Integrand size = 19, antiderivative size = 204 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=-\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3} \]
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Time = 0.20 (sec) , antiderivative size = 204, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.368, Rules used = {378, 1412, 837, 815, 649, 212, 266} \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=-\frac {a-b \sqrt {c+d x}}{2 x^2 \left (a^2-b^2 c\right )}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c x \left (a^2-b^2 c\right )^2}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac {b d^2 \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3} \]
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Rule 212
Rule 266
Rule 378
Rule 649
Rule 815
Rule 837
Rule 1412
Rubi steps \begin{align*} \text {integral}& = d^2 \text {Subst}\left (\int \frac {1}{\left (a+b \sqrt {x}\right ) (-c+x)^3} \, dx,x,c+d x\right ) \\ & = \left (2 d^2\right ) \text {Subst}\left (\int \frac {x}{(a+b x) \left (-c+x^2\right )^3} \, dx,x,\sqrt {c+d x}\right ) \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}+\frac {d^2 \text {Subst}\left (\int \frac {-a b c+3 b^2 c x}{(a+b x) \left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )}{2 c \left (a^2-b^2 c\right )} \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \frac {a b c \left (a^2-5 b^2 c\right )+b^2 c \left (a^2+3 b^2 c\right ) x}{(a+b x) \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2} \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}+\frac {d^2 \text {Subst}\left (\int \left (-\frac {8 a b^5 c^2}{\left (a^2-b^2 c\right ) (a+b x)}-\frac {b c \left (-a^4+6 a^2 b^2 c+3 b^4 c^2-8 a b^3 c x\right )}{\left (-a^2+b^2 c\right ) \left (c-x^2\right )}\right ) \, dx,x,\sqrt {c+d x}\right )}{4 c^2 \left (a^2-b^2 c\right )^2} \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}+\frac {\left (b d^2\right ) \text {Subst}\left (\int \frac {-a^4+6 a^2 b^2 c+3 b^4 c^2-8 a b^3 c x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^3} \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac {\left (2 a b^4 d^2\right ) \text {Subst}\left (\int \frac {x}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3}-\frac {\left (b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2\right ) \text {Subst}\left (\int \frac {1}{c-x^2} \, dx,x,\sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^3} \\ & = -\frac {a-b \sqrt {c+d x}}{2 \left (a^2-b^2 c\right ) x^2}-\frac {b d \left (4 a b c-\left (a^2+3 b^2 c\right ) \sqrt {c+d x}\right )}{4 c \left (a^2-b^2 c\right )^2 x}-\frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{4 c^{3/2} \left (a^2-b^2 c\right )^3}+\frac {a b^4 d^2 \log (x)}{\left (a^2-b^2 c\right )^3}-\frac {2 a b^4 d^2 \log \left (a+b \sqrt {c+d x}\right )}{\left (a^2-b^2 c\right )^3} \\ \end{align*}
Time = 0.38 (sec) , antiderivative size = 198, normalized size of antiderivative = 0.97 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=\frac {b \left (a^4-6 a^2 b^2 c-3 b^4 c^2\right ) d^2 x^2 \text {arctanh}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )+\sqrt {c} \left (\left (a^2-b^2 c\right ) \left (2 a^3 c-2 a b^2 c (c-2 d x)+b^3 c (2 c-3 d x) \sqrt {c+d x}-a^2 b \sqrt {c+d x} (2 c+d x)\right )-4 a b^4 c d^2 x^2 \log (-d x)+8 a b^4 c d^2 x^2 \log \left (a+b \sqrt {c+d x}\right )\right )}{4 c^{3/2} \left (-a^2+b^2 c\right )^3 x^2} \]
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Time = 0.09 (sec) , antiderivative size = 227, normalized size of antiderivative = 1.11
method | result | size |
derivativedivides | \(2 d^{2} \left (-\frac {a \,b^{4} \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{3}}-\frac {\frac {-\frac {b \left (-3 b^{4} c^{2}+2 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\left (-\frac {1}{2} a \,b^{4} c +\frac {1}{2} a^{3} b^{2}\right ) \left (d x +c \right )+\left (\frac {3}{4} a^{2} b^{3} c -\frac {1}{8} b \,a^{4}-\frac {5}{8} c^{2} b^{5}\right ) \sqrt {d x +c}+\frac {3 a \,b^{4} c^{2}}{4}-a^{3} b^{2} c +\frac {a^{5}}{4}}{d^{2} x^{2}}+\frac {b \left (-4 a \,b^{3} c \ln \left (-d x \right )+\frac {\left (-3 b^{4} c^{2}-6 a^{2} b^{2} c +a^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{8 c}}{\left (-b^{2} c +a^{2}\right )^{3}}\right )\) | \(227\) |
