Integrand size = 20, antiderivative size = 152 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \]
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Time = 0.05 (sec) , antiderivative size = 152, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {79, 49, 52, 65, 223, 212} \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\frac {5}{4} a \sqrt {b} (3 a B+4 A b) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right )-\frac {2 (a+b x)^{5/2} (3 a B+4 A b)}{3 a \sqrt {x}}+\frac {5 b \sqrt {x} (a+b x)^{3/2} (3 a B+4 A b)}{6 a}+\frac {5}{4} b \sqrt {x} \sqrt {a+b x} (3 a B+4 A b)-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}} \]
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Rule 49
Rule 52
Rule 65
Rule 79
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {\left (2 \left (2 A b+\frac {3 a B}{2}\right )\right ) \int \frac {(a+b x)^{5/2}}{x^{3/2}} \, dx}{3 a} \\ & = -\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {(5 b (4 A b+3 a B)) \int \frac {(a+b x)^{3/2}}{\sqrt {x}} \, dx}{3 a} \\ & = \frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 b (4 A b+3 a B)) \int \frac {\sqrt {a+b x}}{\sqrt {x}} \, dx \\ & = \frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{8} (5 a b (4 A b+3 a B)) \int \frac {1}{\sqrt {x} \sqrt {a+b x}} \, dx \\ & = \frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 a b (4 A b+3 a B)) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x^2}} \, dx,x,\sqrt {x}\right ) \\ & = \frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {1}{4} (5 a b (4 A b+3 a B)) \text {Subst}\left (\int \frac {1}{1-b x^2} \, dx,x,\frac {\sqrt {x}}{\sqrt {a+b x}}\right ) \\ & = \frac {5}{4} b (4 A b+3 a B) \sqrt {x} \sqrt {a+b x}+\frac {5 b (4 A b+3 a B) \sqrt {x} (a+b x)^{3/2}}{6 a}-\frac {2 (4 A b+3 a B) (a+b x)^{5/2}}{3 a \sqrt {x}}-\frac {2 A (a+b x)^{7/2}}{3 a x^{3/2}}+\frac {5}{4} a \sqrt {b} (4 A b+3 a B) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a+b x}}\right ) \\ \end{align*}
Time = 0.34 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.71 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (6 b^2 x^2 (2 A+B x)-8 a^2 (A+3 B x)+a b x (-56 A+27 B x)\right )}{12 x^{3/2}}+\frac {5}{2} a \sqrt {b} (4 A b+3 a B) \text {arctanh}\left (\frac {\sqrt {b} \sqrt {x}}{-\sqrt {a}+\sqrt {a+b x}}\right ) \]
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Time = 1.41 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.78
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (-6 b^{2} B \,x^{3}-12 A \,b^{2} x^{2}-27 B a b \,x^{2}+56 a A b x +24 a^{2} B x +8 a^{2} A \right )}{12 x^{\frac {3}{2}}}+\frac {5 a \sqrt {b}\, \left (4 A b +3 B a \right ) \ln \left (\frac {\frac {a}{2}+b x}{\sqrt {b}}+\sqrt {b \,x^{2}+a x}\right ) \sqrt {x \left (b x +a \right )}}{8 \sqrt {x}\, \sqrt {b x +a}}\) | \(118\) |
default | \(\frac {\sqrt {b x +a}\, \left (12 B \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, x^{3}+60 A \,b^{2} \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a \,x^{2}+24 A \,b^{\frac {5}{2}} \sqrt {x \left (b x +a \right )}\, x^{2}+45 B b \ln \left (\frac {2 \sqrt {x \left (b x +a \right )}\, \sqrt {b}+2 b x +a}{2 \sqrt {b}}\right ) a^{2} x^{2}+54 B \,b^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, a \,x^{2}-112 A a \,b^{\frac {3}{2}} x \sqrt {x \left (b x +a \right )}-48 B \,a^{2} x \sqrt {x \left (b x +a \right )}\, \sqrt {b}-16 A \,a^{2} \sqrt {x \left (b x +a \right )}\, \sqrt {b}\right )}{24 x^{\frac {3}{2}} \sqrt {x \left (b x +a \right )}\, \sqrt {b}}\) | \(207\) |
