Optimal. Leaf size=64 \[ 3 c \sqrt {b x+c x^2}-\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \]
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Rubi [A]
time = 0.02, antiderivative size = 64, 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 = {676, 678, 634,
212} \begin {gather*} -\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 c \sqrt {b x+c x^2}+3 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right ) \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 634
Rule 676
Rule 678
Rubi steps
\begin {align*} \int \frac {\left (b x+c x^2\right )^{3/2}}{x^3} \, dx &=-\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+(3 c) \int \frac {\sqrt {b x+c x^2}}{x} \, dx\\ &=3 c \sqrt {b x+c x^2}-\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+\frac {1}{2} (3 b c) \int \frac {1}{\sqrt {b x+c x^2}} \, dx\\ &=3 c \sqrt {b x+c x^2}-\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+(3 b c) \text {Subst}\left (\int \frac {1}{1-c x^2} \, dx,x,\frac {x}{\sqrt {b x+c x^2}}\right )\\ &=3 c \sqrt {b x+c x^2}-\frac {2 \left (b x+c x^2\right )^{3/2}}{x^2}+3 b \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {b x+c x^2}}\right )\\ \end {align*}
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Mathematica [A]
time = 0.11, size = 75, normalized size = 1.17 \begin {gather*} \frac {\sqrt {b+c x} \left ((-2 b+c x) \sqrt {b+c x}-3 b \sqrt {c} \sqrt {x} \log \left (-\sqrt {c} \sqrt {x}+\sqrt {b+c x}\right )\right )}{\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. \(124\) vs.
\(2(54)=108\).
time = 0.42, size = 125, normalized size = 1.95
method | result | size |
risch | \(-\frac {\left (c x +b \right ) \left (-c x +2 b \right )}{\sqrt {x \left (c x +b \right )}}+\frac {3 b \sqrt {c}\, \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{2}\) | \(56\) |
default | \(-\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{b \,x^{3}}+\frac {4 c \left (\frac {2 \left (c \,x^{2}+b x \right )^{\frac {5}{2}}}{b \,x^{2}}-\frac {6 c \left (\frac {\left (c \,x^{2}+b x \right )^{\frac {3}{2}}}{3}+\frac {b \left (\frac {\left (2 c x +b \right ) \sqrt {c \,x^{2}+b x}}{4 c}-\frac {b^{2} \ln \left (\frac {\frac {b}{2}+c x}{\sqrt {c}}+\sqrt {c \,x^{2}+b x}\right )}{8 c^{\frac {3}{2}}}\right )}{2}\right )}{b}\right )}{b}\) | \(125\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.30, size = 62, normalized size = 0.97 \begin {gather*} \frac {3}{2} \, b \sqrt {c} \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) - \frac {3 \, \sqrt {c x^{2} + b x} b}{x} + \frac {{\left (c x^{2} + b x\right )}^{\frac {3}{2}}}{x^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.42, size = 116, normalized size = 1.81 \begin {gather*} \left [\frac {3 \, b \sqrt {c} x \log \left (2 \, c x + b + 2 \, \sqrt {c x^{2} + b x} \sqrt {c}\right ) + 2 \, \sqrt {c x^{2} + b x} {\left (c x - 2 \, b\right )}}{2 \, x}, -\frac {3 \, b \sqrt {-c} x \arctan \left (\frac {\sqrt {c x^{2} + b x} \sqrt {-c}}{c x}\right ) - \sqrt {c x^{2} + b x} {\left (c x - 2 \, b\right )}}{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 {\left (x \left (b + c x\right )\right )^{\frac {3}{2}}}{x^{3}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.06, size = 76, normalized size = 1.19 \begin {gather*} -\frac {3}{2} \, b \sqrt {c} \log \left ({\left | -2 \, {\left (\sqrt {c} x - \sqrt {c x^{2} + b x}\right )} \sqrt {c} - b \right |}\right ) + \sqrt {c x^{2} + b x} c + \frac {2 \, b^{2}}{\sqrt {c} x - \sqrt {c x^{2} + b x}} \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 = -0.02 \begin {gather*} \int \frac {{\left (c\,x^2+b\,x\right )}^{3/2}}{x^3} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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