Optimal. Leaf size=155 \[ -\frac {2 a^6 A}{9 x^{9/2}}-\frac {2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac {6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac {10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac {10 a^2 b^3 (4 a B+3 A b)}{\sqrt {x}}+\frac {2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt {x} (5 a B+2 A b)+\frac {2}{5} b^6 B x^{5/2} \]
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Rubi [A] time = 0.08, antiderivative size = 155, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.069, Rules used = {27, 76} \begin {gather*} -\frac {10 a^3 b^2 (3 a B+4 A b)}{3 x^{3/2}}-\frac {10 a^2 b^3 (4 a B+3 A b)}{\sqrt {x}}-\frac {2 a^5 (a B+6 A b)}{7 x^{7/2}}-\frac {6 a^4 b (2 a B+5 A b)}{5 x^{5/2}}-\frac {2 a^6 A}{9 x^{9/2}}+\frac {2}{3} b^5 x^{3/2} (6 a B+A b)+6 a b^4 \sqrt {x} (5 a B+2 A b)+\frac {2}{5} b^6 B x^{5/2} \end {gather*}
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^3}{x^{11/2}} \, dx &=\int \frac {(a+b x)^6 (A+B x)}{x^{11/2}} \, dx\\ &=\int \left (\frac {a^6 A}{x^{11/2}}+\frac {a^5 (6 A b+a B)}{x^{9/2}}+\frac {3 a^4 b (5 A b+2 a B)}{x^{7/2}}+\frac {5 a^3 b^2 (4 A b+3 a B)}{x^{5/2}}+\frac {5 a^2 b^3 (3 A b+4 a B)}{x^{3/2}}+\frac {3 a b^4 (2 A b+5 a B)}{\sqrt {x}}+b^5 (A b+6 a B) \sqrt {x}+b^6 B x^{3/2}\right ) \, dx\\ &=-\frac {2 a^6 A}{9 x^{9/2}}-\frac {2 a^5 (6 A b+a B)}{7 x^{7/2}}-\frac {6 a^4 b (5 A b+2 a B)}{5 x^{5/2}}-\frac {10 a^3 b^2 (4 A b+3 a B)}{3 x^{3/2}}-\frac {10 a^2 b^3 (3 A b+4 a B)}{\sqrt {x}}+6 a b^4 (2 A b+5 a B) \sqrt {x}+\frac {2}{3} b^5 (A b+6 a B) x^{3/2}+\frac {2}{5} b^6 B x^{5/2}\\ \end {align*}
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Mathematica [A] time = 0.04, size = 123, normalized size = 0.79 \begin {gather*} -\frac {2 \left (5 a^6 (7 A+9 B x)+54 a^5 b x (5 A+7 B x)+315 a^4 b^2 x^2 (3 A+5 B x)+2100 a^3 b^3 x^3 (A+3 B x)+4725 a^2 b^4 x^4 (A-B x)-630 a b^5 x^5 (3 A+B x)-21 b^6 x^6 (5 A+3 B x)\right )}{315 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 0.10, size = 151, normalized size = 0.97 \begin {gather*} \frac {2 \left (-35 a^6 A-45 a^6 B x-270 a^5 A b x-378 a^5 b B x^2-945 a^4 A b^2 x^2-1575 a^4 b^2 B x^3-2100 a^3 A b^3 x^3-6300 a^3 b^3 B x^4-4725 a^2 A b^4 x^4+4725 a^2 b^4 B x^5+1890 a A b^5 x^5+630 a b^5 B x^6+105 A b^6 x^6+63 b^6 B x^7\right )}{315 x^{9/2}} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.42, size = 147, normalized size = 0.95 \begin {gather*} \frac {2 \, {\left (63 \, B b^{6} x^{7} - 35 \, A a^{6} + 105 \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{6} + 945 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} x^{5} - 1575 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} - 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} - 189 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} - 45 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.16, size = 148, normalized size = 0.95 \begin {gather*} \frac {2}{5} \, B b^{6} x^{\frac {5}{2}} + 4 \, B a b^{5} x^{\frac {3}{2}} + \frac {2}{3} \, A b^{6} x^{\frac {3}{2}} + 30 \, B a^{2} b^{4} \sqrt {x} + 12 \, A a b^{5} \sqrt {x} - \frac {2 \, {\left (6300 \, B a^{3} b^{3} x^{4} + 4725 \, A a^{2} b^{4} x^{4} + 1575 \, B a^{4} b^{2} x^{3} + 2100 \, A a^{3} b^{3} x^{3} + 378 \, B a^{5} b x^{2} + 945 \, A a^{4} b^{2} x^{2} + 45 \, B a^{6} x + 270 \, A a^{5} b x + 35 \, A a^{6}\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 148, normalized size = 0.95 \begin {gather*} -\frac {2 \left (-63 B \,b^{6} x^{7}-105 A \,b^{6} x^{6}-630 x^{6} B a \,b^{5}-1890 A a \,b^{5} x^{5}-4725 x^{5} B \,a^{2} b^{4}+4725 A \,a^{2} b^{4} x^{4}+6300 x^{4} B \,a^{3} b^{3}+2100 A \,a^{3} b^{3} x^{3}+1575 B \,a^{4} b^{2} x^{3}+945 A \,a^{4} b^{2} x^{2}+378 x^{2} B \,a^{5} b +270 A \,a^{5} b x +45 x B \,a^{6}+35 A \,a^{6}\right )}{315 x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.46, size = 148, normalized size = 0.95 \begin {gather*} \frac {2}{5} \, B b^{6} x^{\frac {5}{2}} + \frac {2}{3} \, {\left (6 \, B a b^{5} + A b^{6}\right )} x^{\frac {3}{2}} + 6 \, {\left (5 \, B a^{2} b^{4} + 2 \, A a b^{5}\right )} \sqrt {x} - \frac {2 \, {\left (35 \, A a^{6} + 1575 \, {\left (4 \, B a^{3} b^{3} + 3 \, A a^{2} b^{4}\right )} x^{4} + 525 \, {\left (3 \, B a^{4} b^{2} + 4 \, A a^{3} b^{3}\right )} x^{3} + 189 \, {\left (2 \, B a^{5} b + 5 \, A a^{4} b^{2}\right )} x^{2} + 45 \, {\left (B a^{6} + 6 \, A a^{5} b\right )} x\right )}}{315 \, x^{\frac {9}{2}}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 1.16, size = 141, normalized size = 0.91 \begin {gather*} x^{3/2}\,\left (\frac {2\,A\,b^6}{3}+4\,B\,a\,b^5\right )-\frac {x\,\left (\frac {2\,B\,a^6}{7}+\frac {12\,A\,b\,a^5}{7}\right )+\frac {2\,A\,a^6}{9}+x^2\,\left (\frac {12\,B\,a^5\,b}{5}+6\,A\,a^4\,b^2\right )+x^3\,\left (10\,B\,a^4\,b^2+\frac {40\,A\,a^3\,b^3}{3}\right )+x^4\,\left (40\,B\,a^3\,b^3+30\,A\,a^2\,b^4\right )}{x^{9/2}}+\frac {2\,B\,b^6\,x^{5/2}}{5}+6\,a\,b^4\,\sqrt {x}\,\left (2\,A\,b+5\,B\,a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 12.62, size = 204, normalized size = 1.32 \begin {gather*} - \frac {2 A a^{6}}{9 x^{\frac {9}{2}}} - \frac {12 A a^{5} b}{7 x^{\frac {7}{2}}} - \frac {6 A a^{4} b^{2}}{x^{\frac {5}{2}}} - \frac {40 A a^{3} b^{3}}{3 x^{\frac {3}{2}}} - \frac {30 A a^{2} b^{4}}{\sqrt {x}} + 12 A a b^{5} \sqrt {x} + \frac {2 A b^{6} x^{\frac {3}{2}}}{3} - \frac {2 B a^{6}}{7 x^{\frac {7}{2}}} - \frac {12 B a^{5} b}{5 x^{\frac {5}{2}}} - \frac {10 B a^{4} b^{2}}{x^{\frac {3}{2}}} - \frac {40 B a^{3} b^{3}}{\sqrt {x}} + 30 B a^{2} b^{4} \sqrt {x} + 4 B a b^{5} x^{\frac {3}{2}} + \frac {2 B b^{6} x^{\frac {5}{2}}}{5} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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