Integrand size = 20, antiderivative size = 169 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}-\frac {2 a^2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}} \]
[Out]
Time = 0.09 (sec) , antiderivative size = 169, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {90, 52, 65, 214} \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=-\frac {2 a^2 (b c-a d)^{5/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}}+\frac {2 a^2 \sqrt {c+d x} (b c-a d)^2}{b^5}+\frac {2 a^2 (c+d x)^{3/2} (b c-a d)}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (c+d x)^{7/2} (a d+b c)}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2} \]
[In]
[Out]
Rule 52
Rule 65
Rule 90
Rule 214
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {(-b c-a d) (c+d x)^{5/2}}{b^2 d}+\frac {a^2 (c+d x)^{5/2}}{b^2 (a+b x)}+\frac {(c+d x)^{7/2}}{b d}\right ) \, dx \\ & = -\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {a^2 \int \frac {(c+d x)^{5/2}}{a+b x} \, dx}{b^2} \\ & = \frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)\right ) \int \frac {(c+d x)^{3/2}}{a+b x} \, dx}{b^3} \\ & = \frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)^2\right ) \int \frac {\sqrt {c+d x}}{a+b x} \, dx}{b^4} \\ & = \frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (a^2 (b c-a d)^3\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{b^5} \\ & = \frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}+\frac {\left (2 a^2 (b c-a 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 )}{b^5 d} \\ & = \frac {2 a^2 (b c-a d)^2 \sqrt {c+d x}}{b^5}+\frac {2 a^2 (b c-a d) (c+d x)^{3/2}}{3 b^4}+\frac {2 a^2 (c+d x)^{5/2}}{5 b^3}-\frac {2 (b c+a d) (c+d x)^{7/2}}{7 b^2 d^2}+\frac {2 (c+d x)^{9/2}}{9 b d^2}-\frac {2 a^2 (b c-a d)^{5/2} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{b^{11/2}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.94 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \sqrt {c+d x} \left (315 a^4 d^4-45 a b^3 d (c+d x)^3-5 b^4 (2 c-7 d x) (c+d x)^3-105 a^3 b d^3 (7 c+d x)+21 a^2 b^2 d^2 \left (23 c^2+11 c d x+3 d^2 x^2\right )\right )}{315 b^5 d^2}-\frac {2 a^2 (-b c+a d)^{5/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{b^{11/2}} \]
[In]
[Out]
Time = 0.58 (sec) , antiderivative size = 165, normalized size of antiderivative = 0.98
method | result | size |
pseudoelliptic | \(-\frac {2 \left (-\sqrt {\left (a d -b c \right ) b}\, \left (-\frac {2 \left (-\frac {7 d x}{2}+c \right ) \left (d x +c \right )^{3} b^{4}}{63}-\frac {a \,b^{3} d \left (d x +c \right )^{3}}{7}+\frac {23 d^{2} \left (\frac {3}{23} d^{2} x^{2}+\frac {11}{23} c d x +c^{2}\right ) a^{2} b^{2}}{15}-\frac {7 \left (\frac {d x}{7}+c \right ) d^{3} a^{3} b}{3}+a^{4} d^{4}\right ) \sqrt {d x +c}+a^{2} d^{2} \left (a d -b c \right )^{3} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )\right )}{\sqrt {\left (a d -b c \right ) b}\, d^{2} b^{5}}\) | \(165\) |
derivativedivides | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{4} d^{4} \sqrt {d x +c}-2 a^{3} b c \,d^{3} \sqrt {d x +c}+a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{5}}-\frac {2 a^{2} d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{5} \sqrt {\left (a d -b c \right ) b}}}{d^{2}}\) | \(236\) |
default | \(\frac {\frac {2 \left (\frac {\left (d x +c \right )^{\frac {9}{2}} b^{4}}{9}-\frac {a \,b^{3} d \left (d x +c \right )^{\frac {7}{2}}}{7}-\frac {b^{4} c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {a^{2} d^{2} \left (d x +c \right )^{\frac {5}{2}} b^{2}}{5}-\frac {a^{3} b \,d^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+\frac {a^{2} b^{2} c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}}{3}+a^{4} d^{4} \sqrt {d x +c}-2 a^{3} b c \,d^{3} \sqrt {d x +c}+a^{2} b^{2} c^{2} d^{2} \sqrt {d x +c}\right )}{b^{5}}-\frac {2 a^{2} d^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{5} \sqrt {\left (a d -b c \right ) b}}}{d^{2}}\) | \(236\) |
risch | \(\frac {2 \left (35 d^{4} x^{4} b^{4}-45 a \,b^{3} d^{4} x^{3}+95 b^{4} c \,d^{3} x^{3}+63 a^{2} b^{2} d^{4} x^{2}-135 a \,b^{3} c \,d^{3} x^{2}+75 b^{4} c^{2} d^{2} x^{2}-105 a^{3} b \,d^{4} x +231 a^{2} b^{2} c \,d^{3} x -135 a \,b^{3} c^{2} d^{2} x +5 b^{4} c^{3} d x +315 a^{4} d^{4}-735 a^{3} b c \,d^{3}+483 a^{2} b^{2} c^{2} d^{2}-45 a \,b^{3} c^{3} d -10 b^{4} c^{4}\right ) \sqrt {d x +c}}{315 d^{2} b^{5}}-\frac {2 a^{2} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{b^{5} \sqrt {\left (a d -b c \right ) b}}\) | \(268\) |
