Optimal. Leaf size=73 \[ \frac {2 a (A b-a B)}{9 b^3 \left (a+b x^3\right )^{3/2}}-\frac {2 (A b-2 a B)}{3 b^3 \sqrt {a+b x^3}}+\frac {2 B \sqrt {a+b x^3}}{3 b^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 73, 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 = {457, 78}
\begin {gather*} -\frac {2 (A b-2 a B)}{3 b^3 \sqrt {a+b x^3}}+\frac {2 a (A b-a B)}{9 b^3 \left (a+b x^3\right )^{3/2}}+\frac {2 B \sqrt {a+b x^3}}{3 b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^5 \left (A+B x^3\right )}{\left (a+b x^3\right )^{5/2}} \, dx &=\frac {1}{3} \text {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx,x,x^3\right )\\ &=\frac {1}{3} \text {Subst}\left (\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^{5/2}}+\frac {A b-2 a B}{b^2 (a+b x)^{3/2}}+\frac {B}{b^2 \sqrt {a+b x}}\right ) \, dx,x,x^3\right )\\ &=\frac {2 a (A b-a B)}{9 b^3 \left (a+b x^3\right )^{3/2}}-\frac {2 (A b-2 a B)}{3 b^3 \sqrt {a+b x^3}}+\frac {2 B \sqrt {a+b x^3}}{3 b^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.77 \begin {gather*} \frac {2 \left (-2 a A b+8 a^2 B-3 A b^2 x^3+12 a b B x^3+3 b^2 B x^6\right )}{9 b^3 \left (a+b x^3\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.33, size = 113, normalized size = 1.55
method | result | size |
gosper | \(-\frac {2 \left (-3 b^{2} B \,x^{6}+3 A \,b^{2} x^{3}-12 B a b \,x^{3}+2 a b A -8 a^{2} B \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} b^{3}}\) | \(53\) |
trager | \(-\frac {2 \left (-3 b^{2} B \,x^{6}+3 A \,b^{2} x^{3}-12 B a b \,x^{3}+2 a b A -8 a^{2} B \right )}{9 \left (b \,x^{3}+a \right )^{\frac {3}{2}} b^{3}}\) | \(53\) |
risch | \(\frac {2 B \sqrt {b \,x^{3}+a}}{3 b^{3}}-\frac {2 \left (3 A \,b^{2} x^{3}-6 B a b \,x^{3}+2 a b A -5 a^{2} B \right )}{9 b^{3} \left (b \,x^{3}+a \right )^{\frac {3}{2}}}\) | \(60\) |
elliptic | \(\frac {2 \left (A b -B a \right ) a \sqrt {b \,x^{3}+a}}{9 b^{5} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2 \left (A b -2 B a \right )}{3 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 B \sqrt {b \,x^{3}+a}}{3 b^{3}}\) | \(77\) |
default | \(B \left (-\frac {2 a^{2} \sqrt {b \,x^{3}+a}}{9 b^{5} \left (x^{3}+\frac {a}{b}\right )^{2}}+\frac {4 a}{3 b^{3} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}+\frac {2 \sqrt {b \,x^{3}+a}}{3 b^{3}}\right )+A \left (\frac {2 a \sqrt {b \,x^{3}+a}}{9 b^{4} \left (x^{3}+\frac {a}{b}\right )^{2}}-\frac {2}{3 b^{2} \sqrt {\left (x^{3}+\frac {a}{b}\right ) b}}\right )\) | \(113\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.28, size = 84, normalized size = 1.15 \begin {gather*} \frac {2}{9} \, B {\left (\frac {3 \, \sqrt {b x^{3} + a}}{b^{3}} + \frac {6 \, a}{\sqrt {b x^{3} + a} b^{3}} - \frac {a^{2}}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{3}}\right )} - \frac {2}{9} \, A {\left (\frac {3}{\sqrt {b x^{3} + a} b^{2}} - \frac {a}{{\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{2}}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.83, size = 75, normalized size = 1.03 \begin {gather*} \frac {2 \, {\left (3 \, B b^{2} x^{6} + 3 \, {\left (4 \, B a b - A b^{2}\right )} x^{3} + 8 \, B a^{2} - 2 \, A a b\right )} \sqrt {b x^{3} + a}}{9 \, {\left (b^{5} x^{6} + 2 \, a b^{4} x^{3} + a^{2} b^{3}\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. 240 vs.
\(2 (70) = 140\).
time = 0.52, size = 240, normalized size = 3.29 \begin {gather*} \begin {cases} - \frac {4 A a b}{9 a b^{3} \sqrt {a + b x^{3}} + 9 b^{4} x^{3} \sqrt {a + b x^{3}}} - \frac {6 A b^{2} x^{3}}{9 a b^{3} \sqrt {a + b x^{3}} + 9 b^{4} x^{3} \sqrt {a + b x^{3}}} + \frac {16 B a^{2}}{9 a b^{3} \sqrt {a + b x^{3}} + 9 b^{4} x^{3} \sqrt {a + b x^{3}}} + \frac {24 B a b x^{3}}{9 a b^{3} \sqrt {a + b x^{3}} + 9 b^{4} x^{3} \sqrt {a + b x^{3}}} + \frac {6 B b^{2} x^{6}}{9 a b^{3} \sqrt {a + b x^{3}} + 9 b^{4} x^{3} \sqrt {a + b x^{3}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{6}}{6} + \frac {B x^{9}}{9}}{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 = 1.38, size = 63, normalized size = 0.86 \begin {gather*} \frac {2 \, \sqrt {b x^{3} + a} B}{3 \, b^{3}} + \frac {2 \, {\left (6 \, {\left (b x^{3} + a\right )} B a - B a^{2} - 3 \, {\left (b x^{3} + a\right )} A b + A a b\right )}}{9 \, {\left (b x^{3} + a\right )}^{\frac {3}{2}} b^{3}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Mupad [B]
time = 2.76, size = 60, normalized size = 0.82 \begin {gather*} \frac {6\,B\,{\left (b\,x^3+a\right )}^2-2\,B\,a^2-6\,A\,b\,\left (b\,x^3+a\right )+12\,B\,a\,\left (b\,x^3+a\right )+2\,A\,a\,b}{9\,b^3\,{\left (b\,x^3+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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