Optimal. Leaf size=68 \[ \frac {a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {A b-2 a B}{b^3 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^3} \]
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Rubi [A]
time = 0.04, antiderivative size = 68, 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 {A b-2 a B}{b^3 \sqrt {a+b x^2}}+\frac {a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}+\frac {B \sqrt {a+b x^2}}{b^3} \end {gather*}
Antiderivative was successfully verified.
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Rule 78
Rule 457
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^{5/2}} \, dx &=\frac {1}{2} \text {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^{5/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \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^2\right )\\ &=\frac {a (A b-a B)}{3 b^3 \left (a+b x^2\right )^{3/2}}-\frac {A b-2 a B}{b^3 \sqrt {a+b x^2}}+\frac {B \sqrt {a+b x^2}}{b^3}\\ \end {align*}
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Mathematica [A]
time = 0.04, size = 56, normalized size = 0.82 \begin {gather*} \frac {-2 a A b+8 a^2 B-3 A b^2 x^2+12 a b B x^2+3 b^2 B x^4}{3 b^3 \left (a+b x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Maple [A]
time = 0.10, size = 95, normalized size = 1.40
method | result | size |
gosper | \(-\frac {-3 b^{2} B \,x^{4}+3 A \,b^{2} x^{2}-12 B a b \,x^{2}+2 a b A -8 a^{2} B}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}\) | \(53\) |
trager | \(-\frac {-3 b^{2} B \,x^{4}+3 A \,b^{2} x^{2}-12 B a b \,x^{2}+2 a b A -8 a^{2} B}{3 \left (b \,x^{2}+a \right )^{\frac {3}{2}} b^{3}}\) | \(53\) |
risch | \(\frac {B \sqrt {b \,x^{2}+a}}{b^{3}}-\frac {\sqrt {b \,x^{2}+a}\, \left (3 A \,b^{2} x^{2}-6 B a b \,x^{2}+2 a b A -5 a^{2} B \right )}{3 b^{3} \left (b^{2} x^{4}+2 a b \,x^{2}+a^{2}\right )}\) | \(79\) |
default | \(B \left (\frac {x^{4}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {4 a \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )}{b}\right )+A \left (-\frac {x^{2}}{b \left (b \,x^{2}+a \right )^{\frac {3}{2}}}-\frac {2 a}{3 b^{2} \left (b \,x^{2}+a \right )^{\frac {3}{2}}}\right )\) | \(95\) |
Verification of antiderivative is not currently implemented for this CAS.
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Maxima [A]
time = 0.31, size = 89, normalized size = 1.31 \begin {gather*} \frac {B x^{4}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {4 \, B a x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} - \frac {A x^{2}}{{\left (b x^{2} + a\right )}^{\frac {3}{2}} b} + \frac {8 \, B a^{2}}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{3}} - \frac {2 \, A a}{3 \, {\left (b x^{2} + a\right )}^{\frac {3}{2}} b^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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Fricas [A]
time = 1.86, size = 75, normalized size = 1.10 \begin {gather*} \frac {{\left (3 \, B b^{2} x^{4} + 8 \, B a^{2} - 2 \, A a b + 3 \, {\left (4 \, B a b - A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{5} x^{4} + 2 \, a b^{4} x^{2} + 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.
\(2 (60) = 120\).
time = 0.39, size = 240, normalized size = 3.53 \begin {gather*} \begin {cases} - \frac {2 A a b}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} - \frac {3 A b^{2} x^{2}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} + \frac {8 B a^{2}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} + \frac {12 B a b x^{2}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} + \frac {3 B b^{2} x^{4}}{3 a b^{3} \sqrt {a + b x^{2}} + 3 b^{4} x^{2} \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{4}}{4} + \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 = 1.34, size = 62, normalized size = 0.91 \begin {gather*} \frac {\sqrt {b x^{2} + a} B}{b^{3}} + \frac {6 \, {\left (b x^{2} + a\right )} B a - B a^{2} - 3 \, {\left (b x^{2} + a\right )} A b + A a b}{3 \, {\left (b x^{2} + 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 = 0.34, size = 59, normalized size = 0.87 \begin {gather*} \frac {3\,B\,{\left (b\,x^2+a\right )}^2-B\,a^2-3\,A\,b\,\left (b\,x^2+a\right )+6\,B\,a\,\left (b\,x^2+a\right )+A\,a\,b}{3\,b^3\,{\left (b\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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