Optimal. Leaf size=207 \[ -\frac {256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac {128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac {32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac {16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \]
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Rubi [A] time = 0.20, antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, 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 {256 b^4 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{765765 c^6 x^{7/2}}+\frac {128 b^3 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{109395 c^5 x^{5/2}}-\frac {32 b^2 \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{12155 c^4 x^{3/2}}+\frac {16 b \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{3315 c^3 \sqrt {x}}-\frac {2 \sqrt {x} \left (b x+c x^2\right )^{7/2} (10 b B-17 A c)}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c} \end {gather*}
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) \left (b x+c x^2\right )^{5/2} \, dx &=\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac {\left (2 \left (\frac {3}{2} (-b B+A c)+\frac {7}{2} (-b B+2 A c)\right )\right ) \int x^{3/2} \left (b x+c x^2\right )^{5/2} \, dx}{17 c}\\ &=-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac {(8 b (10 b B-17 A c)) \int \sqrt {x} \left (b x+c x^2\right )^{5/2} \, dx}{255 c^2}\\ &=\frac {16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt {x}}-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {\left (16 b^2 (10 b B-17 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{\sqrt {x}} \, dx}{1105 c^3}\\ &=-\frac {32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac {16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt {x}}-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}+\frac {\left (64 b^3 (10 b B-17 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{3/2}} \, dx}{12155 c^4}\\ &=\frac {128 b^3 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{109395 c^5 x^{5/2}}-\frac {32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac {16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt {x}}-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}-\frac {\left (128 b^4 (10 b B-17 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^{5/2}} \, dx}{109395 c^5}\\ &=-\frac {256 b^4 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{765765 c^6 x^{7/2}}+\frac {128 b^3 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{109395 c^5 x^{5/2}}-\frac {32 b^2 (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{12155 c^4 x^{3/2}}+\frac {16 b (10 b B-17 A c) \left (b x+c x^2\right )^{7/2}}{3315 c^3 \sqrt {x}}-\frac {2 (10 b B-17 A c) \sqrt {x} \left (b x+c x^2\right )^{7/2}}{255 c^2}+\frac {2 B x^{3/2} \left (b x+c x^2\right )^{7/2}}{17 c}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 120, normalized size = 0.58 \begin {gather*} \frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (128 b^4 c (17 A+35 B x)-224 b^3 c^2 x (34 A+45 B x)+336 b^2 c^3 x^2 (51 A+55 B x)-462 b c^4 x^3 (68 A+65 B x)+3003 c^5 x^4 (17 A+15 B x)-1280 b^5 B\right )}{765765 c^6 \sqrt {x}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.71, size = 131, normalized size = 0.63 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{7/2} \left (2176 A b^4 c-7616 A b^3 c^2 x+17136 A b^2 c^3 x^2-31416 A b c^4 x^3+51051 A c^5 x^4-1280 b^5 B+4480 b^4 B c x-10080 b^3 B c^2 x^2+18480 b^2 B c^3 x^3-30030 b B c^4 x^4+45045 B c^5 x^5\right )}{765765 c^6 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 199, normalized size = 0.96 \begin {gather*} \frac {2 \, {\left (45045 \, B c^{8} x^{8} - 1280 \, B b^{8} + 2176 \, A b^{7} c + 3003 \, {\left (35 \, B b c^{7} + 17 \, A c^{8}\right )} x^{7} + 231 \, {\left (275 \, B b^{2} c^{6} + 527 \, A b c^{7}\right )} x^{6} + 63 \, {\left (5 \, B b^{3} c^{5} + 1207 \, A b^{2} c^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{4} c^{4} - 17 \, A b^{3} c^{5}\right )} x^{4} + 40 \, {\left (10 \, B b^{5} c^{3} - 17 \, A b^{4} c^{4}\right )} x^{3} - 48 \, {\left (10 \, B b^{6} c^{2} - 17 \, A b^{5} c^{3}\right )} x^{2} + 64 \, {\left (10 \, B b^{7} c - 17 \, A b^{6} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{765765 \, c^{6} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.31, size = 560, normalized size = 2.71 \begin {gather*} \frac {2}{109395} \, B c^{2} {\left (\frac {2048 \, b^{\frac {17}{2}}}{c^{8}} + \frac {6435 \, {\left (c x + b\right )}^{\frac {17}{2}} - 51051 \, {\left (c x + b\right )}^{\frac {15}{2}} b + 176715 \, {\left (c x + b\right )}^{\frac {13}{2}} b^{2} - 348075 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{3} + 425425 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{4} - 328185 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{5} + 153153 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{6} - 36465 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{7}}{c^{8}}\right )} - \frac {4}{45045} \, B b c {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} - \frac {2}{45045} \, A c^{2} {\left (\frac {1024 \, b^{\frac {15}{2}}}{c^{7}} - \frac {3003 \, {\left (c x + b\right )}^{\frac {15}{2}} - 20790 \, {\left (c x + b\right )}^{\frac {13}{2}} b + 61425 \, {\left (c x + b\right )}^{\frac {11}{2}} b^{2} - 100100 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{3} + 96525 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{4} - 54054 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{5} + 15015 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{6}}{c^{7}}\right )} + \frac {2}{9009} \, B b^{2} {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} + \frac {4}{9009} \, A b c {\left (\frac {256 \, b^{\frac {13}{2}}}{c^{6}} + \frac {693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}}{c^{6}}\right )} - \frac {2}{3465} \, A b^{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 )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 131, normalized size = 0.63 \begin {gather*} \frac {2 \left (c x +b \right ) \left (45045 B \,x^{5} c^{5}+51051 A \,c^{5} x^{4}-30030 B b \,c^{4} x^{4}-31416 A b \,c^{4} x^{3}+18480 B \,b^{2} c^{3} x^{3}+17136 A \,b^{2} c^{3} x^{2}-10080 B \,b^{3} c^{2} x^{2}-7616 A \,b^{3} c^{2} x +4480 B \,b^{4} c x +2176 A \,b^{4} c -1280 B \,b^{5}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{765765 c^{6} x^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.85, size = 507, normalized size = 2.45 \begin {gather*} \frac {2 \, {\left ({\left (3003 \, c^{7} x^{7} + 231 \, b c^{6} x^{6} - 252 \, b^{2} c^{5} x^{5} + 280 \, b^{3} c^{4} x^{4} - 320 \, b^{4} c^{3} x^{3} + 384 \, b^{5} c^{2} x^{2} - 512 \, b^{6} c x + 1024 \, b^{7}\right )} x^{6} + 10 \, {\left (693 \, b c^{6} x^{7} + 63 \, b^{2} c^{5} x^{6} - 70 \, b^{3} c^{4} x^{5} + 80 \, b^{4} c^{3} x^{4} - 96 \, b^{5} c^{2} x^{3} + 128 \, b^{6} c x^{2} - 256 \, b^{7} x\right )} x^{5} + 13 \, {\left (315 \, b^{2} c^{5} x^{7} + 35 \, b^{3} c^{4} x^{6} - 40 \, b^{4} c^{3} x^{5} + 48 \, b^{5} c^{2} x^{4} - 64 \, b^{6} c x^{3} + 128 \, b^{7} x^{2}\right )} x^{4}\right )} \sqrt {c x + b} A}{45045 \, c^{5} x^{6}} + \frac {2 \, {\left (7 \, {\left (6435 \, c^{8} x^{8} + 429 \, b c^{7} x^{7} - 462 \, b^{2} c^{6} x^{6} + 504 \, b^{3} c^{5} x^{5} - 560 \, b^{4} c^{4} x^{4} + 640 \, b^{5} c^{3} x^{3} - 768 \, b^{6} c^{2} x^{2} + 1024 \, b^{7} c x - 2048 \, b^{8}\right )} x^{7} + 34 \, {\left (3003 \, b c^{7} x^{8} + 231 \, b^{2} c^{6} x^{7} - 252 \, b^{3} c^{5} x^{6} + 280 \, b^{4} c^{4} x^{5} - 320 \, b^{5} c^{3} x^{4} + 384 \, b^{6} c^{2} x^{3} - 512 \, b^{7} c x^{2} + 1024 \, b^{8} x\right )} x^{6} + 85 \, {\left (693 \, b^{2} c^{6} x^{8} + 63 \, b^{3} c^{5} x^{7} - 70 \, b^{4} c^{4} x^{6} + 80 \, b^{5} c^{3} x^{5} - 96 \, b^{6} c^{2} x^{4} + 128 \, b^{7} c x^{3} - 256 \, b^{8} x^{2}\right )} x^{5}\right )} \sqrt {c x + b} B}{765765 \, c^{6} x^{7}} \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 = -0.00 \begin {gather*} \int x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{5/2}\,\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(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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