Optimal. Leaf size=109 \[ -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{3 x^{3/2}}-\frac {2 \left (2 a B c+2 A b c+b^2 B\right )}{\sqrt {x}}-\frac {2 a (a B+2 A b)}{5 x^{5/2}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2} \]
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Rubi [A] time = 0.06, antiderivative size = 109, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {765} \begin {gather*} -\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 \left (A \left (2 a c+b^2\right )+2 a b B\right )}{3 x^{3/2}}-\frac {2 \left (2 a B c+2 A b c+b^2 B\right )}{\sqrt {x}}-\frac {2 a (a B+2 A b)}{5 x^{5/2}}+2 c \sqrt {x} (A c+2 b B)+\frac {2}{3} B c^2 x^{3/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a+b x+c x^2\right )^2}{x^{9/2}} \, dx &=\int \left (\frac {a^2 A}{x^{9/2}}+\frac {a (2 A b+a B)}{x^{7/2}}+\frac {2 a b B+A \left (b^2+2 a c\right )}{x^{5/2}}+\frac {b^2 B+2 A b c+2 a B c}{x^{3/2}}+\frac {c (2 b B+A c)}{\sqrt {x}}+B c^2 \sqrt {x}\right ) \, dx\\ &=-\frac {2 a^2 A}{7 x^{7/2}}-\frac {2 a (2 A b+a B)}{5 x^{5/2}}-\frac {2 \left (2 a b B+A \left (b^2+2 a c\right )\right )}{3 x^{3/2}}-\frac {2 \left (b^2 B+2 A b c+2 a B c\right )}{\sqrt {x}}+2 c (2 b B+A c) \sqrt {x}+\frac {2}{3} B c^2 x^{3/2}\\ \end {align*}
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Mathematica [A] time = 0.08, size = 95, normalized size = 0.87 \begin {gather*} -\frac {2 \left (3 a^2 (5 A+7 B x)+14 a x (A (3 b+5 c x)+5 B x (b+3 c x))+35 x^2 \left (A \left (b^2+6 b c x-3 c^2 x^2\right )-B x \left (-3 b^2+6 b c x+c^2 x^2\right )\right )\right )}{105 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.09, size = 105, normalized size = 0.96 \begin {gather*} \frac {2 \left (-15 a^2 A-21 a^2 B x-42 a A b x-70 a A c x^2-70 a b B x^2-210 a B c x^3-35 A b^2 x^2-210 A b c x^3+105 A c^2 x^4-105 b^2 B x^3+210 b B c x^4+35 B c^2 x^5\right )}{105 x^{7/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.41, size = 93, normalized size = 0.85 \begin {gather*} \frac {2 \, {\left (35 \, B c^{2} x^{5} + 105 \, {\left (2 \, B b c + A c^{2}\right )} x^{4} - 105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} - 15 \, A a^{2} - 35 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} - 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 102, normalized size = 0.94 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 4 \, B b c \sqrt {x} + 2 \, A c^{2} \sqrt {x} - \frac {2 \, {\left (105 \, B b^{2} x^{3} + 210 \, B a c x^{3} + 210 \, A b c x^{3} + 70 \, B a b x^{2} + 35 \, A b^{2} x^{2} + 70 \, A a c x^{2} + 21 \, B a^{2} x + 42 \, A a b x + 15 \, A a^{2}\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 102, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-35 B \,c^{2} x^{5}-105 A \,c^{2} x^{4}-210 x^{4} b B c +210 x^{3} A b c +210 B a c \,x^{3}+105 B \,b^{2} x^{3}+70 A a c \,x^{2}+35 A \,b^{2} x^{2}+70 B a b \,x^{2}+42 A a b x +21 B \,a^{2} x +15 A \,a^{2}\right )}{105 x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.48, size = 94, normalized size = 0.86 \begin {gather*} \frac {2}{3} \, B c^{2} x^{\frac {3}{2}} + 2 \, {\left (2 \, B b c + A c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (105 \, {\left (B b^{2} + 2 \, {\left (B a + A b\right )} c\right )} x^{3} + 15 \, A a^{2} + 35 \, {\left (2 \, B a b + A b^{2} + 2 \, A a c\right )} x^{2} + 21 \, {\left (B a^{2} + 2 \, A a b\right )} x\right )}}{105 \, x^{\frac {7}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.28, size = 94, normalized size = 0.86 \begin {gather*} \sqrt {x}\,\left (2\,A\,c^2+4\,B\,b\,c\right )-\frac {\frac {2\,A\,a^2}{7}+x^2\,\left (\frac {2\,A\,b^2}{3}+\frac {4\,B\,a\,b}{3}+\frac {4\,A\,a\,c}{3}\right )+x^3\,\left (2\,B\,b^2+4\,A\,c\,b+4\,B\,a\,c\right )+x\,\left (\frac {2\,B\,a^2}{5}+\frac {4\,A\,b\,a}{5}\right )}{x^{7/2}}+\frac {2\,B\,c^2\,x^{3/2}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 5.88, size = 153, normalized size = 1.40 \begin {gather*} - \frac {2 A a^{2}}{7 x^{\frac {7}{2}}} - \frac {4 A a b}{5 x^{\frac {5}{2}}} - \frac {4 A a c}{3 x^{\frac {3}{2}}} - \frac {2 A b^{2}}{3 x^{\frac {3}{2}}} - \frac {4 A b c}{\sqrt {x}} + 2 A c^{2} \sqrt {x} - \frac {2 B a^{2}}{5 x^{\frac {5}{2}}} - \frac {4 B a b}{3 x^{\frac {3}{2}}} - \frac {4 B a c}{\sqrt {x}} - \frac {2 B b^{2}}{\sqrt {x}} + 4 B b c \sqrt {x} + \frac {2 B c^{2} x^{\frac {3}{2}}}{3} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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