Optimal. Leaf size=93 \[ \frac {8 c (b+2 c x) (5 b B-6 A c)}{15 b^4 \sqrt {b x+c x^2}}-\frac {2 (5 b B-6 A c)}{15 b^2 x \sqrt {b x+c x^2}}-\frac {2 A}{5 b x^2 \sqrt {b x+c x^2}} \]
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Rubi [A] time = 0.08, antiderivative size = 93, 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, 613} \begin {gather*} \frac {8 c (b+2 c x) (5 b B-6 A c)}{15 b^4 \sqrt {b x+c x^2}}-\frac {2 (5 b B-6 A c)}{15 b^2 x \sqrt {b x+c x^2}}-\frac {2 A}{5 b x^2 \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 658
Rule 792
Rubi steps
\begin {align*} \int \frac {A+B x}{x^2 \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 A}{5 b x^2 \sqrt {b x+c x^2}}+\frac {\left (2 \left (\frac {1}{2} (b B-2 A c)-2 (-b B+A c)\right )\right ) \int \frac {1}{x \left (b x+c x^2\right )^{3/2}} \, dx}{5 b}\\ &=-\frac {2 A}{5 b x^2 \sqrt {b x+c x^2}}-\frac {2 (5 b B-6 A c)}{15 b^2 x \sqrt {b x+c x^2}}-\frac {(4 c (5 b B-6 A c)) \int \frac {1}{\left (b x+c x^2\right )^{3/2}} \, dx}{15 b^2}\\ &=-\frac {2 A}{5 b x^2 \sqrt {b x+c x^2}}-\frac {2 (5 b B-6 A c)}{15 b^2 x \sqrt {b x+c x^2}}+\frac {8 c (5 b B-6 A c) (b+2 c x)}{15 b^4 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 75, normalized size = 0.81 \begin {gather*} -\frac {2 \left (3 A \left (b^3-2 b^2 c x+8 b c^2 x^2+16 c^3 x^3\right )+5 b B x \left (b^2-4 b c x-8 c^2 x^2\right )\right )}{15 b^4 x^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.36, size = 91, normalized size = 0.98 \begin {gather*} \frac {2 \sqrt {b x+c x^2} \left (-3 A b^3+6 A b^2 c x-24 A b c^2 x^2-48 A c^3 x^3-5 b^3 B x+20 b^2 B c x^2+40 b B c^2 x^3\right )}{15 b^4 x^3 (b+c x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 93, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (3 \, A b^{3} - 8 \, {\left (5 \, B b c^{2} - 6 \, A c^{3}\right )} x^{3} - 4 \, {\left (5 \, B b^{2} c - 6 \, A b c^{2}\right )} x^{2} + {\left (5 \, B b^{3} - 6 \, A b^{2} c\right )} x\right )} \sqrt {c x^{2} + b x}}{15 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {B x + A}{{\left (c x^{2} + b x\right )}^{\frac {3}{2}} x^{2}}\,{d x} \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 = 0.92 \begin {gather*} -\frac {2 \left (c x +b \right ) \left (48 A \,c^{3} x^{3}-40 B b \,c^{2} x^{3}+24 A b \,c^{2} x^{2}-20 B \,b^{2} c \,x^{2}-6 A \,b^{2} c x +5 B \,b^{3} x +3 A \,b^{3}\right )}{15 \left (c \,x^{2}+b x \right )^{\frac {3}{2}} b^{4} x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.92, size = 142, normalized size = 1.53 \begin {gather*} \frac {16 \, B c^{2} x}{3 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {32 \, A c^{3} x}{5 \, \sqrt {c x^{2} + b x} b^{4}} + \frac {8 \, B c}{3 \, \sqrt {c x^{2} + b x} b^{2}} - \frac {16 \, A c^{2}}{5 \, \sqrt {c x^{2} + b x} b^{3}} - \frac {2 \, B}{3 \, \sqrt {c x^{2} + b x} b x} + \frac {4 \, A c}{5 \, \sqrt {c x^{2} + b x} b^{2} x} - \frac {2 \, A}{5 \, \sqrt {c x^{2} + b x} b x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.23, size = 87, normalized size = 0.94 \begin {gather*} -\frac {2\,\sqrt {c\,x^2+b\,x}\,\left (5\,B\,b^3\,x+3\,A\,b^3-20\,B\,b^2\,c\,x^2-6\,A\,b^2\,c\,x-40\,B\,b\,c^2\,x^3+24\,A\,b\,c^2\,x^2+48\,A\,c^3\,x^3\right )}{15\,b^4\,x^3\,\left (b+c\,x\right )} \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^{2} \left (x \left (b + c x\right )\right )^{\frac {3}{2}}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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