Optimal. Leaf size=78 \[ \frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}} \]
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Rubi [A]
time = 0.02, antiderivative size = 78, 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 = {678, 626, 634,
212} \begin {gather*} -\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}}+\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 626
Rule 634
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x} \, dx &=\frac {1}{3} \left (b x+c x^2\right )^{3/2}+\frac {1}{2} b \int \sqrt {b x+c x^2} \, dx\\ &=\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \int \frac {1}{\sqrt {b x+c x^2}} \, dx}{16 c}\\ &=\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )}{8 c}\\ &=\frac {b (b+2 c x) \sqrt {b x+c x^2}}{8 c}+\frac {1}{3} \left (b x+c x^2\right )^{3/2}-\frac {b^3 \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )}{8 c^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 88, normalized size = 1.13 \begin {gather*} \frac {\sqrt {x (b+c x)} \left (\sqrt {c} \left (3 b^2+14 b c x+8 c^2 x^2\right )+\frac {3 b^3 \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )}{\sqrt {x} \sqrt {b+c x}}\right )}{24 c^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.41, size = 73, normalized size = 0.94
method | result | size |
default | \(\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2}\) | \(73\) |
risch | \(\frac {\left (8 c^{2} x^{2}+14 b c x +3 b^{2}\right ) x \left (c x +b \right )}{24 c \sqrt {x \left (c x +b \right )}}-\frac {b^{3} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{16 c^{\frac {3}{2}}}\) | \(73\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.27, size = 79, normalized size = 1.01 \begin {gather*} \frac {1}{4} \, \sqrt {c x^{2} + b x} b x - \frac {b^{3} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right )}{16 \, c^{\frac {3}{2}}} + \frac {1}{3} \, {\left (c x^{2} + b x\right )}^{\frac {3}{2}} + \frac {\sqrt {c x^{2} + b x} b^{2}}{8 \, c} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.25, size = 147, normalized size = 1.88 \begin {gather*} \left [\frac {3 \, b^{3} \sqrt {c} \log \left (2 \, c x + b - 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{48 \, c^{2}}, \frac {3 \, b^{3} \sqrt {-c} \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) + {\left (8 \, c^{3} x^{2} + 14 \, b c^{2} x + 3 \, b^{2} c\right )} \sqrt {c x^{2} + b x}}{24 \, 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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.54, size = 72, normalized size = 0.92 \begin {gather*} \frac {b^{3} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right )}{16 \, c^{\frac {3}{2}}} + \frac {1}{24} \, \sqrt {c x^{2} + b x} {\left (2 \, {\left (4 \, c x + 7 \, b\right )} x + \frac {3 \, b^{2}}{c}\right )} \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 {{\left (c\,x^2+b\,x\right )}^{3/2}}{x} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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