Integrand size = 22, antiderivative size = 174 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=-\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \]
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Time = 0.10 (sec) , antiderivative size = 174, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.227, Rules used = {100, 148, 65, 223, 212} \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (c \left (3 a^2 d^2-22 a b c d+15 b^2 c^2\right )+d x (5 b c-3 a d) (b c-a d)\right )}{3 b d^3 \sqrt {c+d x} (b c-a d)^2}-\frac {(a d+5 b c) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}}-\frac {2 c x^2 \sqrt {a+b x}}{3 d (c+d x)^{3/2} (b c-a d)} \]
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Rule 65
Rule 100
Rule 148
Rule 212
Rule 223
Rubi steps \begin{align*} \text {integral}& = -\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {2 \int \frac {x \left (2 a c+\frac {1}{2} (5 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^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \int \frac {1}{\sqrt {a+b x} \sqrt {c+d x}} \, dx}{2 b d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \text {Subst}\left (\int \frac {1}{\sqrt {c-\frac {a d}{b}+\frac {d x^2}{b}}} \, dx,x,\sqrt {a+b x}\right )}{b^2 d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \text {Subst}\left (\int \frac {1}{1-\frac {d x^2}{b}} \, dx,x,\frac {\sqrt {a+b x}}{\sqrt {c+d x}}\right )}{b^2 d^3} \\ & = -\frac {2 c x^2 \sqrt {a+b x}}{3 d (b c-a d) (c+d x)^{3/2}}+\frac {\sqrt {a+b x} \left (c \left (15 b^2 c^2-22 a b c d+3 a^2 d^2\right )+d (5 b c-3 a d) (b c-a d) x\right )}{3 b d^3 (b c-a d)^2 \sqrt {c+d x}}-\frac {(5 b c+a d) \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \\ \end{align*}
Time = 0.32 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.90 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {\sqrt {a+b x} \left (3 a^2 d^2 (c+d x)^2-2 a b c d \left (11 c^2+15 c d x+3 d^2 x^2\right )+b^2 c^2 \left (15 c^2+20 c d x+3 d^2 x^2\right )\right )}{3 b d^3 (b c-a d)^2 (c+d x)^{3/2}}-\frac {(5 b c+a d) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{b^{3/2} d^{7/2}} \]
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Leaf count of result is larger than twice the leaf count of optimal. \(927\) vs. \(2(150)=300\).
Time = 1.67 (sec) , antiderivative size = 928, normalized size of antiderivative = 5.33
method | result | size |
default | \(-\frac {\sqrt {b x +a}\, \left (3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} d^{5} x^{2}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b c \,d^{4} x^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{2} d^{3} x^{2}+15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{3} d^{2} x^{2}+6 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} c \,d^{4} x +18 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,c^{2} d^{3} x -54 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{3} d^{2} x +30 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{4} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} d^{4} x^{2}+12 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b c \,d^{3} x^{2}-6 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{2} d^{2} x^{2}+3 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{3} c^{2} d^{3}+9 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a^{2} b \,c^{3} d^{2}-27 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) a \,b^{2} c^{4} d +15 \ln \left (\frac {2 b d x +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}+a d +b c}{2 \sqrt {b d}}\right ) b^{3} c^{5}-12 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c \,d^{3} x +60 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{2} d^{2} x -40 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{3} d x -6 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, a^{2} c^{2} d^{2}+44 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, a b \,c^{3} d -30 \sqrt {b d}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, b^{2} c^{4}\right )}{6 \sqrt {b d}\, b \left (a d -b c \right )^{2} \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, d^{3} \left (d x +c \right )^{\frac {3}{2}}}\) | \(928\) |
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Leaf count of result is larger than twice the leaf count of optimal. 441 vs. \(2 (150) = 300\).
Time = 0.38 (sec) , antiderivative size = 896, normalized size of antiderivative = 5.15 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\left [\frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, 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 (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{12 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}, \frac {3 \, {\left (5 \, b^{3} c^{5} - 9 \, a b^{2} c^{4} d + 3 \, a^{2} b c^{3} d^{2} + a^{3} c^{2} d^{3} + {\left (5 \, b^{3} c^{3} d^{2} - 9 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4} + a^{3} d^{5}\right )} x^{2} + 2 \, {\left (5 \, b^{3} c^{4} d - 9 \, a b^{2} c^{3} d^{2} + 3 \, 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 (15 \, b^{3} c^{4} d - 22 \, a b^{2} c^{3} d^{2} + 3 \, a^{2} b c^{2} d^{3} + 3 \, {\left (b^{3} c^{2} d^{3} - 2 \, a b^{2} c d^{4} + a^{2} b d^{5}\right )} x^{2} + 2 \, {\left (10 \, b^{3} c^{3} d^{2} - 15 \, a b^{2} c^{2} d^{3} + 3 \, a^{2} b c d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{6 \, {\left (b^{4} c^{4} d^{4} - 2 \, a b^{3} c^{3} d^{5} + a^{2} b^{2} c^{2} d^{6} + {\left (b^{4} c^{2} d^{6} - 2 \, a b^{3} c d^{7} + a^{2} b^{2} d^{8}\right )} x^{2} + 2 \, {\left (b^{4} c^{3} d^{5} - 2 \, a b^{3} c^{2} d^{6} + a^{2} b^{2} c d^{7}\right )} x\right )}}\right ] \]
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\[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^{3}}{\sqrt {a + b x} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 373 vs. \(2 (150) = 300\).
Time = 0.37 (sec) , antiderivative size = 373, normalized size of antiderivative = 2.14 \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\frac {{\left ({\left (b x + a\right )} {\left (\frac {3 \, {\left (b^{6} c^{2} d^{4} {\left | b \right |} - 2 \, a b^{5} c d^{5} {\left | b \right |} + a^{2} b^{4} d^{6} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}} + \frac {2 \, {\left (10 \, b^{7} c^{3} d^{3} {\left | b \right |} - 18 \, a b^{6} c^{2} d^{4} {\left | b \right |} + 9 \, a^{2} b^{5} c d^{5} {\left | b \right |} - 3 \, a^{3} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} + \frac {3 \, {\left (5 \, b^{8} c^{4} d^{2} {\left | b \right |} - 14 \, a b^{7} c^{3} d^{3} {\left | b \right |} + 12 \, a^{2} b^{6} c^{2} d^{4} {\left | b \right |} - 4 \, a^{3} b^{5} c d^{5} {\left | b \right |} + a^{4} b^{4} d^{6} {\left | b \right |}\right )}}{b^{7} c^{2} d^{5} - 2 \, a b^{6} c d^{6} + a^{2} b^{5} d^{7}}\right )} \sqrt {b x + a}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} + \frac {{\left (5 \, b c {\left | b \right |} + a d {\left | b \right |}\right )} \log \left ({\left | -\sqrt {b d} \sqrt {b x + a} + \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d} \right |}\right )}{\sqrt {b d} b^{2} d^{3}} \]
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Timed out. \[ \int \frac {x^3}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx=\int \frac {x^3}{\sqrt {a+b\,x}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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