Optimal. Leaf size=316 \[ \frac {20 a^2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 b^4 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {10 a b^3 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 a^4 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac {2 a^3 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x} \]
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Rubi [A] time = 0.12, antiderivative size = 316, 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} \[ \frac {2 b^4 x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (5 a B+A b)}{11 (a+b x)}+\frac {10 a b^3 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (2 a B+A b)}{9 (a+b x)}+\frac {20 a^2 b^2 x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a^3 b x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+2 A b)}{a+b x}+\frac {2 a^4 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+5 A b)}{3 (a+b x)}+\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)} \]
Antiderivative was successfully verified.
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Rule 76
Rule 770
Rubi steps
\begin {align*} \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{5/2}}{\sqrt {x}} \, dx &=\frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \frac {\left (a b+b^2 x\right )^5 (A+B x)}{\sqrt {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 (\frac {a^5 A b^5}{\sqrt {x}}+a^4 b^5 (5 A b+a B) \sqrt {x}+5 a^3 b^6 (2 A b+a B) x^{3/2}+10 a^2 b^7 (A b+a B) x^{5/2}+5 a b^8 (A b+2 a B) x^{7/2}+b^9 (A b+5 a B) x^{9/2}+b^{10} B x^{11/2}\right ) \, dx}{b^4 \left (a b+b^2 x\right )}\\ &=\frac {2 a^5 A \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 a^4 (5 A b+a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^3 b (2 A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {20 a^2 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 {10 a b^3 (A b+2 a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b^4 (A b+5 a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b^5 B x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}\\ \end {align*}
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Mathematica [A] time = 0.06, size = 126, normalized size = 0.40 \[ \frac {2 \sqrt {x} \sqrt {(a+b x)^2} \left (3003 a^5 (3 A+B x)+3003 a^4 b x (5 A+3 B x)+2574 a^3 b^2 x^2 (7 A+5 B x)+1430 a^2 b^3 x^3 (9 A+7 B x)+455 a b^4 x^4 (11 A+9 B x)+63 b^5 x^5 (13 A+11 B x)\right )}{9009 (a+b x)} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.92, size = 119, normalized size = 0.38 \[ \frac {2}{9009} \, {\left (693 \, B b^{5} x^{6} + 9009 \, A a^{5} + 819 \, {\left (5 \, B a b^{4} + A b^{5}\right )} x^{5} + 5005 \, {\left (2 \, B a^{2} b^{3} + A a b^{4}\right )} x^{4} + 12870 \, {\left (B a^{3} b^{2} + A a^{2} b^{3}\right )} x^{3} + 9009 \, {\left (B a^{4} b + 2 \, A a^{3} b^{2}\right )} x^{2} + 3003 \, {\left (B a^{5} + 5 \, A a^{4} b\right )} x\right )} \sqrt {x} \]
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 \[ \frac {2}{13} \, B b^{5} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{11} \, B a b^{4} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{11} \, A b^{5} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{9} \, B a^{2} b^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{9} \, A a b^{4} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, B a^{3} b^{2} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {20}{7} \, A a^{2} b^{3} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a^{4} b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 4 \, A a^{3} b^{2} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a^{5} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {10}{3} \, A a^{4} b x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, A a^{5} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) \]
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 \[ \frac {2 \left (693 B \,b^{5} x^{6}+819 A \,b^{5} x^{5}+4095 B a \,b^{4} x^{5}+5005 A a \,b^{4} x^{4}+10010 B \,a^{2} b^{3} x^{4}+12870 A \,a^{2} b^{3} x^{3}+12870 B \,a^{3} b^{2} x^{3}+18018 A \,a^{3} b^{2} x^{2}+9009 B \,a^{4} b \,x^{2}+15015 A \,a^{4} b x +3003 B \,a^{5} x +9009 A \,a^{5}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {5}{2}} \sqrt {x}}{9009 \left (b x +a \right )^{5}} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [A] time = 0.77, size = 240, normalized size = 0.76 \[ \frac {2}{3465} \, {\left (35 \, {\left (9 \, b^{5} x^{2} + 11 \, a b^{4} x\right )} x^{\frac {7}{2}} + 220 \, {\left (7 \, a b^{4} x^{2} + 9 \, a^{2} b^{3} x\right )} x^{\frac {5}{2}} + 594 \, {\left (5 \, a^{2} b^{3} x^{2} + 7 \, a^{3} b^{2} x\right )} x^{\frac {3}{2}} + 924 \, {\left (3 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x\right )} \sqrt {x} + \frac {1155 \, {\left (a^{4} b x^{2} + 3 \, a^{5} x\right )}}{\sqrt {x}}\right )} A + \frac {2}{45045} \, {\left (315 \, {\left (11 \, b^{5} x^{2} + 13 \, a b^{4} x\right )} x^{\frac {9}{2}} + 1820 \, {\left (9 \, a b^{4} x^{2} + 11 \, a^{2} b^{3} x\right )} x^{\frac {7}{2}} + 4290 \, {\left (7 \, a^{2} b^{3} x^{2} + 9 \, a^{3} b^{2} x\right )} x^{\frac {5}{2}} + 5148 \, {\left (5 \, a^{3} b^{2} x^{2} + 7 \, a^{4} b x\right )} x^{\frac {3}{2}} + 3003 \, {\left (3 \, a^{4} b x^{2} + 5 \, a^{5} x\right )} \sqrt {x}\right )} B \]
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 \[ \int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{5/2}}{\sqrt {x}} \,d x \]
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 \[ \int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {5}{2}}}{\sqrt {x}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
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