Integrand size = 22, antiderivative size = 258 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}} \]
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Time = 0.16 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {100, 155, 152, 65, 223, 212} \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\left (3 a^2 d^2+10 a b c d+35 b^2 c^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (9 a^3 d^3-2 b d x \left (3 a^2 d^2-46 a b c d+35 b^2 c^2\right )+15 a^2 b c d^2-145 a b^2 c^2 d+105 b^3 c^3\right )}{12 b^2 d^4 (b c-a d)^2}-\frac {2 c x^2 \sqrt {a+b x} (7 b c-9 a d)}{3 d^2 \sqrt {c+d x} (b c-a d)^2}-\frac {2 c x^3 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rule 65
Rule 100
Rule 152
Rule 155
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 \int \frac {x^2 \left (3 a c+\frac {1}{2} (7 b c-3 a d) x\right )}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 d (b c-a d)} \\ & = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {4 \int \frac {x \left (-a c (7 b c-9 a d)+\frac {1}{4} \left (-35 b^2 c^2+46 a b c d-3 a^2 d^2\right ) x\right )}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{3 d^2 (b c-a d)^2} \\ & = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{8 b^2 d^4} \\ & = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\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 )}{4 b^3 d^4} \\ & = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\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 )}{4 b^3 d^4} \\ & = -\frac {2 c x^3 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}-\frac {2 c (7 b c-9 a d) x^2 \sqrt {a+b x}}{3 d^2 (b c-a d)^2 \sqrt {c+d x}}-\frac {\sqrt {a+b x} \sqrt {c+d x} \left (105 b^3 c^3-145 a b^2 c^2 d+15 a^2 b c d^2+9 a^3 d^3-2 b d \left (35 b^2 c^2-46 a b c d+3 a^2 d^2\right ) x\right )}{12 b^2 d^4 (b c-a d)^2}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}} \\ \end{align*}
Time = 0.48 (sec) , antiderivative size = 221, normalized size of antiderivative = 0.86 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {\sqrt {a+b x} \left (9 a^3 d^3 (c+d x)^2+3 a^2 b d^2 (5 c-2 d x) (c+d x)^2-a b^2 c d \left (145 c^3+198 c^2 d x+33 c d^2 x^2-12 d^3 x^3\right )+b^3 c^2 \left (105 c^3+140 c^2 d x+21 c d^2 x^2-6 d^3 x^3\right )\right )}{12 b^2 d^4 (b c-a d)^2 (c+d x)^{3/2}}+\frac {\left (35 b^2 c^2+10 a b c d+3 a^2 d^2\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{4 b^{5/2} d^{9/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(1286\) vs. \(2(226)=452\).
Time = 0.58 (sec) , antiderivative size = 1287, normalized size of antiderivative = 4.99
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Leaf count of result is larger than twice the leaf count of optimal. 565 vs. \(2 (226) = 452\).