default | \(2 d^{2} \left (-\frac {a \,b^{4} \ln \left (a +b \sqrt {d x +c}\right )}{\left (-b^{2} c +a^{2}\right )^{3}}-\frac {\frac {-\frac {b \left (-3 b^{4} c^{2}+2 a^{2} b^{2} c +a^{4}\right ) \left (d x +c \right )^{\frac {3}{2}}}{8 c}+\left (-\frac {1}{2} a \,b^{4} c +\frac {1}{2} a^{3} b^{2}\right ) \left (d x +c \right )+\left (\frac {3}{4} a^{2} b^{3} c -\frac {1}{8} b \,a^{4}-\frac {5}{8} c^{2} b^{5}\right ) \sqrt {d x +c}+\frac {3 a \,b^{4} c^{2}}{4}-a^{3} b^{2} c +\frac {a^{5}}{4}}{d^{2} x^{2}}+\frac {b \left (-4 a \,b^{3} c \ln \left (-d x \right )+\frac {\left (-3 b^{4} c^{2}-6 a^{2} b^{2} c +a^{4}\right ) \operatorname {arctanh}\left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{\sqrt {c}}\right )}{8 c}}{\left (-b^{2} c +a^{2}\right )^{3}}\right )\) | \(227\) |
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Time = 0.68 (sec) , antiderivative size = 534, normalized size of antiderivative = 2.62 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=\left [\frac {16 \, a b^{4} c^{2} d^{2} x^{2} \log \left (\sqrt {d x + c} b + a\right ) - 8 \, a b^{4} c^{2} d^{2} x^{2} \log \left (x\right ) + 4 \, a b^{4} c^{4} - 8 \, a^{3} b^{2} c^{3} + 4 \, a^{5} c^{2} + {\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \sqrt {c} d^{2} x^{2} \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) - 8 \, {\left (a b^{4} c^{3} - a^{3} b^{2} c^{2}\right )} d x - 2 \, {\left (2 \, b^{5} c^{4} - 4 \, a^{2} b^{3} c^{3} + 2 \, a^{4} b c^{2} - {\left (3 \, b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} - a^{4} b c\right )} d x\right )} \sqrt {d x + c}}{8 \, {\left (b^{6} c^{5} - 3 \, a^{2} b^{4} c^{4} + 3 \, a^{4} b^{2} c^{3} - a^{6} c^{2}\right )} x^{2}}, \frac {8 \, a b^{4} c^{2} d^{2} x^{2} \log \left (\sqrt {d x + c} b + a\right ) - 4 \, a b^{4} c^{2} d^{2} x^{2} \log \left (x\right ) + 2 \, a b^{4} c^{4} - 4 \, a^{3} b^{2} c^{3} + 2 \, a^{5} c^{2} + {\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \sqrt {-c} d^{2} x^{2} \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) - 4 \, {\left (a b^{4} c^{3} - a^{3} b^{2} c^{2}\right )} d x - {\left (2 \, b^{5} c^{4} - 4 \, a^{2} b^{3} c^{3} + 2 \, a^{4} b c^{2} - {\left (3 \, b^{5} c^{3} - 2 \, a^{2} b^{3} c^{2} - a^{4} b c\right )} d x\right )} \sqrt {d x + c}}{4 \, {\left (b^{6} c^{5} - 3 \, a^{2} b^{4} c^{4} + 3 \, a^{4} b^{2} c^{3} - a^{6} c^{2}\right )} x^{2}}\right ] \]
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\[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=\int \frac {1}{x^{3} \left (a + b \sqrt {c + d x}\right )}\, dx \]
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Time = 0.30 (sec) , antiderivative size = 367, normalized size of antiderivative = 1.80 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=-\frac {1}{8} \, {\left (\frac {8 \, a b^{4} \log \left (d x\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} - \frac {16 \, a b^{4} \log \left (\sqrt {d x + c} b + a\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} - \frac {{\left (3 \, b^{5} c^{2} + 6 \, a^{2} b^{3} c - a^{4} b\right )} \log \left (\frac {\sqrt {d x + c} - \sqrt {c}}{\sqrt {d x + c} + \sqrt {c}}\right )}{{\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt {c}} + \frac {2 \, {\left (4 \, {\left (d x + c\right )} a b^{2} c - 6 \, a b^{2} c^{2} + 2 \, a^{3} c - {\left (3 \, b^{3} c + a^{2} b\right )} {\left (d x + c\right )}^{\frac {3}{2}} + {\left (5 \, b^{3} c^{2} - a^{2} b c\right )} \sqrt {d x + c}\right )}}{b^{4} c^{5} - 2 \, a^{2} b^{2} c^{4} + a^{4} c^{3} + {\left (b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c\right )} {\left (d x + c\right )}^{2} - 2 \, {\left (b^{4} c^{4} - 2 \, a^{2} b^{2} c^{3} + a^{4} c^{2}\right )} {\left (d x + c\right )}}\right )} d^{2} \]
