Optimal. Leaf size=207 \[ -\frac {256 b^4 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac {128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt {x}}-\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \]
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Rubi [A] time = 0.19, 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 )^{3/2} (10 b B-13 A c)}{45045 c^6 x^{3/2}}+\frac {128 b^3 \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{15015 c^5 \sqrt {x}}-\frac {32 b^2 \sqrt {x} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{3003 c^4}+\frac {16 b x^{3/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{1287 c^3}-\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2} (10 b B-13 A c)}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 648
Rule 656
Rule 794
Rubi steps
\begin {align*} \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx &=\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac {\left (2 \left (\frac {7}{2} (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int x^{7/2} \sqrt {b x+c x^2} \, dx}{13 c}\\ &=-\frac {2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac {(8 b (10 b B-13 A c)) \int x^{5/2} \sqrt {b x+c x^2} \, dx}{143 c^2}\\ &=\frac {16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac {2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {\left (16 b^2 (10 b B-13 A c)\right ) \int x^{3/2} \sqrt {b x+c x^2} \, dx}{429 c^3}\\ &=-\frac {32 b^2 (10 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac {16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac {2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}+\frac {\left (64 b^3 (10 b B-13 A c)\right ) \int \sqrt {x} \sqrt {b x+c x^2} \, dx}{3003 c^4}\\ &=\frac {128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt {x}}-\frac {32 b^2 (10 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac {16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac {2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {\left (128 b^4 (10 b B-13 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{\sqrt {x}} \, dx}{15015 c^5}\\ &=-\frac {256 b^4 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{45045 c^6 x^{3/2}}+\frac {128 b^3 (10 b B-13 A c) \left (b x+c x^2\right )^{3/2}}{15015 c^5 \sqrt {x}}-\frac {32 b^2 (10 b B-13 A c) \sqrt {x} \left (b x+c x^2\right )^{3/2}}{3003 c^4}+\frac {16 b (10 b B-13 A c) x^{3/2} \left (b x+c x^2\right )^{3/2}}{1287 c^3}-\frac {2 (10 b B-13 A c) x^{5/2} \left (b x+c x^2\right )^{3/2}}{143 c^2}+\frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}\\ \end {align*}
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Mathematica [A] time = 0.09, size = 113, normalized size = 0.55 \begin {gather*} \frac {2 (x (b+c x))^{3/2} \left (128 b^4 c (13 A+15 B x)-96 b^3 c^2 x (26 A+25 B x)+80 b^2 c^3 x^2 (39 A+35 B x)-70 b c^4 x^3 (52 A+45 B x)+315 c^5 x^4 (13 A+11 B x)-1280 b^5 B\right )}{45045 c^6 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.16, size = 131, normalized size = 0.63 \begin {gather*} \frac {2 \left (b x+c x^2\right )^{3/2} \left (1664 A b^4 c-2496 A b^3 c^2 x+3120 A b^2 c^3 x^2-3640 A b c^4 x^3+4095 A c^5 x^4-1280 b^5 B+1920 b^4 B c x-2400 b^3 B c^2 x^2+2800 b^2 B c^3 x^3-3150 b B c^4 x^4+3465 B c^5 x^5\right )}{45045 c^6 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 150, normalized size = 0.72 \begin {gather*} \frac {2 \, {\left (3465 \, B c^{6} x^{6} - 1280 \, B b^{6} + 1664 \, A b^{5} c + 315 \, {\left (B b c^{5} + 13 \, A c^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{2} c^{4} - 13 \, A b c^{5}\right )} x^{4} + 40 \, {\left (10 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} - 48 \, {\left (10 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} + 64 \, {\left (10 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, c^{6} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 158, normalized size = 0.76 \begin {gather*} \frac {2}{9009} \, B {\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 {\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 (3465 B \,x^{5} c^{5}+4095 A \,c^{5} x^{4}-3150 B b \,c^{4} x^{4}-3640 A b \,c^{4} x^{3}+2800 B \,b^{2} c^{3} x^{3}+3120 A \,b^{2} c^{3} x^{2}-2400 B \,b^{3} c^{2} x^{2}-2496 A \,b^{3} c^{2} x +1920 B \,b^{4} c x +1664 A \,b^{4} c -1280 B \,b^{5}\right ) \sqrt {c \,x^{2}+b x}}{45045 c^{6} \sqrt {x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.65, size = 142, normalized size = 0.69 \begin {gather*} \frac {2 \, {\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 )} \sqrt {c x + b} A}{3465 \, c^{5}} + \frac {2 \, {\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} \sqrt {c x + b} B}{9009 \, c^{6}} \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^{7/2}\,\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 x^{\frac {7}{2}} \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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