Optimal. Leaf size=157 \[ -\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {16 b^{5/2} \sqrt [4]{1+\frac {a}{b x^2}} \sqrt {c x} E\left (\left .\frac {1}{2} \cot ^{-1}\left (\frac {\sqrt {b} x}{\sqrt {a}}\right )\right |2\right )}{3 a^{7/2} c^6 \sqrt [4]{a+b x^2}} \]
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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 = 4, integrand size = 19, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {292, 290, 342,
202} \begin {gather*} \frac {16 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 )}{3 a^{7/2} c^6 \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {2}{9 a c (c x)^{9/2} \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)^{11/2} \left (a+b x^2\right )^{5/4}} \, dx &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}-\frac {(10 b) \int \frac {1}{(c x)^{7/2} \left (a+b x^2\right )^{5/4}} \, dx}{9 a c^2}\\ &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}+\frac {\left (4 b^2\right ) \int \frac {1}{(c x)^{3/2} \left (a+b x^2\right )^{5/4}} \, dx}{3 a^2 c^4}\\ &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {\left (8 b^3\right ) \int \frac {\sqrt {c x}}{\left (a+b x^2\right )^{5/4}} \, dx}{3 a^3 c^6}\\ &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}-\frac {\left (8 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}{3 a^3 c^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {\left (8 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 )}{3 a^3 c^6 \sqrt [4]{a+b x^2}}\\ &=-\frac {2}{9 a c (c x)^{9/2} \sqrt [4]{a+b x^2}}+\frac {4 b}{9 a^2 c^3 (c x)^{5/2} \sqrt [4]{a+b x^2}}-\frac {8 b^2}{3 a^3 c^5 \sqrt {c x} \sqrt [4]{a+b x^2}}+\frac {16 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 )}{3 a^{7/2} 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 = 59, normalized size = 0.38 \begin {gather*} -\frac {2 x \sqrt [4]{1+\frac {b x^2}{a}} \, _2F_1\left (-\frac {9}{4},\frac {5}{4};-\frac {5}{4};-\frac {b x^2}{a}\right )}{9 a (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 \[\int \frac {1}{\left (c x \right )^{\frac {11}{2}} \left (b \,x^{2}+a \right )^{\frac {5}{4}}}\, dx\]
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 [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: SystemError} \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 )}^{5/4}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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