Optimal. Leaf size=133 \[ -\frac {16 b^2 \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^{3/2}}+\frac {8 b \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{105 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
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Rubi [A] time = 0.09, antiderivative size = 133, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {794, 656, 648} \[ -\frac {16 b^2 \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{315 c^4 x^{3/2}}+\frac {8 b \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{105 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2} (2 b B-3 A c)}{21 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c} \]
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac {\left (2 \left (\frac {3}{2} (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{9 c}\\ &=-\frac {2 (2 b B-3 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}+\frac {(4 b (2 b B-3 A c)) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{21 c^2}\\ &=\frac {8 b (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {2 (2 b B-3 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {\left (8 b^2 (2 b B-3 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{105 c^3}\\ &=-\frac {16 b^2 (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{315 c^4 x^{3/2}}+\frac {8 b (2 b B-3 A c) \left (b x+c x^2\right )^{3/2}}{105 c^3 \sqrt {x}}-\frac {2 (2 b B-3 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{21 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 72, normalized size = 0.54 \[ \frac {2 (x (b+c x))^{3/2} \left (24 b^2 c (A+B x)-6 b c^2 x (6 A+5 B x)+5 c^3 x^2 (9 A+7 B x)-16 b^3 B\right )}{315 c^4 x^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.83, size = 102, normalized size = 0.77 \[ \frac {2 \, {\left (35 \, B c^{4} x^{4} - 16 \, B b^{4} + 24 \, A b^{3} c + 5 \, {\left (B b c^{3} + 9 \, A c^{4}\right )} x^{3} - 3 \, {\left (2 \, B b^{2} c^{2} - 3 \, A b c^{3}\right )} x^{2} + 4 \, {\left (2 \, B b^{3} c - 3 \, A b^{2} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{315 \, c^{4} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 110, normalized size = 0.83 \[ \frac {2}{315} \, B {\left (\frac {16 \, b^{\frac {9}{2}}}{c^{4}} + \frac {35 \, {\left (c x + b\right )}^{\frac {9}{2}} - 135 \, {\left (c x + b\right )}^{\frac {7}{2}} b + 189 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{2} - 105 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{3}}{c^{4}}\right )} - \frac {2}{105} \, A {\left (\frac {8 \, b^{\frac {7}{2}}}{c^{3}} - \frac {15 \, {\left (c x + b\right )}^{\frac {7}{2}} - 42 \, {\left (c x + b\right )}^{\frac {5}{2}} b + 35 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{2}}{c^{3}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.08, size = 83, normalized size = 0.62 \[ \frac {2 \left (c x +b \right ) \left (35 B \,c^{3} x^{3}+45 A \,c^{3} x^{2}-30 B b \,c^{2} x^{2}-36 A b \,c^{2} x +24 B \,b^{2} c x +24 A \,b^{2} c -16 b^{3} B \right ) \sqrt {c \,x^{2}+b x}}{315 c^{4} \sqrt {x}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 98, normalized size = 0.74 \[ \frac {2 \, {\left (15 \, c^{3} x^{3} + 3 \, b c^{2} x^{2} - 4 \, b^{2} c x + 8 \, b^{3}\right )} \sqrt {c x + b} A}{105 \, c^{3}} + \frac {2 \, {\left (35 \, c^{4} x^{4} + 5 \, b c^{3} x^{3} - 6 \, b^{2} c^{2} x^{2} + 8 \, b^{3} c x - 16 \, b^{4}\right )} \sqrt {c x + b} B}{315 \, c^{4}} \]
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 \[ \int x^{3/2}\,\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right ) \,d x \]
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 \[ \int x^{\frac {3}{2}} \sqrt {x \left (b + c x\right )} \left (A + B x\right )\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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