[In]
[Out]
none
Time = 0.24 (sec) , antiderivative size = 552, normalized size of antiderivative = 3.27 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\left [\frac {315 \, {\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right ) + 2 \, {\left (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \, {\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}}{315 \, b^{5} d^{2}}, -\frac {2 \, {\left (315 \, {\left (a^{2} b^{2} c^{2} d^{2} - 2 \, a^{3} b c d^{3} + a^{4} d^{4}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right ) - {\left (35 \, b^{4} d^{4} x^{4} - 10 \, b^{4} c^{4} - 45 \, a b^{3} c^{3} d + 483 \, a^{2} b^{2} c^{2} d^{2} - 735 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} + 5 \, {\left (19 \, b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 3 \, {\left (25 \, b^{4} c^{2} d^{2} - 45 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} + {\left (5 \, b^{4} c^{3} d - 135 \, a b^{3} c^{2} d^{2} + 231 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {d x + c}\right )}}{315 \, b^{5} d^{2}}\right ] \]
[In]
[Out]
Time = 2.39 (sec) , antiderivative size = 236, normalized size of antiderivative = 1.40 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\begin {cases} \frac {2 \left (\frac {a^{2} d^{3} \left (c + d x\right )^{\frac {5}{2}}}{5 b^{3}} - \frac {a^{2} d^{3} \left (a d - b c\right )^{3} \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{b^{6} \sqrt {\frac {a d - b c}{b}}} + \frac {d \left (c + d x\right )^{\frac {9}{2}}}{9 b} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (- a d^{2} - b c d\right )}{7 b^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (- a^{3} d^{4} + a^{2} b c d^{3}\right )}{3 b^{4}} + \frac {\sqrt {c + d x} \left (a^{4} d^{5} - 2 a^{3} b c d^{4} + a^{2} b^{2} c^{2} d^{3}\right )}{b^{5}}\right )}{d^{3}} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a^{2} \left (\begin {cases} \frac {x}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x \right )}}{b} & \text {otherwise} \end {cases}\right )}{b^{2}} - \frac {a x}{b^{2}} + \frac {x^{2}}{2 b}\right ) & \text {otherwise} \end {cases} \]
[In]
[Out]
Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
none
Time = 0.30 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.49 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\frac {2 \, {\left (a^{2} b^{3} c^{3} - 3 \, a^{3} b^{2} c^{2} d + 3 \, a^{4} b c d^{2} - a^{5} d^{3}\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{5}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{8} d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{8} c d^{16} - 45 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{7} d^{17} + 63 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{6} d^{18} + 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{6} c d^{18} + 315 \, \sqrt {d x + c} a^{2} b^{6} c^{2} d^{18} - 105 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b^{5} d^{19} - 630 \, \sqrt {d x + c} a^{3} b^{5} c d^{19} + 315 \, \sqrt {d x + c} a^{4} b^{4} d^{20}\right )}}{315 \, b^{9} d^{18}} \]
[In]
[Out]
Time = 0.54 (sec) , antiderivative size = 397, normalized size of antiderivative = 2.35 \[ \int \frac {x^2 (c+d x)^{5/2}}{a+b x} \, dx=\left (\frac {2\,c^2}{5\,b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{5\,b\,d^2}\right )\,{\left (c+d\,x\right )}^{5/2}-\left (\frac {4\,c}{7\,b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{7\,b^2\,d^4}\right )\,{\left (c+d\,x\right )}^{7/2}+\frac {2\,{\left (c+d\,x\right )}^{9/2}}{9\,b\,d^2}-\frac {2\,a^2\,\mathrm {atan}\left (\frac {a^2\,\sqrt {b}\,{\left (a\,d-b\,c\right )}^{5/2}\,\sqrt {c+d\,x}}{a^5\,d^3-3\,a^4\,b\,c\,d^2+3\,a^3\,b^2\,c^2\,d-a^2\,b^3\,c^3}\right )\,{\left (a\,d-b\,c\right )}^{5/2}}{b^{11/2}}-\frac {\left (\frac {2\,c^2}{b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{b\,d^2}\right )\,\left (a\,d^3-b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,b\,d^2}+\frac {\left (\frac {2\,c^2}{b\,d^2}+\frac {\left (\frac {4\,c}{b\,d^2}+\frac {2\,\left (a\,d^3-b\,c\,d^2\right )}{b^2\,d^4}\right )\,\left (a\,d^3-b\,c\,d^2\right )}{b\,d^2}\right )\,{\left (a\,d^3-b\,c\,d^2\right )}^2\,\sqrt {c+d\,x}}{b^2\,d^4} \]
[In]
[Out]