Optimal. Leaf size=90 \[ -\frac {8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.02, antiderivative size = 90, 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 = {638, 613} \begin {gather*} -\frac {8 (b+2 c x) (b B-2 A c)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 (-2 a B-x (b B-2 A c)+A b)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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Rule 613
Rule 638
Rubi steps
\begin {align*} \int \frac {A+B x}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {(4 (b B-2 A c)) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {8 (b B-2 A c) (b+2 c x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.10, size = 99, normalized size = 1.10 \begin {gather*} -\frac {2 \left (B \left (8 a^2 c+2 a b (b+6 c x)+b x \left (3 b^2+12 b c x+8 c^2 x^2\right )\right )+A (b+2 c x) \left (-4 c \left (3 a+2 c x^2\right )+b^2-8 b c x\right )\right )}{3 \left (b^2-4 a c\right )^2 (a+x (b+c x))^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.01, size = 123, normalized size = 1.37 \begin {gather*} -\frac {2 \left (8 a^2 B c-12 a A b c-24 a A c^2 x+2 a b^2 B+12 a b B c x+A b^3-6 A b^2 c x-24 A b c^2 x^2-16 A c^3 x^3+3 b^3 B x+12 b^2 B c x^2+8 b B c^2 x^3\right )}{3 \left (b^2-4 a c\right )^2 \left (a+b x+c x^2\right )^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [B] time = 1.10, size = 245, normalized size = 2.72 \begin {gather*} -\frac {2 \, {\left (2 \, B a b^{2} + A b^{3} + 8 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x^{3} + 12 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )} x^{2} + 4 \, {\left (2 \, B a^{2} - 3 \, A a b\right )} c + 3 \, {\left (B b^{3} - 8 \, A a c^{2} + 2 \, {\left (2 \, B a b - A b^{2}\right )} c\right )} x\right )} \sqrt {c x^{2} + b x + a}}{3 \, {\left (a^{2} b^{4} - 8 \, a^{3} b^{2} c + 16 \, a^{4} c^{2} + {\left (b^{4} c^{2} - 8 \, a b^{2} c^{3} + 16 \, a^{2} c^{4}\right )} x^{4} + 2 \, {\left (b^{5} c - 8 \, a b^{3} c^{2} + 16 \, a^{2} b c^{3}\right )} x^{3} + {\left (b^{6} - 6 \, a b^{4} c + 32 \, a^{3} c^{3}\right )} x^{2} + 2 \, {\left (a b^{5} - 8 \, a^{2} b^{3} c + 16 \, a^{3} b c^{2}\right )} x\right )}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [B] time = 0.30, size = 193, normalized size = 2.14 \begin {gather*} -\frac {2 \, {\left ({\left (4 \, {\left (\frac {2 \, {\left (B b c^{2} - 2 \, A c^{3}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (B b^{2} c - 2 \, A b c^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (B b^{3} + 4 \, B a b c - 2 \, A b^{2} c - 8 \, A a c^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, B a b^{2} + A b^{3} + 8 \, B a^{2} c - 12 \, A a b c}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )}}{3 \, {\left (c x^{2} + b x + a\right )}^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.01, size = 132, normalized size = 1.47 \begin {gather*} \frac {\frac {32}{3} A \,c^{3} x^{3}-\frac {16}{3} B b \,c^{2} x^{3}+16 A b \,c^{2} x^{2}-8 B \,b^{2} c \,x^{2}+16 A a \,c^{2} x +4 A \,b^{2} c x -8 B a b c x -2 B \,b^{3} x +8 A a b c -\frac {2}{3} A \,b^{3}-\frac {16}{3} B \,a^{2} c -\frac {4}{3} B a \,b^{2}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a \,b^{2} c +b^{4}\right )} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F(-2)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 121, normalized size = 1.34 \begin {gather*} -\frac {2\,\left (8\,B\,a^2\,c+2\,B\,a\,b^2+12\,B\,a\,b\,c\,x-12\,A\,a\,b\,c-24\,A\,a\,c^2\,x+3\,B\,b^3\,x+A\,b^3+12\,B\,b^2\,c\,x^2-6\,A\,b^2\,c\,x+8\,B\,b\,c^2\,x^3-24\,A\,b\,c^2\,x^2-16\,A\,c^3\,x^3\right )}{3\,{\left (4\,a\,c-b^2\right )}^2\,{\left (c\,x^2+b\,x+a\right )}^{3/2}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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