Optimal. Leaf size=42 \[ \frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt {c+a^2 c x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 42, normalized size of antiderivative = 1.00, number of steps
used = 1, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {5783}
\begin {gather*} \frac {2 \sqrt {a^2 x^2+1} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt {a^2 c x^2+c}} \end {gather*}
Antiderivative was successfully verified.
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Rule 5783
Rubi steps
\begin {align*} \int \frac {\sinh ^{-1}(a x)^{3/2}}{\sqrt {c+a^2 c x^2}} \, dx &=\frac {\sqrt {1+a^2 x^2} \int \frac {\sinh ^{-1}(a x)^{3/2}}{\sqrt {1+a^2 x^2}} \, dx}{\sqrt {c+a^2 c x^2}}\\ &=\frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt {c+a^2 c x^2}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 42, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {1+a^2 x^2} \sinh ^{-1}(a x)^{5/2}}{5 a \sqrt {c+a^2 c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 1.74, size = 36, normalized size = 0.86
method | result | size |
default | \(\frac {2 \arcsinh \left (a x \right )^{\frac {5}{2}} \sqrt {a^{2} x^{2}+1}}{5 a \sqrt {c \left (a^{2} x^{2}+1\right )}}\) | \(36\) |
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \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 {\operatorname {asinh}^{\frac {3}{2}}{\left (a x \right )}}{\sqrt {c \left (a^{2} x^{2} + 1\right )}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \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 {{\mathrm {asinh}\left (a\,x\right )}^{3/2}}{\sqrt {c\,a^2\,x^2+c}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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