Optimal. Leaf size=57 \[ -\frac {2 \left (b x+c x^2\right )^{7/2} (9 b B-2 A c)}{63 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{9 b x^8} \]
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Rubi [A] time = 0.05, antiderivative size = 57, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {792, 650} \begin {gather*} -\frac {2 \left (b x+c x^2\right )^{7/2} (9 b B-2 A c)}{63 b^2 x^7}-\frac {2 A \left (b x+c x^2\right )^{7/2}}{9 b x^8} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rule 792
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^{5/2}}{x^8} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{9 b x^8}+\frac {\left (2 \left (-8 (-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^7} \, dx}{9 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{7/2}}{9 b x^8}-\frac {2 (9 b B-2 A c) \left (b x+c x^2\right )^{7/2}}{63 b^2 x^7}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 43, normalized size = 0.75 \begin {gather*} -\frac {2 (b+c x)^3 \sqrt {x (b+c x)} (7 A b-2 A c x+9 b B x)}{63 b^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.41, size = 108, normalized size = 1.89 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-7 A b^4-19 A b^3 c x-15 A b^2 c^2 x^2-A b c^3 x^3+2 A c^4 x^4-9 b^4 B x-27 b^3 B c x^2-27 b^2 B c^2 x^3-9 b B c^3 x^4\right )}{63 b^2 x^5} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 0.40, size = 102, normalized size = 1.79 \begin {gather*} -\frac {2 \, {\left (7 \, A b^{4} + {\left (9 \, B b c^{3} - 2 \, A c^{4}\right )} x^{4} + {\left (27 \, B b^{2} c^{2} + A b c^{3}\right )} x^{3} + 3 \, {\left (9 \, B b^{3} c + 5 \, A b^{2} c^{2}\right )} x^{2} + {\left (9 \, B b^{4} + 19 \, A b^{3} c\right )} x\right )} \sqrt {c x^{2} + b x}}{63 \, b^{2} x^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 431, normalized size = 7.56 \begin {gather*} \frac {2 \, {\left (63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{8} B c^{3} + 189 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} B b c^{\frac {5}{2}} + 63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7} A c^{\frac {7}{2}} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} B b^{2} c^{2} + 273 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{6} A b c^{3} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B b^{3} c^{\frac {3}{2}} + 567 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} A b^{2} c^{\frac {5}{2}} + 189 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b^{4} c + 693 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A b^{3} c^{2} + 63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{5} \sqrt {c} + 525 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b^{4} c^{\frac {3}{2}} + 9 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{6} + 243 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{5} c + 63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{6} \sqrt {c} + 7 \, A b^{7}\right )}}{63 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{9}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.04, size = 40, normalized size = 0.70 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-2 A c x +9 B b x +7 A b \right ) \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{63 b^{2} x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [B] time = 1.00, size = 258, normalized size = 4.53 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} B c^{3}}{7 \, b x} + \frac {4 \, \sqrt {c x^{2} + b x} A c^{4}}{63 \, b^{2} x} + \frac {\sqrt {c x^{2} + b x} B c^{2}}{7 \, x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} A c^{3}}{63 \, b x^{2}} - \frac {3 \, \sqrt {c x^{2} + b x} B b c}{28 \, x^{3}} + \frac {\sqrt {c x^{2} + b x} A c^{2}}{42 \, x^{3}} - \frac {15 \, \sqrt {c x^{2} + b x} B b^{2}}{28 \, x^{4}} - \frac {5 \, \sqrt {c x^{2} + b x} A b c}{252 \, x^{4}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} B b}{4 \, x^{5}} - \frac {5 \, \sqrt {c x^{2} + b x} A b^{2}}{36 \, x^{5}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} B}{x^{6}} + \frac {5 \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} A b}{12 \, x^{6}} - \frac {{\left (c x^{2} + b x\right )}^{\frac {5}{2}} A}{2 \, x^{7}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.89, size = 188, normalized size = 3.30 \begin {gather*} \frac {4\,A\,c^4\,\sqrt {c\,x^2+b\,x}}{63\,b^2\,x}-\frac {10\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{21\,x^3}-\frac {2\,B\,b^2\,\sqrt {c\,x^2+b\,x}}{7\,x^4}-\frac {6\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{7\,x^2}-\frac {2\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{63\,b\,x^2}-\frac {2\,A\,b^2\,\sqrt {c\,x^2+b\,x}}{9\,x^5}-\frac {2\,B\,c^3\,\sqrt {c\,x^2+b\,x}}{7\,b\,x}-\frac {38\,A\,b\,c\,\sqrt {c\,x^2+b\,x}}{63\,x^4}-\frac {6\,B\,b\,c\,\sqrt {c\,x^2+b\,x}}{7\,x^3} \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^{8}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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