Optimal. Leaf size=130 \[ -\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 130, 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 = {327, 226}
\begin {gather*} \frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \text {ArcTan}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}}-\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 327
Rubi steps
\begin {align*} \int \frac {x^8}{\sqrt {a+b x^4}} \, dx &=\frac {x^5 \sqrt {a+b x^4}}{7 b}-\frac {(5 a) \int \frac {x^4}{\sqrt {a+b x^4}} \, dx}{7 b}\\ &=-\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {\left (5 a^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{21 b^2}\\ &=-\frac {5 a x \sqrt {a+b x^4}}{21 b^2}+\frac {x^5 \sqrt {a+b x^4}}{7 b}+\frac {5 a^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{42 b^{9/4} \sqrt {a+b x^4}}\\ \end {align*}
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Mathematica [C] Result contains higher order function than in optimal. Order 5 vs. order 4 in
optimal.
time = 10.03, size = 79, normalized size = 0.61 \begin {gather*} \frac {-5 a^2 x-2 a b x^5+3 b^2 x^9+5 a^2 x \sqrt {1+\frac {b x^4}{a}} \, _2F_1\left (\frac {1}{4},\frac {1}{2};\frac {5}{4};-\frac {b x^4}{a}\right )}{21 b^2 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Maple [C] Result contains complex when optimal does not.
time = 0.14, size = 111, normalized size = 0.85
method | result | size |
risch | \(-\frac {x \left (-3 b \,x^{4}+5 a \right ) \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(103\) |
default | \(\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(111\) |
elliptic | \(\frac {x^{5} \sqrt {b \,x^{4}+a}}{7 b}-\frac {5 a x \sqrt {b \,x^{4}+a}}{21 b^{2}}+\frac {5 a^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 b^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(111\) |
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 [A]
time = 0.08, size = 56, normalized size = 0.43 \begin {gather*} \frac {5 \, a \sqrt {b} \left (-\frac {a}{b}\right )^{\frac {3}{4}} F(\arcsin \left (\frac {\left (-\frac {a}{b}\right )^{\frac {1}{4}}}{x}\right )\,|\,-1) + {\left (3 \, b x^{5} - 5 \, a x\right )} \sqrt {b x^{4} + a}}{21 \, b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [C] Result contains complex when optimal does not.
time = 0.47, size = 37, normalized size = 0.28 \begin {gather*} \frac {x^{9} \Gamma \left (\frac {9}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} \frac {1}{2}, \frac {9}{4} \\ \frac {13}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} \Gamma \left (\frac {13}{4}\right )} \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 = -0.01 \begin {gather*} \int \frac {x^8}{\sqrt {b\,x^4+a}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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