Optimal. Leaf size=125 \[ -\frac {16 c^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac {8 c \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac {2 \sqrt {b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4} \]
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Rubi [A] time = 0.11, antiderivative size = 125, normalized size of antiderivative = 1.00, number of steps used = 4, 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 {16 c^2 \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^4 x}+\frac {8 c \sqrt {b x+c x^2} (7 b B-6 A c)}{105 b^3 x^2}-\frac {2 \sqrt {b x+c x^2} (7 b B-6 A c)}{35 b^2 x^3}-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4} \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}{x^4 \sqrt {b x+c x^2}} \, dx &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}+\frac {\left (2 \left (-4 (-b B+A c)+\frac {1}{2} (-b B+2 A c)\right )\right ) \int \frac {1}{x^3 \sqrt {b x+c x^2}} \, dx}{7 b}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}-\frac {(4 c (7 b B-6 A c)) \int \frac {1}{x^2 \sqrt {b x+c x^2}} \, dx}{35 b^2}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}+\frac {8 c (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^3 x^2}+\frac {\left (8 c^2 (7 b B-6 A c)\right ) \int \frac {1}{x \sqrt {b x+c x^2}} \, dx}{105 b^3}\\ &=-\frac {2 A \sqrt {b x+c x^2}}{7 b x^4}-\frac {2 (7 b B-6 A c) \sqrt {b x+c x^2}}{35 b^2 x^3}+\frac {8 c (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^3 x^2}-\frac {16 c^2 (7 b B-6 A c) \sqrt {b x+c x^2}}{105 b^4 x}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 79, normalized size = 0.63 \begin {gather*} -\frac {2 \sqrt {x (b+c x)} \left (3 A \left (5 b^3-6 b^2 c x+8 b c^2 x^2-16 c^3 x^3\right )+7 b B x \left (3 b^2-4 b c x+8 c^2 x^2\right )\right )}{105 b^4 x^4} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.35, size = 84, normalized size = 0.67 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-15 A b^3+18 A b^2 c x-24 A b c^2 x^2+48 A c^3 x^3-21 b^3 B x+28 b^2 B c x^2-56 b B c^2 x^3\right )}{105 b^4 x^4} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 82, normalized size = 0.66 \begin {gather*} -\frac {2 \, {\left (15 \, A b^{3} + 8 \, {\left (7 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} - 4 \, {\left (7 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + 3 \, {\left (7 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{105 \, b^{4} x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 191, normalized size = 1.53 \begin {gather*} \frac {2 \, {\left (140 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{4} B c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} B b \sqrt {c} + 210 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{3} A c^{\frac {3}{2}} + 21 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} B b^{2} + 252 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )}^{2} A b c + 105 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} A b^{2} \sqrt {c} + 15 \, A b^{3}\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 = 86, normalized size = 0.69 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (-48 A \,c^{3} x^{3}+56 B b \,c^{2} x^{3}+24 A b \,c^{2} x^{2}-28 B \,b^{2} c \,x^{2}-18 A \,b^{2} c x +21 B \,b^{3} x +15 A \,b^{3}\right )}{105 \sqrt {c \,x^{2}+b x}\, b^{4} x^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.93, size = 152, normalized size = 1.22 \begin {gather*} -\frac {16 \, \sqrt {c x^{2} + b x} B c^{2}}{15 \, b^{3} x} + \frac {32 \, \sqrt {c x^{2} + b x} A c^{3}}{35 \, b^{4} x} + \frac {8 \, \sqrt {c x^{2} + b x} B c}{15 \, b^{2} x^{2}} - \frac {16 \, \sqrt {c x^{2} + b x} A c^{2}}{35 \, b^{3} x^{2}} - \frac {2 \, \sqrt {c x^{2} + b x} B}{5 \, b x^{3}} + \frac {12 \, \sqrt {c x^{2} + b x} A c}{35 \, b^{2} x^{3}} - \frac {2 \, \sqrt {c x^{2} + b x} A}{7 \, b x^{4}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.12, size = 113, normalized size = 0.90 \begin {gather*} \frac {\sqrt {c\,x^2+b\,x}\,\left (96\,A\,c^3-112\,B\,b\,c^2\right )}{105\,b^4\,x}-\frac {\left (48\,A\,c^2-56\,B\,b\,c\right )\,\sqrt {c\,x^2+b\,x}}{105\,b^3\,x^2}-\frac {2\,A\,\sqrt {c\,x^2+b\,x}}{7\,b\,x^4}+\frac {\sqrt {c\,x^2+b\,x}\,\left (12\,A\,c-14\,B\,b\right )}{35\,b^2\,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 {A + B x}{x^{4} \sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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