Time = 0.53 (sec) , antiderivative size = 1144, normalized size of antiderivative = 4.43 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (35 \, b^{4} c^{6} - 60 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} + 4 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4} + {\left (35 \, b^{4} c^{4} d^{2} - 60 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} + 4 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{5} d - 60 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} + 4 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\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 (105 \, b^{4} c^{5} d - 145 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} + 9 \, a^{3} b c^{2} d^{4} - 6 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x^{3} + 3 \, {\left (7 \, b^{4} c^{3} d^{3} - 11 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + 3 \, a^{3} b d^{6}\right )} x^{2} + 2 \, {\left (70 \, b^{4} c^{4} d^{2} - 99 \, a b^{3} c^{3} d^{3} + 12 \, a^{2} b^{2} c^{2} d^{4} + 9 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{48 \, {\left (b^{5} c^{4} d^{5} - 2 \, a b^{4} c^{3} d^{6} + a^{2} b^{3} c^{2} d^{7} + {\left (b^{5} c^{2} d^{7} - 2 \, a b^{4} c d^{8} + a^{2} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (b^{5} c^{3} d^{6} - 2 \, a b^{4} c^{2} d^{7} + a^{2} b^{3} c d^{8}\right )} x\right )}}, -\frac {3 \, {\left (35 \, b^{4} c^{6} - 60 \, a b^{3} c^{5} d + 18 \, a^{2} b^{2} c^{4} d^{2} + 4 \, a^{3} b c^{3} d^{3} + 3 \, a^{4} c^{2} d^{4} + {\left (35 \, b^{4} c^{4} d^{2} - 60 \, a b^{3} c^{3} d^{3} + 18 \, a^{2} b^{2} c^{2} d^{4} + 4 \, a^{3} b c d^{5} + 3 \, a^{4} d^{6}\right )} x^{2} + 2 \, {\left (35 \, b^{4} c^{5} d - 60 \, a b^{3} c^{4} d^{2} + 18 \, a^{2} b^{2} c^{3} d^{3} + 4 \, a^{3} b c^{2} d^{4} + 3 \, a^{4} c d^{5}\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 (105 \, b^{4} c^{5} d - 145 \, a b^{3} c^{4} d^{2} + 15 \, a^{2} b^{2} c^{3} d^{3} + 9 \, a^{3} b c^{2} d^{4} - 6 \, {\left (b^{4} c^{2} d^{4} - 2 \, a b^{3} c d^{5} + a^{2} b^{2} d^{6}\right )} x^{3} + 3 \, {\left (7 \, b^{4} c^{3} d^{3} - 11 \, a b^{3} c^{2} d^{4} + a^{2} b^{2} c d^{5} + 3 \, a^{3} b d^{6}\right )} x^{2} + 2 \, {\left (70 \, b^{4} c^{4} d^{2} - 99 \, a b^{3} c^{3} d^{3} + 12 \, a^{2} b^{2} c^{2} d^{4} + 9 \, a^{3} b c d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{24 \, {\left (b^{5} c^{4} d^{5} - 2 \, a b^{4} c^{3} d^{6} + a^{2} b^{3} c^{2} d^{7} + {\left (b^{5} c^{2} d^{7} - 2 \, a b^{4} c d^{8} + a^{2} b^{3} d^{9}\right )} x^{2} + 2 \, {\left (b^{5} c^{3} d^{6} - 2 \, a b^{4} c^{2} d^{7} + a^{2} b^{3} c d^{8}\right )} x\right )}}\right ] \]
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\[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{4}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (226) = 452\).
Time = 0.39 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.97 \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {{\left ({\left (3 \, {\left (b x + a\right )} {\left (\frac {2 \, {\left (b^{7} c^{2} d^{6} - 2 \, a b^{6} c d^{7} + a^{2} b^{5} d^{8}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}} - \frac {7 \, b^{8} c^{3} d^{5} - 5 \, a b^{7} c^{2} d^{6} - 11 \, a^{2} b^{6} c d^{7} + 9 \, a^{3} b^{5} d^{8}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} - \frac {4 \, {\left (35 \, b^{9} c^{4} d^{4} - 60 \, a b^{8} c^{3} d^{5} + 18 \, a^{2} b^{7} c^{2} d^{6} + 12 \, a^{3} b^{6} c d^{7} - 9 \, a^{4} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} {\left (b x + a\right )} - \frac {3 \, {\left (35 \, b^{10} c^{5} d^{3} - 95 \, a b^{9} c^{4} d^{4} + 78 \, a^{2} b^{8} c^{3} d^{5} - 14 \, a^{3} b^{7} c^{2} d^{6} - 9 \, a^{4} b^{6} c d^{7} + 5 \, a^{5} b^{5} d^{8}\right )}}{b^{7} c^{2} d^{7} {\left | b \right |} - 2 \, a b^{6} c d^{8} {\left | b \right |} + a^{2} b^{5} d^{9} {\left | b \right |}}\right )} \sqrt {b x + a}}{12 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {{\left (35 \, b^{2} c^{2} + 10 \, a b c d + 3 \, a^{2} d^{2}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{4 \, \sqrt {b d} b d^{4} {\left | b \right |}} \]
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Timed out. \[ \int \frac {x^4}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^4}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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