Optimal. Leaf size=151 \[ \frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}-\frac {b^3 (b+2 c x) \sqrt {b x+c x^2} (7 b B-12 A c)}{512 c^4}+\frac {b (b+2 c x) \left (b x+c x^2\right )^{3/2} (7 b B-12 A c)}{192 c^3}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]
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Rubi [A] time = 0.07, antiderivative size = 151, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {779, 612, 620, 206} \[ -\frac {b^3 (b+2 c x) \sqrt {b x+c x^2} (7 b B-12 A c)}{512 c^4}+\frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}+\frac {b (b+2 c x) \left (b x+c x^2\right )^{3/2} (7 b B-12 A c)}{192 c^3}-\frac {\left (b x+c x^2\right )^{5/2} (-12 A c+7 b B-10 B c x)}{60 c^2} \]
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) \left (b x+c x^2\right )^{3/2} \, dx &=-\frac {(7 b B-12 A c-10 B c x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {(b (7 b B-12 A c)) \int \left (b x+c x^2\right )^{3/2} \, dx}{24 c^2}\\ &=\frac {b (7 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (b x+c x^2\right )^{5/2}}{60 c^2}-\frac {\left (b^3 (7 b B-12 A c)\right ) \int \sqrt {b x+c x^2} \, dx}{128 c^3}\\ &=-\frac {b^3 (7 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {b (7 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (b^5 (7 b B-12 A c)\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{1024 c^4}\\ &=-\frac {b^3 (7 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {b (7 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {\left (b^5 (7 b B-12 A c)\right ) \operatorname {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{512 c^4}\\ &=-\frac {b^3 (7 b B-12 A c) (b+2 c x) \sqrt {b x+c x^2}}{512 c^4}+\frac {b (7 b B-12 A c) (b+2 c x) \left (b x+c x^2\right )^{3/2}}{192 c^3}-\frac {(7 b B-12 A c-10 B c x) \left (b x+c x^2\right )^{5/2}}{60 c^2}+\frac {b^5 (7 b B-12 A c) \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{512 c^{9/2}}\\ \end {align*}
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Mathematica [A] time = 0.29, size = 166, normalized size = 1.10 \[ \frac {\sqrt {x (b+c x)} \left (\frac {15 b^{9/2} (7 b B-12 A c) \sinh ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {b}}\right )}{\sqrt {x} \sqrt {\frac {c x}{b}+1}}+\sqrt {c} \left (10 b^4 c (18 A+7 B x)-8 b^3 c^2 x (15 A+7 B x)+48 b^2 c^3 x^2 (2 A+B x)+64 b c^4 x^3 (33 A+26 B x)+256 c^5 x^4 (6 A+5 B x)-105 b^5 B\right )\right )}{7680 c^{9/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.90, size = 350, normalized size = 2.32 \[ \left [-\frac {15 \, {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, {\left (1280 \, B c^{6} x^{5} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 48 \, {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{15360 \, c^{5}}, -\frac {15 \, {\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (1280 \, B c^{6} x^{5} - 105 \, B b^{5} c + 180 \, A b^{4} c^{2} + 128 \, {\left (13 \, B b c^{5} + 12 \, A c^{6}\right )} x^{4} + 48 \, {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )} x^{3} - 8 \, {\left (7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}\right )} x^{2} + 10 \, {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )} x\right )} \sqrt {c x^{2} + b x}}{7680 \, c^{5}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 194, normalized size = 1.28 \[ \frac {1}{7680} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, {\left (2 \, {\left (8 \, {\left (10 \, B c x + \frac {13 \, B b c^{5} + 12 \, A c^{6}}{c^{5}}\right )} x + \frac {3 \, {\left (B b^{2} c^{4} + 44 \, A b c^{5}\right )}}{c^{5}}\right )} x - \frac {7 \, B b^{3} c^{3} - 12 \, A b^{2} c^{4}}{c^{5}}\right )} x + \frac {5 \, {\left (7 \, B b^{4} c^{2} - 12 \, A b^{3} c^{3}\right )}}{c^{5}}\right )} x - \frac {15 \, {\left (7 \, B b^{5} c - 12 \, A b^{4} c^{2}\right )}}{c^{5}}\right )} - \frac {{\left (7 \, B b^{6} - 12 \, A b^{5} c\right )} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{1024 \, c^{\frac {9}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [B] time = 0.05, size = 283, normalized size = 1.87 \[ -\frac {3 A \,b^{5} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{256 c^{\frac {7}{2}}}+\frac {7 B \,b^{6} \ln \left (\frac {c x +\frac {b}{2}}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{1024 c^{\frac {9}{2}}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{3} x}{64 c^{2}}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{4} x}{256 c^{3}}+\frac {3 \sqrt {c \,x^{2}+b x}\, A \,b^{4}}{128 c^{3}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A b x}{8 c}-\frac {7 \sqrt {c \,x^{2}+b x}\, B \,b^{5}}{512 c^{4}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{2} x}{96 c^{2}}-\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}} A \,b^{2}}{16 c^{2}}+\frac {7 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} B \,b^{3}}{192 c^{3}}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} B x}{6 c}+\frac {\left (c \,x^{2}+b x \right )^{\frac {5}{2}} A}{5 c}-\frac {7 \left (c \,x^{2}+b x \right )^{\frac {5}{2}} B b}{60 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.95, size = 280, normalized size = 1.85 \[ -\frac {7 \, \sqrt {c x^{2} + b x} B b^{4} x}{256 \, c^{3}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{2} x}{96 \, c^{2}} + \frac {3 \, \sqrt {c x^{2} + b x} A b^{3} x}{64 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B x}{6 \, c} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b x}{8 \, c} + \frac {7 \, B b^{6} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{1024 \, c^{\frac {9}{2}}} - \frac {3 \, A b^{5} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{256 \, c^{\frac {7}{2}}} - \frac {7 \, \sqrt {c x^{2} + b x} B b^{5}}{512 \, c^{4}} + \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b^{3}}{192 \, c^{3}} + \frac {3 \, \sqrt {c x^{2} + b x} A b^{4}}{128 \, c^{3}} - \frac {7 \, {\left (c x^{2} + b x\right )}^{\frac {5}{2}} B b}{60 \, c^{2}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b^{2}}{16 \, c^{2}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{5 \, c} \]
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 \[ \int x\,{\left (c\,x^2+b\,x\right )}^{3/2}\,\left (A+B\,x\right ) \,d x \]
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 \[ \int x \left (x \left (b + c x\right )\right )^{\frac {3}{2}} \left (A + B x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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