Integrand size = 20, antiderivative size = 220 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \]
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Time = 0.15 (sec) , antiderivative size = 220, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {99, 158, 152, 52, 65, 214} \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}}+\frac {a^2 \sqrt {c+d x} (6 b c-11 a d) (b c-a d)}{b^6}+\frac {a^2 (c+d x)^{3/2} (6 b c-11 a d)}{3 b^5}-\frac {(c+d x)^{5/2} \left (-693 a^2 d^2-5 b d x (10 b c-99 a d)+180 a b c d+20 b^2 c^2\right )}{315 b^4 d^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2} \]
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Rule 52
Rule 65
Rule 99
Rule 152
Rule 158
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {\int \frac {x^2 (c+d x)^{3/2} \left (3 c+\frac {11 d x}{2}\right )}{a+b x} \, dx}{b} \\ & = \frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}+\frac {2 \int \frac {x (c+d x)^{3/2} \left (-11 a c d+\frac {1}{4} d (10 b c-99 a d) x\right )}{a+b x} \, dx}{9 b^2 d} \\ & = \frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{2 b^4} \\ & = \frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a d)\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{2 b^5} \\ & = \frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a d)^2\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{2 b^6} \\ & = \frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}+\frac {\left (a^2 (6 b c-11 a d) (b c-a 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 )}{b^6 d} \\ & = \frac {a^2 (6 b c-11 a d) (b c-a d) \sqrt {c+d x}}{b^6}+\frac {a^2 (6 b c-11 a d) (c+d x)^{3/2}}{3 b^5}+\frac {11 x^2 (c+d x)^{5/2}}{9 b^2}-\frac {x^3 (c+d x)^{5/2}}{b (a+b x)}-\frac {(c+d x)^{5/2} \left (20 b^2 c^2+180 a b c d-693 a^2 d^2-5 b d (10 b c-99 a d) x\right )}{315 b^4 d^2}-\frac {a^2 (6 b c-11 a d) (b c-a d)^{3/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{13/2}} \\ \end{align*}
Time = 0.46 (sec) , antiderivative size = 222, normalized size of antiderivative = 1.01 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {\sqrt {c+d x} \left (3465 a^5 d^4+10 b^5 x (c+d x)^3 (-2 c+7 d x)+210 a^4 b d^3 (-31 c+11 d x)-10 a b^4 (c+d x)^3 (2 c+11 d x)-21 a^3 b^2 d^2 \left (-153 c^2+214 c d x+22 d^2 x^2\right )+18 a^2 b^3 d \left (-10 c^3+131 c^2 d x+47 c d^2 x^2+11 d^3 x^3\right )\right )}{315 b^6 d^2 (a+b x)}+\frac {a^2 (6 b c-11 a d) (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{13/2}} \]
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Time = 0.66 (sec) , antiderivative size = 228, normalized size of antiderivative = 1.04
method | result | size |
pseudoelliptic | \(-\frac {11 \left (\left (a d -b c \right )^{2} \left (a d -\frac {6 b c}{11}\right ) d^{2} \left (b x +a \right ) a^{2} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )-\left (-\frac {4 x \left (-\frac {7 d x}{2}+c \right ) \left (d x +c \right )^{3} b^{5}}{693}-\frac {4 \left (d x +c \right )^{3} a \left (\frac {11 d x}{2}+c \right ) b^{4}}{693}-\frac {4 \left (-\frac {11}{10} d^{3} x^{3}-\frac {47}{10} c \,d^{2} x^{2}-\frac {131}{10} c^{2} d x +c^{3}\right ) d \,a^{2} b^{3}}{77}+\frac {51 \left (-\frac {22}{153} d^{2} x^{2}-\frac {214}{153} c d x +c^{2}\right ) d^{2} a^{3} b^{2}}{55}-\frac {62 d^{3} \left (-\frac {11 d x}{31}+c \right ) a^{4} b}{33}+a^{5} d^{4}\right ) \sqrt {d x +c}\, \sqrt {\left (a d -b c \right ) b}\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{2} b^{6} \left (b x +a \right )}\) | \(228\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {2 a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {3 a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+5 a^{4} d^{4} \sqrt {d x +c}-8 a^{3} b c \,d^{3} \sqrt {d x +c}+3 a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{6}}-\frac {2 a^{2} d^{2} \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (11 a^{3} d^{3}-28 a^{2} b c \,d^{2}+23 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}}{d^{2}}\) | \(294\) |
default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {2 a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {3 a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {4 a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}+5 a^{4} d^{4} \sqrt {d x +c}-8 a^{3} b c \,d^{3} \sqrt {d x +c}+3 a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{6}}-\frac {2 a^{2} d^{2} \left (\frac {\left (-\frac {1}{2} a^{3} d^{3}+a^{2} b c \,d^{2}-\frac {1}{2} a \,b^{2} c^{2} d \right ) \sqrt {d x +c}}{\left (d x +c \right ) b +a d -b c}+\frac {\left (11 a^{3} d^{3}-28 a^{2} b c \,d^{2}+23 a \,b^{2} c^{2} d -6 b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}}{d^{2}}\) | \(294\) |
risch | \(\frac {2 \left (35 d^{4} x^{4} b^{4}-90 a \,b^{3} d^{4} x^{3}+95 b^{4} c \,d^{3} x^{3}+189 a^{2} b^{2} d^{4} x^{2}-270 a \,b^{3} c \,d^{3} x^{2}+75 b^{4} c^{2} d^{2} x^{2}-420 a^{3} b \,d^{4} x +693 a^{2} b^{2} c \,d^{3} x -270 a \,b^{3} c^{2} d^{2} x +5 b^{4} c^{3} d x +1575 a^{4} d^{4}-2940 a^{3} b c \,d^{3}+1449 a^{2} b^{2} c^{2} d^{2}-90 a \,b^{3} c^{3} d -10 b^{4} c^{4}\right ) \sqrt {d x +c}}{315 d^{2} b^{6}}-\frac {a^{2} \left (2 a^{2} d^{2}-4 a b c d +2 b^{2} c^{2}\right ) \left (-\frac {a d \sqrt {d x +c}}{2 \left (\left (d x +c \right ) b +a d -b c \right )}+\frac {\left (11 a d -6 b c \right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{2 \sqrt {\left (a d -b c \right ) b}}\right )}{b^{6}}\) | \(295\) |
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Leaf count of result is larger than twice the leaf count of optimal. 389 vs. \(2 (194) = 388\).
