Optimal. Leaf size=160 \[ \frac {32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac {4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac {2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \]
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Rubi [A] time = 0.16, antiderivative size = 160, normalized size of antiderivative = 1.00, number of steps used = 5, 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 {32 c^3 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{45045 b^5 x^7}-\frac {16 c^2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{6435 b^4 x^8}+\frac {4 c \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{715 b^3 x^9}-\frac {2 \left (b x+c x^2\right )^{7/2} (15 b B-8 A c)}{195 b^2 x^{10}}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}} \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^{11}} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}+\frac {\left (2 \left (-11 (-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^{10}} \, dx}{15 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}-\frac {(2 c (15 b B-8 A c)) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^9} \, dx}{65 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}+\frac {\left (8 c^2 (15 b B-8 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^8} \, dx}{715 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac {16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}-\frac {\left (16 c^3 (15 b B-8 A c)\right ) \int \frac {\left (b x+c x^2\right )^{5/2}}{x^7} \, dx}{6435 b^4}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{15 b x^{11}}-\frac {2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{195 b^2 x^{10}}+\frac {4 c (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{715 b^3 x^9}-\frac {16 c^2 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{6435 b^4 x^8}+\frac {32 c^3 (15 b B-8 A c) \left (b x+c x^2\right )^{7/2}}{45045 b^5 x^7}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 107, normalized size = 0.67 \begin {gather*} \frac {2 (b+c x)^3 \sqrt {x (b+c x)} \left (A \left (-3003 b^4+1848 b^3 c x-1008 b^2 c^2 x^2+448 b c^3 x^3-128 c^4 x^4\right )+15 b B x \left (-231 b^3+126 b^2 c x-56 b c^2 x^2+16 c^3 x^3\right )\right )}{45045 b^5 x^8} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.55, size = 180, normalized size = 1.12 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (3003 A b^7+7161 A b^6 c x+4473 A b^5 c^2 x^2+35 A b^4 c^3 x^3-40 A b^3 c^4 x^4+48 A b^2 c^5 x^5-64 A b c^6 x^6+128 A c^7 x^7+3465 b^7 B x+8505 b^6 B c x^2+5565 b^5 B c^2 x^3+75 b^4 B c^3 x^4-90 b^3 B c^4 x^5+120 b^2 B c^5 x^6-240 b B c^6 x^7\right )}{45045 b^5 x^8} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 177, normalized size = 1.11 \begin {gather*} -\frac {2 \, {\left (3003 \, A b^{7} - 16 \, {\left (15 \, B b c^{6} - 8 \, A c^{7}\right )} x^{7} + 8 \, {\left (15 \, B b^{2} c^{5} - 8 \, A b c^{6}\right )} x^{6} - 6 \, {\left (15 \, B b^{3} c^{4} - 8 \, A b^{2} c^{5}\right )} x^{5} + 5 \, {\left (15 \, B b^{4} c^{3} - 8 \, A b^{3} c^{4}\right )} x^{4} + 35 \, {\left (159 \, B b^{5} c^{2} + A b^{4} c^{3}\right )} x^{3} + 63 \, {\left (135 \, B b^{6} c + 71 \, A b^{5} c^{2}\right )} x^{2} + 231 \, {\left (15 \, B b^{7} + 31 \, A b^{6} c\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, b^{5} x^{8}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.24, size = 611, normalized size = 3.82 \begin {gather*} \frac {2 \, {\left (90090 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{11} B c^{\frac {9}{2}} + 540540 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} B b c^{4} + 144144 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{10} A c^{5} + 1486485 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} B b^{2} c^{\frac {7}{2}} + 960960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9} A b c^{\frac {9}{2}} + 2425995 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B b^{3} c^{3} + 2934360 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} A b^{2} c^{4} + 2567565 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b^{4} c^{\frac {5}{2}} + 5360355 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A b^{3} c^{\frac {7}{2}} + 1816815 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{5} c^{2} + 6451445 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b^{4} c^{3} + 855855 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{6} c^{\frac {3}{2}} + 5324319 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{5} c^{\frac {5}{2}} + 257985 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{7} c + 3042585 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{6} c^{2} + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{8} \sqrt {c} + 1186185 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{7} c^{\frac {3}{2}} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{9} + 301455 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{8} c + 45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{9} \sqrt {c} + 3003 \, A b^{10}\right )}}{45045 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{15}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 110, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-240 B b \,c^{3} x^{4}-448 A b \,c^{3} x^{3}+840 B \,b^{2} c^{2} x^{3}+1008 A \,b^{2} c^{2} x^{2}-1890 B \,b^{3} c \,x^{2}-1848 A \,b^{3} c x +3465 b^{4} B x +3003 A \,b^{4}\right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{45045 b^{5} x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 0.93, size = 396, normalized size = 2.48 \begin {gather*} \frac {32 \, \sqrt {c x^{2} + b x} B c^{6}}{3003 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{7}}{45045 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{5}}{3003 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{6}}{45045 \, b^{4} x^{2}} + \frac {4 \, \sqrt {c x^{2} + b x} B c^{4}}{1001 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{5}}{15015 \, b^{3} x^{3}} - \frac {10 \, \sqrt {c x^{2} + b x} B c^{3}}{3003 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c^{4}}{9009 \, b^{2} x^{4}} + \frac {5 \, \sqrt {c x^{2} + b x} B c^{2}}{1716 \, x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{1287 \, b x^{5}} - \frac {3 \, \sqrt {c x^{2} + b x} B b c}{1144 \, x^{6}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{715 \, x^{6}} - \frac {3 \, \sqrt {c x^{2} + b x} B b^{2}}{104 \, x^{7}} - \frac {\sqrt {c x^{2} + b x} A b c}{780 \, x^{7}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{8 \, x^{8}} - \frac {\sqrt {c x^{2} + b x} A b^{2}}{60 \, x^{8}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{4 \, x^{9}} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{12 \, x^{9}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{5 \, x^{10}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.76, size = 326, normalized size = 2.04 \begin {gather*} \frac {16\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{9009\,b^2\,x^4}-\frac {142\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{715\,x^6}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{13\,x^7}-\frac {106\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{429\,x^5}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{1287\,b\,x^5}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{15\,x^8}-\frac {32\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{15015\,b^3\,x^3}+\frac {128\,A\,c^6\,\sqrt {c\,x^2+b\,x}}{45045\,b^4\,x^2}-\frac {256\,A\,c^7\,\sqrt {c\,x^2+b\,x}}{45045\,b^5\,x}-\frac {10\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{3003\,b\,x^4}+\frac {4\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{1001\,b^2\,x^3}-\frac {16\,B\,c^5\,\sqrt {c\,x^2+b\,x}}{3003\,b^3\,x^2}+\frac {32\,B\,c^6\,\sqrt {c\,x^2+b\,x}}{3003\,b^4\,x}-\frac {62\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{195\,x^7}-\frac {54\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{143\,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^{11}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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