Optimal. Leaf size=90 \[ \frac {4 c \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{693 b^3 x^7}-\frac {2 \left (b x+c x^2\right )^{7/2} (11 b B-4 A c)}{99 b^2 x^8}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{11 b x^9} \]
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Rubi [A] time = 0.09, 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 = {792, 658, 650} \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 650
Rule 658
Rule 792
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 = 0.02, size = 63, normalized size = 0.70 \begin {gather*} \frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (A \left (-63 b^2+28 b c x-8 c^2 x^2\right )+11 b B x (2 c x-7 b)\right )}{693 b^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.47, size = 132, normalized size = 1.47 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (63 A b^5+161 A b^4 c x+113 A b^3 c^2 x^2+3 A b^2 c^3 x^3-4 A b c^4 x^4+8 A c^5 x^5+77 b^5 B x+209 b^4 B c x^2+165 b^3 B c^2 x^3+11 b^2 B c^3 x^4-22 b B c^4 x^5\right )}{693 b^3 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, 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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giac [B] time = 0.22, 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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maple [A] time = 0.05, size = 62, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (8 A \,c^{2} x^{2}-22 B b c \,x^{2}-28 A b c x +77 B \,b^{2} x +63 A \,b^{2}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{693 b^{3} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.97, 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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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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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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