Integrand size = 17, antiderivative size = 173 \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=-\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}+\frac {35 \sqrt {b} d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}} \]
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Time = 0.05 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {44, 53, 65, 214} \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\frac {35 \sqrt {b} d^3 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}}-\frac {35 d^3}{8 \sqrt {c+d x} (b c-a d)^4}-\frac {35 d^2}{24 (a+b x) \sqrt {c+d x} (b c-a d)^3}+\frac {7 d}{12 (a+b x)^2 \sqrt {c+d x} (b c-a d)^2}-\frac {1}{3 (a+b x)^3 \sqrt {c+d x} (b c-a d)} \]
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Rule 44
Rule 53
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}-\frac {(7 d) \int \frac {1}{(a+b x)^3 (c+d x)^{3/2}} \, dx}{6 (b c-a d)} \\ & = -\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}+\frac {\left (35 d^2\right ) \int \frac {1}{(a+b x)^2 (c+d x)^{3/2}} \, dx}{24 (b c-a d)^2} \\ & = -\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 d^3\right ) \int \frac {1}{(a+b x) (c+d x)^{3/2}} \, dx}{16 (b c-a d)^3} \\ & = -\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 b d^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{16 (b c-a d)^4} \\ & = -\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}-\frac {\left (35 b d^2\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{8 (b c-a d)^4} \\ & = -\frac {35 d^3}{8 (b c-a d)^4 \sqrt {c+d x}}-\frac {1}{3 (b c-a d) (a+b x)^3 \sqrt {c+d x}}+\frac {7 d}{12 (b c-a d)^2 (a+b x)^2 \sqrt {c+d x}}-\frac {35 d^2}{24 (b c-a d)^3 (a+b x) \sqrt {c+d x}}+\frac {35 \sqrt {b} d^3 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{8 (b c-a d)^{9/2}} \\ \end{align*}
Time = 0.64 (sec) , antiderivative size = 170, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\frac {-48 a^3 d^3-3 a^2 b d^2 (29 c+77 d x)-2 a b^2 d \left (-19 c^2+49 c d x+140 d^2 x^2\right )-b^3 \left (8 c^3-14 c^2 d x+35 c d^2 x^2+105 d^3 x^3\right )}{24 (b c-a d)^4 (a+b x)^3 \sqrt {c+d x}}-\frac {35 \sqrt {b} d^3 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{8 (-b c+a d)^{9/2}} \]
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Time = 0.34 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90
method | result | size |
derivativedivides | \(2 d^{3} \left (-\frac {1}{\left (a d -b c \right )^{4} \sqrt {d x +c}}-\frac {b \left (\frac {\frac {19 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {17 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {29}{16} a^{2} d^{2}-\frac {29}{8} a b c d +\frac {29}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {35 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\right )\) | \(156\) |
default | \(2 d^{3} \left (-\frac {1}{\left (a d -b c \right )^{4} \sqrt {d x +c}}-\frac {b \left (\frac {\frac {19 \left (d x +c \right )^{\frac {5}{2}} b^{2}}{16}+\frac {17 \left (a d -b c \right ) b \left (d x +c \right )^{\frac {3}{2}}}{6}+\left (\frac {29}{16} a^{2} d^{2}-\frac {29}{8} a b c d +\frac {29}{16} b^{2} c^{2}\right ) \sqrt {d x +c}}{\left (\left (d x +c \right ) b +a d -b c \right )^{3}}+\frac {35 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16 \sqrt {\left (a d -b c \right ) b}}\right )}{\left (a d -b c \right )^{4}}\right )\) | \(156\) |
pseudoelliptic | \(-\frac {2 \left (\frac {35 b \,d^{3} \sqrt {d x +c}\, \left (b x +a \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{16}+\sqrt {\left (a d -b c \right ) b}\, \left (\left (\frac {35}{16} b^{3} x^{3}+\frac {35}{6} a \,b^{2} x^{2}+\frac {77}{16} a^{2} b x +a^{3}\right ) d^{3}+\frac {29 b c \left (\frac {35}{87} b^{2} x^{2}+\frac {98}{87} a b x +a^{2}\right ) d^{2}}{16}-\frac {19 b^{2} c^{2} \left (\frac {7 b x}{19}+a \right ) d}{24}+\frac {b^{3} c^{3}}{6}\right )\right )}{\sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{3} \left (a d -b c \right )^{4}}\) | \(175\) |
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Leaf count of result is larger than twice the leaf count of optimal. 597 vs. \(2 (145) = 290\).
