Optimal. Leaf size=33 \[ -\frac {2 (A b-(b B-2 A c) x)}{b^2 \sqrt {b x+c x^2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 33, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {650}
\begin {gather*} -\frac {2 (A b-x (b B-2 A c))}{b^2 \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 650
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 (A b-(b B-2 A c) x)}{b^2 \sqrt {b x+c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.09, size = 30, normalized size = 0.91 \begin {gather*} \frac {2 b B x-2 A (b+2 c x)}{b^2 \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(67\) vs.
\(2(31)=62\).
time = 0.54, size = 68, normalized size = 2.06
method | result | size |
gosper | \(-\frac {2 x \left (c x +b \right ) \left (2 A c x -b B x +A b \right )}{b^{2} \left (c \,x^{2}+b x \right )^{\frac {3}{2}}}\) | \(37\) |
trager | \(-\frac {2 \left (2 A c x -b B x +A b \right ) \sqrt {c \,x^{2}+b x}}{\left (c x +b \right ) b^{2} x}\) | \(41\) |
risch | \(-\frac {2 A \left (c x +b \right )}{b^{2} \sqrt {x \left (c x +b \right )}}-\frac {2 \left (A c -B b \right ) x}{\sqrt {x \left (c x +b \right )}\, b^{2}}\) | \(45\) |
default | \(B \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )-\frac {2 A \left (2 c x +b \right )}{b^{2} \sqrt {c \,x^{2}+b x}}\) | \(68\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 55, normalized size = 1.67 \begin {gather*} \frac {2 \, B x}{\sqrt {c x^{2} + b x} b} - \frac {4 \, A c x}{\sqrt {c x^{2} + b x} b^{2}} - \frac {2 \, A}{\sqrt {c x^{2} + b x} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 3.32, size = 44, normalized size = 1.33 \begin {gather*} -\frac {2 \, \sqrt {c x^{2} + b x} {\left (A b - {\left (B b - 2 \, A c\right )} x\right )}}{b^{2} c x^{2} + b^{3} 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 {A + B x}{\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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Giac [A]
time = 0.98, size = 33, normalized size = 1.00 \begin {gather*} -\frac {2 \, {\left (\frac {A}{b} - \frac {{\left (B b - 2 \, A c\right )} x}{b^{2}}\right )}}{\sqrt {c x^{2} + b x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 1.12, size = 31, normalized size = 0.94 \begin {gather*} -\frac {2\,A\,b+4\,A\,c\,x-2\,B\,b\,x}{b^2\,\sqrt {c\,x^2+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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