Optimal. Leaf size=38 \[ \frac {\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2} \]
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Rubi [A] time = 0.02, antiderivative size = 38, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {266, 43} \[ \frac {\left (a+c x^4\right )^{5/2}}{10 c^2}-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \sqrt {a+c x^4} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x \sqrt {a+c x} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {a \sqrt {a+c x}}{c}+\frac {(a+c x)^{3/2}}{c}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \left (a+c x^4\right )^{3/2}}{6 c^2}+\frac {\left (a+c x^4\right )^{5/2}}{10 c^2}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 28, normalized size = 0.74 \[ \frac {\left (a+c x^4\right )^{3/2} \left (3 c x^4-2 a\right )}{30 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.62, size = 34, normalized size = 0.89 \[ \frac {{\left (3 \, c^{2} x^{8} + a c x^{4} - 2 \, a^{2}\right )} \sqrt {c x^{4} + a}}{30 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 29, normalized size = 0.76 \[ \frac {3 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} - 5 \, {\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{30 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 25, normalized size = 0.66 \[ -\frac {\left (c \,x^{4}+a \right )^{\frac {3}{2}} \left (-3 c \,x^{4}+2 a \right )}{30 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.36, size = 30, normalized size = 0.79 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}}}{10 \, c^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {3}{2}} a}{6 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 33, normalized size = 0.87 \[ \sqrt {c\,x^4+a}\,\left (\frac {x^8}{10}-\frac {a^2}{15\,c^2}+\frac {a\,x^4}{30\,c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.90, size = 61, normalized size = 1.61 \[ \begin {cases} - \frac {a^{2} \sqrt {a + c x^{4}}}{15 c^{2}} + \frac {a x^{4} \sqrt {a + c x^{4}}}{30 c} + \frac {x^{8} \sqrt {a + c x^{4}}}{10} & \text {for}\: c \neq 0 \\\frac {\sqrt {a} x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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