Integrand size = 18, antiderivative size = 107 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=\frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}-\frac {3 b (A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Time = 0.03 (sec) , antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 43, 52, 65, 214} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=-\frac {3 b (4 a B+A b) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}}-\frac {(a+b x)^{3/2} (4 a B+A b)}{4 a x}+\frac {3 b \sqrt {a+b x} (4 a B+A b)}{4 a}-\frac {A (a+b x)^{5/2}}{2 a x^2} \]
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Rule 43
Rule 52
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {\left (\frac {A b}{2}+2 a B\right ) \int \frac {(a+b x)^{3/2}}{x^2} \, dx}{2 a} \\ & = -\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {(3 b (A b+4 a B)) \int \frac {\sqrt {a+b x}}{x} \, dx}{8 a} \\ & = \frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {1}{8} (3 b (A b+4 a B)) \int \frac {1}{x \sqrt {a+b x}} \, dx \\ & = \frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}+\frac {1}{4} (3 (A b+4 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right ) \\ & = \frac {3 b (A b+4 a B) \sqrt {a+b x}}{4 a}-\frac {(A b+4 a B) (a+b x)^{3/2}}{4 a x}-\frac {A (a+b x)^{5/2}}{2 a x^2}-\frac {3 b (A b+4 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}} \\ \end{align*}
Time = 0.15 (sec) , antiderivative size = 72, normalized size of antiderivative = 0.67 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=-\frac {\sqrt {a+b x} (b x (5 A-8 B x)+2 a (A+2 B x))}{4 x^2}-\frac {3 b (A b+4 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{4 \sqrt {a}} \]
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Time = 0.52 (sec) , antiderivative size = 68, normalized size of antiderivative = 0.64
method | result | size |
pseudoelliptic | \(-\frac {3 \left (b \,x^{2} \left (A b +4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )+\frac {5 \sqrt {b x +a}\, \left (\frac {2 \left (2 B x +A \right ) a^{\frac {3}{2}}}{5}+b x \sqrt {a}\, \left (-\frac {8 B x}{5}+A \right )\right )}{3}\right )}{4 \sqrt {a}\, x^{2}}\) | \(68\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (5 A b x +4 B a x +2 A a \right )}{4 x^{2}}+\frac {b \left (16 B \sqrt {b x +a}-\frac {2 \left (3 A b +12 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{8}\) | \(69\) |
derivativedivides | \(2 b \left (B \sqrt {b x +a}-\frac {\left (\frac {5 A b}{8}+\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (-\frac {1}{2} a^{2} B -\frac {3}{8} a b A \right ) \sqrt {b x +a}}{b^{2} x^{2}}-\frac {3 \left (A b +4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\) | \(85\) |
default | \(2 b \left (B \sqrt {b x +a}-\frac {\left (\frac {5 A b}{8}+\frac {B a}{2}\right ) \left (b x +a \right )^{\frac {3}{2}}+\left (-\frac {1}{2} a^{2} B -\frac {3}{8} a b A \right ) \sqrt {b x +a}}{b^{2} x^{2}}-\frac {3 \left (A b +4 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 \sqrt {a}}\right )\) | \(85\) |
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Time = 0.23 (sec) , antiderivative size = 177, normalized size of antiderivative = 1.65 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=\left [\frac {3 \, {\left (4 \, B a b + A b^{2}\right )} \sqrt {a} x^{2} \log \left (\frac {b x - 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, B a b x^{2} - 2 \, A a^{2} - {\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt {b x + a}}{8 \, a x^{2}}, \frac {3 \, {\left (4 \, B a b + A b^{2}\right )} \sqrt {-a} x^{2} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) + {\left (8 \, B a b x^{2} - 2 \, A a^{2} - {\left (4 \, B a^{2} + 5 \, A a b\right )} x\right )} \sqrt {b x + a}}{4 \, a x^{2}}\right ] \]
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Time = 35.71 (sec) , antiderivative size = 228, normalized size of antiderivative = 2.13 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=- \frac {A a^{2}}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 A a \sqrt {b}}{4 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {A b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} - \frac {A b^{\frac {3}{2}}}{4 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {3 A b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 \sqrt {a}} - B \sqrt {a} b \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )} - \frac {B a \sqrt {b} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} + B b \left (\begin {cases} \frac {2 a \operatorname {atan}{\left (\frac {\sqrt {a + b x}}{\sqrt {- a}} \right )}}{\sqrt {- a}} + 2 \sqrt {a + b x} & \text {for}\: b \neq 0 \\\sqrt {a} \log {\left (x \right )} & \text {otherwise} \end {cases}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 130, normalized size of antiderivative = 1.21 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=\frac {1}{8} \, b^{2} {\left (\frac {16 \, \sqrt {b x + a} B}{b} + \frac {3 \, {\left (4 \, B a + A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{\sqrt {a} b} - \frac {2 \, {\left ({\left (4 \, B a + 5 \, A b\right )} {\left (b x + a\right )}^{\frac {3}{2}} - {\left (4 \, B a^{2} + 3 \, A a b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{2} b - 2 \, {\left (b x + a\right )} a b + a^{2} b}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.11 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=\frac {8 \, \sqrt {b x + a} B b^{2} + \frac {3 \, {\left (4 \, B a b^{2} + A b^{3}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} B a b^{2} - 4 \, \sqrt {b x + a} B a^{2} b^{2} + 5 \, {\left (b x + a\right )}^{\frac {3}{2}} A b^{3} - 3 \, \sqrt {b x + a} A a b^{3}}{b^{2} x^{2}}}{4 \, b} \]
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Time = 0.16 (sec) , antiderivative size = 127, normalized size of antiderivative = 1.19 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^3} \, dx=\frac {\left (B\,a^2\,b+\frac {3\,A\,a\,b^2}{4}\right )\,\sqrt {a+b\,x}-\left (\frac {5\,A\,b^2}{4}+B\,a\,b\right )\,{\left (a+b\,x\right )}^{3/2}}{{\left (a+b\,x\right )}^2-2\,a\,\left (a+b\,x\right )+a^2}+2\,B\,b\,\sqrt {a+b\,x}-\frac {3\,b\,\mathrm {atanh}\left (\frac {3\,b\,\left (A\,b+4\,B\,a\right )\,\sqrt {a+b\,x}}{2\,\sqrt {a}\,\left (\frac {3\,A\,b^2}{2}+6\,B\,a\,b\right )}\right )\,\left (A\,b+4\,B\,a\right )}{4\,\sqrt {a}} \]
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