Integrand size = 31, antiderivative size = 220 \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 a^3 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^2 (3 A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {6 a b (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b^2 (A b+3 a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b^3 B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.065, Rules used = {784, 77} \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 b^2 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{9 (a+b x)}+\frac {6 a b x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{7 (a+b x)}+\frac {2 a^2 x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{5 (a+b x)}+\frac {2 b^3 B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 a^3 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)} \]
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Rule 77
Rule 784
Rubi steps \begin{align*} \text {integral}& = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \sqrt {x} \left (a b+b^2 x\right )^3 (A+B x) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (a^3 A b^3 \sqrt {x}+a^2 b^3 (3 A b+a B) x^{3/2}+3 a b^4 (A b+a B) x^{5/2}+b^5 (A b+3 a B) x^{7/2}+b^6 B x^{9/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {2 a^3 A x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 a^2 (3 A b+a B) x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac {6 a b (A b+a B) x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 b^2 (A b+3 a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {2 b^3 B x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.40 \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 x^{3/2} \sqrt {(a+b x)^2} \left (231 a^3 (5 A+3 B x)+297 a^2 b x (7 A+5 B x)+165 a b^2 x^2 (9 A+7 B x)+35 b^3 x^3 (11 A+9 B x)\right )}{3465 (a+b x)} \]
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Time = 0.13 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {2 x^{\frac {3}{2}} \left (315 x^{4} B \,b^{3}+385 A \,b^{3} x^{3}+1155 B a \,b^{2} x^{3}+1485 A a \,b^{2} x^{2}+1485 B \,a^{2} b \,x^{2}+2079 A \,a^{2} b x +693 a^{3} B x +1155 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3465 \left (b x +a \right )^{3}}\) | \(92\) |
default | \(\frac {2 x^{\frac {3}{2}} \left (315 x^{4} B \,b^{3}+385 A \,b^{3} x^{3}+1155 B a \,b^{2} x^{3}+1485 A a \,b^{2} x^{2}+1485 B \,a^{2} b \,x^{2}+2079 A \,a^{2} b x +693 a^{3} B x +1155 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{3465 \left (b x +a \right )^{3}}\) | \(92\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {3}{2}} \left (315 x^{4} B \,b^{3}+385 A \,b^{3} x^{3}+1155 B a \,b^{2} x^{3}+1485 A a \,b^{2} x^{2}+1485 B \,a^{2} b \,x^{2}+2079 A \,a^{2} b x +693 a^{3} B x +1155 A \,a^{3}\right )}{3465 \left (b x +a \right )}\) | \(92\) |
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Time = 0.40 (sec) , antiderivative size = 76, normalized size of antiderivative = 0.35 \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{3465} \, {\left (315 \, B b^{3} x^{5} + 1155 \, A a^{3} x + 385 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{4} + 1485 \, {\left (B a^{2} b + A a b^{2}\right )} x^{3} + 693 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{2}\right )} \sqrt {x} \]
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\[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \sqrt {x} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.22 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.62 \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{315} \, {\left (5 \, {\left (7 \, b^{3} x^{2} + 9 \, a b^{2} x\right )} x^{\frac {5}{2}} + 18 \, {\left (5 \, a b^{2} x^{2} + 7 \, a^{2} b x\right )} x^{\frac {3}{2}} + 21 \, {\left (3 \, a^{2} b x^{2} + 5 \, a^{3} x\right )} \sqrt {x}\right )} A + \frac {2}{3465} \, {\left (35 \, {\left (9 \, b^{3} x^{2} + 11 \, a b^{2} x\right )} x^{\frac {7}{2}} + 110 \, {\left (7 \, a b^{2} x^{2} + 9 \, a^{2} b x\right )} x^{\frac {5}{2}} + 99 \, {\left (5 \, a^{2} b x^{2} + 7 \, a^{3} x\right )} x^{\frac {3}{2}}\right )} B \]
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Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.57 \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{11} \, B b^{3} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, B a b^{2} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, A b^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, B a^{2} b x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{7} \, A a b^{2} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{5} \, B a^{3} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{5} \, A a^{2} b x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A a^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) \]
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Timed out. \[ \int \sqrt {x} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int \sqrt {x}\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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