Optimal. Leaf size=117 \[ -\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {15 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 117, normalized size of antiderivative = 1.00, number of steps
used = 5, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {686, 680, 674,
213} \begin {gather*} -\frac {15 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 213
Rule 674
Rule 680
Rule 686
Rubi steps
\begin {align*} \int \frac {1}{x^{3/2} \left (b x+c x^2\right )^{3/2}} \, dx &=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}-\frac {(5 c) \int \frac {1}{\sqrt {x} \left (b x+c x^2\right )^{3/2}} \, dx}{4 b}\\ &=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \int \frac {\sqrt {x}}{\left (b x+c x^2\right )^{3/2}} \, dx}{8 b^2}\\ &=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^3}\\ &=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}+\frac {\left (15 c^2\right ) \text {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^3}\\ &=-\frac {1}{2 b x^{3/2} \sqrt {b x+c x^2}}+\frac {5 c}{4 b^2 \sqrt {x} \sqrt {b x+c x^2}}+\frac {15 c^2 \sqrt {x}}{4 b^3 \sqrt {b x+c x^2}}-\frac {15 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{7/2}}\\ \end {align*}
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Mathematica [A]
time = 0.10, size = 84, normalized size = 0.72 \begin {gather*} \frac {\sqrt {b} \left (-2 b^2+5 b c x+15 c^2 x^2\right )-15 c^2 x^2 \sqrt {b+c x} \tanh ^{-1}\left (\frac {\sqrt {b+c x}}{\sqrt {b}}\right )}{4 b^{7/2} x^{3/2} \sqrt {x (b+c x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 76, normalized size = 0.65
method | result | size |
default | \(-\frac {\sqrt {x \left (c x +b \right )}\, \left (15 \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right ) \sqrt {c x +b}\, c^{2} x^{2}-5 b^{\frac {3}{2}} c x -15 c^{2} x^{2} \sqrt {b}+2 b^{\frac {5}{2}}\right )}{4 x^{\frac {5}{2}} \left (c x +b \right ) b^{\frac {7}{2}}}\) | \(76\) |
risch | \(-\frac {\left (c x +b \right ) \left (-7 c x +2 b \right )}{4 b^{3} x^{\frac {3}{2}} \sqrt {x \left (c x +b \right )}}+\frac {c^{2} \left (-\frac {30 \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )}{\sqrt {b}}+\frac {16}{\sqrt {c x +b}}\right ) \sqrt {c x +b}\, \sqrt {x}}{8 b^{3} \sqrt {x \left (c x +b \right )}}\) | \(86\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.38, size = 218, normalized size = 1.86 \begin {gather*} \left [\frac {15 \, {\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \sqrt {b} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}, \frac {15 \, {\left (c^{3} x^{4} + b c^{2} x^{3}\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (15 \, b c^{2} x^{2} + 5 \, b^{2} c x - 2 \, b^{3}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, {\left (b^{4} c x^{4} + b^{5} x^{3}\right )}}\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 {1}{x^{\frac {3}{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 [A]
time = 1.09, size = 80, normalized size = 0.68 \begin {gather*} \frac {15 \, c^{2} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{4 \, \sqrt {-b} b^{3}} + \frac {2 \, c^{2}}{\sqrt {c x + b} b^{3}} + \frac {7 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{2} - 9 \, \sqrt {c x + b} b c^{2}}{4 \, b^{3} c^{2} x^{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 {1}{x^{3/2}\,{\left (c\,x^2+b\,x\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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