Optimal. Leaf size=450 \[ -\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}-\frac {24\ 3^{3/4} b^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{935 a^2 c^{23/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}} \]
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Rubi [A]
time = 0.57, antiderivative size = 450, 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 = {283, 331, 335,
247, 231} \begin {gather*} -\frac {24\ 3^{3/4} b^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}+c^{4/3}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\text {ArcCos}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{b x^2+a}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{935 a^2 c^{23/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 231
Rule 247
Rule 283
Rule 331
Rule 335
Rubi steps
\begin {align*} \int \frac {\left (a+b x^2\right )^{4/3}}{(c x)^{20/3}} \, dx &=-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}+\frac {(8 b) \int \frac {\sqrt [3]{a+b x^2}}{(c x)^{14/3}} \, dx}{17 c^2}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}+\frac {\left (16 b^2\right ) \int \frac {1}{(c x)^{8/3} \left (a+b x^2\right )^{2/3}} \, dx}{187 c^4}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}-\frac {\left (48 b^3\right ) \int \frac {1}{(c x)^{2/3} \left (a+b x^2\right )^{2/3}} \, dx}{935 a c^6}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}-\frac {\left (144 b^3\right ) \text {Subst}\left (\int \frac {1}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{935 a c^7}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}-\frac {\left (144 b^3\right ) \text {Subst}\left (\int \frac {1}{\sqrt {1-\frac {b x^6}{c^2}}} \, dx,x,\frac {\sqrt [3]{c x}}{\sqrt [6]{a+b x^2}}\right )}{935 a c^7 \sqrt {\frac {a}{a+b x^2}} \sqrt {a+b x^2}}\\ &=-\frac {24 b \sqrt [3]{a+b x^2}}{187 c^3 (c x)^{11/3}}-\frac {48 b^2 \sqrt [3]{a+b x^2}}{935 a c^5 (c x)^{5/3}}-\frac {3 \left (a+b x^2\right )^{4/3}}{17 c (c x)^{17/3}}-\frac {24\ 3^{3/4} b^3 \sqrt [3]{c x} \sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right ) \sqrt {\frac {c^{4/3}+\frac {b^{2/3} (c x)^{4/3}}{\left (a+b x^2\right )^{2/3}}+\frac {\sqrt [3]{b} c^{2/3} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{\left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^2}} F\left (\cos ^{-1}\left (\frac {c^{2/3}-\frac {\left (1-\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}{c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}}\right )|\frac {1}{4} \left (2+\sqrt {3}\right )\right )}{935 a^2 c^{23/3} \sqrt {-\frac {\sqrt [3]{b} (c x)^{2/3} \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{\sqrt [3]{a+b x^2} \left (c^{2/3}-\frac {\left (1+\sqrt {3}\right ) \sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )^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 = 10.03, size = 57, normalized size = 0.13 \begin {gather*} -\frac {3 a x \sqrt [3]{a+b x^2} \, _2F_1\left (-\frac {17}{6},-\frac {4}{3};-\frac {11}{6};-\frac {b x^2}{a}\right )}{17 (c x)^{20/3} \sqrt [3]{1+\frac {b x^2}{a}}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (b \,x^{2}+a \right )^{\frac {4}{3}}}{\left (c x \right )^{\frac {20}{3}}}\, dx\]
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(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \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 = -0.00 \begin {gather*} \int \frac {{\left (b\,x^2+a\right )}^{4/3}}{{\left (c\,x\right )}^{20/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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