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Time = 0.36 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.84 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=\frac {2 \, a b^{5} d^{2} \log \left ({\left | \sqrt {d x + c} b + a \right |}\right )}{b^{7} c^{3} - 3 \, a^{2} b^{5} c^{2} + 3 \, a^{4} b^{3} c - a^{6} b} - \frac {a b^{4} d^{2} \log \left (d x\right )}{b^{6} c^{3} - 3 \, a^{2} b^{4} c^{2} + 3 \, a^{4} b^{2} c - a^{6}} + \frac {{\left (3 \, b^{5} c^{2} d^{2} + 6 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )} \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right )}{4 \, {\left (b^{6} c^{4} - 3 \, a^{2} b^{4} c^{3} + 3 \, a^{4} b^{2} c^{2} - a^{6} c\right )} \sqrt {-c}} + \frac {6 \, a b^{4} c^{3} d^{2} - 8 \, a^{3} b^{2} c^{2} d^{2} + 2 \, a^{5} c d^{2} + {\left (3 \, b^{5} c^{2} d^{2} - 2 \, a^{2} b^{3} c d^{2} - a^{4} b d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 4 \, {\left (a b^{4} c^{2} d^{2} - a^{3} b^{2} c d^{2}\right )} {\left (d x + c\right )} - {\left (5 \, b^{5} c^{3} d^{2} - 6 \, a^{2} b^{3} c^{2} d^{2} + a^{4} b c d^{2}\right )} \sqrt {d x + c}}{4 \, {\left (b^{2} c - a^{2}\right )}^{3} c d^{2} x^{2}} \]
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Time = 19.57 (sec) , antiderivative size = 1094, normalized size of antiderivative = 5.36 \[ \int \frac {1}{x^3 \left (a+b \sqrt {c+d x}\right )} \, dx=\frac {\ln \left (\frac {b^5\,d^4\,{\left (a^2+3\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {a\,b^4\,d^4\,\left (-a^4+2\,a^2\,b^2\,c+15\,b^4\,c^2\right )}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {b\,d^2\,\sqrt {c^3}\,\left (\frac {b^2\,d^2\,\left (3\,b^2\,c-a^2\right )}{4\,c\,\left (b^2\,c-a^2\right )}+\frac {b^2\,d^2\,\sqrt {c^3}\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (3\,b^4\,c^2-a^4+6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^3}-\frac {a\,b^3\,d^2\,\left (9\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (3\,b^4\,c^2-a^4+6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{8\,c^3\,{\left (b^2\,c-a^2\right )}^3}\right )\,\left (8\,a\,b^4\,c^3\,d^2-a^4\,b\,d^2\,\sqrt {c^3}+3\,b^5\,c^2\,d^2\,\sqrt {c^3}+6\,a^2\,b^3\,c\,d^2\,\sqrt {c^3}\right )}{8\,\left (a^6\,c^3-3\,a^4\,b^2\,c^4+3\,a^2\,b^4\,c^5-b^6\,c^6\right )}-\frac {\frac {a^3\,d^2-3\,a\,b^2\,c\,d^2}{2\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}-\frac {\left (a^2\,b\,d^2+3\,c\,b^3\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{4\,c\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {b\,d^2\,\left (5\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{4\,\left (a^4-2\,a^2\,b^2\,c+b^4\,c^2\right )}+\frac {a\,b^2\,d^2\,\left (c+d\,x\right )}{a^4-2\,a^2\,b^2\,c+b^4\,c^2}}{{\left (c+d\,x\right )}^2-2\,c\,\left (c+d\,x\right )+c^2}+\frac {\ln \left (\frac {b^5\,d^4\,{\left (a^2+3\,c\,b^2\right )}^2\,\sqrt {c+d\,x}}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {a\,b^4\,d^4\,\left (-a^4+2\,a^2\,b^2\,c+15\,b^4\,c^2\right )}{16\,c^2\,{\left (b^2\,c-a^2\right )}^4}-\frac {b\,d^2\,\sqrt {c^3}\,\left (\frac {b^2\,d^2\,\left (3\,b^2\,c-a^2\right )}{4\,c\,\left (b^2\,c-a^2\right )}+\frac {b^2\,d^2\,\sqrt {c^3}\,\left (a^2\,\sqrt {c+d\,x}+4\,a\,b\,c+3\,b^2\,c\,\sqrt {c+d\,x}\right )\,\left (a^4-3\,b^4\,c^2-6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{4\,c^3\,{\left (b^2\,c-a^2\right )}^3}-\frac {a\,b^3\,d^2\,\left (9\,b^2\,c-a^2\right )\,\sqrt {c+d\,x}}{2\,c\,{\left (b^2\,c-a^2\right )}^2}\right )\,\left (a^4-3\,b^4\,c^2-6\,a^2\,b^2\,c+8\,a\,b^3\,\sqrt {c^3}\right )}{8\,c^3\,{\left (b^2\,c-a^2\right )}^3}\right )\,\left (8\,a\,b^4\,c^3\,d^2+a^4\,b\,d^2\,\sqrt {c^3}-3\,b^5\,c^2\,d^2\,\sqrt {c^3}-6\,a^2\,b^3\,c\,d^2\,\sqrt {c^3}\right )}{8\,\left (a^6\,c^3-3\,a^4\,b^2\,c^4+3\,a^2\,b^4\,c^5-b^6\,c^6\right )}+\frac {2\,a\,b^4\,d^2\,\ln \left (a+b\,\sqrt {c+d\,x}\right )}{{\left (b^2\,c-a^2\right )}^3} \]
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