Optimal. Leaf size=90 \[ -\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac {2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}+\frac {4 c (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{693 b^3 x^7} \]
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Rubi [A]
time = 0.06, antiderivative size = 90, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {4 c \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{693 b^3 x^7}-\frac {2 \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{99 b^2 x^8}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9} \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^9} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}+\frac {\left (2 \left (-9 (-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^8} \, dx}{11 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac {2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}-\frac {(2 c (11 b B-4 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{99 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9}-\frac {2 (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{99 b^2 x^8}+\frac {4 c (11 b B-4 A c) \left (b x+c x^2\right )^{7/2}}{693 b^3 x^7}\\ \end {align*}
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Mathematica [A]
time = 0.31, size = 63, normalized size = 0.70 \begin {gather*} -\frac {2 (b+c x) (x (b+c x))^{5/2} \left (63 A b^2+77 b^2 B x-28 A b c x-22 b B c x^2+8 A c^2 x^2\right )}{693 b^3 x^8} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.53, size = 112, normalized size = 1.24
method | result | size |
gosper | \(-\frac {2 \left (c x +b \right ) \left (8 A \,c^{2} x^{2}-22 b B \,x^{2} c -28 A b c x +77 b^{2} B x +63 b^{2} A \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{693 b^{3} x^{8}}\) | \(62\) |
default | \(A \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{11 b \,x^{9}}-\frac {4 c \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )}{11 b}\right )+B \left (-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{9 b \,x^{8}}+\frac {4 c \left (c \,x^{2}+b x \right )^{\frac {7}{2}}}{63 b^{2} x^{7}}\right )\) | \(112\) |
trager | \(-\frac {2 \left (8 A \,c^{5} x^{5}-22 B b \,c^{4} x^{5}-4 A b \,c^{4} x^{4}+11 B \,b^{2} c^{3} x^{4}+3 A \,b^{2} c^{3} x^{3}+165 B \,b^{3} c^{2} x^{3}+113 A \,b^{3} c^{2} x^{2}+209 B \,b^{4} c \,x^{2}+161 A \,b^{4} c x +77 b^{5} B x +63 b^{5} A \right ) \sqrt {c \,x^{2}+b x}}{693 b^{3} x^{6}}\) | \(129\) |
risch | \(-\frac {2 \left (c x +b \right ) \left (8 A \,c^{5} x^{5}-22 B b \,c^{4} x^{5}-4 A b \,c^{4} x^{4}+11 B \,b^{2} c^{3} x^{4}+3 A \,b^{2} c^{3} x^{3}+165 B \,b^{3} c^{2} x^{3}+113 A \,b^{3} c^{2} x^{2}+209 B \,b^{4} c \,x^{2}+161 A \,b^{4} c x +77 b^{5} B x +63 b^{5} A \right )}{693 x^{5} \sqrt {x \left (c x +b \right )}\, b^{3}}\) | \(132\) |
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. 304 vs.
\(2 (78) = 156\).
time = 0.28, size = 304, normalized size = 3.38 \begin {gather*} \frac {4 \, \sqrt {c x^{2} + b x} B c^{4}}{63 \, b^{2} x} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{5}}{693 \, b^{3} x} - \frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{63 \, b x^{2}} + \frac {8 \, \sqrt {c x^{2} + b x} A c^{4}}{693 \, b^{2} x^{2}} + \frac {\sqrt {c x^{2} + b x} B c^{2}}{42 \, x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{231 \, b x^{3}} - \frac {5 \, \sqrt {c x^{2} + b x} B b c}{252 \, x^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, x^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} B b^{2}}{36 \, x^{5}} - \frac {5 \, \sqrt {c x^{2} + b x} A b c}{792 \, x^{5}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{12 \, x^{6}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{88 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{2 \, x^{7}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{24 \, x^{7}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{3 \, x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.03, size = 127, normalized size = 1.41 \begin {gather*} -\frac {2 \, {\left (63 \, A b^{5} - 2 \, {\left (11 \, B b c^{4} - 4 \, A c^{5}\right )} x^{5} + {\left (11 \, B b^{2} c^{3} - 4 \, A b c^{4}\right )} x^{4} + 3 \, {\left (55 \, B b^{3} c^{2} + A b^{2} c^{3}\right )} x^{3} + {\left (209 \, B b^{4} c + 113 \, A b^{3} c^{2}\right )} x^{2} + 7 \, {\left (11 \, B b^{5} + 23 \, A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{693 \, b^{3} x^{6}} \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^{9}}\, 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. 491 vs.
\(2 (78) = 156\).
time = 0.75, size = 491, normalized size = 5.46 \begin {gather*} \frac {2 \, {\left (693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B c^{\frac {7}{2}} + 3003 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b c^{3} + 924 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A c^{4} + 6237 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{2} c^{\frac {5}{2}} + 4851 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b c^{\frac {7}{2}} + 7623 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{3} c^{2} + 11781 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{2} c^{3} + 5775 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{4} c^{\frac {3}{2}} + 16863 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{3} c^{\frac {5}{2}} + 2673 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{5} c + 15345 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{4} c^{2} + 693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{6} \sqrt {c} + 9009 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{5} c^{\frac {3}{2}} + 77 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{7} + 3311 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{6} c + 693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{7} \sqrt {c} + 63 \, A b^{8}\right )}}{693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 3.46, size = 234, normalized size = 2.60 \begin {gather*} \frac {8\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^2}-\frac {226\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,x^4}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {10\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{21\,x^3}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{231\,b\,x^3}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {16\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{693\,b^3\,x}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^2}+\frac {4\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{63\,b^2\,x}-\frac {46\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{99\,x^5}-\frac {38\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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