Optimal. Leaf size=90 \[ \frac {4 c \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{105 b^3 x^3}-\frac {2 \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^4}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]
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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 )^{3/2} (7 b B-4 A c)}{105 b^3 x^3}-\frac {2 \left (b x+c x^2\right )^{3/2} (7 b B-4 A c)}{35 b^2 x^4}-\frac {2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5} \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^5} \, dx &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5}+\frac {\left (2 \left (-5 (-b B+A c)+\frac {3}{2} (-b B+2 A c)\right )\right ) \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx}{7 b}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5}-\frac {2 (7 b B-4 A c) \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac {(2 c (7 b B-4 A c)) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{35 b^2}\\ &=-\frac {2 A \left (b x+c x^2\right )^{3/2}}{7 b x^5}-\frac {2 (7 b B-4 A c) \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}+\frac {4 c (7 b B-4 A c) \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 56, normalized size = 0.62 \begin {gather*} -\frac {2 (x (b+c x))^{3/2} \left (A \left (15 b^2-12 b c x+8 c^2 x^2\right )+7 b B x (3 b-2 c x)\right )}{105 b^3 x^5} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.34, size = 84, normalized size = 0.93 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (15 A b^3+3 A b^2 c x-4 A b c^2 x^2+8 A c^3 x^3+21 b^3 B x+7 b^2 B c x^2-14 b B c^2 x^3\right )}{105 b^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 80, normalized size = 0.89 \begin {gather*} -\frac {2 \, {\left (15 \, A b^{3} - 2 \, {\left (7 \, B b c^{2} - 4 \, A c^{3}\right )} x^{3} + {\left (7 \, B b^{2} c - 4 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} + A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.23, size = 251, normalized size = 2.79 \begin {gather*} \frac {2 \, {\left (105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{5} B c^{\frac {3}{2}} + 175 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B b c + 140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} A c^{2} + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b^{2} \sqrt {c} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A b c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{3} + 273 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b^{2} c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{3} \sqrt {c} + 15 \, A b^{4}\right )}}{105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{7}} \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}-14 B b c \,x^{2}-12 A b c x +21 B \,b^{2} x +15 A \,b^{2}\right ) \sqrt {c \,x^{2}+b x}}{105 b^{3} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 146, normalized size = 1.62 \begin {gather*} \frac {4 \, \sqrt {c x^{2} + b x} B c^{2}}{15 \, b^{2} x} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{3}}{105 \, b^{3} x} - \frac {2 \, \sqrt {c x^{2} + b x} B c}{15 \, b x^{2}} + \frac {8 \, \sqrt {c x^{2} + b x} A c^{2}}{105 \, b^{2} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{5 \, x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A c}{35 \, b x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{7 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.67, size = 146, normalized size = 1.62 \begin {gather*} \frac {8\,A\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^2}-\frac {2\,B\,\sqrt {c\,x^2+b\,x}}{5\,x^3}-\frac {2\,A\,c\,\sqrt {c\,x^2+b\,x}}{35\,b\,x^3}-\frac {2\,B\,c\,\sqrt {c\,x^2+b\,x}}{15\,b\,x^2}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{7\,x^4}-\frac {16\,A\,c^3\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x}+\frac {4\,B\,c^2\,\sqrt {c\,x^2+b\,x}}{15\,b^2\,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^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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