Optimal. Leaf size=132 \[ -\frac {\sqrt {a+b x^4}}{7 a x^7}+\frac {5 b \sqrt {a+b x^4}}{21 a^2 x^3}+\frac {5 b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{42 a^{9/4} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.03, antiderivative size = 132, 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 = {331, 226}
\begin {gather*} \frac {5 b^{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 a^{9/4} \sqrt {a+b x^4}}+\frac {5 b \sqrt {a+b x^4}}{21 a^2 x^3}-\frac {\sqrt {a+b x^4}}{7 a x^7} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^8 \sqrt {a+b x^4}} \, dx &=-\frac {\sqrt {a+b x^4}}{7 a x^7}-\frac {(5 b) \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx}{7 a}\\ &=-\frac {\sqrt {a+b x^4}}{7 a x^7}+\frac {5 b \sqrt {a+b x^4}}{21 a^2 x^3}+\frac {\left (5 b^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{21 a^2}\\ &=-\frac {\sqrt {a+b x^4}}{7 a x^7}+\frac {5 b \sqrt {a+b x^4}}{21 a^2 x^3}+\frac {5 b^{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 a^{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 = 10.02, size = 51, normalized size = 0.39 \begin {gather*} -\frac {\sqrt {1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {7}{4},\frac {1}{2};-\frac {3}{4};-\frac {b x^4}{a}\right )}{7 x^7 \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 = 0.16, size = 113, normalized size = 0.86
method | result | size |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-5 b \,x^{4}+3 a \right )}{21 a^{2} x^{7}}+\frac {5 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(105\) |
default | \(-\frac {\sqrt {b \,x^{4}+a}}{7 a \,x^{7}}+\frac {5 b \sqrt {b \,x^{4}+a}}{21 a^{2} x^{3}}+\frac {5 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(113\) |
elliptic | \(-\frac {\sqrt {b \,x^{4}+a}}{7 a \,x^{7}}+\frac {5 b \sqrt {b \,x^{4}+a}}{21 a^{2} x^{3}}+\frac {5 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{21 a^{2} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(113\) |
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 = 0.09, size = 60, normalized size = 0.45 \begin {gather*} -\frac {5 \, \sqrt {a} b x^{7} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (5 \, b x^{4} - 3 \, a\right )} \sqrt {b x^{4} + a}}{21 \, a^{2} x^{7}} \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 = 0.55, size = 44, normalized size = 0.33 \begin {gather*} \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {1}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 \sqrt {a} x^{7} \Gamma \left (- \frac {3}{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 {1}{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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