Optimal. Leaf size=320 \[ \frac {20 a^2 b^2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 b^4 x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac {10 a b^3 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac {2 b^5 B x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)}+\frac {2 a^5 A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 a^4 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac {10 a^3 b x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)} \]
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Rubi [A] time = 0.12, antiderivative size = 320, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {770, 76} \begin {gather*} \frac {2 b^4 x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{15 (a+b x)}+\frac {10 a b^3 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{13 (a+b x)}+\frac {20 a^2 b^2 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {10 a^3 b x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{9 (a+b x)}+\frac {2 a^4 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{7 (a+b x)}+\frac {2 a^5 A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 b^5 B x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int x^{3/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int x^{3/2} \left (a b+b^2 x\right )^5 (A+B x) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^5 A b^5 x^{3/2}+a^4 b^5 (5 A b+a B) x^{5/2}+5 a^3 b^6 (2 A b+a B) x^{7/2}+10 a^2 b^7 (A b+a B) x^{9/2}+5 a b^8 (A b+2 a B) x^{11/2}+b^9 (A b+5 a B) x^{13/2}+b^{10} B x^{15/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 a^5 A x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {2 a^4 (5 A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {10 a^3 b (2 A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {20 a^2 b^2 (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {10 a b^3 (A b+2 a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)}+\frac {2 b^5 B x^{17/2} \sqrt {a^2+2 a b x+b^2 x^2}}{17 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.05, size = 127, normalized size = 0.40 \begin {gather*} \frac {2 x^{5/2} \sqrt {(a+b x)^2} \left (21879 a^5 (7 A+5 B x)+60775 a^4 b x (9 A+7 B x)+77350 a^3 b^2 x^2 (11 A+9 B x)+53550 a^2 b^3 x^3 (13 A+11 B x)+19635 a b^4 x^4 (15 A+13 B x)+3003 b^5 x^5 (17 A+15 B x)\right )}{765765 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 13.34, size = 171, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (153153 a^5 A x^{5/2}+109395 a^5 B x^{7/2}+546975 a^4 A b x^{7/2}+425425 a^4 b B x^{9/2}+850850 a^3 A b^2 x^{9/2}+696150 a^3 b^2 B x^{11/2}+696150 a^2 A b^3 x^{11/2}+589050 a^2 b^3 B x^{13/2}+294525 a A b^4 x^{13/2}+255255 a b^4 B x^{15/2}+51051 A b^5 x^{15/2}+45045 b^5 B x^{17/2}\right )}{765765 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 124, normalized size = 0.39 \begin {gather*} \frac {2}{765765} \, {\left (45045 \, B b^{5} x^{8} + 153153 \, A a^{5} x^{2} + 51051 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{7} + 294525 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{6} + 696150 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{5} + 425425 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{4} + 109395 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{3}\right )} \sqrt {x} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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giac [A] time = 0.17, size = 197, normalized size = 0.62 \begin {gather*} \frac {2}{17} \, B b^{5} x^{\frac {17}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a b^{4} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{15} \, A b^{5} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{13} \, B a^{2} b^{3} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{13} \, A a b^{4} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{11} \, B a^{3} b^{2} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{11} \, A a^{2} b^{3} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, B a^{4} b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, A a^{3} b^{2} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, B a^{5} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, A a^{4} b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, A a^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.05, size = 140, normalized size = 0.44 \begin {gather*} \frac {2 \left (45045 B \,b^{5} x^{6}+51051 A \,b^{5} x^{5}+255255 B a \,b^{4} x^{5}+294525 A a \,b^{4} x^{4}+589050 B \,a^{2} b^{3} x^{4}+696150 A \,a^{2} b^{3} x^{3}+696150 B \,a^{3} b^{2} x^{3}+850850 A \,a^{3} b^{2} x^{2}+425425 B \,a^{4} b \,x^{2}+546975 A \,a^{4} b x +109395 B \,a^{5} x +153153 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{\frac {5}{2}}}{765765 \left (b x +a \right )^{5}} \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.69, size = 241, normalized size = 0.75 \begin {gather*} \frac {2}{45045} \, {\left (231 \, {\left (13 \, b^{5} x^{2} + 15 \, a b^{4} x\right )} x^{\frac {11}{2}} + 1260 \, {\left (11 \, a b^{4} x^{2} + 13 \, a^{2} b^{3} x\right )} x^{\frac {9}{2}} + 2730 \, {\left (9 \, a^{2} b^{3} x^{2} + 11 \, a^{3} b^{2} x\right )} x^{\frac {7}{2}} + 2860 \, {\left (7 \, a^{3} b^{2} x^{2} + 9 \, a^{4} b x\right )} x^{\frac {5}{2}} + 1287 \, {\left (5 \, a^{4} b x^{2} + 7 \, a^{5} x\right )} x^{\frac {3}{2}}\right )} A + \frac {2}{765765} \, {\left (3003 \, {\left (15 \, b^{5} x^{2} + 17 \, a b^{4} x\right )} x^{\frac {13}{2}} + 15708 \, {\left (13 \, a b^{4} x^{2} + 15 \, a^{2} b^{3} x\right )} x^{\frac {11}{2}} + 32130 \, {\left (11 \, a^{2} b^{3} x^{2} + 13 \, a^{3} b^{2} x\right )} x^{\frac {9}{2}} + 30940 \, {\left (9 \, a^{3} b^{2} x^{2} + 11 \, a^{4} b x\right )} x^{\frac {7}{2}} + 12155 \, {\left (7 \, a^{4} b x^{2} + 9 \, a^{5} x\right )} x^{\frac {5}{2}}\right )} B \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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mupad [F] time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int x^{3/2}\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2} \,d x \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] 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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