Integrand size = 22, antiderivative size = 319 \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}} \]
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Time = 0.26 (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} \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=-\frac {5 (b c-a d) \left (a^2 d^2-14 a b c d+21 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}}+\frac {5 \sqrt {a+b x} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{8 d^5}-\frac {5 (a+b x)^{3/2} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{12 d^4 (b c-a d)}+\frac {(a+b x)^{5/2} \sqrt {c+d x} \left (a^2 d^2-14 a b c d+21 b^2 c^2\right )}{3 d^3 (b c-a d)^2}+\frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (c+d x)^{3/2} (b c-a d)}-\frac {4 c (a+b x)^{7/2} (5 b c-3 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2} \]
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Rule 52
Rule 65
Rule 79
Rule 91
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {2 \int \frac {(a+b x)^{5/2} \left (\frac {1}{2} c (7 b c-3 a d)-\frac {3}{2} d (b c-a d) x\right )}{(c+d x)^{3/2}} \, dx}{3 d^2 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \int \frac {(a+b x)^{5/2}}{\sqrt {c+d x}} \, dx}{d^2 (b c-a d)^2} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {(a+b x)^{3/2}}{\sqrt {c+d x}} \, dx}{6 d^3 (b c-a d)} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}+\frac {\left (5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {\sqrt {a+b x}}{\sqrt {c+d x}} \, dx}{8 d^4} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{16 d^5} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+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 d^5} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {\left (5 (b c-a d) \left (21 b^2 c^2-14 a b c d+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 d^5} \\ & = \frac {2 c^2 (a+b x)^{7/2}}{3 d^2 (b c-a d) (c+d x)^{3/2}}-\frac {4 c (5 b c-3 a d) (a+b x)^{7/2}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}+\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \sqrt {a+b x} \sqrt {c+d x}}{8 d^5}-\frac {5 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{3/2} \sqrt {c+d x}}{12 d^4 (b c-a d)}+\frac {\left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (a+b x)^{5/2} \sqrt {c+d x}}{3 d^3 (b c-a d)^2}-\frac {5 (b c-a d) \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{8 \sqrt {b} d^{11/2}} \\ \end{align*}
Time = 11.07 (sec) , antiderivative size = 282, normalized size of antiderivative = 0.88 \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {-8 c^2 (a+b x)^4+\frac {16 c (5 b c-3 a d) (a+b x)^4 (c+d x)}{b c-a d}+\frac {15 (b c-a d)^3 \left (21 b^2 c^2-14 a b c d+a^2 d^2\right ) (c+d x)^2 \left (\frac {2 d (a+b x)}{b c-a d}-\frac {4 d^2 (a+b x)^2}{3 (b c-a d)^2}+\frac {16 d^3 (a+b x)^3}{15 (b c-a d)^3}-\frac {2 \sqrt {d} \sqrt {a+b x} \text {arcsinh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}}\right )}{4 d^4 (-b c+a d)}}{12 d^2 (-b c+a d) \sqrt {a+b x} (c+d x)^{3/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1001\) vs. \(2(275)=550\).
Time = 1.62 (sec) , antiderivative size = 1002, normalized size of antiderivative = 3.14
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Time = 0.67 (sec) , antiderivative size = 810, normalized size of antiderivative = 2.54 \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\left [-\frac {15 \, {\left (21 \, b^{3} c^{5} - 35 \, a b^{2} c^{4} d + 15 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (21 \, b^{3} c^{3} d^{2} - 35 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (21 \, b^{3} c^{4} d - 35 \, a b^{2} c^{3} d^{2} + 15 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\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^{3} d^{5} x^{4} + 315 \, b^{3} c^{4} d - 420 \, a b^{2} c^{3} d^{2} + 113 \, a^{2} b c^{2} d^{3} - 2 \, {\left (9 \, b^{3} c d^{4} - 13 \, a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{3} c^{2} d^{3} - 32 \, a b^{2} c d^{4} + 11 \, a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (210 \, b^{3} c^{3} d^{2} - 287 \, a b^{2} c^{2} d^{3} + 81 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{96 \, {\left (b d^{8} x^{2} + 2 \, b c d^{7} x + b c^{2} d^{6}\right )}}, \frac {15 \, {\left (21 \, b^{3} c^{5} - 35 \, a b^{2} c^{4} d + 15 \, a^{2} b c^{3} d^{2} - a^{3} c^{2} d^{3} + {\left (21 \, b^{3} c^{3} d^{2} - 35 \, a b^{2} c^{2} d^{3} + 15 \, a^{2} b c d^{4} - a^{3} d^{5}\right )} x^{2} + 2 \, {\left (21 \, b^{3} c^{4} d - 35 \, a b^{2} c^{3} d^{2} + 15 \, a^{2} b c^{2} d^{3} - a^{3} c d^{4}\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^{3} d^{5} x^{4} + 315 \, b^{3} c^{4} d - 420 \, a b^{2} c^{3} d^{2} + 113 \, a^{2} b c^{2} d^{3} - 2 \, {\left (9 \, b^{3} c d^{4} - 13 \, a b^{2} d^{5}\right )} x^{3} + 3 \, {\left (21 \, b^{3} c^{2} d^{3} - 32 \, a b^{2} c d^{4} + 11 \, a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (210 \, b^{3} c^{3} d^{2} - 287 \, a b^{2} c^{2} d^{3} + 81 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b d^{8} x^{2} + 2 \, b c d^{7} x + b c^{2} d^{6}\right )}}\right ] \]
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\[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^{2} \left (a + b x\right )^{\frac {5}{2}}}{\left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Time = 0.41 (sec) , antiderivative size = 514, normalized size of antiderivative = 1.61 \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\frac {{\left ({\left ({\left (2 \, {\left (b x + a\right )} {\left (\frac {4 \, {\left (b^{6} c d^{8} - a b^{5} d^{9}\right )} {\left (b x + a\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}} - \frac {3 \, {\left (3 \, b^{7} c^{2} d^{7} - 2 \, a b^{6} c d^{8} - a^{2} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} + \frac {3 \, {\left (21 \, b^{8} c^{3} d^{6} - 35 \, a b^{7} c^{2} d^{7} + 15 \, a^{2} b^{6} c d^{8} - a^{3} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {20 \, {\left (21 \, b^{9} c^{4} d^{5} - 56 \, a b^{8} c^{3} d^{6} + 50 \, a^{2} b^{7} c^{2} d^{7} - 16 \, a^{3} b^{6} c d^{8} + a^{4} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} {\left (b x + a\right )} + \frac {15 \, {\left (21 \, b^{10} c^{5} d^{4} - 77 \, a b^{9} c^{4} d^{5} + 106 \, a^{2} b^{8} c^{3} d^{6} - 66 \, a^{3} b^{7} c^{2} d^{7} + 17 \, a^{4} b^{6} c d^{8} - a^{5} b^{5} d^{9}\right )}}{b^{4} c d^{9} {\left | b \right |} - a b^{3} d^{10} {\left | b \right |}}\right )} \sqrt {b x + a}}{24 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {5 \, {\left (21 \, b^{4} c^{3} - 35 \, a b^{3} c^{2} d + 15 \, a^{2} b^{2} c d^{2} - a^{3} b d^{3}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{8 \, \sqrt {b d} d^{5} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^2 (a+b x)^{5/2}}{(c+d x)^{5/2}} \, dx=\int \frac {x^2\,{\left (a+b\,x\right )}^{5/2}}{{\left (c+d\,x\right )}^{5/2}} \,d x \]
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