Optimal. Leaf size=59 \[ -\frac {\sqrt {a x^2+b x^5}}{3 a x^4}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^5}}\right )}{3 a^{3/2}} \]
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Rubi [A]
time = 0.04, antiderivative size = 59, normalized size of antiderivative = 1.00, number of steps
used = 3, 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 {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^5}}\right )}{3 a^{3/2}}-\frac {\sqrt {a x^2+b x^5}}{3 a x^4} \end {gather*}
Antiderivative was successfully verified.
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Rule 212
Rule 2033
Rule 2050
Rubi steps
\begin {align*} \int \frac {1}{x^3 \sqrt {a x^2+b x^5}} \, dx &=-\frac {\sqrt {a x^2+b x^5}}{3 a x^4}-\frac {b \int \frac {1}{\sqrt {a x^2+b x^5}} \, dx}{2 a}\\ &=-\frac {\sqrt {a x^2+b x^5}}{3 a x^4}+\frac {b \text {Subst}\left (\int \frac {1}{1-a x^2} \, dx,x,\frac {x}{\sqrt {a x^2+b x^5}}\right )}{3 a}\\ &=-\frac {\sqrt {a x^2+b x^5}}{3 a x^4}+\frac {b \tanh ^{-1}\left (\frac {\sqrt {a} x}{\sqrt {a x^2+b x^5}}\right )}{3 a^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 76, normalized size = 1.29 \begin {gather*} \frac {-\sqrt {a} \left (a+b x^3\right )+b x^3 \sqrt {a+b x^3} \tanh ^{-1}\left (\frac {\sqrt {a+b x^3}}{\sqrt {a}}\right )}{3 a^{3/2} x^2 \sqrt {x^2 \left (a+b x^3\right )}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.43, size = 66, normalized size = 1.12
method | result | size |
default | \(\frac {\sqrt {b \,x^{3}+a}\, \left (b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) a \,x^{3}-\sqrt {b \,x^{3}+a}\, a^{\frac {3}{2}}\right )}{3 x^{2} \sqrt {b \,x^{5}+a \,x^{2}}\, a^{\frac {5}{2}}}\) | \(66\) |
risch | \(-\frac {b \,x^{3}+a}{3 a \,x^{2} \sqrt {x^{2} \left (b \,x^{3}+a \right )}}+\frac {b \arctanh \left (\frac {\sqrt {b \,x^{3}+a}}{\sqrt {a}}\right ) \sqrt {b \,x^{3}+a}\, x}{3 a^{\frac {3}{2}} \sqrt {x^{2} \left (b \,x^{3}+a \right )}}\) | \(73\) |
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 = 1.78, size = 127, normalized size = 2.15 \begin {gather*} \left [\frac {\sqrt {a} b x^{4} \log \left (\frac {b x^{4} + 2 \, a x + 2 \, \sqrt {b x^{5} + a x^{2}} \sqrt {a}}{x^{4}}\right ) - 2 \, \sqrt {b x^{5} + a x^{2}} a}{6 \, a^{2} x^{4}}, -\frac {\sqrt {-a} b x^{4} \arctan \left (\frac {\sqrt {b x^{5} + a x^{2}} \sqrt {-a}}{a x}\right ) + \sqrt {b x^{5} + a x^{2}} a}{3 \, a^{2} x^{4}}\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^{3} \sqrt {x^{2} \left (a + b x^{3}\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 2.05, size = 55, normalized size = 0.93 \begin {gather*} -\frac {\frac {b^{2} \arctan \left (\frac {\sqrt {b x^{3} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} + \frac {\sqrt {b x^{3} + a} b}{a x^{3}}}{3 \, b \mathrm {sgn}\left (x\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 = -0.02 \begin {gather*} \int \frac {1}{x^3\,\sqrt {b\,x^5+a\,x^2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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