Optimal. Leaf size=198 \[ \frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac {20 a^3 c^{19/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{27 \sqrt {3} b^{11/3}}+\frac {10 a^3 c^{19/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{27 b^{11/3}} \]
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Rubi [A]
time = 0.13, antiderivative size = 198, 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 = {327, 335, 281,
337} \begin {gather*} \frac {20 a^3 c^{19/3} \text {ArcTan}\left (\frac {\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}+1}{\sqrt {3}}\right )}{27 \sqrt {3} b^{11/3}}+\frac {10 a^3 c^{19/3} \log \left (\sqrt [3]{b} (c x)^{2/3}-c^{2/3} \sqrt [3]{a+b x^2}\right )}{27 b^{11/3}}+\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b} \end {gather*}
Antiderivative was successfully verified.
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Rule 281
Rule 327
Rule 335
Rule 337
Rubi steps
\begin {align*} \int \frac {(c x)^{19/3}}{\left (a+b x^2\right )^{2/3}} \, dx &=\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (8 a c^2\right ) \int \frac {(c x)^{13/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{9 b}\\ &=-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac {\left (20 a^2 c^4\right ) \int \frac {(c x)^{7/3}}{\left (a+b x^2\right )^{2/3}} \, dx}{27 b^2}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (40 a^3 c^6\right ) \int \frac {\sqrt [3]{c x}}{\left (a+b x^2\right )^{2/3}} \, dx}{81 b^3}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (40 a^3 c^5\right ) \text {Subst}\left (\int \frac {x^3}{\left (a+\frac {b x^6}{c^2}\right )^{2/3}} \, dx,x,\sqrt [3]{c x}\right )}{27 b^3}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (20 a^3 c^5\right ) \text {Subst}\left (\int \frac {x}{\left (a+\frac {b x^3}{c^2}\right )^{2/3}} \, dx,x,(c x)^{2/3}\right )}{27 b^3}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (20 a^3 c^5\right ) \text {Subst}\left (\int \frac {x}{1-\frac {b x^3}{c^2}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{27 b^3}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}-\frac {\left (20 a^3 c^{17/3}\right ) \text {Subst}\left (\int \frac {1}{1-\frac {\sqrt [3]{b} x}{c^{2/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{10/3}}+\frac {\left (20 a^3 c^{17/3}\right ) \text {Subst}\left (\int \frac {1-\frac {\sqrt [3]{b} x}{c^{2/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{10/3}}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac {20 a^3 c^{19/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}+\frac {\left (10 a^3 c^{17/3}\right ) \text {Subst}\left (\int \frac {1}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{27 b^{10/3}}-\frac {\left (10 a^3 c^{19/3}\right ) \text {Subst}\left (\int \frac {\frac {\sqrt [3]{b}}{c^{2/3}}+\frac {2 b^{2/3} x}{c^{4/3}}}{1+\frac {\sqrt [3]{b} x}{c^{2/3}}+\frac {b^{2/3} x^2}{c^{4/3}}} \, dx,x,\frac {(c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac {20 a^3 c^{19/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac {10 a^3 c^{19/3} \log \left (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}}\right )}{81 b^{11/3}}-\frac {\left (20 a^3 c^{19/3}\right ) \text {Subst}\left (\int \frac {1}{-3-x^2} \, dx,x,1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}\right )}{27 b^{11/3}}\\ &=\frac {10 a^2 c^5 (c x)^{4/3} \sqrt [3]{a+b x^2}}{27 b^3}-\frac {2 a c^3 (c x)^{10/3} \sqrt [3]{a+b x^2}}{9 b^2}+\frac {c (c x)^{16/3} \sqrt [3]{a+b x^2}}{6 b}+\frac {20 a^3 c^{19/3} \tan ^{-1}\left (\frac {1+\frac {2 \sqrt [3]{b} (c x)^{2/3}}{c^{2/3} \sqrt [3]{a+b x^2}}}{\sqrt {3}}\right )}{27 \sqrt {3} b^{11/3}}+\frac {20 a^3 c^{19/3} \log \left (c^{2/3}-\frac {\sqrt [3]{b} (c x)^{2/3}}{\sqrt [3]{a+b x^2}}\right )}{81 b^{11/3}}-\frac {10 a^3 c^{19/3} \log \left (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}}\right )}{81 b^{11/3}}\\ \end {align*}
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Mathematica [A]
time = 5.52, size = 233, normalized size = 1.18 \begin {gather*} \frac {c^6 \sqrt [3]{c x} \left (60 a^2 b^{2/3} x^{4/3} \sqrt [3]{a+b x^2}-36 a b^{5/3} x^{10/3} \sqrt [3]{a+b x^2}+27 b^{8/3} x^{16/3} \sqrt [3]{a+b x^2}+40 \sqrt {3} a^3 \tan ^{-1}\left (\frac {\sqrt {3} \sqrt [3]{b} x^{2/3}}{\sqrt [3]{b} x^{2/3}+2 \sqrt [3]{a+b x^2}}\right )+40 a^3 \log \left (-\sqrt [3]{b} x^{2/3}+\sqrt [3]{a+b x^2}\right )-20 a^3 \log \left (b^{2/3} x^{4/3}+\sqrt [3]{b} x^{2/3} \sqrt [3]{a+b x^2}+\left (a+b x^2\right )^{2/3}\right )\right )}{162 b^{11/3} \sqrt [3]{x}} \end {gather*}
Antiderivative was successfully verified.
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Maple [F]
time = 0.01, size = 0, normalized size = 0.00 \[\int \frac {\left (c x \right )^{\frac {19}{3}}}{\left (b \,x^{2}+a \right )^{\frac {2}{3}}}\, dx\]
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(-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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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 = -0.01 \begin {gather*} \int \frac {{\left (c\,x\right )}^{19/3}}{{\left (b\,x^2+a\right )}^{2/3}} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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