Optimal. Leaf size=74 \[ -\frac {16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac {8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5} \]
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Rubi [A] time = 0.03, antiderivative size = 74, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.118, Rules used = {658, 650} \begin {gather*} -\frac {16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}+\frac {8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac {2 \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
Rubi steps
\begin {align*} \int \frac {\sqrt {b x+c x^2}}{x^5} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}-\frac {(4 c) \int \frac {\sqrt {b x+c x^2}}{x^4} \, dx}{7 b}\\ &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}+\frac {8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}+\frac {\left (8 c^2\right ) \int \frac {\sqrt {b x+c x^2}}{x^3} \, dx}{35 b^2}\\ &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{7 b x^5}+\frac {8 c \left (b x+c x^2\right )^{3/2}}{35 b^2 x^4}-\frac {16 c^2 \left (b x+c x^2\right )^{3/2}}{105 b^3 x^3}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 51, normalized size = 0.69 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (15 b^3+3 b^2 c x-4 b c^2 x^2+8 c^3 x^3\right )}{105 b^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.14, size = 53, normalized size = 0.72 \begin {gather*} -\frac {2 \sqrt {b x+c x^2} \left (15 b^3+3 b^2 c x-4 b c^2 x^2+8 c^3 x^3\right )}{105 b^3 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.39, size = 49, normalized size = 0.66 \begin {gather*} -\frac {2 \, {\left (8 \, c^{3} x^{3} - 4 \, b c^{2} x^{2} + 3 \, b^{2} c x + 15 \, b^{3}\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 = 136, normalized size = 1.84 \begin {gather*} \frac {2 \, {\left (140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} c^{2} + 315 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} b c^{\frac {3}{2}} + 273 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} b^{2} c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} b^{3} \sqrt {c} + 15 \, 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 = 0.04, size = 44, normalized size = 0.59 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (8 c^{2} x^{2}-12 b c x +15 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 = 1.24, size = 81, normalized size = 1.09 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} c^{3}}{105 \, b^{3} x} + \frac {8 \, \sqrt {c x^{2} + b x} c^{2}}{105 \, b^{2} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} c}{35 \, b x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x}}{7 \, x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.42, size = 81, normalized size = 1.09 \begin {gather*} \frac {8\,c^2\,\sqrt {c\,x^2+b\,x}}{105\,b^2\,x^2}-\frac {2\,\sqrt {c\,x^2+b\,x}}{7\,x^4}-\frac {16\,c^3\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x}-\frac {2\,c\,\sqrt {c\,x^2+b\,x}}{35\,b\,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 {\sqrt {x \left (b + c x\right )}}{x^{5}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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