Optimal. Leaf size=320 \[ \frac {20 a^2 b^2 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac {2 b^4 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{13 (a+b x)}+\frac {10 a b^3 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac {2 b^5 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)}+\frac {2 a^5 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^4 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 (a+b x)}+\frac {10 a^3 b x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)} \]
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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^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{13 (a+b x)}+\frac {10 a b^3 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{11 (a+b x)}+\frac {20 a^2 b^2 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{9 (a+b x)}+\frac {10 a^3 b x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{7 (a+b x)}+\frac {2 a^4 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{5 (a+b x)}+\frac {2 a^5 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^5 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \sqrt {x} (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 \sqrt {x} \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 \sqrt {x}+a^4 b^5 (5 A b+a B) x^{3/2}+5 a^3 b^6 (2 A b+a B) x^{5/2}+10 a^2 b^7 (A b+a B) x^{7/2}+5 a b^8 (A b+2 a B) x^{9/2}+b^9 (A b+5 a B) x^{11/2}+b^{10} B x^{13/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 a^5 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^4 (5 A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {10 a^3 b (2 A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {20 a^2 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 {10 a b^3 (A b+2 a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 b^5 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (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^{3/2} \sqrt {(a+b x)^2} \left (3003 a^5 (5 A+3 B x)+6435 a^4 b x (7 A+5 B x)+7150 a^3 b^2 x^2 (9 A+7 B x)+4550 a^2 b^3 x^3 (11 A+9 B x)+1575 a b^4 x^4 (13 A+11 B x)+231 b^5 x^5 (15 A+13 B x)\right )}{45045 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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IntegrateAlgebraic [A] time = 11.35, size = 171, normalized size = 0.53 \begin {gather*} \frac {2 \sqrt {(a+b x)^2} \left (15015 a^5 A x^{3/2}+9009 a^5 B x^{5/2}+45045 a^4 A b x^{5/2}+32175 a^4 b B x^{7/2}+64350 a^3 A b^2 x^{7/2}+50050 a^3 b^2 B x^{9/2}+50050 a^2 A b^3 x^{9/2}+40950 a^2 b^3 B x^{11/2}+20475 a A b^4 x^{11/2}+17325 a b^4 B x^{13/2}+3465 A b^5 x^{13/2}+3003 b^5 B x^{15/2}\right )}{45045 (a+b x)} \end {gather*}
Antiderivative was successfully verified.
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fricas [A] time = 0.43, size = 122, normalized size = 0.38 \begin {gather*} \frac {2}{45045} \, {\left (3003 \, B b^{5} x^{7} + 15015 \, A a^{5} x + 3465 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{6} + 20475 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{5} + 50050 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{4} + 32175 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{3} + 9009 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x^{2}\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}{15} \, B b^{5} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{13} \, B a b^{4} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{13} \, A b^{5} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{11} \, B a^{2} b^{3} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, A a b^{4} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, B a^{3} b^{2} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, A a^{2} b^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{7} \, B a^{4} b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, A a^{3} b^{2} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, B a^{5} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{4} b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A a^{5} x^{\frac {3}{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 (3003 B \,b^{5} x^{6}+3465 A \,b^{5} x^{5}+17325 B a \,b^{4} x^{5}+20475 A a \,b^{4} x^{4}+40950 B \,a^{2} b^{3} x^{4}+50050 A \,a^{2} b^{3} x^{3}+50050 B \,a^{3} b^{2} x^{3}+64350 A \,a^{3} b^{2} x^{2}+32175 B \,a^{4} b \,x^{2}+45045 A \,a^{4} b x +9009 B \,a^{5} x +15015 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} x^{\frac {3}{2}}}{45045 \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 = 0.78, size = 241, normalized size = 0.75 \begin {gather*} \frac {2}{45045} \, {\left (315 \, {\left (11 \, b^{5} x^{2} + 13 \, a b^{4} x\right )} x^{\frac {9}{2}} + 1820 \, {\left (9 \, a b^{4} x^{2} + 11 \, a^{2} b^{3} x\right )} x^{\frac {7}{2}} + 4290 \, {\left (7 \, a^{2} b^{3} x^{2} + 9 \, a^{3} b^{2} x\right )} x^{\frac {5}{2}} + 5148 \, {\left (5 \, a^{3} b^{2} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {3}{2}} + 3003 \, {\left (3 \, a^{4} b x^{2} + 5 \, a^{5} x\right )} \sqrt {x}\right )} A + \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 )} 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 \sqrt {x}\,\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] time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \sqrt {x} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}\, dx \end {gather*}
Verification of antiderivative is not currently implemented for this CAS.
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