Optimal. Leaf size=89 \[ -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}} \]
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Rubi [A] time = 0.03, antiderivative size = 89, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 3, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.158, Rules used = {672, 660, 207} \[ -\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}} \]
Antiderivative was successfully verified.
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Rule 207
Rule 660
Rule 672
Rubi steps
\begin {align*} \int \frac {1}{x^{5/2} \sqrt {b x+c x^2}} \, dx &=-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}-\frac {(3 c) \int \frac {1}{x^{3/2} \sqrt {b x+c x^2}} \, dx}{4 b}\\ &=-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}+\frac {\left (3 c^2\right ) \int \frac {1}{\sqrt {x} \sqrt {b x+c x^2}} \, dx}{8 b^2}\\ &=-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}+\frac {\left (3 c^2\right ) \operatorname {Subst}\left (\int \frac {1}{-b+x^2} \, dx,x,\frac {\sqrt {b x+c x^2}}{\sqrt {x}}\right )}{4 b^2}\\ &=-\frac {\sqrt {b x+c x^2}}{2 b x^{5/2}}+\frac {3 c \sqrt {b x+c x^2}}{4 b^2 x^{3/2}}-\frac {3 c^2 \tanh ^{-1}\left (\frac {\sqrt {b x+c x^2}}{\sqrt {b} \sqrt {x}}\right )}{4 b^{5/2}}\\ \end {align*}
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Mathematica [C] time = 0.01, size = 40, normalized size = 0.45 \[ -\frac {2 c^2 \sqrt {x (b+c x)} \, _2F_1\left (\frac {1}{2},3;\frac {3}{2};\frac {c x}{b}+1\right )}{b^3 \sqrt {x}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 152, normalized size = 1.71 \[ \left [\frac {3 \, \sqrt {b} c^{2} x^{3} \log \left (-\frac {c x^{2} + 2 \, b x - 2 \, \sqrt {c x^{2} + b x} \sqrt {b} \sqrt {x}}{x^{2}}\right ) + 2 \, {\left (3 \, b c x - 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{8 \, b^{3} x^{3}}, \frac {3 \, \sqrt {-b} c^{2} x^{3} \arctan \left (\frac {\sqrt {-b} \sqrt {x}}{\sqrt {c x^{2} + b x}}\right ) + {\left (3 \, b c x - 2 \, b^{2}\right )} \sqrt {c x^{2} + b x} \sqrt {x}}{4 \, b^{3} x^{3}}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.22, size = 69, normalized size = 0.78 \[ \frac {\frac {3 \, c^{3} \arctan \left (\frac {\sqrt {c x + b}}{\sqrt {-b}}\right )}{\sqrt {-b} b^{2}} + \frac {3 \, {\left (c x + b\right )}^{\frac {3}{2}} c^{3} - 5 \, \sqrt {c x + b} b c^{3}}{b^{2} c^{2} x^{2}}}{4 \, c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 72, normalized size = 0.81 \[ -\frac {\sqrt {\left (c x +b \right ) x}\, \left (3 c^{2} x^{2} \arctanh \left (\frac {\sqrt {c x +b}}{\sqrt {b}}\right )-3 \sqrt {c x +b}\, \sqrt {b}\, c x +2 \sqrt {c x +b}\, b^{\frac {3}{2}}\right )}{4 \sqrt {c x +b}\, b^{\frac {5}{2}} x^{\frac {5}{2}}} \]
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 \[ \int \frac {1}{\sqrt {c x^{2} + b x} x^{\frac {5}{2}}}\,{d x} \]
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 \[ \int \frac {1}{x^{5/2}\,\sqrt {c\,x^2+b\,x}} \,d x \]
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 \[ \int \frac {1}{x^{\frac {5}{2}} \sqrt {x \left (b + c x\right )}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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