Optimal. Leaf size=76 \[ -\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 76, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {684, 654, 634,
212} \begin {gather*} \frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 654
Rule 684
Rubi steps
\begin {align*} \int \frac {x^2}{\sqrt {b x+c x^2}} \, dx &=\frac {x \sqrt {b x+c x^2}}{2 c}-\frac {(3 b) \int \frac {x}{\sqrt {b x+c x^2}} \, dx}{4 c}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{8 c^2}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{4 c^2}\\ &=-\frac {3 b \sqrt {b x+c x^2}}{4 c^2}+\frac {x \sqrt {b x+c x^2}}{2 c}+\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{4 c^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.07, size = 89, normalized size = 1.17 \begin {gather*} \frac {\sqrt {c} x \left (-3 b^2-b c x+2 c^2 x^2\right )-3 b^2 \sqrt {x} \sqrt {b+c x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{4 c^{5/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 71, normalized size = 0.93
method | result | size |
risch | \(-\frac {\left (-2 c x +3 b \right ) x \left (c x +b \right )}{4 c^{2} \sqrt {x \left (c x +b \right )}}+\frac {3 b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {5}{2}}}\) | \(62\) |
default | \(\frac {x \sqrt {c \,x^{2}+b x}}{2 c}-\frac {3 b \left (\frac {\sqrt {c \,x^{2}+b x}}{c}-\frac {b \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2 c^{\frac {3}{2}}}\right )}{4 c}\) | \(71\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 66, normalized size = 0.87 \begin {gather*} \frac {\sqrt {c x^{2} + b x} x}{2 \, c} + \frac {3 \, b^{2} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{8 \, c^{\frac {5}{2}}} - \frac {3 \, \sqrt {c x^{2} + b x} b}{4 \, c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.71, size = 126, normalized size = 1.66 \begin {gather*} \left [\frac {3 \, b^{2} \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (2 \, c^{2} x - 3 \, b c\right )} \sqrt {c x^{2} + b x}}{8 \, c^{3}}, -\frac {3 \, b^{2} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - {\left (2 \, c^{2} x - 3 \, b c\right )} \sqrt {c x^{2} + b x}}{4 \, c^{3}}\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}}{\sqrt {x \left (b + c x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 0.85, size = 65, normalized size = 0.86 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x} {\left (\frac {2 \, x}{c} - \frac {3 \, b}{c^{2}}\right )} - \frac {3 \, b^{2} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{8 \, c^{\frac {5}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {x^2}{\sqrt {c\,x^2+b\,x}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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