Integrand size = 15, antiderivative size = 91 \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\frac {\frac {15 b^2}{4 a^3}-\frac {1}{2 a x^2}+\frac {5 b}{4 a^2 x}}{\sqrt {a+b x}}+\frac {15 b^2 \log \left (\frac {-\sqrt {a}+\sqrt {a+b x}}{\sqrt {a}+\sqrt {a+b x}}\right )}{8 a^{5/2}} \]
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Time = 0.07 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.38, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1973, 44, 53, 65, 214} \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\frac {15 b^2 (a+b x)}{4 a^3 \sqrt {(a+b x)^3}}-\frac {15 b^2 \left (\frac {b x}{a}+1\right )^{3/2} \text {arctanh}\left (\sqrt {\frac {b x}{a}+1}\right )}{4 a^2 \sqrt {(a+b x)^3}}+\frac {5 b (a+b x)}{4 a^2 x \sqrt {(a+b x)^3}}-\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rule 1973
Rubi steps \begin{align*} \text {integral}& = \frac {\left (1+\frac {b x}{a}\right )^{3/2} \int \frac {1}{x^3 \left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{\sqrt {(a+b x)^3}} \\ & = -\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}}-\frac {\left (5 b \left (1+\frac {b x}{a}\right )^{3/2}\right ) \int \frac {1}{x^2 \left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{4 a \sqrt {(a+b x)^3}} \\ & = -\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}}+\frac {5 b (a+b x)}{4 a^2 x \sqrt {(a+b x)^3}}+\frac {\left (15 b^2 \left (1+\frac {b x}{a}\right )^{3/2}\right ) \int \frac {1}{x \left (1+\frac {b x}{a}\right )^{3/2}} \, dx}{8 a^2 \sqrt {(a+b x)^3}} \\ & = \frac {15 b^2 (a+b x)}{4 a^3 \sqrt {(a+b x)^3}}-\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}}+\frac {5 b (a+b x)}{4 a^2 x \sqrt {(a+b x)^3}}+\frac {\left (15 b^2 \left (1+\frac {b x}{a}\right )^{3/2}\right ) \int \frac {1}{x \sqrt {1+\frac {b x}{a}}} \, dx}{8 a^2 \sqrt {(a+b x)^3}} \\ & = \frac {15 b^2 (a+b x)}{4 a^3 \sqrt {(a+b x)^3}}-\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}}+\frac {5 b (a+b x)}{4 a^2 x \sqrt {(a+b x)^3}}+\frac {\left (15 b \left (1+\frac {b x}{a}\right )^{3/2}\right ) \text {Subst}\left (\int \frac {1}{-\frac {a}{b}+\frac {a x^2}{b}} \, dx,x,\sqrt {1+\frac {b x}{a}}\right )}{4 a \sqrt {(a+b x)^3}} \\ & = \frac {15 b^2 (a+b x)}{4 a^3 \sqrt {(a+b x)^3}}-\frac {a+b x}{2 a x^2 \sqrt {(a+b x)^3}}+\frac {5 b (a+b x)}{4 a^2 x \sqrt {(a+b x)^3}}-\frac {15 b^2 \left (1+\frac {b x}{a}\right )^{3/2} \text {arctanh}\left (\sqrt {1+\frac {b x}{a}}\right )}{4 a^2 \sqrt {(a+b x)^3}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.96 \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=-\frac {(a+b x) \left (\sqrt {a} \left (2 a^2-5 a b x-15 b^2 x^2\right )+15 b^2 x^2 \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{4 a^{7/2} x^2 \sqrt {(a+b x)^3}} \]
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Time = 0.19 (sec) , antiderivative size = 74, normalized size of antiderivative = 0.81
method | result | size |
default | \(-\frac {\left (b x +a \right ) \left (15 \sqrt {b x +a}\, \operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right ) b^{2} x^{2}-5 a^{\frac {3}{2}} b x -15 b^{2} x^{2} \sqrt {a}+2 a^{\frac {5}{2}}\right )}{4 \sqrt {\left (b x +a \right )^{3}}\, a^{\frac {7}{2}} x^{2}}\) | \(74\) |
risch | \(-\frac {\left (b x +a \right )^{2} \left (-7 b x +2 a \right )}{4 a^{3} x^{2} \sqrt {\left (b x +a \right )^{3}}}+\frac {b^{2} \left (\frac {16}{\sqrt {b x +a}}-\frac {30 \,\operatorname {arctanh}\left (\frac {\sqrt {b x +a}}{\sqrt {a}}\right )}{\sqrt {a}}\right ) \left (b x +a \right )^{\frac {3}{2}}}{8 a^{3} \sqrt {\left (b x +a \right )^{3}}}\) | \(85\) |
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Leaf count of result is larger than twice the leaf count of optimal. 164 vs. \(2 (72) = 144\).
Time = 0.26 (sec) , antiderivative size = 347, normalized size of antiderivative = 3.81 \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\left [\frac {15 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {a} \log \left (\frac {b^{2} x^{2} + 3 \, a b x + 2 \, a^{2} - 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {a}}{b x^{2} + a x}\right ) + 2 \, \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )}}{8 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}, \frac {15 \, {\left (b^{4} x^{4} + 2 \, a b^{3} x^{3} + a^{2} b^{2} x^{2}\right )} \sqrt {-a} \arctan \left (\frac {\sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} \sqrt {-a}}{a b x + a^{2}}\right ) + \sqrt {b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}} {\left (15 \, a b^{2} x^{2} + 5 \, a^{2} b x - 2 \, a^{3}\right )}}{4 \, {\left (a^{4} b^{2} x^{4} + 2 \, a^{5} b x^{3} + a^{6} x^{2}\right )}}\right ] \]
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\[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\int \frac {1}{x^{3} \sqrt {\left (a + b x\right )^{3}}}\, dx \]
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\[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\int { \frac {1}{\sqrt {{\left (b x + a\right )}^{3}} x^{3}} \,d x } \]
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none
Time = 0.26 (sec) , antiderivative size = 80, normalized size of antiderivative = 0.88 \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\frac {15 \, b^{2} \arctan \left (\frac {\sqrt {b x + a}}{\sqrt {-a}}\right )}{4 \, \sqrt {-a} a^{3}} + \frac {2 \, b^{2}}{\sqrt {b x + a} a^{3}} + \frac {7 \, {\left (b x + a\right )}^{\frac {3}{2}} b^{2} - 9 \, \sqrt {b x + a} a b^{2}}{4 \, a^{3} b^{2} x^{2}} \]
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Timed out. \[ \int \frac {1}{x^3 \sqrt {(a+b x)^3}} \, dx=\int \frac {1}{x^3\,\sqrt {{\left (a+b\,x\right )}^3}} \,d x \]
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