Integrand size = 22, antiderivative size = 189 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {4 a^2 c}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 c \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 c \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \sqrt {a+b x}}{3 b d (b c-a d)^4 \sqrt {c+d x}} \]
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Time = 0.11 (sec) , antiderivative size = 189, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {96, 91, 79, 37} \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {4 c \sqrt {a+b x} \left (-3 a^2 d^2-6 a b c d+b^2 c^2\right )}{3 b d \sqrt {c+d x} (b c-a d)^4}-\frac {4 c \sqrt {a+b x} \left (3 a^2 d^2+b^2 c^2\right )}{3 b^2 d (c+d x)^{3/2} (b c-a d)^3}-\frac {4 a^2 c}{b^2 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2}-\frac {2 x^3}{3 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)} \]
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Rule 37
Rule 79
Rule 91
Rule 96
Rubi steps \begin{align*} \text {integral}& = -\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {(2 c) \int \frac {x^2}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{b c-a d} \\ & = -\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {4 a^2 c}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {(4 c) \int \frac {-\frac {1}{2} a (b c+3 a d)+\frac {1}{2} b (b c-a d) x}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b^2 (b c-a d)^2} \\ & = -\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {4 a^2 c}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 c \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {\left (2 c \left (b^2 c^2-6 a b c d-3 a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 b d (b c-a d)^3} \\ & = -\frac {2 x^3}{3 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {4 a^2 c}{b^2 (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}-\frac {4 c \left (b^2 c^2+3 a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 d (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 c \left (b^2 c^2-6 a b c d-3 a^2 d^2\right ) \sqrt {a+b x}}{3 b d (b c-a d)^4 \sqrt {c+d x}} \\ \end{align*}
Time = 0.21 (sec) , antiderivative size = 92, normalized size of antiderivative = 0.49 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (c^3-\frac {9 a c^2 (c+d x)}{a+b x}-\frac {9 a^2 c (c+d x)^2}{(a+b x)^2}+\frac {a^3 (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \]
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Time = 0.60 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.75
method | result | size |
default | \(-\frac {2 \left (-x^{3} a^{3} d^{3}+9 x^{3} a^{2} b c \,d^{2}+9 x^{3} a \,b^{2} c^{2} d -x^{3} b^{3} c^{3}+6 a^{3} c \,d^{2} x^{2}+36 a^{2} b \,c^{2} d \,x^{2}+6 a \,b^{2} c^{3} x^{2}+24 a^{3} c^{2} d x +24 a^{2} b \,c^{3} x +16 c^{3} a^{3}\right )}{3 \left (a d -b c \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}\) | \(141\) |
gosper | \(-\frac {2 \left (-x^{3} a^{3} d^{3}+9 x^{3} a^{2} b c \,d^{2}+9 x^{3} a \,b^{2} c^{2} d -x^{3} b^{3} c^{3}+6 a^{3} c \,d^{2} x^{2}+36 a^{2} b \,c^{2} d \,x^{2}+6 a \,b^{2} c^{3} x^{2}+24 a^{3} c^{2} d x +24 a^{2} b \,c^{3} x +16 c^{3} a^{3}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(182\) |
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Leaf count of result is larger than twice the leaf count of optimal. 448 vs. \(2 (167) = 334\).
Time = 0.59 (sec) , antiderivative size = 448, normalized size of antiderivative = 2.37 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 \, {\left (16 \, a^{3} c^{3} - {\left (b^{3} c^{3} - 9 \, a b^{2} c^{2} d - 9 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{3} + 6 \, {\left (a b^{2} c^{3} + 6 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x^{2} + 24 \, {\left (a^{2} b c^{3} + a^{3} c^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
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\[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{3}}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 771 vs. \(2 (167) = 334\).
Time = 0.55 (sec) , antiderivative size = 771, normalized size of antiderivative = 4.08 \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {{\left (b^{7} c^{6} d {\left | b \right |} - 12 \, a b^{6} c^{5} d^{2} {\left | b \right |} + 30 \, a^{2} b^{5} c^{4} d^{3} {\left | b \right |} - 28 \, a^{3} b^{4} c^{3} d^{4} {\left | b \right |} + 9 \, a^{4} b^{3} c^{2} d^{5} {\left | b \right |}\right )} {\left (b x + a\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}} - \frac {9 \, {\left (a b^{7} c^{6} d {\left | b \right |} - 4 \, a^{2} b^{6} c^{5} d^{2} {\left | b \right |} + 6 \, a^{3} b^{5} c^{4} d^{3} {\left | b \right |} - 4 \, a^{4} b^{4} c^{3} d^{4} {\left | b \right |} + a^{5} b^{3} c^{2} d^{5} {\left | b \right |}\right )}}{b^{9} c^{7} d - 7 \, a b^{8} c^{6} d^{2} + 21 \, a^{2} b^{7} c^{5} d^{3} - 35 \, a^{3} b^{6} c^{4} d^{4} + 35 \, a^{4} b^{5} c^{3} d^{5} - 21 \, a^{5} b^{4} c^{2} d^{6} + 7 \, a^{6} b^{3} c d^{7} - a^{7} b^{2} d^{8}}\right )}}{3 \, {\left (b^{2} c + {\left (b x + a\right )} b d - a b d\right )}^{\frac {3}{2}}} - \frac {4 \, {\left (9 \, \sqrt {b d} a^{2} b^{5} c^{3} - 19 \, \sqrt {b d} a^{3} b^{4} c^{2} d + 11 \, \sqrt {b d} a^{4} b^{3} c d^{2} - \sqrt {b d} a^{5} b^{2} d^{3} - 18 \, \sqrt {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} a^{2} b^{3} c^{2} + 18 \, \sqrt {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} a^{3} b^{2} c d + 9 \, \sqrt {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 )}^{4} a^{2} b c - 3 \, \sqrt {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 )}^{4} a^{3} d\right )}}{3 \, {\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} {\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}} \]
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Timed out. \[ \int \frac {x^3}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^3}{{\left (a+b\,x\right )}^{5/2}\,{\left (c+d\,x\right )}^{5/2}} \,d x \]
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