Integrand size = 31, antiderivative size = 220 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 a^3 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 a^2 (3 A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {6 a b (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b^2 (A b+3 a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 b^3 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \]
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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} \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 b^2 x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{13 (a+b x)}+\frac {6 a b x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{11 (a+b x)}+\frac {2 a^2 x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{9 (a+b x)}+\frac {2 b^3 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)}+\frac {2 a^3 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)} \]
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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 x^{5/2} \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 x^{5/2}+a^2 b^3 (3 A b+a B) x^{7/2}+3 a b^4 (A b+a B) x^{9/2}+b^5 (A b+3 a B) x^{11/2}+b^6 B x^{13/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {2 a^3 A x^{7/2} \sqrt {a^2+2 a b x+b^2 x^2}}{7 (a+b x)}+\frac {2 a^2 (3 A b+a B) x^{9/2} \sqrt {a^2+2 a b x+b^2 x^2}}{9 (a+b x)}+\frac {6 a b (A b+a B) x^{11/2} \sqrt {a^2+2 a b x+b^2 x^2}}{11 (a+b x)}+\frac {2 b^2 (A b+3 a B) x^{13/2} \sqrt {a^2+2 a b x+b^2 x^2}}{13 (a+b x)}+\frac {2 b^3 B x^{15/2} \sqrt {a^2+2 a b x+b^2 x^2}}{15 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.40 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2 x^{7/2} \sqrt {(a+b x)^2} \left (715 a^3 (9 A+7 B x)+1365 a^2 b x (11 A+9 B x)+945 a b^2 x^2 (13 A+11 B x)+231 b^3 x^3 (15 A+13 B x)\right )}{45045 (a+b x)} \]
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Time = 0.15 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.42
method | result | size |
gosper | \(\frac {2 x^{\frac {7}{2}} \left (3003 x^{4} B \,b^{3}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 A a \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 A \,a^{2} b x +5005 a^{3} B x +6435 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 \left (b x +a \right )^{3}}\) | \(92\) |
default | \(\frac {2 x^{\frac {7}{2}} \left (3003 x^{4} B \,b^{3}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 A a \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 A \,a^{2} b x +5005 a^{3} B x +6435 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{45045 \left (b x +a \right )^{3}}\) | \(92\) |
risch | \(\frac {2 \sqrt {\left (b x +a \right )^{2}}\, x^{\frac {7}{2}} \left (3003 x^{4} B \,b^{3}+3465 A \,b^{3} x^{3}+10395 B a \,b^{2} x^{3}+12285 A a \,b^{2} x^{2}+12285 B \,a^{2} b \,x^{2}+15015 A \,a^{2} b x +5005 a^{3} B x +6435 A \,a^{3}\right )}{45045 \left (b x +a \right )}\) | \(92\) |
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Time = 0.39 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.35 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{45045} \, {\left (3003 \, B b^{3} x^{7} + 6435 \, A a^{3} x^{3} + 3465 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{6} + 12285 \, {\left (B a^{2} b + A a b^{2}\right )} x^{5} + 5005 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x^{4}\right )} \sqrt {x} \]
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\[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int x^{\frac {5}{2}} \left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}\, dx \]
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Time = 0.21 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.62 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{9009} \, {\left (63 \, {\left (11 \, b^{3} x^{2} + 13 \, a b^{2} x\right )} x^{\frac {9}{2}} + 182 \, {\left (9 \, a b^{2} x^{2} + 11 \, a^{2} b x\right )} x^{\frac {7}{2}} + 143 \, {\left (7 \, a^{2} b x^{2} + 9 \, a^{3} x\right )} x^{\frac {5}{2}}\right )} A + \frac {2}{6435} \, {\left (33 \, {\left (13 \, b^{3} x^{2} + 15 \, a b^{2} x\right )} x^{\frac {11}{2}} + 90 \, {\left (11 \, a b^{2} x^{2} + 13 \, a^{2} b x\right )} x^{\frac {9}{2}} + 65 \, {\left (9 \, a^{2} b x^{2} + 11 \, a^{3} x\right )} x^{\frac {7}{2}}\right )} B \]
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Time = 0.27 (sec) , antiderivative size = 125, normalized size of antiderivative = 0.57 \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\frac {2}{15} \, B b^{3} x^{\frac {15}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{13} \, B a b^{2} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{13} \, A b^{3} x^{\frac {13}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{11} \, B a^{2} b x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {6}{11} \, A a b^{2} x^{\frac {11}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{9} \, B a^{3} x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A a^{2} b x^{\frac {9}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{7} \, A a^{3} x^{\frac {7}{2}} \mathrm {sgn}\left (b x + a\right ) \]
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Timed out. \[ \int x^{5/2} (A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2} \, dx=\int x^{5/2}\,\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2} \,d x \]
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