Integrand size = 31, antiderivative size = 216 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=-\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {6 a b (A b+a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^2 (A b+3 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \]
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Time = 0.06 (sec) , antiderivative size = 216, 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} \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=-\frac {2 a^2 \sqrt {a^2+2 a b x+b^2 x^2} (a B+3 A b)}{\sqrt {x} (a+b x)}+\frac {6 a b \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2} (a B+A b)}{a+b x}+\frac {2 b^2 x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2} (3 a B+A b)}{3 (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)}-\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)} \]
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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 \frac {\left (a b+b^2 x\right )^3 (A+B x)}{x^{5/2}} \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = \frac {\sqrt {a^2+2 a b x+b^2 x^2} \int \left (\frac {a^3 A b^3}{x^{5/2}}+\frac {a^2 b^3 (3 A b+a B)}{x^{3/2}}+\frac {3 a b^4 (A b+a B)}{\sqrt {x}}+b^5 (A b+3 a B) \sqrt {x}+b^6 B x^{3/2}\right ) \, dx}{b^2 \left (a b+b^2 x\right )} \\ & = -\frac {2 a^3 A \sqrt {a^2+2 a b x+b^2 x^2}}{3 x^{3/2} (a+b x)}-\frac {2 a^2 (3 A b+a B) \sqrt {a^2+2 a b x+b^2 x^2}}{\sqrt {x} (a+b x)}+\frac {6 a b (A b+a B) \sqrt {x} \sqrt {a^2+2 a b x+b^2 x^2}}{a+b x}+\frac {2 b^2 (A b+3 a B) x^{3/2} \sqrt {a^2+2 a b x+b^2 x^2}}{3 (a+b x)}+\frac {2 b^3 B x^{5/2} \sqrt {a^2+2 a b x+b^2 x^2}}{5 (a+b x)} \\ \end{align*}
Time = 0.02 (sec) , antiderivative size = 84, normalized size of antiderivative = 0.39 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=-\frac {2 \sqrt {(a+b x)^2} \left (45 a^2 b x (A-B x)-15 a b^2 x^2 (3 A+B x)+5 a^3 (A+3 B x)-b^3 x^3 (5 A+3 B x)\right )}{15 x^{3/2} (a+b x)} \]
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Time = 0.14 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.43
method | result | size |
gosper | \(-\frac {2 \left (-3 x^{4} B \,b^{3}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 A a \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 A \,a^{2} b x +15 a^{3} B x +5 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15 x^{\frac {3}{2}} \left (b x +a \right )^{3}}\) | \(92\) |
default | \(-\frac {2 \left (-3 x^{4} B \,b^{3}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 A a \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 A \,a^{2} b x +15 a^{3} B x +5 A \,a^{3}\right ) \left (\left (b x +a \right )^{2}\right )^{\frac {3}{2}}}{15 x^{\frac {3}{2}} \left (b x +a \right )^{3}}\) | \(92\) |
risch | \(-\frac {2 \sqrt {\left (b x +a \right )^{2}}\, \left (-3 x^{4} B \,b^{3}-5 A \,b^{3} x^{3}-15 B a \,b^{2} x^{3}-45 A a \,b^{2} x^{2}-45 B \,a^{2} b \,x^{2}+45 A \,a^{2} b x +15 a^{3} B x +5 A \,a^{3}\right )}{15 \left (b x +a \right ) x^{\frac {3}{2}}}\) | \(92\) |
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Time = 0.29 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.34 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2 \, {\left (3 \, B b^{3} x^{4} - 5 \, A a^{3} + 5 \, {\left (3 \, B a b^{2} + A b^{3}\right )} x^{3} + 45 \, {\left (B a^{2} b + A a b^{2}\right )} x^{2} - 15 \, {\left (B a^{3} + 3 \, A a^{2} b\right )} x\right )}}{15 \, x^{\frac {3}{2}}} \]
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\[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=\int \frac {\left (A + B x\right ) \left (\left (a + b x\right )^{2}\right )^{\frac {3}{2}}}{x^{\frac {5}{2}}}\, dx \]
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Time = 0.20 (sec) , antiderivative size = 130, normalized size of antiderivative = 0.60 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2}{15} \, {\left ({\left (3 \, b^{3} x^{2} + 5 \, a b^{2} x\right )} \sqrt {x} + \frac {10 \, {\left (a b^{2} x^{2} + 3 \, a^{2} b x\right )}}{\sqrt {x}} + \frac {15 \, {\left (a^{2} b x^{2} - a^{3} x\right )}}{x^{\frac {3}{2}}}\right )} B + \frac {2}{3} \, A {\left (\frac {b^{3} x^{2} + 3 \, a b^{2} x}{\sqrt {x}} + \frac {6 \, {\left (a b^{2} x^{2} - a^{2} b x\right )}}{x^{\frac {3}{2}}} - \frac {3 \, a^{2} b x^{2} + a^{3} x}{x^{\frac {5}{2}}}\right )} \]
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Time = 0.27 (sec) , antiderivative size = 123, normalized size of antiderivative = 0.57 \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=\frac {2}{5} \, B b^{3} x^{\frac {5}{2}} \mathrm {sgn}\left (b x + a\right ) + 2 \, B a b^{2} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + \frac {2}{3} \, A b^{3} x^{\frac {3}{2}} \mathrm {sgn}\left (b x + a\right ) + 6 \, B a^{2} b \sqrt {x} \mathrm {sgn}\left (b x + a\right ) + 6 \, A a b^{2} \sqrt {x} \mathrm {sgn}\left (b x + a\right ) - \frac {2 \, {\left (3 \, B a^{3} x \mathrm {sgn}\left (b x + a\right ) + 9 \, A a^{2} b x \mathrm {sgn}\left (b x + a\right ) + A a^{3} \mathrm {sgn}\left (b x + a\right )\right )}}{3 \, x^{\frac {3}{2}}} \]
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Timed out. \[ \int \frac {(A+B x) \left (a^2+2 a b x+b^2 x^2\right )^{3/2}}{x^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a^2+2\,a\,b\,x+b^2\,x^2\right )}^{3/2}}{x^{5/2}} \,d x \]
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