Optimal. Leaf size=35 \[ \frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \]
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Rubi [A] time = 0.02, antiderivative size = 35, 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} \[ \frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2} \]
Antiderivative was successfully verified.
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Rule 43
Rule 266
Rubi steps
\begin {align*} \int \frac {x^7}{\left (a+b x^4\right )^{5/4}} \, dx &=\frac {1}{4} \operatorname {Subst}\left (\int \frac {x}{(a+b x)^{5/4}} \, dx,x,x^4\right )\\ &=\frac {1}{4} \operatorname {Subst}\left (\int \left (-\frac {a}{b (a+b x)^{5/4}}+\frac {1}{b \sqrt [4]{a+b x}}\right ) \, dx,x,x^4\right )\\ &=\frac {a}{b^2 \sqrt [4]{a+b x^4}}+\frac {\left (a+b x^4\right )^{3/4}}{3 b^2}\\ \end {align*}
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Mathematica [A] time = 0.01, size = 27, normalized size = 0.77 \[ \frac {4 a+b x^4}{3 b^2 \sqrt [4]{a+b x^4}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.84, size = 35, normalized size = 1.00 \[ \frac {{\left (b x^{4} + 4 \, a\right )} {\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, {\left (b^{3} x^{4} + a b^{2}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.19, size = 34, normalized size = 0.97 \[ \frac {\frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{b} + \frac {3 \, a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b}}{3 \, b} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 24, normalized size = 0.69 \[ \frac {b \,x^{4}+4 a}{3 \left (b \,x^{4}+a \right )^{\frac {1}{4}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.32, size = 29, normalized size = 0.83 \[ \frac {{\left (b x^{4} + a\right )}^{\frac {3}{4}}}{3 \, b^{2}} + \frac {a}{{\left (b x^{4} + a\right )}^{\frac {1}{4}} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.15, size = 23, normalized size = 0.66 \[ \frac {b\,x^4+4\,a}{3\,b^2\,{\left (b\,x^4+a\right )}^{1/4}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.10, size = 44, normalized size = 1.26 \[ \begin {cases} \frac {4 a}{3 b^{2} \sqrt [4]{a + b x^{4}}} + \frac {x^{4}}{3 b \sqrt [4]{a + b x^{4}}} & \text {for}\: b \neq 0 \\\frac {x^{8}}{8 a^{\frac {5}{4}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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