Integrand size = 18, antiderivative size = 143 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=-\frac {5 b^2 (7 A b-6 a B)}{8 a^4 \sqrt {a+b x}}-\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}+\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{9/2}} \]
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Time = 0.04 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.278, Rules used = {79, 44, 53, 65, 214} \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{9/2}}-\frac {5 b^2 (7 A b-6 a B)}{8 a^4 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {A}{3 a x^3 \sqrt {a+b x}} \]
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Rule 44
Rule 53
Rule 65
Rule 79
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {\left (-\frac {7 A b}{2}+3 a B\right ) \int \frac {1}{x^3 (a+b x)^{3/2}} \, dx}{3 a} \\ & = -\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}+\frac {(5 b (7 A b-6 a B)) \int \frac {1}{x^2 (a+b x)^{3/2}} \, dx}{24 a^2} \\ & = -\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}-\frac {\left (5 b^2 (7 A b-6 a B)\right ) \int \frac {1}{x (a+b x)^{3/2}} \, dx}{16 a^3} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{8 a^4 \sqrt {a+b x}}-\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}-\frac {\left (5 b^2 (7 A b-6 a B)\right ) \int \frac {1}{x \sqrt {a+b x}} \, dx}{16 a^4} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{8 a^4 \sqrt {a+b x}}-\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}-\frac {(5 b (7 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^4} \\ & = -\frac {5 b^2 (7 A b-6 a B)}{8 a^4 \sqrt {a+b x}}-\frac {A}{3 a x^3 \sqrt {a+b x}}+\frac {7 A b-6 a B}{12 a^2 x^2 \sqrt {a+b x}}-\frac {5 b (7 A b-6 a B)}{24 a^3 x \sqrt {a+b x}}+\frac {5 b^2 (7 A b-6 a B) \tanh ^{-1}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{9/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 112, normalized size of antiderivative = 0.78 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=\frac {-105 A b^3 x^3-4 a^3 (2 A+3 B x)+2 a^2 b x (7 A+15 B x)+5 a b^2 x^2 (-7 A+18 B x)}{24 a^4 x^3 \sqrt {a+b x}}+\frac {5 b^2 (7 A b-6 a B) \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )}{8 a^{9/2}} \]
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Time = 0.56 (sec) , antiderivative size = 106, normalized size of antiderivative = 0.74
method | result | size |
pseudoelliptic | \(\frac {\frac {35 x^{3} \left (A b -\frac {6 B a}{7}\right ) \sqrt {b x +a}\, b^{2} \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{8}-\frac {35 x^{2} b^{2} \left (-\frac {18 B x}{7}+A \right ) a^{\frac {3}{2}}}{24}+\frac {7 b x \left (\frac {15 B x}{7}+A \right ) a^{\frac {5}{2}}}{12}+\frac {\left (-3 B x -2 A \right ) a^{\frac {7}{2}}}{6}-\frac {35 A \sqrt {a}\, b^{3} x^{3}}{8}}{a^{\frac {9}{2}} \sqrt {b x +a}\, x^{3}}\) | \(106\) |
risch | \(-\frac {\sqrt {b x +a}\, \left (57 A \,b^{2} x^{2}-42 B a b \,x^{2}-22 a A b x +12 a^{2} B x +8 a^{2} A \right )}{24 a^{4} x^{3}}-\frac {b^{2} \left (-\frac {2 \left (-16 A b +16 B a \right )}{\sqrt {b x +a}}-\frac {2 \left (35 A b -30 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right )}{16 a^{4}}\) | \(107\) |
derivativedivides | \(2 b^{2} \left (\frac {-\frac {\left (\frac {19 A b}{16}-\frac {7 B a}{8}\right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {17}{6} a b A +2 a^{2} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {29}{16} a^{2} b A -\frac {9}{8} a^{3} B \right ) \sqrt {b x +a}}{b^{3} x^{3}}+\frac {5 \left (7 A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{4}}-\frac {A b -B a}{a^{4} \sqrt {b x +a}}\right )\) | \(126\) |
default | \(2 b^{2} \left (\frac {-\frac {\left (\frac {19 A b}{16}-\frac {7 B a}{8}\right ) \left (b x +a \right )^{\frac {5}{2}}+\left (-\frac {17}{6} a b A +2 a^{2} B \right ) \left (b x +a \right )^{\frac {3}{2}}+\left (\frac {29}{16} a^{2} b A -\frac {9}{8} a^{3} B \right ) \sqrt {b x +a}}{b^{3} x^{3}}+\frac {5 \left (7 A b -6 B a \right ) \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{16 \sqrt {a}}}{a^{4}}-\frac {A b -B a}{a^{4} \sqrt {b x +a}}\right )\) | \(126\) |
