Integrand size = 18, antiderivative size = 112 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=\frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}+\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}} \]
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Time = 0.03 (sec) , antiderivative size = 112, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {79, 43, 65, 214} \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}}+\frac {(a+b x)^{3/2} (A b-6 a B)}{12 a x^2}+\frac {b \sqrt {a+b x} (A b-6 a B)}{8 a x}-\frac {A (a+b x)^{5/2}}{3 a x^3} \]
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Rule 43
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A (a+b x)^{5/2}}{3 a x^3}+\frac {\left (-\frac {A b}{2}+3 a B\right ) \int \frac {(a+b x)^{3/2}}{x^3} \, dx}{3 a} \\ & = \frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {(b (A b-6 a B)) \int \frac {\sqrt {a+b x}}{x^2} \, dx}{8 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {\left (b^2 (A b-6 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}-\frac {(b (A b-6 a B)) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {x^2}{b}} \, dx,x,\sqrt {a+b x}\right )}{8 a} \\ & = \frac {b (A b-6 a B) \sqrt {a+b x}}{8 a x}+\frac {(A b-6 a B) (a+b x)^{3/2}}{12 a x^2}-\frac {A (a+b x)^{5/2}}{3 a x^3}+\frac {b^2 (A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}} \\ \end{align*}
Time = 0.18 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.82 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=-\frac {\sqrt {a+b x} \left (3 A b^2 x^2+4 a^2 (2 A+3 B x)+2 a b x (7 A+15 B x)\right )}{24 a x^3}+\frac {b^2 (A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{3/2}} \]
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Time = 0.52 (sec) , antiderivative size = 82, normalized size of antiderivative = 0.73
method | result | size |
risch | \(-\frac {\sqrt {b x +a}\, \left (3 A \,b^{2} x^{2}+30 B a b \,x^{2}+14 a A b x +12 a^{2} B x +8 a^{2} A \right )}{24 x^{3} a}+\frac {b^{2} \left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8 a^{\frac {3}{2}}}\) | \(82\) |
pseudoelliptic | \(-\frac {-\frac {3 b^{2} x^{3} \left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8}+\left (\frac {7 x \left (\frac {15 B x}{7}+A \right ) b \,a^{\frac {3}{2}}}{4}+\left (\frac {3 B x}{2}+A \right ) a^{\frac {5}{2}}+\frac {3 A \sqrt {a}\, b^{2} x^{2}}{8}\right ) \sqrt {b x +a}}{3 a^{\frac {3}{2}} x^{3}}\) | \(82\) |
derivativedivides | \(2 b^{2} \left (-\frac {\frac {\left (A b +10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a}+\left (\frac {A b}{6}-B a \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {3}{8} a^{2} B -\frac {1}{16} a b A \right ) \sqrt {b x +a}}{b^{3} x^{3}}+\frac {\left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {3}{2}}}\right )\) | \(98\) |
default | \(2 b^{2} \left (-\frac {\frac {\left (A b +10 B a \right ) \left (b x +a \right )^{\frac {5}{2}}}{16 a}+\left (\frac {A b}{6}-B a \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {3}{8} a^{2} B -\frac {1}{16} a b A \right ) \sqrt {b x +a}}{b^{3} x^{3}}+\frac {\left (A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 a^{\frac {3}{2}}}\right )\) | \(98\) |
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Time = 0.24 (sec) , antiderivative size = 210, normalized size of antiderivative = 1.88 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=\left [-\frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {a} x^{3} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, A a^{3} + 3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{48 \, a^{2} x^{3}}, \frac {3 \, {\left (6 \, B a b^{2} - A b^{3}\right )} \sqrt {-a} x^{3} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (8 \, A a^{3} + 3 \, {\left (10 \, B a^{2} b + A a b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{3} + 7 \, A a^{2} b\right )} x\right )} \sqrt {b x + a}}{24 \, a^{2} x^{3}}\right ] \]
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Leaf count of result is larger than twice the leaf count of optimal. 264 vs. \(2 (97) = 194\).
Time = 55.86 (sec) , antiderivative size = 264, normalized size of antiderivative = 2.36 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=- \frac {A a^{2}}{3 \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {11 A a \sqrt {b}}{12 x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {17 A b^{\frac {3}{2}}}{24 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {A b^{\frac {5}{2}}}{8 a \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {A b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {3}{2}}} - \frac {B a^{2}}{2 \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {3 B a \sqrt {b}}{4 x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {B b^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}}{\sqrt {x}} - \frac {B b^{\frac {3}{2}}}{4 \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {3 B b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 \sqrt {a}} \]
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Time = 0.28 (sec) , antiderivative size = 158, normalized size of antiderivative = 1.41 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=-\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (3 \, {\left (10 \, B a + A b\right )} {\left (b x + a\right )}^{\frac {5}{2}} - 8 \, {\left (6 \, B a^{2} - A a b\right )} {\left (b x + a\right )}^{\frac {3}{2}} + 3 \, {\left (6 \, B a^{3} - A a^{2} b\right )} \sqrt {b x + a}\right )}}{{\left (b x + a\right )}^{3} a b - 3 \, {\left (b x + a\right )}^{2} a^{2} b + 3 \, {\left (b x + a\right )} a^{3} b - a^{4} b} - \frac {3 \, {\left (6 \, B a - A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {3}{2}} b}\right )} \]
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Time = 0.30 (sec) , antiderivative size = 145, normalized size of antiderivative = 1.29 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=\frac {\frac {3 \, {\left (6 \, B a b^{3} - A b^{4}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a} - \frac {30 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{3} - 48 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{3} + 18 \, \sqrt {b x + a} B a^{3} b^{3} + 3 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{4} + 8 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{4} - 3 \, \sqrt {b x + a} A a^{2} b^{4}}{a b^{3} x^{3}}}{24 \, b} \]
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Time = 0.48 (sec) , antiderivative size = 138, normalized size of antiderivative = 1.23 \[ \int \frac {(a+b x)^{3/2} (A+B x)}{x^4} \, dx=\frac {\left (\frac {A\,b^3}{3}-2\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{3/2}+\left (\frac {3\,B\,a^2\,b^2}{4}-\frac {A\,a\,b^3}{8}\right )\,\sqrt {a+b\,x}+\frac {\left (A\,b^3+10\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^{5/2}}{8\,a}}{3\,a\,{\left (a+b\,x\right )}^2-3\,a^2\,\left (a+b\,x\right )-{\left (a+b\,x\right )}^3+a^3}+\frac {b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (A\,b-6\,B\,a\right )}{8\,a^{3/2}} \]
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