Optimal. Leaf size=87 \[ -\frac {\sqrt {a x^2+b x^3}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3}}{4 a^2 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{5/2}} \]
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Rubi [A]
time = 0.06, antiderivative size = 87, 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 = {2050, 2033,
212} \begin {gather*} -\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{5/2}}+\frac {3 b \sqrt {a x^2+b x^3}}{4 a^2 x^2}-\frac {\sqrt {a x^2+b x^3}}{2 a x^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2050
Rubi steps
\begin {align*} \int \frac {1}{x^2 \sqrt {a x^2+b x^3}} \, dx &=-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}-\frac {(3 b) \int \frac {1}{x \sqrt {a x^2+b x^3}} \, dx}{4 a}\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3}}{4 a^2 x^2}+\frac {\left (3 b^2\right ) \int \frac {1}{\sqrt {a x^2+b x^3}} \, dx}{8 a^2}\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3}}{4 a^2 x^2}-\frac {\left (3 b^2\right ) \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^3}}\right )}{4 a^2}\\ &=-\frac {\sqrt {a x^2+b x^3}}{2 a x^3}+\frac {3 b \sqrt {a x^2+b x^3}}{4 a^2 x^2}-\frac {3 b^2 \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^3}}\right )}{4 a^{5/2}}\\ \end {align*}
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Mathematica [A]
time = 0.08, size = 83, normalized size = 0.95 \begin {gather*} \frac {\sqrt {a} \left (-2 a^2+a b x+3 b^2 x^2\right )-3 b^2 x^2 \sqrt {a+b x} \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 a^{5/2} x \sqrt {x^2 (a+b x)}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.38, size = 77, normalized size = 0.89
method | result | size |
risch | \(-\frac {\left (b x +a \right ) \left (-3 b x +2 a \right )}{4 a^{2} x \sqrt {x^{2} \left (b x +a \right )}}-\frac {3 b^{2} \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) \sqrt {b x +a}\, x}{4 a^{\frac {5}{2}} \sqrt {x^{2} \left (b x +a \right )}}\) | \(73\) |
default | \(-\frac {\sqrt {b x +a}\, \left (-3 \sqrt {b x +a}\, a^{\frac {3}{2}} b x +3 \arctanh \left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) a \,b^{2} x^{2}+2 \sqrt {b x +a}\, a^{\frac {5}{2}}\right )}{4 x \sqrt {b \,x^{3}+a \,x^{2}}\, a^{\frac {7}{2}}}\) | \(77\) |
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 = 1.29, size = 153, normalized size = 1.76 \begin {gather*} \left [\frac {3 \, \sqrt {a} b^{2} x^{3} \log \left (\frac {b x^{2} + 2 \, a x - 2 \, \sqrt {b x^{3} + a x^{2}} \sqrt {a}}{x^{2}}\right ) + 2 \, \sqrt {b x^{3} + a x^{2}} {\left (3 \, a b x - 2 \, a^{2}\right )}}{8 \, a^{3} x^{3}}, \frac {3 \, \sqrt {-a} b^{2} x^{3} \arctan \left (\frac {\sqrt {b x^{3} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{3} + a x^{2}} {\left (3 \, a b x - 2 \, a^{2}\right )}}{4 \, a^{3} x^{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 {1}{x^{2} \sqrt {x^{2} \left (a + b x\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.11, size = 73, normalized size = 0.84 \begin {gather*} \frac {\frac {3 \, b^{3} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a^{2}} + \frac {3 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{3} - 5 \, \sqrt {b x + a} a b^{3}}{a^{2} b^{2} x^{2}}}{4 \, b \mathrm {sgn}\left (x\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 5.41, size = 44, normalized size = 0.51 \begin {gather*} -\frac {2\,\sqrt {\frac {a}{b\,x}+1}\,{{}}_2{\mathrm {F}}_1\left (\frac {1}{2},\frac {5}{2};\ \frac {7}{2};\ -\frac {a}{b\,x}\right )}{5\,x\,\sqrt {b\,x^3+a\,x^2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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