Optimal. Leaf size=125 \[ -\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac {2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}+\frac {8 c (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{1287 b^3 x^8}-\frac {16 c^2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{9009 b^4 x^7} \]
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Rubi [A]
time = 0.08, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {806, 672, 664}
\begin {gather*} -\frac {16 c^2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{9009 b^4 x^7}+\frac {8 c \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{1287 b^3 x^8}-\frac {2 \left (b x+c x^2\right )^{7/2} (13 b B-6 A c)}{143 b^2 x^9}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}} \end {gather*}
Antiderivative was successfully verified.
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Rule 664
Rule 672
Rule 806
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^{10}} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}+\frac {\left (2 \left (-10 (-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^9} \, dx}{13 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac {2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}-\frac {(4 c (13 b B-6 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{143 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac {2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}+\frac {8 c (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{1287 b^3 x^8}+\frac {\left (8 c^2 (13 b B-6 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{1287 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{13 b x^{10}}-\frac {2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{143 b^2 x^9}+\frac {8 c (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{1287 b^3 x^8}-\frac {16 c^2 (13 b B-6 A c) \left (b x+c x^2\right )^{7/2}}{9009 b^4 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.39, size = 87, normalized size = 0.70 \begin {gather*} -\frac {2 (b+c x) (x (b+c x))^{5/2} \left (693 A b^3+819 b^3 B x-378 A b^2 c x-364 b^2 B c x^2+168 A b c^2 x^2+104 b B c^2 x^3-48 A c^3 x^3\right )}{9009 b^4 x^9} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 164, normalized size = 1.31
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+104 B b \,c^{2} x^{3}+168 A b \,c^{2} x^{2}-364 B \,b^{2} c \,x^{2}-378 A \,b^{2} c x +819 B \,b^{3} x +693 A \,b^{3}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{9009 b^{4} x^{9}}\) | \(86\) |
trager | \(-\frac {2 \left (-48 A \,c^{6} x^{6}+104 B b \,c^{5} x^{6}+24 A b \,c^{5} x^{5}-52 B \,b^{2} c^{4} x^{5}-18 A \,b^{2} c^{4} x^{4}+39 B \,b^{3} c^{3} x^{4}+15 A \,b^{3} c^{3} x^{3}+1469 B \,b^{4} c^{2} x^{3}+1113 A \,b^{4} c^{2} x^{2}+2093 B \,b^{5} c \,x^{2}+1701 A \,b^{5} c x +819 b^{6} B x +693 A \,b^{6}\right ) \sqrt {c \,x^{2}+b x}}{9009 b^{4} x^{7}}\) | \(153\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (-48 A \,c^{6} x^{6}+104 B b \,c^{5} x^{6}+24 A b \,c^{5} x^{5}-52 B \,b^{2} c^{4} x^{5}-18 A \,b^{2} c^{4} x^{4}+39 B \,b^{3} c^{3} x^{4}+15 A \,b^{3} c^{3} x^{3}+1469 B \,b^{4} c^{2} x^{3}+1113 A \,b^{4} c^{2} x^{2}+2093 B \,b^{5} c \,x^{2}+1701 A \,b^{5} c x +819 b^{6} B x +693 A \,b^{6}\right )}{9009 x^{6} \sqrt {x \left (c x +b \right )}\, b^{4}}\) | \(156\) |
default | \(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{13 b \,x^{10}}-\frac {6 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )}{13 b}\right )+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )\) | \(164\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [B] Leaf count of result is larger than twice the leaf count of optimal. 350 vs.
\(2 (109) = 218\).
time = 0.28, size = 350, normalized size = 2.80 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} B c^{5}}{693 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{6}}{3003 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c^{4}}{693 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{5}}{3003 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{231 \, b x^{3}} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{4}}{1001 \, b^{2} x^{3}} + \frac {5 \, \sqrt {c x^{2} + b x} B c^{2}}{693 \, x^{4}} - \frac {10 \, \sqrt {c x^{2} + b x} A c^{3}}{3003 \, b x^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} B b c}{792 \, x^{5}} + \frac {5 \, \sqrt {c x^{2} + b x} A c^{2}}{1716 \, x^{5}} - \frac {5 \, \sqrt {c x^{2} + b x} B b^{2}}{88 \, x^{6}} - \frac {3 \, \sqrt {c x^{2} + b x} A b c}{1144 \, x^{6}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{24 \, x^{7}} - \frac {3 \, \sqrt {c x^{2} + b x} A b^{2}}{104 \, x^{7}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{3 \, x^{8}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{8 \, x^{8}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{4 \, x^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.24, size = 153, normalized size = 1.22 \begin {gather*} -\frac {2 \, {\left (693 \, A b^{6} + 8 \, {\left (13 \, B b c^{5} - 6 \, A c^{6}\right )} x^{6} - 4 \, {\left (13 \, B b^{2} c^{4} - 6 \, A b c^{5}\right )} x^{5} + 3 \, {\left (13 \, B b^{3} c^{3} - 6 \, A b^{2} c^{4}\right )} x^{4} + {\left (1469 \, B b^{4} c^{2} + 15 \, A b^{3} c^{3}\right )} x^{3} + 7 \, {\left (299 \, B b^{5} c + 159 \, A b^{4} c^{2}\right )} x^{2} + 63 \, {\left (13 \, B b^{6} + 27 \, A b^{5} c\right )} x\right )} \sqrt {c x^{2} + b x}}{9009 \, b^{4} x^{7}} \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^{10}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 551 vs.
\(2 (109) = 218\).
time = 0.68, size = 551, normalized size = 4.41 \begin {gather*} \frac {2 \, {\left (12012 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B c^{4} + 63063 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b c^{\frac {7}{2}} + 18018 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A c^{\frac {9}{2}} + 153153 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{2} c^{3} + 108108 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b c^{4} + 219219 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{3} c^{\frac {5}{2}} + 297297 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{2} c^{\frac {7}{2}} + 199485 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{4} c^{2} + 485199 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{3} c^{3} + 117117 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{5} c^{\frac {3}{2}} + 513513 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{4} c^{\frac {5}{2}} + 43043 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{6} c + 363363 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{5} c^{2} + 9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{7} \sqrt {c} + 171171 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{6} c^{\frac {3}{2}} + 819 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{8} + 51597 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{7} c + 9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{8} \sqrt {c} + 693 \, A b^{9}\right )}}{9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{13}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 4.09, size = 280, normalized size = 2.24 \begin {gather*} \frac {4\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{1001\,b^2\,x^3}-\frac {106\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{429\,x^5}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {226\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,x^4}-\frac {10\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{3003\,b\,x^4}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {16\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{3003\,b^3\,x^2}+\frac {32\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{3003\,b^4\,x}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^3}+\frac {8\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^2}-\frac {16\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x}-\frac {54\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{143\,x^6}-\frac {46\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{99\,x^5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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