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{1920 \, c^{5}}\right ] \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 160, normalized size = 0.97 \begin {gather*} \frac {1}{1920} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (6 \, {\left (8 \, B x + \frac {B b c^{3} + 10 \, A c^{4}}{c^{4}}\right )} x - \frac {7 \, B b^{2} c^{2} - 10 \, A b c^{3}}{c^{4}}\right )} x + \frac {5 \, {\left (7 \, B b^{3} c - 10 \, A b^{2} c^{2}\right )}}{c^{4}}\right )} x - \frac {15 \, {\left (7 \, B b^{4} - 10 \, A b^{3} c\right )}}{c^{4}}\right )} - \frac {{\left (7 \, B b^{5} - 10 \, A b^{4} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{256 \, c^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 245, normalized size = 1.48 \begin {gather*} -\frac {5 A \,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 {7 B \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {9}{2}}}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{2} x}{32 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{3} x}{64 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,x^{2}}{5 c}+\frac {5 \sqrt {c \,x^{2}+b x}\, A \,b^{3}}{64 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A x}{4 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4}}{128 c^{4}}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B b x}{40 c^{2}}-\frac {5 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b}{24 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2}}{48 c^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.86, size = 242, normalized size = 1.47 \begin {gather*} \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B x^{2}}{5 \, c} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{3} x}{64 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b x}{40 \, c^{2}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{2} x}{32 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A x}{4 \, c} + \frac {7 \, B b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {9}{2}}} - \frac {5 \, A b^{4} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{128 \, c^{\frac {7}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{4}}{128 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2}}{48 \, c^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} A b^{3}}{64 \, c^{3}} - \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{24 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.48, size = 215, normalized size = 1.30 \begin {gather*} \frac {A\,x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,A\,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 {7\,B\,b\,\left (\frac {x\,{\left (c\,x^2+b\,x\right )}^{3/2}}{4\,c}-\frac {5\,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}\right )}{10\,c}+\frac {B\,x^2\,{\left (c\,x^2+b\,x\right )}^{3/2}}{5\,c} \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^{2} \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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