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Time = 0.25 (sec) , antiderivative size = 330, normalized size of antiderivative = 2.31 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=\left [-\frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3}\right )} \sqrt {a} \log \left (\frac {b x + 2 \, \sqrt {b x + a} \sqrt {a} + 2 \, a}{x}\right ) + 2 \, {\left (8 \, A a^{4} - 15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{48 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}}, \frac {15 \, {\left ({\left (6 \, B a b^{3} - 7 \, A b^{4}\right )} x^{4} + {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b x + a} \sqrt {-a}}{a}\right ) - {\left (8 \, A a^{4} - 15 \, {\left (6 \, B a^{2} b^{2} - 7 \, A a b^{3}\right )} x^{3} - 5 \, {\left (6 \, B a^{3} b - 7 \, A a^{2} b^{2}\right )} x^{2} + 2 \, {\left (6 \, B a^{4} - 7 \, A a^{3} b\right )} x\right )} \sqrt {b x + a}}{24 \, {\left (a^{5} b x^{4} + a^{6} x^{3}\right )}}\right ] \]
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Time = 77.13 (sec) , antiderivative size = 246, normalized size of antiderivative = 1.72 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=A \left (- \frac {1}{3 a \sqrt {b} x^{\frac {7}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {7 \sqrt {b}}{12 a^{2} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {3}{2}}}{24 a^{3} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} - \frac {35 b^{\frac {5}{2}}}{8 a^{4} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} + \frac {35 b^{3} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{8 a^{\frac {9}{2}}}\right ) + B \left (- \frac {1}{2 a \sqrt {b} x^{\frac {5}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {5 \sqrt {b}}{4 a^{2} x^{\frac {3}{2}} \sqrt {\frac {a}{b x} + 1}} + \frac {15 b^{\frac {3}{2}}}{4 a^{3} \sqrt {x} \sqrt {\frac {a}{b x} + 1}} - \frac {15 b^{2} \operatorname {asinh}{\left (\frac {\sqrt {a}}{\sqrt {b} \sqrt {x}} \right )}}{4 a^{\frac {7}{2}}}\right ) \]
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Time = 0.29 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.27 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=-\frac {1}{48} \, b^{3} {\left (\frac {2 \, {\left (48 \, B a^{4} - 48 \, A a^{3} b - 15 \, {\left (6 \, B a - 7 \, A b\right )} {\left (b x + a\right )}^{3} + 40 \, {\left (6 \, B a^{2} - 7 \, A a b\right )} {\left (b x + a\right )}^{2} - 33 \, {\left (6 \, B a^{3} - 7 \, A a^{2} b\right )} {\left (b x + a\right )}\right )}}{{\left (b x + a\right )}^{\frac {7}{2}} a^{4} b - 3 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{5} b + 3 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{6} b - \sqrt {b x + a} a^{7} b} - \frac {15 \, {\left (6 \, B a - 7 \, A b\right )} \log \left (\frac {\sqrt {b x + a} - \sqrt {a}}{\sqrt {b x + a} + \sqrt {a}}\right )}{a^{\frac {9}{2}} b}\right )} \]
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Time = 0.28 (sec) , antiderivative size = 165, normalized size of antiderivative = 1.15 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=\frac {5 \, {\left (6 \, B a b^{2} - 7 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{8 \, \sqrt {-a} a^{4}} + \frac {2 \, {\left (B a b^{2} - A b^{3}\right )}}{\sqrt {b x + a} a^{4}} + \frac {42 \, {\left (b x + a\right )}^{\frac {5}{2}} B a b^{2} - 96 \, {\left (b x + a\right )}^{\frac {3}{2}} B a^{2} b^{2} + 54 \, \sqrt {b x + a} B a^{3} b^{2} - 57 \, {\left (b x + a\right )}^{\frac {5}{2}} A b^{3} + 136 \, {\left (b x + a\right )}^{\frac {3}{2}} A a b^{3} - 87 \, \sqrt {b x + a} A a^{2} b^{3}}{24 \, a^{4} b^{3} x^{3}} \]
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Time = 0.12 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.20 \[ \int \frac {A+B x}{x^4 (a+b x)^{3/2}} \, dx=\frac {5\,b^2\,\mathrm {atanh}\left (\frac {\sqrt {a+b\,x}}{\sqrt {a}}\right )\,\left (7\,A\,b-6\,B\,a\right )}{8\,a^{9/2}}-\frac {\frac {2\,\left (A\,b^3-B\,a\,b^2\right )}{a}-\frac {11\,\left (7\,A\,b^3-6\,B\,a\,b^2\right )\,\left (a+b\,x\right )}{8\,a^2}+\frac {5\,\left (7\,A\,b^3-6\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^2}{3\,a^3}-\frac {5\,\left (7\,A\,b^3-6\,B\,a\,b^2\right )\,{\left (a+b\,x\right )}^3}{8\,a^4}}{3\,a\,{\left (a+b\,x\right )}^{5/2}-{\left (a+b\,x\right )}^{7/2}+a^3\,\sqrt {a+b\,x}-3\,a^2\,{\left (a+b\,x\right )}^{3/2}} \]
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