Integrand size = 17, antiderivative size = 172 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {3 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}} \]
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Time = 0.05 (sec) , antiderivative size = 172, 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 = {43, 44, 65, 214} \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=-\frac {3 d^4 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (a+b x) (b c-a d)^2}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (a+b x)^2 (b c-a d)}-\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{(a+b x)^4} \, dx}{8 b} \\ & = -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {d^2 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{16 b^2} \\ & = -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {\left (3 d^3\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{64 b^2 (b c-a d)} \\ & = -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{128 b^2 (b c-a d)^2} \\ & = -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}+\frac {\left (3 d^3\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{64 b^2 (b c-a d)^2} \\ & = -\frac {d \sqrt {c+d x}}{8 b^2 (a+b x)^3}-\frac {d^2 \sqrt {c+d x}}{32 b^2 (b c-a d) (a+b x)^2}+\frac {3 d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)}-\frac {(c+d x)^{3/2}}{4 b (a+b x)^4}-\frac {3 d^4 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{64 b^{5/2} (b c-a d)^{5/2}} \\ \end{align*}
Time = 0.83 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.99 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=-\frac {\sqrt {c+d x} \left (3 a^3 d^3+a^2 b d^2 (2 c+11 d x)-a b^2 d \left (24 c^2+44 c d x+11 d^2 x^2\right )+b^3 \left (16 c^3+24 c^2 d x+2 c d^2 x^2-3 d^3 x^3\right )\right )}{64 b^2 (b c-a d)^2 (a+b x)^4}+\frac {3 d^4 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{64 b^{5/2} (-b c+a d)^{5/2}} \]
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Time = 0.33 (sec) , antiderivative size = 145, normalized size of antiderivative = 0.84
method | result | size |
pseudoelliptic | \(\frac {\frac {3 d^{4} \left (b x +a \right )^{4} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{64}-\frac {3 \sqrt {d x +c}\, \left (\left (d^{2} x^{2}-\frac {8}{3} c d x -\frac {8}{3} c^{2}\right ) b^{2}+\frac {8 d \left (\frac {7 d x}{4}+c \right ) a b}{3}+a^{2} d^{2}\right ) \left (\left (-d x -2 c \right ) b +a d \right ) \sqrt {\left (a d -b c \right ) b}}{64}}{\sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{4} \left (a d -b c \right )^{2} b^{2}}\) | \(145\) |
derivativedivides | \(2 d^{4} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{128 \left (a d -b c \right )}-\frac {11 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{128 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(172\) |
default | \(2 d^{4} \left (\frac {\frac {3 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {11 \left (d x +c \right )^{\frac {5}{2}}}{128 \left (a d -b c \right )}-\frac {11 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{128 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{4}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(172\) |
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Leaf count of result is larger than twice the leaf count of optimal. 515 vs. \(2 (144) = 288\).
