Optimal. Leaf size=93 \[ -\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {4 \sqrt {b} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 93, normalized size of antiderivative = 1.00, number of steps
used = 4, number of rules used = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {292, 290, 342,
202} \begin {gather*} \frac {4 \sqrt {b} \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 )}{a^{3/2} c^2 \sqrt [4]{a+b x^2}}-\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 202
Rule 290
Rule 292
Rule 342
Rubi steps
\begin {align*} \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {(2 b) \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{a c^2}\\ &=-\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {\left (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}{a c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {\left (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 )}{a c^2 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{a c \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {4 \sqrt {b} \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 )}{a^{3/2} c^2 \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 = 57, normalized size = 0.61 \begin {gather*} -\frac {2 x \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {1}{4},\frac {5}{4};\frac {3}{4};-\frac {b x^2}{a}\right )}{a (c x)^{3/2} \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 \[\int \frac {1}{\left (c x \right )^{\frac {3}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
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 = 3.20, size = 48, normalized size = 0.52 \begin {gather*} \frac {\Gamma \left (- \frac {1}{4}\right ) {{}_{2}F_{1}\left (\begin {matrix} - \frac {1}{4}, \frac {5}{4} \\ \frac {3}{4} \end {matrix}\middle | {\frac {b x^{2} e^{i \pi }}{a}} \right )}}{2 a^{\frac {5}{4}} c^{\frac {3}{2}} \sqrt {x} \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}{{\left (c\,x\right )}^{3/2}\,{\left (b\,x^2+a\right )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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