Optimal. Leaf size=157 \[ -\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac {8 b^{5/2} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.05, antiderivative size = 157, normalized size of antiderivative = 1.00, number of steps
used = 6, number of rules used = 5, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.263, Rules used = {331, 322, 290,
342, 202} \begin {gather*} \frac {8 b^{5/2} \sqrt {c x} \sqrt [4]{\frac {a}{b x^2}+1} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a+b x^2}}-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 290
Rule 322
Rule 331
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{11/2} \sqrt [4]{a+b x^2}} \, dx &=-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}-\frac {(2 b) \int \frac {1}{(c x)^{7/2} \sqrt [4]{a+b x^2}} \, dx}{3 a c^2}\\ &=-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac {\left (4 b^2\right ) \int \frac {1}{(c x)^{3/2} \sqrt [4]{a+b x^2}} \, dx}{15 a^2 c^4}\\ &=-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac {\left (4 b^3\right ) \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{15 a^2 c^6}\\ &=-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}-\frac {\left (4 b^2 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \int \frac {1}{\left (1+\frac {a}{b x^2}\right )^{5/4} x^2} \, dx}{15 a^2 c^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac {\left (4 b^2 \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x}\right ) \text {Subst}\left (\int \frac {1}{\left (1+\frac {a x^2}{b}\right )^{5/4}} \, dx,x,\frac {1}{x}\right )}{15 a^2 c^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {8 b^2}{15 a^2 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {2 \left (a+b x^2\right )^{3/4}}{9 a c (c x)^{9/2}}+\frac {4 b \left (a+b x^2\right )^{3/4}}{15 a^2 c^3 (c x)^{5/2}}+\frac {8 b^{5/2} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{15 a^{5/2} c^6 \sqrt [4]{a+b x^2}}\\ \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 = 56, normalized size = 0.36 \begin {gather*} -\frac {2 x \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {9}{4},\frac {1}{4};-\frac {5}{4};-\frac {b x^2}{a}\right )}{9 (c x)^{11/2} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x \right )^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {1}{4}}}\, dx\]
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 [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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Sympy [C] Result contains complex when optimal does not.
time = 73.09, size = 34, normalized size = 0.22 \begin {gather*} - \frac {{{}_{2}F_{1}\left (\begin {matrix} \frac {1}{4}, \frac {5}{2} \\ \frac {7}{2} \end {matrix}\middle | {\frac {a e^{i \pi }}{b x^{2}}} \right )}}{5 \sqrt [4]{b} c^{\frac {11}{2}} x^{5}} \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}{{\left (c\,x\right )}^{11/2}\,{\left (b\,x^2+a\right )}^{1/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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