Optimal. Leaf size=67 \[ \frac {\sqrt {a+b x^2} (A b-2 a B)}{b^3}+\frac {a (A b-a B)}{b^3 \sqrt {a+b x^2}}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^3} \]
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Rubi [A] time = 0.06, antiderivative size = 67, 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 = {446, 77} \[ \frac {\sqrt {a+b x^2} (A b-2 a B)}{b^3}+\frac {a (A b-a B)}{b^3 \sqrt {a+b x^2}}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^3} \]
Antiderivative was successfully verified.
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Rule 77
Rule 446
Rubi steps
\begin {align*} \int \frac {x^3 \left (A+B x^2\right )}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int \frac {x (A+B x)}{(a+b x)^{3/2}} \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (\frac {a (-A b+a B)}{b^2 (a+b x)^{3/2}}+\frac {A b-2 a B}{b^2 \sqrt {a+b x}}+\frac {B \sqrt {a+b x}}{b^2}\right ) \, dx,x,x^2\right )\\ &=\frac {a (A b-a B)}{b^3 \sqrt {a+b x^2}}+\frac {(A b-2 a B) \sqrt {a+b x^2}}{b^3}+\frac {B \left (a+b x^2\right )^{3/2}}{3 b^3}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 55, normalized size = 0.82 \[ \frac {-8 a^2 B+a \left (6 A b-4 b B x^2\right )+b^2 x^2 \left (3 A+B x^2\right )}{3 b^3 \sqrt {a+b x^2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.72, size = 63, normalized size = 0.94 \[ \frac {{\left (B b^{2} x^{4} - 8 \, B a^{2} + 6 \, A a b - {\left (4 \, B a b - 3 \, A b^{2}\right )} x^{2}\right )} \sqrt {b x^{2} + a}}{3 \, {\left (b^{4} x^{2} + a b^{3}\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.38, size = 77, normalized size = 1.15 \[ -\frac {B a^{2} - A a b}{\sqrt {b x^{2} + a} b^{3}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} B b^{6} - 6 \, \sqrt {b x^{2} + a} B a b^{6} + 3 \, \sqrt {b x^{2} + a} A b^{7}}{3 \, b^{9}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 52, normalized size = 0.78 \[ \frac {B \,b^{2} x^{4}+3 A \,b^{2} x^{2}-4 B a b \,x^{2}+6 a b A -8 a^{2} B}{3 \sqrt {b \,x^{2}+a}\, b^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 1.04, size = 89, normalized size = 1.33 \[ \frac {B x^{4}}{3 \, \sqrt {b x^{2} + a} b} - \frac {4 \, B a x^{2}}{3 \, \sqrt {b x^{2} + a} b^{2}} + \frac {A x^{2}}{\sqrt {b x^{2} + a} b} - \frac {8 \, B a^{2}}{3 \, \sqrt {b x^{2} + a} b^{3}} + \frac {2 \, A a}{\sqrt {b x^{2} + a} b^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.68, size = 59, normalized size = 0.88 \[ \frac {B\,{\left (b\,x^2+a\right )}^2-3\,B\,a^2+3\,A\,b\,\left (b\,x^2+a\right )-6\,B\,a\,\left (b\,x^2+a\right )+3\,A\,a\,b}{3\,b^3\,\sqrt {b\,x^2+a}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 2.02, size = 117, normalized size = 1.75 \[ \begin {cases} \frac {2 A a}{b^{2} \sqrt {a + b x^{2}}} + \frac {A x^{2}}{b \sqrt {a + b x^{2}}} - \frac {8 B a^{2}}{3 b^{3} \sqrt {a + b x^{2}}} - \frac {4 B a x^{2}}{3 b^{2} \sqrt {a + b x^{2}}} + \frac {B x^{4}}{3 b \sqrt {a + b x^{2}}} & \text {for}\: b \neq 0 \\\frac {\frac {A x^{4}}{4} + \frac {B x^{6}}{6}}{a^{\frac {3}{2}}} & \text {otherwise} \end {cases} \]
Verification of antiderivative is not currently implemented for this CAS.
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