Optimal. Leaf size=114 \[ \frac {2 (b+2 c x) \left (4 a B c-4 A b c+b^2 B\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}} \]
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Rubi [A] time = 0.04, antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.095, Rules used = {777, 613} \begin {gather*} \frac {2 (b+2 c x) \left (4 a B c-4 A b c+b^2 B\right )}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 \left (x \left (-2 a B c-A b c+b^2 B\right )+a (b B-2 A c)\right )}{3 c \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 777
Rubi steps
\begin {align*} \int \frac {x (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}-\frac {\left (b^2 B-4 A b c+4 a B c\right ) \int \frac {1}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 c \left (b^2-4 a c\right )}\\ &=-\frac {2 \left (a (b B-2 A c)+\left (b^2 B-A b c-2 a B c\right ) x\right )}{3 c \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {2 \left (b^2 B-4 A b c+4 a B c\right ) (b+2 c x)}{3 c \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}\\ \end {align*}
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Mathematica [A] time = 0.17, size = 114, normalized size = 1.00 \begin {gather*} \frac {2 \left (8 a^2 (b B-A c)-2 a A b (b+6 c x)+4 a B x \left (3 b^2+3 b c x+2 c^2 x^2\right )+b x \left (b B x (3 b+2 c x)-A \left (3 b^2+12 b c x+8 c^2 x^2\right )\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 = 0.99, size = 130, normalized size = 1.14 \begin {gather*} -\frac {2 \left (8 a^2 A c-8 a^2 b B+2 a A b^2+12 a A b c x-12 a b^2 B x-12 a b B c x^2-8 a B c^2 x^3+3 A b^3 x+12 A b^2 c x^2+8 A b c^2 x^3-3 b^3 B x^2-2 b^2 B c 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 = 1.20, size = 244, normalized size = 2.14 \begin {gather*} \frac {2 \, {\left (8 \, B a^{2} b - 2 \, A a b^{2} - 8 \, A a^{2} c + 2 \, {\left (B b^{2} c + 4 \, {\left (B a - A b\right )} c^{2}\right )} x^{3} + 3 \, {\left (B b^{3} + 4 \, {\left (B a b - A b^{2}\right )} c\right )} x^{2} + 3 \, {\left (4 \, B a b^{2} - A b^{3} - 4 \, A a b 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 [A] time = 0.25, size = 196, normalized size = 1.72 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {2 \, {\left (B b^{2} c + 4 \, B a c^{2} - 4 \, A b c^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} + \frac {3 \, {\left (B b^{3} + 4 \, B a b c - 4 \, A b^{2} c\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {3 \, {\left (4 \, B a b^{2} - A b^{3} - 4 \, A a b c\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x + \frac {2 \, {\left (4 \, B a^{2} b - A a b^{2} - 4 \, A a^{2} c\right )}}{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 = 0.05, size = 138, normalized size = 1.21 \begin {gather*} -\frac {2 \left (8 A b \,c^{2} x^{3}-8 B a \,c^{2} x^{3}-2 B \,b^{2} c \,x^{3}+12 A \,b^{2} c \,x^{2}-12 B a b c \,x^{2}-3 B \,b^{3} x^{2}+12 A a b c x +3 A \,b^{3} x -12 B a \,b^{2} x +8 A \,a^{2} c +2 A a \,b^{2}-8 B \,a^{2} b \right )}{3 \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 = 1.57, size = 128, normalized size = 1.12 \begin {gather*} \frac {2\,\left (8\,B\,a^2\,b-8\,A\,a^2\,c+12\,B\,a\,b^2\,x-2\,A\,a\,b^2+12\,B\,a\,b\,c\,x^2-12\,A\,a\,b\,c\,x+8\,B\,a\,c^2\,x^3+3\,B\,b^3\,x^2-3\,A\,b^3\,x+2\,B\,b^2\,c\,x^3-12\,A\,b^2\,c\,x^2-8\,A\,b\,c^2\,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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