Optimal. Leaf size=81 \[ -\frac {2 A b^3}{3 x^{3/2}}-\frac {2 b^2 (3 A c+b B)}{\sqrt {x}}+\frac {2}{3} c^2 x^{3/2} (A c+3 b B)+6 b c \sqrt {x} (A c+b B)+\frac {2}{5} B c^3 x^{5/2} \]
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Rubi [A] time = 0.04, antiderivative size = 81, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.045, Rules used = {765} \begin {gather*} -\frac {2 b^2 (3 A c+b B)}{\sqrt {x}}-\frac {2 A b^3}{3 x^{3/2}}+\frac {2}{3} c^2 x^{3/2} (A c+3 b B)+6 b c \sqrt {x} (A c+b B)+\frac {2}{5} B c^3 x^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 765
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (b x+c x^2\right )^3}{x^{11/2}} \, dx &=\int \left (\frac {A b^3}{x^{5/2}}+\frac {b^2 (b B+3 A c)}{x^{3/2}}+\frac {3 b c (b B+A c)}{\sqrt {x}}+c^2 (3 b B+A c) \sqrt {x}+B c^3 x^{3/2}\right ) \, dx\\ &=-\frac {2 A b^3}{3 x^{3/2}}-\frac {2 b^2 (b B+3 A c)}{\sqrt {x}}+6 b c (b B+A c) \sqrt {x}+\frac {2}{3} c^2 (3 b B+A c) x^{3/2}+\frac {2}{5} B c^3 x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.02, size = 74, normalized size = 0.91 \begin {gather*} \frac {6 B x \left (-5 b^3+15 b^2 c x+5 b c^2 x^2+c^3 x^3\right )-10 A \left (b^3+9 b^2 c x-9 b c^2 x^2-c^3 x^3\right )}{15 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.05, size = 79, normalized size = 0.98 \begin {gather*} \frac {2 \left (-5 A b^3-45 A b^2 c x+45 A b c^2 x^2+5 A c^3 x^3-15 b^3 B x+45 b^2 B c x^2+15 b B c^2 x^3+3 B c^3 x^4\right )}{15 x^{3/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.40, size = 73, normalized size = 0.90 \begin {gather*} \frac {2 \, {\left (3 \, B c^{3} x^{4} - 5 \, A b^{3} + 5 \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{3} + 45 \, {\left (B b^{2} c + A b c^{2}\right )} x^{2} - 15 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )}}{15 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 75, normalized size = 0.93 \begin {gather*} \frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + 2 \, B b c^{2} x^{\frac {3}{2}} + \frac {2}{3} \, A c^{3} x^{\frac {3}{2}} + 6 \, B b^{2} c \sqrt {x} + 6 \, A b c^{2} \sqrt {x} - \frac {2 \, {\left (3 \, B b^{3} x + 9 \, A b^{2} c x + A b^{3}\right )}}{3 \, x^{\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 = 76, normalized size = 0.94 \begin {gather*} -\frac {2 \left (-3 B \,c^{3} x^{4}-5 A \,c^{3} x^{3}-15 B b \,c^{2} x^{3}-45 A b \,c^{2} x^{2}-45 B \,b^{2} c \,x^{2}+45 A \,b^{2} c x +15 B \,b^{3} x +5 A \,b^{3}\right )}{15 x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.57, size = 73, normalized size = 0.90 \begin {gather*} \frac {2}{5} \, B c^{3} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (3 \, B b c^{2} + A c^{3}\right )} x^{\frac {3}{2}} + 6 \, {\left (B b^{2} c + A b c^{2}\right )} \sqrt {x} - \frac {2 \, {\left (A b^{3} + 3 \, {\left (B b^{3} + 3 \, A b^{2} c\right )} x\right )}}{3 \, x^{\frac {3}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.06, size = 70, normalized size = 0.86 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,c^3}{3}+2\,B\,b\,c^2\right )-\frac {x\,\left (2\,B\,b^3+6\,A\,c\,b^2\right )+\frac {2\,A\,b^3}{3}}{x^{3/2}}+\frac {2\,B\,c^3\,x^{5/2}}{5}+6\,b\,c\,\sqrt {x}\,\left (A\,c+B\,b\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 10.34, size = 105, normalized size = 1.30 \begin {gather*} - \frac {2 A b^{3}}{3 x^{\frac {3}{2}}} - \frac {6 A b^{2} c}{\sqrt {x}} + 6 A b c^{2} \sqrt {x} + \frac {2 A c^{3} x^{\frac {3}{2}}}{3} - \frac {2 B b^{3}}{\sqrt {x}} + 6 B b^{2} c \sqrt {x} + 2 B b c^{2} x^{\frac {3}{2}} + \frac {2 B c^{3} x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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