Integrand size = 22, antiderivative size = 319 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{11/2} \sqrt {d}} \]
[Out]
Time = 0.23 (sec) , antiderivative size = 319, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {91, 79, 52, 65, 223, 212} \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {5 (b c-a d) \left (21 a^2 d^2-14 a b c d+b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{11/2} \sqrt {d}}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (a+b x)^{3/2} (b c-a d)}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{8 b^5}+\frac {5 \sqrt {a+b x} (c+d x)^{3/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{12 b^4 (b c-a d)}+\frac {\sqrt {a+b x} (c+d x)^{5/2} \left (21 a^2 d^2-14 a b c d+b^2 c^2\right )}{3 b^3 (b c-a d)^2}+\frac {4 a (c+d x)^{7/2} (3 b c-5 a d)}{3 b^2 \sqrt {a+b x} (b c-a d)^2} \]
[In]
[Out]
Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {2 \int \frac {(c+d x)^{5/2} \left (-\frac {1}{2} a (3 b c-7 a d)+\frac {3}{2} b (b c-a d) x\right )}{(a+b x)^{3/2}} \, dx}{3 b^2 (b c-a d)} \\ & = -\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \int \frac {(c+d x)^{5/2}}{\sqrt {a+b x}} \, dx}{b^2 (b c-a d)^2} \\ & = \frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \int \frac {(c+d x)^{3/2}}{\sqrt {a+b x}} \, dx}{6 b^3 (b c-a d)} \\ & = \frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \int \frac {\sqrt {c+d x}}{\sqrt {a+b x}} \, dx}{8 b^4} \\ & = \frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 b^5} \\ & = \frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{8 b^6} \\ & = \frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {\left (5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right )\right ) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{8 b^6} \\ & = \frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 b^5}+\frac {5 \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{3/2}}{12 b^4 (b c-a d)}+\frac {\left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \sqrt {a+b x} (c+d x)^{5/2}}{3 b^3 (b c-a d)^2}-\frac {2 a^2 (c+d x)^{7/2}}{3 b^2 (b c-a d) (a+b x)^{3/2}}+\frac {4 a (3 b c-5 a d) (c+d x)^{7/2}}{3 b^2 (b c-a d)^2 \sqrt {a+b x}}+\frac {5 (b c-a d) \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 b^{11/2} \sqrt {d}} \\ \end{align*}
Time = 10.52 (sec) , antiderivative size = 218, normalized size of antiderivative = 0.68 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {\sqrt {c+d x} \left (\frac {315 a^4 d^2+420 a^3 b d (-c+d x)-6 a b^3 x \left (-27 c^2+16 c d x+3 d^2 x^2\right )+b^4 x^2 \left (33 c^2+26 c d x+8 d^2 x^2\right )+a^2 b^2 \left (113 c^2-574 c d x+63 d^2 x^2\right )}{(a+b x)^{3/2}}+\frac {15 \sqrt {b c-a d} \left (b^2 c^2-14 a b c d+21 a^2 d^2\right ) \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{24 b^5} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(275)=550\).
Time = 0.60 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.14
[In]
[Out]
none
Time = 0.49 (sec) , antiderivative size = 804, normalized size of antiderivative = 2.52 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 35 \, a^{4} b c d^{2} - 21 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 35 \, a^{2} b^{3} c d^{2} - 21 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 15 \, a^{2} b^{3} c^{2} d + 35 \, a^{3} b^{2} c d^{2} - 21 \, a^{4} b d^{3}\right )} x\right )} \sqrt {b d} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} - 4 \, {\left (2 \, b d x + b c + a d\right )} \sqrt {b d} \sqrt {b x + a} \sqrt {d x + c} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) - 4 \, {\left (8 \, b^{5} d^{3} x^{4} + 113 \, a^{2} b^{3} c^{2} d - 420 \, a^{3} b^{2} c d^{2} + 315 \, a^{4} b d^{3} + 2 \, {\left (13 \, b^{5} c d^{2} - 9 \, a b^{4} d^{3}\right )} x^{3} + 3 \, {\left (11 \, b^{5} c^{2} d - 32 \, a b^{4} c d^{2} + 21 \, a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (81 \, a b^{4} c^{2} d - 287 \, a^{2} b^{3} c d^{2} + 210 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}}, -\frac {15 \, {\left (a^{2} b^{3} c^{3} - 15 \, a^{3} b^{2} c^{2} d + 35 \, a^{4} b c d^{2} - 21 \, a^{5} d^{3} + {\left (b^{5} c^{3} - 15 \, a b^{4} c^{2} d + 35 \, a^{2} b^{3} c d^{2} - 21 \, a^{3} b^{2} d^{3}\right )} x^{2} + 2 \, {\left (a b^{4} c^{3} - 15 \, a^{2} b^{3} c^{2} d + 35 \, a^{3} b^{2} c d^{2} - 21 \, a^{4} b d^{3}\right )} x\right )} \sqrt {-b d} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {-b d} \sqrt {b x + a} \sqrt {d x + c}}{2 \, {\left (b^{2} d^{2} x^{2} + a b c d + {\left (b^{2} c d + a b d^{2}\right )} x\right )}}\right ) - 2 \, {\left (8 \, b^{5} d^{3} x^{4} + 113 \, a^{2} b^{3} c^{2} d - 420 \, a^{3} b^{2} c d^{2} + 315 \, a^{4} b d^{3} + 2 \, {\left (13 \, b^{5} c d^{2} - 9 \, a b^{4} d^{3}\right )} x^{3} + 3 \, {\left (11 \, b^{5} c^{2} d - 32 \, a b^{4} c d^{2} + 21 \, a^{2} b^{3} d^{3}\right )} x^{2} + 2 \, {\left (81 \, a b^{4} c^{2} d - 287 \, a^{2} b^{3} c d^{2} + 210 \, a^{3} b^{2} d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{8} d x^{2} + 2 \, a b^{7} d x + a^{2} b^{6} d\right )}}\right ] \]
[In]
[Out]
\[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^{2} \left (c + d x\right )^{\frac {5}{2}}}{\left (a + b x\right )^{\frac {5}{2}}}\, dx \]
[In]
[Out]
Exception generated. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 848 vs. \(2 (275) = 550\).
Time = 0.57 (sec) , antiderivative size = 848, normalized size of antiderivative = 2.66 \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\frac {1}{24} \, \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \sqrt {b x + a} {\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b x + a\right )} d^{2} {\left | b \right |}}{b^{7}} + \frac {13 \, b^{21} c d^{5} {\left | b \right |} - 25 \, a b^{20} d^{6} {\left | b \right |}}{b^{27} d^{4}}\right )} + \frac {3 \, {\left (11 \, b^{22} c^{2} d^{4} {\left | b \right |} - 58 \, a b^{21} c d^{5} {\left | b \right |} + 55 \, a^{2} b^{20} d^{6} {\left | b \right |}\right )}}{b^{27} d^{4}}\right )} - \frac {5 \, {\left (b^{3} c^{3} {\left | b \right |} - 15 \, a b^{2} c^{2} d {\left | b \right |} + 35 \, a^{2} b c d^{2} {\left | b \right |} - 21 \, a^{3} d^{3} {\left | b \right |}\right )} \log \left ({\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}{16 \, \sqrt {b d} b^{6}} + \frac {4 \, {\left (6 \, a b^{7} c^{5} d {\left | b \right |} - 37 \, a^{2} b^{6} c^{4} d^{2} {\left | b \right |} + 88 \, a^{3} b^{5} c^{3} d^{3} {\left | b \right |} - 102 \, a^{4} b^{4} c^{2} d^{4} {\left | b \right |} + 58 \, a^{5} b^{3} c d^{5} {\left | b \right |} - 13 \, a^{6} b^{2} d^{6} {\left | b \right |} - 12 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{4} d {\left | b \right |} + 60 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c^{3} d^{2} {\left | b \right |} - 108 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} c^{2} d^{3} {\left | b \right |} + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{4} b^{2} c d^{4} {\left | b \right |} - 24 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{5} b d^{5} {\left | b \right |} + 6 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c^{3} d {\left | b \right |} - 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} c^{2} d^{2} {\left | b \right |} + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{3} b c d^{3} {\left | b \right |} - 15 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{4} d^{4} {\left | b \right |}\right )}}{3 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3} \sqrt {b d} b^{5}} \]
[In]
[Out]
Timed out. \[ \int \frac {x^2 (c+d x)^{5/2}}{(a+b x)^{5/2}} \, dx=\int \frac {x^2\,{\left (c+d\,x\right )}^{5/2}}{{\left (a+b\,x\right )}^{5/2}} \,d x \]
[In]
[Out]