Time = 0.28 (sec) , antiderivative size = 788, normalized size of antiderivative = 3.58 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\left [\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{630 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}, -\frac {315 \, {\left (6 \, a^{3} b^{2} c^{2} d^{2} - 17 \, a^{4} b c d^{3} + 11 \, a^{5} d^{4} + {\left (6 \, a^{2} b^{3} c^{2} d^{2} - 17 \, a^{3} b^{2} c d^{3} + 11 \, a^{4} b d^{4}\right )} x\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (70 \, b^{5} d^{4} x^{5} - 20 \, a b^{4} c^{4} - 180 \, a^{2} b^{3} c^{3} d + 3213 \, a^{3} b^{2} c^{2} d^{2} - 6510 \, a^{4} b c d^{3} + 3465 \, a^{5} d^{4} + 10 \, {\left (19 \, b^{5} c d^{3} - 11 \, a b^{4} d^{4}\right )} x^{4} + 2 \, {\left (75 \, b^{5} c^{2} d^{2} - 175 \, a b^{4} c d^{3} + 99 \, a^{2} b^{3} d^{4}\right )} x^{3} + 2 \, {\left (5 \, b^{5} c^{3} d - 195 \, a b^{4} c^{2} d^{2} + 423 \, a^{2} b^{3} c d^{3} - 231 \, a^{3} b^{2} d^{4}\right )} x^{2} - 2 \, {\left (10 \, b^{5} c^{4} + 85 \, a b^{4} c^{3} d - 1179 \, a^{2} b^{3} c^{2} d^{2} + 2247 \, a^{3} b^{2} c d^{3} - 1155 \, a^{4} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (b^{7} d^{2} x + a b^{6} d^{2}\right )}}\right ] \]
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Timed out. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\text {Exception raised: ValueError} \]
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none
Time = 0.28 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.47 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx=\frac {{\left (6 \, a^{2} b^{3} c^{3} - 23 \, a^{3} b^{2} c^{2} d + 28 \, a^{4} b c d^{2} - 11 \, a^{5} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{6}} + \frac {\sqrt {d x + c} a^{3} b^{2} c^{2} d - 2 \, \sqrt {d x + c} a^{4} b c d^{2} + \sqrt {d x + c} a^{5} d^{3}}{{\left ({\left (d x + c\right )} b - b c + a d\right )} b^{6}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{16} d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{16} c d^{16} - 90 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{15} d^{17} + 189 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{14} d^{18} + 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{14} c d^{18} + 945 \, \sqrt {d x + c} a^{2} b^{14} c^{2} d^{18} - 420 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{13} d^{19} - 2520 \, \sqrt {d x + c} a^{3} b^{13} c d^{19} + 1575 \, \sqrt {d x + c} a^{4} b^{12} d^{20}\right )}}{315 \, b^{18} d^{18}} \]
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Time = 0.15 (sec) , antiderivative size = 679, normalized size of antiderivative = 3.09 \[ \int \frac {x^3 (c+d x)^{5/2}}{(a+b x)^2} \, dx={\left (c+d\,x\right )}^{5/2}\,\left (\frac {6\,c^2}{5\,b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{5\,b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{5\,b}\right )-\left (\frac {{\left (a\,d-b\,c\right )}^2\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b^2}-\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {2\,c^3}{b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{b}\right )}{b}\right )\,\sqrt {c+d\,x}-{\left (c+d\,x\right )}^{3/2}\,\left (\frac {2\,c^3}{3\,b^2\,d^2}-\frac {\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,{\left (a\,d-b\,c\right )}^2}{3\,b^2}+\frac {2\,\left (a\,d-b\,c\right )\,\left (\frac {6\,c^2}{b^2\,d^2}-\frac {2\,{\left (a\,d-b\,c\right )}^2}{b^4\,d^2}+\frac {2\,\left (\frac {6\,c}{b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{b^3\,d^2}\right )\,\left (a\,d-b\,c\right )}{b}\right )}{3\,b}\right )-\left (\frac {6\,c}{7\,b^2\,d^2}+\frac {4\,\left (a\,d-b\,c\right )}{7\,b^3\,d^2}\right )\,{\left (c+d\,x\right )}^{7/2}+\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,b^2\,d^2}+\frac {\sqrt {c+d\,x}\,\left (a^5\,d^3-2\,a^4\,b\,c\,d^2+a^3\,b^2\,c^2\,d\right )}{b^7\,\left (c+d\,x\right )-b^7\,c+a\,b^6\,d}-\frac {a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )\,\sqrt {c+d\,x}}{11\,a^5\,d^3-28\,a^4\,b\,c\,d^2+23\,a^3\,b^2\,c^2\,d-6\,a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{3/2}\,\left (11\,a\,d-6\,b\,c\right )}{b^{13/2}} \]
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