Optimal. Leaf size=160 \[ \frac {32 c^3 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac {16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac {4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac {2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \]
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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 )^{3/2} (11 b B-8 A c)}{3465 b^5 x^3}-\frac {16 c^2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{1155 b^4 x^4}+\frac {4 c \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{231 b^3 x^5}-\frac {2 \left (b x+c x^2\right )^{3/2} (11 b B-8 A c)}{99 b^2 x^6}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7} \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) \sqrt {b x+c x^2}}{x^7} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}+\frac {\left (2 \left (-7 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{x^6} \, dx}{11 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}-\frac {(2 c (11 b B-8 A c)) \int \frac {\sqrt {b x+c x^2}}{x^5} \, dx}{33 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}+\frac {\left (8 c^2 (11 b B-8 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx}{231 b^3}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}-\frac {\left (16 c^3 (11 b B-8 A c)\right ) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{1155 b^4}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{11 b x^7}-\frac {2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{99 b^2 x^6}+\frac {4 c (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{231 b^3 x^5}-\frac {16 c^2 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{1155 b^4 x^4}+\frac {32 c^3 (11 b B-8 A c) \left (b x+c x^2\right )^{3/2}}{3465 b^5 x^3}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 100, normalized size = 0.62 \begin {gather*} -\frac {2 (x (b+c x))^{3/2} \left (A \left (315 b^4-280 b^3 c x+240 b^2 c^2 x^2-192 b c^3 x^3+128 c^4 x^4\right )+11 b B x \left (35 b^3-30 b^2 c x+24 b c^2 x^2-16 c^3 x^3\right )\right )}{3465 b^5 x^7} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.40, size = 132, normalized size = 0.82 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (315 A b^5+35 A b^4 c x-40 A b^3 c^2 x^2+48 A b^2 c^3 x^3-64 A b c^4 x^4+128 A c^5 x^5+385 b^5 B x+55 b^4 B c x^2-66 b^3 B c^2 x^3+88 b^2 B c^3 x^4-176 b B c^4 x^5\right )}{3465 b^5 x^6} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 129, normalized size = 0.81 \begin {gather*} -\frac {2 \, {\left (315 \, A b^{5} - 16 \, {\left (11 \, B b c^{4} - 8 \, A c^{5}\right )} x^{5} + 8 \, {\left (11 \, B b^{2} c^{3} - 8 \, A b c^{4}\right )} x^{4} - 6 \, {\left (11 \, B b^{3} c^{2} - 8 \, A b^{2} c^{3}\right )} x^{3} + 5 \, {\left (11 \, B b^{4} c - 8 \, A b^{3} c^{2}\right )} x^{2} + 35 \, {\left (11 \, B b^{5} + A b^{4} c\right )} x\right )} \sqrt {c x^{2} + b x}}{3465 \, b^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.20, size = 371, normalized size = 2.32 \begin {gather*} \frac {2 \, {\left (6930 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B c^{\frac {5}{2}} + 19404 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b c^{2} + 11088 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A c^{3} + 21945 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{2} c^{\frac {3}{2}} + 36960 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b c^{\frac {5}{2}} + 12375 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{3} c + 51480 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{2} c^{2} + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{4} \sqrt {c} + 38115 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{3} c^{\frac {3}{2}} + 385 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{5} + 15785 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{4} c + 3465 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{5} \sqrt {c} + 315 \, A b^{6}\right )}}{3465 \, {\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 = 110, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (128 A \,c^{4} x^{4}-176 B b \,c^{3} x^{4}-192 A b \,c^{3} x^{3}+264 B \,b^{2} c^{2} x^{3}+240 A \,b^{2} c^{2} x^{2}-330 B \,b^{3} c \,x^{2}-280 A \,b^{3} c x +385 b^{4} B x +315 A \,b^{4}\right ) \sqrt {c \,x^{2}+b x}}{3465 b^{5} x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.98, size = 238, normalized size = 1.49 \begin {gather*} \frac {32 \, \sqrt {c x^{2} + b x} B c^{4}}{315 \, b^{4} x} - \frac {256 \, \sqrt {c x^{2} + b x} A c^{5}}{3465 \, b^{5} x} - \frac {16 \, \sqrt {c x^{2} + b x} B c^{3}}{315 \, b^{3} x^{2}} + \frac {128 \, \sqrt {c x^{2} + b x} A c^{4}}{3465 \, b^{4} x^{2}} + \frac {4 \, \sqrt {c x^{2} + b x} B c^{2}}{105 \, b^{2} x^{3}} - \frac {32 \, \sqrt {c x^{2} + b x} A c^{3}}{1155 \, b^{3} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{63 \, b x^{4}} + \frac {16 \, \sqrt {c x^{2} + b x} A c^{2}}{693 \, b^{2} x^{4}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{9 \, x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{99 \, b x^{5}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{11 \, x^{6}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.23, size = 238, normalized size = 1.49 \begin {gather*} \frac {16\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{693\,b^2\,x^4}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{99\,b\,x^5}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^4}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{11\,x^6}-\frac {32\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{1155\,b^3\,x^3}+\frac {128\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{3465\,b^4\,x^2}-\frac {256\,A\,c^5\,\sqrt {c\,x^2+b\,x}}{3465\,b^5\,x}+\frac {4\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^3}-\frac {16\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{315\,b^3\,x^2}+\frac {32\,B\,c^4\,\sqrt {c\,x^2+b\,x}}{315\,b^4\,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 \frac {\sqrt {x \left (b + c x\right )} \left (A + B x\right )}{x^{7}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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