Optimal. Leaf size=77 \[ -\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\sqrt {a} c \sqrt {\frac {c}{a+b x^2}} \sqrt {1+\frac {b x^2}{a}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 77, 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 = {1973, 294, 221}
\begin {gather*} \frac {\sqrt {a} c \sqrt {\frac {b x^2}{a}+1} \sqrt {\frac {c}{a+b x^2}} \sinh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )}{b^{3/2}}-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b} \end {gather*}
Antiderivative was successfully verified.
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Rule 221
Rule 294
Rule 1973
Rubi steps
\begin {align*} \int x^2 \left (\frac {c}{a+b x^2}\right )^{3/2} \, dx &=\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {x^2}{\left (a+b x^2\right )^{3/2}} \, dx\\ &=-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \int \frac {1}{\sqrt {a+b x^2}} \, dx}{b}\\ &=-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {\left (c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2}\right ) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {x}{\sqrt {a+b x^2}}\right )}{b}\\ &=-\frac {c x \sqrt {\frac {c}{a+b x^2}}}{b}+\frac {c \sqrt {\frac {c}{a+b x^2}} \sqrt {a+b x^2} \tanh ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{b^{3/2}}\\ \end {align*}
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Mathematica [A]
time = 0.06, size = 64, normalized size = 0.83 \begin {gather*} -\frac {c \sqrt {\frac {c}{a+b x^2}} \left (\sqrt {b} x+\sqrt {a+b x^2} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )\right )}{b^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.03, size = 59, normalized size = 0.77
method | result | size |
default | \(\frac {\left (\frac {c}{b \,x^{2}+a}\right )^{\frac {3}{2}} \left (b \,x^{2}+a \right ) \left (-x \,b^{\frac {3}{2}}+\ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right ) b \sqrt {b \,x^{2}+a}\right )}{b^{\frac {5}{2}}}\) | \(59\) |
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 = 0.37, size = 141, normalized size = 1.83 \begin {gather*} \left [-\frac {2 \, c x \sqrt {\frac {c}{b x^{2} + a}} - c \sqrt {\frac {c}{b}} \log \left (-2 \, b c x^{2} - a c - 2 \, {\left (b^{2} x^{3} + a b x\right )} \sqrt {\frac {c}{b x^{2} + a}} \sqrt {\frac {c}{b}}\right )}{2 \, b}, -\frac {c x \sqrt {\frac {c}{b x^{2} + a}} + c \sqrt {-\frac {c}{b}} \arctan \left (\frac {b x \sqrt {\frac {c}{b x^{2} + a}} \sqrt {-\frac {c}{b}}}{c}\right )}{b}\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 x^{2} \left (\frac {c}{a + b x^{2}}\right )^{\frac {3}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 5.72, size = 71, normalized size = 0.92 \begin {gather*} -{\left (\frac {c x \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c x^{2} + a c} b} + \frac {c \log \left ({\left | -\sqrt {b c} x + \sqrt {b c x^{2} + a c} \right |}\right ) \mathrm {sgn}\left (b x^{2} + a\right )}{\sqrt {b c} b}\right )} c \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 x^2\,{\left (\frac {c}{b\,x^2+a}\right )}^{3/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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