Optimal. Leaf size=67 \[ \frac {x (a C+2 A c)}{3 a^2 c \sqrt {a+c x^2}}-\frac {a B-x (A c-a C)}{3 a c \left (a+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 67, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 3, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.136, Rules used = {1814, 12, 191} \[ \frac {x (a C+2 A c)}{3 a^2 c \sqrt {a+c x^2}}-\frac {a B-x (A c-a C)}{3 a c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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Rule 12
Rule 191
Rule 1814
Rubi steps
\begin {align*} \int \frac {A+B x+C x^2}{\left (a+c x^2\right )^{5/2}} \, dx &=-\frac {a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}-\frac {\int \frac {-2 A-\frac {a C}{c}}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a}\\ &=-\frac {a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {(2 A c+a C) \int \frac {1}{\left (a+c x^2\right )^{3/2}} \, dx}{3 a c}\\ &=-\frac {a B-(A c-a C) x}{3 a c \left (a+c x^2\right )^{3/2}}+\frac {(2 A c+a C) x}{3 a^2 c \sqrt {a+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.03, size = 50, normalized size = 0.75 \[ \frac {-a^2 B+a c x \left (3 A+C x^2\right )+2 A c^2 x^3}{3 a^2 c \left (a+c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.79, size = 68, normalized size = 1.01 \[ \frac {{\left (3 \, A a c x + {\left (C a c + 2 \, A c^{2}\right )} x^{3} - B a^{2}\right )} \sqrt {c x^{2} + a}}{3 \, {\left (a^{2} c^{3} x^{4} + 2 \, a^{3} c^{2} x^{2} + a^{4} c\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.21, size = 48, normalized size = 0.72 \[ \frac {x {\left (\frac {3 \, A}{a} + \frac {{\left (C a c + 2 \, A c^{2}\right )} x^{2}}{a^{2} c}\right )} - \frac {B}{c}}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.00, size = 47, normalized size = 0.70 \[ \frac {2 A \,c^{2} x^{3}+C a c \,x^{3}+3 A x a c -B \,a^{2}}{3 \left (c \,x^{2}+a \right )^{\frac {3}{2}} a^{2} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.43, size = 83, normalized size = 1.24 \[ \frac {2 \, A x}{3 \, \sqrt {c x^{2} + a} a^{2}} + \frac {A x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} a} - \frac {C x}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} + \frac {C x}{3 \, \sqrt {c x^{2} + a} a c} - \frac {B}{3 \, {\left (c x^{2} + a\right )}^{\frac {3}{2}} c} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 4.22, size = 59, normalized size = 0.88 \[ \frac {2\,A\,c\,x\,\left (c\,x^2+a\right )-C\,a^2\,x-B\,a^2+C\,a\,x\,\left (c\,x^2+a\right )+A\,a\,c\,x}{3\,a^2\,c\,{\left (c\,x^2+a\right )}^{3/2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 17.13, size = 194, normalized size = 2.90 \[ A \left (\frac {3 a x}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} + \frac {2 c x^{3}}{3 a^{\frac {7}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {5}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}}\right ) + B \left (\begin {cases} - \frac {1}{3 a c \sqrt {a + c x^{2}} + 3 c^{2} x^{2} \sqrt {a + c x^{2}}} & \text {for}\: c \neq 0 \\\frac {x^{2}}{2 a^{\frac {5}{2}}} & \text {otherwise} \end {cases}\right ) + \frac {C x^{3}}{3 a^{\frac {5}{2}} \sqrt {1 + \frac {c x^{2}}{a}} + 3 a^{\frac {3}{2}} c x^{2} \sqrt {1 + \frac {c x^{2}}{a}}} \]
Verification of antiderivative is not currently implemented for this CAS.
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