Time = 0.26 (sec) , antiderivative size = 1043, normalized size of antiderivative = 6.06 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=\left [\frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {b^{2} c - a b d} \sqrt {d x + c}}{b x + a}\right ) - 2 \, {\left (16 \, b^{5} c^{4} - 40 \, a b^{4} c^{3} d + 26 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} c d^{3} - 3 \, a^{4} b d^{4} - 3 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (24 \, b^{5} c^{3} d - 68 \, a b^{4} c^{2} d^{2} + 55 \, a^{2} b^{3} c d^{3} - 11 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{128 \, {\left (a^{4} b^{6} c^{3} - 3 \, a^{5} b^{5} c^{2} d + 3 \, a^{6} b^{4} c d^{2} - a^{7} b^{3} d^{3} + {\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} x^{4} + 4 \, {\left (a b^{9} c^{3} - 3 \, a^{2} b^{8} c^{2} d + 3 \, a^{3} b^{7} c d^{2} - a^{4} b^{6} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{3} - 3 \, a^{4} b^{6} c^{2} d + 3 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )} x\right )}}, \frac {3 \, {\left (b^{4} d^{4} x^{4} + 4 \, a b^{3} d^{4} x^{3} + 6 \, a^{2} b^{2} d^{4} x^{2} + 4 \, a^{3} b d^{4} x + a^{4} d^{4}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (\frac {\sqrt {-b^{2} c + a b d} \sqrt {d x + c}}{b d x + b c}\right ) - {\left (16 \, b^{5} c^{4} - 40 \, a b^{4} c^{3} d + 26 \, a^{2} b^{3} c^{2} d^{2} + a^{3} b^{2} c d^{3} - 3 \, a^{4} b d^{4} - 3 \, {\left (b^{5} c d^{3} - a b^{4} d^{4}\right )} x^{3} + {\left (2 \, b^{5} c^{2} d^{2} - 13 \, a b^{4} c d^{3} + 11 \, a^{2} b^{3} d^{4}\right )} x^{2} + {\left (24 \, b^{5} c^{3} d - 68 \, a b^{4} c^{2} d^{2} + 55 \, a^{2} b^{3} c d^{3} - 11 \, a^{3} b^{2} d^{4}\right )} x\right )} \sqrt {d x + c}}{64 \, {\left (a^{4} b^{6} c^{3} - 3 \, a^{5} b^{5} c^{2} d + 3 \, a^{6} b^{4} c d^{2} - a^{7} b^{3} d^{3} + {\left (b^{10} c^{3} - 3 \, a b^{9} c^{2} d + 3 \, a^{2} b^{8} c d^{2} - a^{3} b^{7} d^{3}\right )} x^{4} + 4 \, {\left (a b^{9} c^{3} - 3 \, a^{2} b^{8} c^{2} d + 3 \, a^{3} b^{7} c d^{2} - a^{4} b^{6} d^{3}\right )} x^{3} + 6 \, {\left (a^{2} b^{8} c^{3} - 3 \, a^{3} b^{7} c^{2} d + 3 \, a^{4} b^{6} c d^{2} - a^{5} b^{5} d^{3}\right )} x^{2} + 4 \, {\left (a^{3} b^{7} c^{3} - 3 \, a^{4} b^{6} c^{2} d + 3 \, a^{5} b^{5} c d^{2} - a^{6} b^{4} d^{3}\right )} x\right )}}\right ] \]
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Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.35 (sec) , antiderivative size = 285, normalized size of antiderivative = 1.66 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=\frac {3 \, d^{4} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{64 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} \sqrt {-b^{2} c + a b d}} + \frac {3 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{3} d^{4} - 11 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{3} c d^{4} - 11 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{3} c^{2} d^{4} + 3 \, \sqrt {d x + c} b^{3} c^{3} d^{4} + 11 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{2} d^{5} + 22 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{2} c d^{5} - 9 \, \sqrt {d x + c} a b^{2} c^{2} d^{5} - 11 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b d^{6} + 9 \, \sqrt {d x + c} a^{2} b c d^{6} - 3 \, \sqrt {d x + c} a^{3} d^{7}}{64 \, {\left (b^{4} c^{2} - 2 \, a b^{3} c d + a^{2} b^{2} d^{2}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{4}} \]
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Time = 0.42 (sec) , antiderivative size = 296, normalized size of antiderivative = 1.72 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^5} \, dx=\frac {3\,d^4\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{64\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{5/2}}-\frac {\frac {11\,d^4\,{\left (c+d\,x\right )}^{3/2}}{64\,b}-\frac {11\,d^4\,{\left (c+d\,x\right )}^{5/2}}{64\,\left (a\,d-b\,c\right )}+\frac {3\,d^4\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{64\,b^2}-\frac {3\,b\,d^4\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^2}}{b^4\,{\left (c+d\,x\right )}^4-\left (4\,b^4\,c-4\,a\,b^3\,d\right )\,{\left (c+d\,x\right )}^3-\left (c+d\,x\right )\,\left (-4\,a^3\,b\,d^3+12\,a^2\,b^2\,c\,d^2-12\,a\,b^3\,c^2\,d+4\,b^4\,c^3\right )+a^4\,d^4+b^4\,c^4+{\left (c+d\,x\right )}^2\,\left (6\,a^2\,b^2\,d^2-12\,a\,b^3\,c\,d+6\,b^4\,c^2\right )+6\,a^2\,b^2\,c^2\,d^2-4\,a\,b^3\,c^3\,d-4\,a^3\,b\,c\,d^3} \]
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