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Time = 0.24 (sec) , antiderivative size = 217, normalized size of antiderivative = 1.43 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\left [\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {b} x^{2} \log \left (2 \, b x + 2 \, \sqrt {b x + a} \sqrt {b} \sqrt {x} + a\right ) + 2 \, {\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{24 \, x^{2}}, -\frac {15 \, {\left (3 \, B a^{2} + 4 \, A a b\right )} \sqrt {-b} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-b}}{b \sqrt {x}}\right ) - {\left (6 \, B b^{2} x^{3} - 8 \, A a^{2} + 3 \, {\left (9 \, B a b + 4 \, A b^{2}\right )} x^{2} - 8 \, {\left (3 \, B a^{2} + 7 \, A a b\right )} x\right )} \sqrt {b x + a} \sqrt {x}}{12 \, x^{2}}\right ] \]
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Time = 6.08 (sec) , antiderivative size = 364, normalized size of antiderivative = 2.39 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=- \frac {4 A a^{\frac {3}{2}} b}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + A \sqrt {a} b^{2} \sqrt {x} \sqrt {1 + \frac {b x}{a}} - \frac {4 A \sqrt {a} b^{2} \sqrt {x}}{\sqrt {1 + \frac {b x}{a}}} - \frac {2 A a^{2} \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{3 x} - \frac {2 A a b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{3} + 5 A a b^{\frac {3}{2}} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} - \frac {2 B a^{\frac {5}{2}}}{\sqrt {x} \sqrt {1 + \frac {b x}{a}}} + 2 B a^{\frac {3}{2}} b \sqrt {x} \sqrt {1 + \frac {b x}{a}} - \frac {2 B a^{\frac {3}{2}} b \sqrt {x}}{\sqrt {1 + \frac {b x}{a}}} + 4 B a^{2} \sqrt {b} \operatorname {asinh}{\left (\frac {\sqrt {b} \sqrt {x}}{\sqrt {a}} \right )} + 2 B b^{2} \left (\begin {cases} - \frac {a^{2} \left (\begin {cases} \frac {\log {\left (2 \sqrt {b} \sqrt {a + b x} + 2 b \sqrt {x} \right )}}{\sqrt {b}} & \text {for}\: a \neq 0 \\\frac {\sqrt {x} \log {\left (\sqrt {x} \right )}}{\sqrt {b x}} & \text {otherwise} \end {cases}\right )}{8 b} + \frac {a \sqrt {x} \sqrt {a + b x}}{8 b} + \frac {x^{\frac {3}{2}} \sqrt {a + b x}}{4} & \text {for}\: b \neq 0 \\\frac {\sqrt {a} x^{\frac {3}{2}}}{3} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.20 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.26 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\frac {15}{8} \, B a^{2} \sqrt {b} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) + \frac {5}{2} \, A a b^{\frac {3}{2}} \log \left (2 \, b x + a + 2 \, \sqrt {b x^{2} + a x} \sqrt {b}\right ) - \frac {15 \, \sqrt {b x^{2} + a x} B a^{2}}{4 \, x} - \frac {35 \, \sqrt {b x^{2} + a x} A a b}{6 \, x} + \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} B a}{4 \, x^{2}} - \frac {5 \, \sqrt {b x^{2} + a x} A a^{2}}{6 \, x^{2}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} B}{2 \, x^{3}} - \frac {5 \, {\left (b x^{2} + a x\right )}^{\frac {3}{2}} A a}{6 \, x^{3}} + \frac {{\left (b x^{2} + a x\right )}^{\frac {5}{2}} A}{x^{4}} \]
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Time = 76.94 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.22 \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=-\frac {{\left (\frac {15 \, {\left (3 \, B a^{2} b + 4 \, A a b^{2}\right )} \log \left ({\left | -\sqrt {b x + a} \sqrt {b} + \sqrt {{\left (b x + a\right )} b - a b} \right |}\right )}{\sqrt {b}} - \frac {{\left ({\left (3 \, {\left (2 \, {\left (b x + a\right )} B b^{2} + \frac {3 \, B a^{2} b^{3} + 4 \, A a b^{4}}{a b}\right )} {\left (b x + a\right )} - \frac {20 \, {\left (3 \, B a^{3} b^{3} + 4 \, A a^{2} b^{4}\right )}}{a b}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (3 \, B a^{4} b^{3} + 4 \, A a^{3} b^{4}\right )}}{a b}\right )} \sqrt {b x + a}}{{\left ({\left (b x + a\right )} b - a b\right )}^{\frac {3}{2}}}\right )} b}{12 \, {\left | b \right |}} \]
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Timed out. \[ \int \frac {(a+b x)^{5/2} (A+B x)}{x^{5/2}} \, dx=\int \frac {\left (A+B\,x\right )\,{\left (a+b\,x\right )}^{5/2}}{x^{5/2}} \,d x \]
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