Optimal. Leaf size=153 \[ \frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}+\frac {15 b^{7/4} \left (\sqrt {a}+\sqrt {b} x^2\right ) \sqrt {\frac {a+b x^4}{\left (\sqrt {a}+\sqrt {b} x^2\right )^2}} F\left (2 \tan ^{-1}\left (\frac {\sqrt [4]{b} x}{\sqrt [4]{a}}\right )|\frac {1}{2}\right )}{28 a^{13/4} \sqrt {a+b x^4}} \]
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Rubi [A]
time = 0.04, antiderivative size = 153, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 3, integrand size = 15, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {296, 331, 226}
\begin {gather*} \frac {15 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 )}{28 a^{13/4} \sqrt {a+b x^4}}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {1}{2 a x^7 \sqrt {a+b x^4}} \end {gather*}
Antiderivative was successfully verified.
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Rule 226
Rule 296
Rule 331
Rubi steps
\begin {align*} \int \frac {1}{x^8 \left (a+b x^4\right )^{3/2}} \, dx &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}+\frac {9 \int \frac {1}{x^8 \sqrt {a+b x^4}} \, dx}{2 a}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}-\frac {(45 b) \int \frac {1}{x^4 \sqrt {a+b x^4}} \, dx}{14 a^2}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}+\frac {\left (15 b^2\right ) \int \frac {1}{\sqrt {a+b x^4}} \, dx}{14 a^3}\\ &=\frac {1}{2 a x^7 \sqrt {a+b x^4}}-\frac {9 \sqrt {a+b x^4}}{14 a^2 x^7}+\frac {15 b \sqrt {a+b x^4}}{14 a^3 x^3}+\frac {15 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 )}{28 a^{13/4} \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 = 54, normalized size = 0.35 \begin {gather*} -\frac {\sqrt {1+\frac {b x^4}{a}} \, _2F_1\left (-\frac {7}{4},\frac {3}{2};-\frac {3}{4};-\frac {b x^4}{a}\right )}{7 a 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 = 0.17, size = 135, normalized size = 0.88
method | result | size |
default | \(-\frac {\sqrt {b \,x^{4}+a}}{7 a^{2} x^{7}}+\frac {4 b \sqrt {b \,x^{4}+a}}{7 a^{3} x^{3}}+\frac {b^{2} x}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {15 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{14 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(135\) |
elliptic | \(-\frac {\sqrt {b \,x^{4}+a}}{7 a^{2} x^{7}}+\frac {4 b \sqrt {b \,x^{4}+a}}{7 a^{3} x^{3}}+\frac {b^{2} x}{2 a^{3} \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {15 b^{2} \sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{14 a^{3} \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\) | \(135\) |
risch | \(-\frac {\sqrt {b \,x^{4}+a}\, \left (-4 b \,x^{4}+a \right )}{7 a^{3} x^{7}}+\frac {b^{2} \left (4 b \left (-\frac {x}{2 b \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 b \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )+11 a \left (\frac {x}{2 a \sqrt {\left (x^{4}+\frac {a}{b}\right ) b}}+\frac {\sqrt {1-\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \sqrt {1+\frac {i \sqrt {b}\, x^{2}}{\sqrt {a}}}\, \EllipticF \left (x \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}, i\right )}{2 a \sqrt {\frac {i \sqrt {b}}{\sqrt {a}}}\, \sqrt {b \,x^{4}+a}}\right )\right )}{7 a^{3}}\) | \(228\) |
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 = 93, normalized size = 0.61 \begin {gather*} -\frac {15 \, {\left (b^{2} x^{11} + a b x^{7}\right )} \sqrt {a} \left (-\frac {b}{a}\right )^{\frac {3}{4}} F(\arcsin \left (x \left (-\frac {b}{a}\right )^{\frac {1}{4}}\right )\,|\,-1) - {\left (15 \, b^{2} x^{8} + 6 \, a b x^{4} - 2 \, a^{2}\right )} \sqrt {b x^{4} + a}}{14 \, {\left (a^{3} b x^{11} + a^{4} x^{7}\right )}} \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 = 0.68, size = 44, normalized size = 0.29 \begin {gather*} \frac {\Gamma \left (- \frac {7}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {7}{4}, \frac {3}{2} \\ - \frac {3}{4} \end {matrix}\middle | {\frac {b x^{4} e^{i \pi }}{a}} \right )}}{4 a^{\frac {3}{2}} 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\,{\left (b\,x^4+a\right )}^{3/2}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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