Optimal. Leaf size=53 \[ -\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 B}{3 c^2 \sqrt {a+c x^2}} \]
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Rubi [A] time = 0.02, antiderivative size = 53, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {805, 261} \begin {gather*} -\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 B}{3 c^2 \sqrt {a+c x^2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 261
Rule 805
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {(2 B) \int \frac {x}{\left (a+c x^2\right )^{3/2}} \, dx}{3 c}\\ &=-\frac {x^2 (a B-A c x)}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {2 B}{3 c^2 \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 44, normalized size = 0.83 \begin {gather*} \frac {-2 a^2 B-3 a B c x^2+A c^2 x^3}{3 a c^2 \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.45, size = 44, normalized size = 0.83 \begin {gather*} \frac {-2 a^2 B-3 a B c x^2+A c^2 x^3}{3 a c^2 \left (a+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 63, normalized size = 1.19 \begin {gather*} \frac {{\left (A c^{2} x^{3} - 3 \, B a c x^{2} - 2 \, B a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a c^{4} x^{4} + 2 \, a^{2} c^{3} x^{2} + a^{3} c^{2}\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.20, size = 36, normalized size = 0.68 \begin {gather*} \frac {{\left (\frac {A x}{a} - \frac {3 \, B}{c}\right )} x^{2} - \frac {2 \, B a}{c^{2}}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 41, normalized size = 0.77 \begin {gather*} \frac {A \,c^{2} x^{3}-3 B a c \,x^{2}-2 B \,a^{2}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a \,c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.58, size = 70, normalized size = 1.32 \begin {gather*} -\frac {B x^{2}}{{\left (c x^{2} + a\right )}^{\frac {3}{2}} c} - \frac {A x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {A x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {2 \, B a}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c^{2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.13, size = 51, normalized size = 0.96 \begin {gather*} \frac {B\,a^2-3\,B\,a\,\left (c\,x^2+a\right )+A\,c\,x\,\left (c\,x^2+a\right )-A\,a\,c\,x}{3\,a\,c^2\,{\left (c\,x^2+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [B] time = 13.69, size = 141, normalized size = 2.66 \begin {gather*} \frac {A x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {3}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + B \left (\begin {cases} - \frac {2 a}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} - \frac {3 c x^{2}}{3 a c^{2} \sqrt {a + c x^{2}} + 3 c^{3} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{4}}{4 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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