Optimal. Leaf size=36 \[ \frac {2 c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b} \]
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Rubi [A]
time = 0.01, antiderivative size = 36, 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 = {1605, 15, 30}
\begin {gather*} \frac {2 c \left (a+b x^2\right )^{3/2} \sqrt {c \sqrt {a+b x^2}}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 15
Rule 30
Rule 1605
Rubi steps
\begin {align*} \int x \left (c \sqrt {a+b x^2}\right )^{3/2} \, dx &=\frac {\text {Subst}\left (\int \left (c \sqrt {x}\right )^{3/2} \, dx,x,a+b x^2\right )}{2 b}\\ &=\frac {\left (c \sqrt {c \sqrt {a+b x^2}}\right ) \text {Subst}\left (\int x^{3/4} \, dx,x,a+b x^2\right )}{2 b \sqrt [4]{a+b x^2}}\\ &=\frac {2 c \sqrt {c \sqrt {a+b x^2}} \left (a+b x^2\right )^{3/2}}{7 b}\\ \end {align*}
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Mathematica [A]
time = 0.01, size = 31, normalized size = 0.86 \begin {gather*} \frac {2 \left (c \sqrt {a+b x^2}\right )^{3/2} \left (a+b x^2\right )}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.04, size = 26, normalized size = 0.72
method | result | size |
gosper | \(\frac {2 \left (b \,x^{2}+a \right ) \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{7 b}\) | \(26\) |
derivativedivides | \(\frac {2 \left (b \,x^{2}+a \right ) \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{7 b}\) | \(26\) |
default | \(\frac {2 \left (b \,x^{2}+a \right ) \left (c \sqrt {b \,x^{2}+a}\right )^{\frac {3}{2}}}{7 b}\) | \(26\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.26, size = 25, normalized size = 0.69 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )} \left (\sqrt {b x^{2} + a} c\right )^{\frac {3}{2}}}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 0.38, size = 37, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (b c x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {\sqrt {b x^{2} + a} c}}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [A]
time = 1.70, size = 58, normalized size = 1.61 \begin {gather*} \begin {cases} \frac {2 a \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{7 b} + \frac {2 x^{2} \left (c \sqrt {a + b x^{2}}\right )^{\frac {3}{2}}}{7} & \text {for}\: b \neq 0 \\\frac {x^{2} \left (\sqrt {a} c\right )^{\frac {3}{2}}}{2} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 4.10, size = 17, normalized size = 0.47 \begin {gather*} \frac {2 \, {\left (b x^{2} + a\right )}^{\frac {7}{4}} c^{\frac {3}{2}}}{7 \, b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.77, size = 28, normalized size = 0.78 \begin {gather*} \frac {2\,c\,{\left (b\,x^2+a\right )}^{3/2}\,\sqrt {c\,\sqrt {b\,x^2+a}}}{7\,b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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