Optimal. Leaf size=111 \[ \frac {2}{3} a^4 A x^{3/2}+\frac {2}{5} a^3 x^{5/2} (a B+4 A b)+\frac {4}{7} a^2 b x^{7/2} (2 a B+3 A b)+\frac {2}{11} b^3 x^{11/2} (4 a B+A b)+\frac {4}{9} a b^2 x^{9/2} (3 a B+2 A b)+\frac {2}{13} b^4 B x^{13/2} \]
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Rubi [A] time = 0.05, antiderivative size = 111, 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} \[ \frac {4}{7} a^2 b x^{7/2} (2 a B+3 A b)+\frac {2}{5} a^3 x^{5/2} (a B+4 A b)+\frac {2}{3} a^4 A x^{3/2}+\frac {2}{11} b^3 x^{11/2} (4 a B+A b)+\frac {4}{9} a b^2 x^{9/2} (3 a B+2 A b)+\frac {2}{13} b^4 B x^{13/2} \]
Antiderivative was successfully verified.
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Rule 27
Rule 76
Rubi steps
\begin {align*} \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^2 \, dx &=\int \sqrt {x} (a+b x)^4 (A+B x) \, dx\\ &=\int \left (a^4 A \sqrt {x}+a^3 (4 A b+a B) x^{3/2}+2 a^2 b (3 A b+2 a B) x^{5/2}+2 a b^2 (2 A b+3 a B) x^{7/2}+b^3 (A b+4 a B) x^{9/2}+b^4 B x^{11/2}\right ) \, dx\\ &=\frac {2}{3} a^4 A x^{3/2}+\frac {2}{5} a^3 (4 A b+a B) x^{5/2}+\frac {4}{7} a^2 b (3 A b+2 a B) x^{7/2}+\frac {4}{9} a b^2 (2 A b+3 a B) x^{9/2}+\frac {2}{11} b^3 (A b+4 a B) x^{11/2}+\frac {2}{13} b^4 B x^{13/2}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 81, normalized size = 0.73 \[ \frac {2 \left (\frac {x^{3/2} \left (1155 a^4+2772 a^3 b x+2970 a^2 b^2 x^2+1540 a b^3 x^3+315 b^4 x^4\right ) (13 A b-3 a B)}{3465}+B x^{3/2} (a+b x)^5\right )}{13 b} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.93, size = 102, normalized size = 0.92 \[ \frac {2}{45045} \, {\left (3465 \, B b^{4} x^{6} + 15015 \, A a^{4} x + 4095 \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{5} + 10010 \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{4} + 12870 \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{3} + 9009 \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{2}\right )} \sqrt {x} \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.15, size = 101, normalized size = 0.91 \[ \frac {2}{13} \, B b^{4} x^{\frac {13}{2}} + \frac {8}{11} \, B a b^{3} x^{\frac {11}{2}} + \frac {2}{11} \, A b^{4} x^{\frac {11}{2}} + \frac {4}{3} \, B a^{2} b^{2} x^{\frac {9}{2}} + \frac {8}{9} \, A a b^{3} x^{\frac {9}{2}} + \frac {8}{7} \, B a^{3} b x^{\frac {7}{2}} + \frac {12}{7} \, A a^{2} b^{2} x^{\frac {7}{2}} + \frac {2}{5} \, B a^{4} x^{\frac {5}{2}} + \frac {8}{5} \, A a^{3} b x^{\frac {5}{2}} + \frac {2}{3} \, A a^{4} x^{\frac {3}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.06, size = 100, normalized size = 0.90 \[ \frac {2 \left (3465 b^{4} B \,x^{5}+4095 A \,b^{4} x^{4}+16380 x^{4} B a \,b^{3}+20020 A a \,b^{3} x^{3}+30030 B \,a^{2} b^{2} x^{3}+38610 A \,a^{2} b^{2} x^{2}+25740 B \,a^{3} b \,x^{2}+36036 A \,a^{3} b x +9009 B \,a^{4} x +15015 A \,a^{4}\right ) x^{\frac {3}{2}}}{45045} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.56, size = 99, normalized size = 0.89 \[ \frac {2}{13} \, B b^{4} x^{\frac {13}{2}} + \frac {2}{3} \, A a^{4} x^{\frac {3}{2}} + \frac {2}{11} \, {\left (4 \, B a b^{3} + A b^{4}\right )} x^{\frac {11}{2}} + \frac {4}{9} \, {\left (3 \, B a^{2} b^{2} + 2 \, A a b^{3}\right )} x^{\frac {9}{2}} + \frac {4}{7} \, {\left (2 \, B a^{3} b + 3 \, A a^{2} b^{2}\right )} x^{\frac {7}{2}} + \frac {2}{5} \, {\left (B a^{4} + 4 \, A a^{3} b\right )} x^{\frac {5}{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 0.04, size = 91, normalized size = 0.82 \[ x^{5/2}\,\left (\frac {2\,B\,a^4}{5}+\frac {8\,A\,b\,a^3}{5}\right )+x^{11/2}\,\left (\frac {2\,A\,b^4}{11}+\frac {8\,B\,a\,b^3}{11}\right )+\frac {2\,A\,a^4\,x^{3/2}}{3}+\frac {2\,B\,b^4\,x^{13/2}}{13}+\frac {4\,a^2\,b\,x^{7/2}\,\left (3\,A\,b+2\,B\,a\right )}{7}+\frac {4\,a\,b^2\,x^{9/2}\,\left (2\,A\,b+3\,B\,a\right )}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [A] time = 4.38, size = 124, normalized size = 1.12 \[ \frac {2 A a^{4} x^{\frac {3}{2}}}{3} + \frac {2 B b^{4} x^{\frac {13}{2}}}{13} + \frac {2 x^{\frac {11}{2}} \left (A b^{4} + 4 B a b^{3}\right )}{11} + \frac {2 x^{\frac {9}{2}} \left (4 A a b^{3} + 6 B a^{2} b^{2}\right )}{9} + \frac {2 x^{\frac {7}{2}} \left (6 A a^{2} b^{2} + 4 B a^{3} b\right )}{7} + \frac {2 x^{\frac {5}{2}} \left (4 A a^{3} b + B a^{4}\right )}{5} \]
Verification of antiderivative is not currently implemented for this CAS.
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