Optimal. Leaf size=44 \[ \frac {9 x}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}} \]
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Rubi [A]
time = 0.01, antiderivative size = 44, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {424, 391}
\begin {gather*} \frac {3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}+\frac {9 x}{2 \sqrt [3]{a-b x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 391
Rule 424
Rubi steps
\begin {align*} \int \frac {\left (3 a+b x^2\right )^2}{\left (a-b x^2\right )^{7/3}} \, dx &=\frac {3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}-\frac {3 \int \frac {-12 a^2 b+4 a b^2 x^2}{\left (a-b x^2\right )^{4/3}} \, dx}{8 a b}\\ &=\frac {9 x}{2 \sqrt [3]{a-b x^2}}+\frac {3 x \left (3 a+b x^2\right )}{2 \left (a-b x^2\right )^{4/3}}\\ \end {align*}
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Mathematica [A]
time = 15.04, size = 24, normalized size = 0.55 \begin {gather*} \frac {9 a x-3 b x^3}{\left (a-b x^2\right )^{4/3}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.07, size = 24, normalized size = 0.55
method | result | size |
gosper | \(\frac {3 x \left (-b \,x^{2}+3 a \right )}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}}}\) | \(24\) |
trager | \(\frac {3 x \left (-b \,x^{2}+3 a \right )}{\left (-b \,x^{2}+a \right )^{\frac {4}{3}}}\) | \(24\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.33, size = 33, normalized size = 0.75 \begin {gather*} \frac {3 \, {\left (b x^{3} - 3 \, a x\right )}}{{\left (b x^{2} - a\right )} {\left (-b x^{2} + a\right )}^{\frac {1}{3}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.23, size = 42, normalized size = 0.95 \begin {gather*} -\frac {3 \, {\left (b x^{3} - 3 \, a x\right )} {\left (-b x^{2} + a\right )}^{\frac {2}{3}}}{b^{2} x^{4} - 2 \, a b x^{2} + a^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\left (3 a + b x^{2}\right )^{2}}{\left (a - b x^{2}\right )^{\frac {7}{3}}}\, dx \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 [B]
time = 4.78, size = 27, normalized size = 0.61 \begin {gather*} \frac {3\,x\,\left (a-b\,x^2\right )+6\,a\,x}{{\left (a-b\,x^2\right )}^{4/3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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