Optimal. Leaf size=94 \[ -\frac {2 x^2 (A b-2 a B-(b B-2 A c) x)}{3 \left (b^2-4 a c\right ) \left (a+b x+c x^2\right )^{3/2}}+\frac {8 (A b-2 a B) (2 a+b x)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}} \]
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Rubi [A]
time = 0.03, antiderivative size = 94, normalized size of antiderivative = 1.00, number of steps
used = 2, number of rules used = 2, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {818, 650}
\begin {gather*} \frac {8 (2 a+b x) (A b-2 a B)}{3 \left (b^2-4 a c\right )^2 \sqrt {a+b x+c x^2}}-\frac {2 x^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 650
Rule 818
Rubi steps
\begin {align*} \int \frac {x^2 (A+B x)}{\left (a+b x+c x^2\right )^{5/2}} \, dx &=-\frac {2 x^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 (A b-2 a B)) \int \frac {x}{\left (a+b x+c x^2\right )^{3/2}} \, dx}{3 \left (b^2-4 a c\right )}\\ &=-\frac {2 x^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 (A b-2 a B) (2 a+b 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 = 0.88, size = 110, normalized size = 1.17 \begin {gather*} \frac {2 \left (-16 a^3 B+b^2 x^2 (3 A b+b B x+2 A c x)+8 a^2 (A b-3 B x (b+c x))+2 a x \left (-3 b B x (b+2 c x)+A \left (6 b^2+6 b c x+4 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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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(502\) vs.
\(2(86)=172\).
time = 0.73, size = 503, normalized size = 5.35
method | result | size |
trager | \(\frac {\frac {16}{3} a A \,c^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,x^{3}-8 B a b c \,x^{3}+\frac {2}{3} B \,b^{3} x^{3}+8 A a b c \,x^{2}+2 A \,b^{3} x^{2}-16 a^{2} B c \,x^{2}-4 B a \,b^{2} x^{2}+8 A a \,b^{2} x -16 B \,a^{2} b x +\frac {16}{3} A \,a^{2} b -\frac {32}{3} B \,a^{3}}{\left (4 a c -b^{2}\right )^{2} \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}\) | \(132\) |
gosper | \(\frac {\frac {16}{3} a A \,c^{2} x^{3}+\frac {4}{3} A \,b^{2} c \,x^{3}-8 B a b c \,x^{3}+\frac {2}{3} B \,b^{3} x^{3}+8 A a b c \,x^{2}+2 A \,b^{3} x^{2}-16 a^{2} B c \,x^{2}-4 B a \,b^{2} x^{2}+8 A a \,b^{2} x -16 B \,a^{2} b x +\frac {16}{3} A \,a^{2} b -\frac {32}{3} B \,a^{3}}{\left (c \,x^{2}+b x +a \right )^{\frac {3}{2}} \left (16 a^{2} c^{2}-8 a c \,b^{2}+b^{4}\right )}\) | \(141\) |
default | \(B \left (-\frac {x^{2}}{c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {b \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{2 c}+\frac {2 a \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{c}\right )+A \left (-\frac {x}{2 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (-\frac {1}{3 c \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}-\frac {b \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )}{4 c}+\frac {a \left (\frac {\frac {4 c x}{3}+\frac {2 b}{3}}{\left (4 a c -b^{2}\right ) \left (c \,x^{2}+b x +a \right )^{\frac {3}{2}}}+\frac {16 c \left (2 c x +b \right )}{3 \left (4 a c -b^{2}\right )^{2} \sqrt {c \,x^{2}+b x +a}}\right )}{2 c}\right )\) | \(503\) |
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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 248 vs.
\(2 (87) = 174\).
time = 4.38, size = 248, normalized size = 2.64 \begin {gather*} -\frac {2 \, {\left (16 \, B a^{3} - 8 \, A a^{2} b - {\left (B b^{3} + 8 \, A a c^{2} - 2 \, {\left (6 \, B a b - A b^{2}\right )} c\right )} x^{3} + 3 \, {\left (2 \, B a b^{2} - A b^{3} + 4 \, {\left (2 \, B a^{2} - A a b\right )} c\right )} x^{2} + 12 \, {\left (2 \, B a^{2} b - A a b^{2}\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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Sympy [F(-1)] Timed out
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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 195 vs.
\(2 (87) = 174\).
time = 1.08, size = 195, normalized size = 2.07 \begin {gather*} \frac {2 \, {\left ({\left ({\left (\frac {{\left (B b^{3} - 12 \, B a b c + 2 \, A b^{2} c + 8 \, A a c^{2}\right )} x}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}} - \frac {3 \, {\left (2 \, B a b^{2} - A b^{3} + 8 \, B a^{2} c - 4 \, A a b c\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {12 \, {\left (2 \, B a^{2} b - A a b^{2}\right )}}{b^{4} - 8 \, a b^{2} c + 16 \, a^{2} c^{2}}\right )} x - \frac {8 \, {\left (2 \, B a^{3} - A a^{2} b\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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Mupad [B]
time = 1.73, size = 131, normalized size = 1.39 \begin {gather*} \frac {2\,\left (-16\,B\,a^3-24\,B\,a^2\,b\,x+8\,A\,a^2\,b-24\,B\,a^2\,c\,x^2-6\,B\,a\,b^2\,x^2+12\,A\,a\,b^2\,x-12\,B\,a\,b\,c\,x^3+12\,A\,a\,b\,c\,x^2+8\,A\,a\,c^2\,x^3+B\,b^3\,x^3+3\,A\,b^3\,x^2+2\,A\,b^2\,c\,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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