Optimal. Leaf size=96 \[ \frac {4 b \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{693 c^3 x^{7/2}}-\frac {2 \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{99 c^2 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}} \]
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Rubi [A] time = 0.09, 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 )^{7/2} (4 b B-11 A c)}{99 c^2 x^{5/2}}+\frac {4 b \left (b x+c x^2\right )^{7/2} (4 b B-11 A c)}{693 c^3 x^{7/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx &=\frac {2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}+\frac {\left (2 \left (-\frac {3}{2} (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right )\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx}{11 c}\\ &=-\frac {2 (4 b B-11 A c) \left (b x+c x^2\right )^{7/2}}{99 c^2 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}+\frac {(2 b (4 b B-11 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx}{99 c^2}\\ &=\frac {4 b (4 b B-11 A c) \left (b x+c x^2\right )^{7/2}}{693 c^3 x^{7/2}}-\frac {2 (4 b B-11 A c) \left (b x+c x^2\right )^{7/2}}{99 c^2 x^{5/2}}+\frac {2 B \left (b x+c x^2\right )^{7/2}}{11 c x^{3/2}}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 63, normalized size = 0.66 \begin {gather*} \frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (-2 b c (11 A+14 B x)+7 c^2 x (11 A+9 B x)+8 b^2 B\right )}{693 c^3 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.56, size = 59, normalized size = 0.61 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{7/2} \left (-22 A b c+77 A c^2 x+8 b^2 B-28 b B c x+63 B c^2 x^2\right )}{693 c^3 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 125, normalized size = 1.30 \begin {gather*} \frac {2 \, {\left (63 \, B c^{5} x^{5} + 8 \, B b^{5} - 22 \, A b^{4} c + 7 \, {\left (23 \, B b c^{4} + 11 \, A c^{5}\right )} x^{4} + {\left (113 \, B b^{2} c^{3} + 209 \, A b c^{4}\right )} x^{3} + 3 \, {\left (B b^{3} c^{2} + 55 \, A b^{2} c^{3}\right )} x^{2} - {\left (4 \, B b^{4} c - 11 \, A b^{3} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{693 \, c^{3} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.28, size = 344, normalized size = 3.58 \begin {gather*} -\frac {2}{3465} \, B c^{2} {\left (\frac {128 \, b^{\frac {11}{2}}}{c^{5}} - \frac {315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}}{c^{5}}\right )} + \frac {4}{315} \, B b c {\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}{315} \, A c^{2} {\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} \, B b^{2} {\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 {4}{105} \, A b c {\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 b^{2} {\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 (-63 B \,c^{2} x^{2}-77 A \,c^{2} x +28 B b c x +22 A b c -8 b^{2} B \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{693 c^{3} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.86, size = 305, normalized size = 3.18 \begin {gather*} \frac {2 \, {\left ({\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 )} x^{3} + 6 \, {\left (15 \, b c^{3} x^{4} + 3 \, b^{2} c^{2} x^{3} - 4 \, b^{3} c x^{2} + 8 \, b^{4} x\right )} x^{2} + 21 \, {\left (3 \, b^{2} c^{2} x^{4} + b^{3} c x^{3} - 2 \, b^{4} x^{2}\right )} x\right )} \sqrt {c x + b} A}{315 \, c^{2} x^{3}} + \frac {2 \, {\left ({\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} x^{4} + 22 \, {\left (35 \, b c^{4} x^{5} + 5 \, b^{2} c^{3} x^{4} - 6 \, b^{3} c^{2} x^{3} + 8 \, b^{4} c x^{2} - 16 \, b^{5} x\right )} x^{3} + 33 \, {\left (15 \, b^{2} c^{3} x^{5} + 3 \, b^{3} c^{2} x^{4} - 4 \, b^{4} c x^{3} + 8 \, b^{5} x^{2}\right )} x^{2}\right )} \sqrt {c x + b} B}{3465 \, c^{3} x^{4}} \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 {{\left (c\,x^2+b\,x\right )}^{5/2}\,\left (A+B\,x\right )}{x^{3/2}} \,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 {\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (A + B x\right )}{x^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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