Time = 0.27 (sec) , antiderivative size = 1204, normalized size of antiderivative = 6.96 \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\left [\frac {105 \, {\left (b^{3} d^{4} x^{4} + a^{3} c d^{3} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {\frac {b}{b c - a d}} \log \left (\frac {b d x + 2 \, b c - a d + 2 \, {\left (b c - a d\right )} \sqrt {d x + c} \sqrt {\frac {b}{b c - a d}}}{b x + a}\right ) - 2 \, {\left (105 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 38 \, a b^{2} c^{2} d + 87 \, a^{2} b c d^{2} + 48 \, a^{3} d^{3} + 35 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} - 33 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{48 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}}, \frac {105 \, {\left (b^{3} d^{4} x^{4} + a^{3} c d^{3} + {\left (b^{3} c d^{3} + 3 \, a b^{2} d^{4}\right )} x^{3} + 3 \, {\left (a b^{2} c d^{3} + a^{2} b d^{4}\right )} x^{2} + {\left (3 \, a^{2} b c d^{3} + a^{3} d^{4}\right )} x\right )} \sqrt {-\frac {b}{b c - a d}} \arctan \left (-\frac {{\left (b c - a d\right )} \sqrt {d x + c} \sqrt {-\frac {b}{b c - a d}}}{b d x + b c}\right ) - {\left (105 \, b^{3} d^{3} x^{3} + 8 \, b^{3} c^{3} - 38 \, a b^{2} c^{2} d + 87 \, a^{2} b c d^{2} + 48 \, a^{3} d^{3} + 35 \, {\left (b^{3} c d^{2} + 8 \, a b^{2} d^{3}\right )} x^{2} - 7 \, {\left (2 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} - 33 \, a^{2} b d^{3}\right )} x\right )} \sqrt {d x + c}}{24 \, {\left (a^{3} b^{4} c^{5} - 4 \, a^{4} b^{3} c^{4} d + 6 \, a^{5} b^{2} c^{3} d^{2} - 4 \, a^{6} b c^{2} d^{3} + a^{7} c d^{4} + {\left (b^{7} c^{4} d - 4 \, a b^{6} c^{3} d^{2} + 6 \, a^{2} b^{5} c^{2} d^{3} - 4 \, a^{3} b^{4} c d^{4} + a^{4} b^{3} d^{5}\right )} x^{4} + {\left (b^{7} c^{5} - a b^{6} c^{4} d - 6 \, a^{2} b^{5} c^{3} d^{2} + 14 \, a^{3} b^{4} c^{2} d^{3} - 11 \, a^{4} b^{3} c d^{4} + 3 \, a^{5} b^{2} d^{5}\right )} x^{3} + 3 \, {\left (a b^{6} c^{5} - 3 \, a^{2} b^{5} c^{4} d + 2 \, a^{3} b^{4} c^{3} d^{2} + 2 \, a^{4} b^{3} c^{2} d^{3} - 3 \, a^{5} b^{2} c d^{4} + a^{6} b d^{5}\right )} x^{2} + {\left (3 \, a^{2} b^{5} c^{5} - 11 \, a^{3} b^{4} c^{4} d + 14 \, a^{4} b^{3} c^{3} d^{2} - 6 \, a^{5} b^{2} c^{2} d^{3} - a^{6} b c d^{4} + a^{7} d^{5}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 326 vs. \(2 (145) = 290\).
Time = 0.34 (sec) , antiderivative size = 326, normalized size of antiderivative = 1.88 \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=-\frac {35 \, b d^{3} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{8 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d}} - \frac {2 \, d^{3}}{{\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {d x + c}} - \frac {57 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} d^{3} - 136 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c d^{3} + 87 \, \sqrt {d x + c} b^{3} c^{2} d^{3} + 136 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} d^{4} - 174 \, \sqrt {d x + c} a b^{2} c d^{4} + 87 \, \sqrt {d x + c} a^{2} b d^{5}}{24 \, {\left (b^{4} c^{4} - 4 \, a b^{3} c^{3} d + 6 \, a^{2} b^{2} c^{2} d^{2} - 4 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{3}} \]
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Time = 0.59 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.70 \[ \int \frac {1}{(a+b x)^4 (c+d x)^{3/2}} \, dx=-\frac {\frac {2\,d^3}{a\,d-b\,c}+\frac {35\,b^2\,d^3\,{\left (c+d\,x\right )}^2}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {35\,b^3\,d^3\,{\left (c+d\,x\right )}^3}{8\,{\left (a\,d-b\,c\right )}^4}+\frac {77\,b\,d^3\,\left (c+d\,x\right )}{8\,{\left (a\,d-b\,c\right )}^2}}{\sqrt {c+d\,x}\,\left (a^3\,d^3-3\,a^2\,b\,c\,d^2+3\,a\,b^2\,c^2\,d-b^3\,c^3\right )+b^3\,{\left (c+d\,x\right )}^{7/2}-\left (3\,b^3\,c-3\,a\,b^2\,d\right )\,{\left (c+d\,x\right )}^{5/2}+{\left (c+d\,x\right )}^{3/2}\,\left (3\,a^2\,b\,d^2-6\,a\,b^2\,c\,d+3\,b^3\,c^2\right )}-\frac {35\,\sqrt {b}\,d^3\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}\,\left (a^4\,d^4-4\,a^3\,b\,c\,d^3+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d+b^4\,c^4\right )}{{\left (a\,d-b\,c\right )}^{9/2}}\right )}{8\,{\left (a\,d-b\,c\right )}^{9/2}} \]
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