Optimal. Leaf size=96 \[ \frac {4 b \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{105 c^3 x^{3/2}}-\frac {2 \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{35 c^2 \sqrt {x}}+\frac {2 B \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \]
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Rubi [A] time = 0.08, antiderivative size = 96, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.125, Rules used = {794, 656, 648} \begin {gather*} -\frac {2 \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{35 c^2 \sqrt {x}}+\frac {4 b \left (b x+c x^2\right )^{3/2} (4 b B-7 A c)}{105 c^3 x^{3/2}}+\frac {2 B \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \sqrt {x} (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {2 B \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {\left (2 \left (\frac {1}{2} (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{7 c}\\ &=-\frac {2 (4 b B-7 A c) \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 B \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}+\frac {(2 b (4 b B-7 A c)) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{35 c^2}\\ &=\frac {4 b (4 b B-7 A c) \left (b x+c x^2\right )^{3/2}}{105 c^3 x^{3/2}}-\frac {2 (4 b B-7 A c) \left (b x+c x^2\right )^{3/2}}{35 c^2 \sqrt {x}}+\frac {2 B \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 56, normalized size = 0.58 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (-2 b c (7 A+6 B x)+3 c^2 x (7 A+5 B x)+8 b^2 B\right )}{105 c^3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 59, normalized size = 0.61 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2} \left (-14 A b c+21 A c^2 x+8 b^2 B-12 b B c x+15 B c^2 x^2\right )}{105 c^3 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 78, normalized size = 0.81 \begin {gather*} \frac {2 \, {\left (15 \, B c^{3} x^{3} + 8 \, B b^{3} - 14 \, A b^{2} c + 3 \, {\left (B b c^{2} + 7 \, A c^{3}\right )} x^{2} - {\left (4 \, B b^{2} c - 7 \, A b c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.24, size = 86, normalized size = 0.90 \begin {gather*} -\frac {2}{105} \, B {\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 )} + \frac {2}{15} \, A {\left (\frac {2 \, b^{\frac {5}{2}}}{c^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {5}{2}} - 5 \, {\left (c x + b\right )}^{\frac {3}{2}} b}{c^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 59, normalized size = 0.61 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-15 B \,c^{2} x^{2}-21 A \,c^{2} x +12 B b c x +14 A b c -8 b^{2} B \right ) \sqrt {c \,x^{2}+b x}}{105 c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.61, size = 75, normalized size = 0.78 \begin {gather*} \frac {2 \, {\left (3 \, c^{2} x^{2} + b c x - 2 \, b^{2}\right )} \sqrt {c x + b} A}{15 \, c^{2}} + \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} B}{105 \, c^{3}} \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 \sqrt {x}\,\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right ) \,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 \sqrt {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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