Optimal. Leaf size=48 \[ -\frac {2 x}{c \sqrt {b x+c x^2}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}} \]
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Rubi [A]
time = 0.01, antiderivative size = 48, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 3, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.176, Rules used = {666, 634, 212}
\begin {gather*} \frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}-\frac {2 x}{c \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 666
Rubi steps
\begin {align*} \int \frac {x^2}{\left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {2 x}{c \sqrt {b x+c x^2}}+\frac {\int \frac {1}{\sqrt {b x+c x^2}} \, dx}{c}\\ &=-\frac {2 x}{c \sqrt {b x+c x^2}}+\frac {2 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{c}\\ &=-\frac {2 x}{c \sqrt {b x+c x^2}}+\frac {2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 64, normalized size = 1.33 \begin {gather*} -\frac {2 \left (\sqrt {c} x+\sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{c^{3/2} \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. \(93\) vs.
\(2(40)=80\).
time = 0.42, size = 94, normalized size = 1.96
method | result | size |
default | \(-\frac {x}{c \sqrt {c \,x^{2}+b x}}-\frac {b \left (-\frac {1}{c \sqrt {c \,x^{2}+b x}}+\frac {2 c x +b}{b c \sqrt {c \,x^{2}+b x}}\right )}{2 c}+\frac {\ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{c^{\frac {3}{2}}}\) | \(94\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 45, normalized size = 0.94 \begin {gather*} -\frac {2 \, x}{\sqrt {c x^{2} + b x} c} + \frac {\log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{c^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.91, size = 126, normalized size = 2.62 \begin {gather*} \left [\frac {{\left (c x + b\right )} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - 2 \, \sqrt {c x^{2} + b x} c}{c^{3} x + b c^{2}}, -\frac {2 \, {\left ({\left (c x + b\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + \sqrt {c x^{2} + b x} c\right )}}{c^{3} x + b c^{2}}\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 {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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Giac [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 0.20, size = 46, normalized size = 0.96 \begin {gather*} \frac {\ln \left (\frac {\frac {b}{2}+c\,x}{\sqrt {c}}+\sqrt {c\,x^2+b\,x}\right )}{c^{3/2}}-\frac {2\,x}{c\,\sqrt {c\,x^2+b\,x}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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