Optimal. Leaf size=46 \[ -\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}} \]
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Rubi [A]
time = 0.03, antiderivative size = 46, normalized size of antiderivative = 1.00, number of steps
used = 3, number of rules used = 2, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {455, 45}
\begin {gather*} -\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}} \end {gather*}
Antiderivative was successfully verified.
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Rule 45
Rule 455
Rubi steps
\begin {align*} \int \frac {x^2 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {A+B x}{(a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {A b-a B}{b (a+b x)^{5/2}}+\frac {B}{b (a+b x)^{3/2}}\right ) \, dx,x,x^3\right )\\ &=-\frac {2 (A b-a B)}{9 b^2 \left (a+b x^3\right )^{3/2}}-\frac {2 B}{3 b^2 \sqrt {a+b x^3}}\\ \end {align*}
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Mathematica [A]
time = 0.03, size = 33, normalized size = 0.72 \begin {gather*} -\frac {2 \left (A b+2 a B+3 b B x^3\right )}{9 b^2 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 64, normalized size = 1.39
method | result | size |
gosper | \(-\frac {2 \left (3 b B \,x^{3}+A b +2 B a \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} b^{2}}\) | \(30\) |
trager | \(-\frac {2 \left (3 b B \,x^{3}+A b +2 B a \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} b^{2}}\) | \(30\) |
elliptic | \(-\frac {2 \left (A b -B a \right ) \sqrt {b \,x^{3}+a}}{9 b^{4} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2 B}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}\) | \(54\) |
default | \(B \left (\frac {2 a \sqrt {b \,x^{3}+a}}{9 b^{4} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}\right )-\frac {2 A}{9 b \left (b \,x^{3}+a \right )^{\frac {3}{2}}}\) | \(64\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.29, size = 49, normalized size = 1.07 \begin {gather*} -\frac {2}{9} \, B {\left (\frac {3}{\sqrt {b x^{3} + a} b^{2}} - \frac {a}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}}\right )} - \frac {2 \, A}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 2.18, size = 52, normalized size = 1.13 \begin {gather*} -\frac {2 \, {\left (3 \, B b x^{3} + 2 \, B a + A b\right )} \sqrt {b x^{3} + a}}{9 \, {\left (b^{4} x^{6} + 2 \, a b^{3} x^{3} + a^{2} b^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 144 vs.
\(2 (44) = 88\).
time = 0.40, size = 144, normalized size = 3.13 \begin {gather*} \begin {cases} - \frac {2 A b}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} - \frac {4 B a}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} - \frac {6 B b x^{3}}{9 a b^{2} \sqrt {a + b x^{3}} + 9 b^{3} x^{3} \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{3}}{3} + \frac {B x^{6}}{6}}{a^{\frac {5}{2}}} & \text {otherwise} \end {cases} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Giac [A]
time = 1.48, size = 32, normalized size = 0.70 \begin {gather*} -\frac {2 \, {\left (3 \, {\left (b x^{3} + a\right )} B - B a + A b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.68, size = 33, normalized size = 0.72 \begin {gather*} -\frac {2\,A\,b-2\,B\,a+6\,B\,\left (b\,x^3+a\right )}{9\,b^2\,{\left (b\,x^3+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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