Optimal. Leaf size=38 \[ \frac {\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac {a \left (a+c x^4\right )^{5/2}}{10 c^2} \]
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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} \[ \frac {\left (a+c x^4\right )^{7/2}}{14 c^2}-\frac {a \left (a+c x^4\right )^{5/2}}{10 c^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int x^7 \left (a+c x^4\right )^{3/2} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int x (a+c x)^{3/2} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {a (a+c x)^{3/2}}{c}+\frac {(a+c x)^{5/2}}{c}\right ) \, dx,x,x^4\right )\\ &=-\frac {a \left (a+c x^4\right )^{5/2}}{10 c^2}+\frac {\left (a+c x^4\right )^{7/2}}{14 c^2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 28, normalized size = 0.74 \[ \frac {\left (a+c x^4\right )^{5/2} \left (5 c x^4-2 a\right )}{70 c^2} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.50, size = 45, normalized size = 1.18 \[ \frac {{\left (5 \, c^{3} x^{12} + 8 \, a c^{2} x^{8} + a^{2} c x^{4} - 2 \, a^{3}\right )} \sqrt {c x^{4} + a}}{70 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 29, normalized size = 0.76 \[ \frac {5 \, {\left (c x^{4} + a\right )}^{\frac {7}{2}} - 7 \, {\left (c x^{4} + a\right )}^{\frac {5}{2}} a}{70 \, c^{2}} \]
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 \[ -\frac {\left (c \,x^{4}+a \right )^{\frac {5}{2}} \left (-5 c \,x^{4}+2 a \right )}{70 c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 30, normalized size = 0.79 \[ \frac {{\left (c x^{4} + a\right )}^{\frac {7}{2}}}{14 \, c^{2}} - \frac {{\left (c x^{4} + a\right )}^{\frac {5}{2}} a}{10 \, c^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.14, size = 42, normalized size = 1.11 \[ \sqrt {c\,x^4+a}\,\left (\frac {4\,a\,x^8}{35}+\frac {c\,x^{12}}{14}-\frac {a^3}{35\,c^2}+\frac {a^2\,x^4}{70\,c}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.42, size = 83, normalized size = 2.18 \[ \begin {cases} - \frac {a^{3} \sqrt {a + c x^{4}}}{35 c^{2}} + \frac {a^{2} x^{4} \sqrt {a + c x^{4}}}{70 c} + \frac {4 a x^{8} \sqrt {a + c x^{4}}}{35} + \frac {c x^{12} \sqrt {a + c x^{4}}}{14} & \text {for}\: c \neq 0 \\\frac {a^{\frac {3}{2}} x^{8}}{8} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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