Integrand size = 22, antiderivative size = 186 \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=-\frac {2 a^2}{3 b^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 a c}{b (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {a+b x}}{3 b (b c-a d)^4 \sqrt {c+d x}} \]
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Time = 0.12 (sec) , antiderivative size = 186, 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 = {91, 79, 47, 37} \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {4 \sqrt {a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b \sqrt {c+d x} (b c-a d)^4}+\frac {2 \sqrt {a+b x} \left (a^2 d^2+6 a b c d+b^2 c^2\right )}{3 b^2 (c+d x)^{3/2} (b c-a d)^3}-\frac {2 a^2}{3 b^2 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)}+\frac {4 a c}{b \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^2} \]
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Rule 37
Rule 47
Rule 79
Rule 91
Rubi steps \begin{align*} \text {integral}& = -\frac {2 a^2}{3 b^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {2 \int \frac {-\frac {3}{2} a (b c+a d)+\frac {3}{2} b (b c-a d) x}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{3 b^2 (b c-a d)} \\ & = -\frac {2 a^2}{3 b^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 a c}{b (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\left (b^2 c^2+6 a b c d+a^2 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{b^2 (b c-a d)^2} \\ & = -\frac {2 a^2}{3 b^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 a c}{b (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 (b c-a d)^3 (c+d x)^{3/2}}+\frac {\left (2 \left (b^2 c^2+6 a b c d+a^2 d^2\right )\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 b (b c-a d)^3} \\ & = -\frac {2 a^2}{3 b^2 (b c-a d) (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {4 a c}{b (b c-a d)^2 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {2 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {a+b x}}{3 b^2 (b c-a d)^3 (c+d x)^{3/2}}+\frac {4 \left (b^2 c^2+6 a b c d+a^2 d^2\right ) \sqrt {a+b x}}{3 b (b c-a d)^4 \sqrt {c+d x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.72 \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 (a+b x)^{3/2} \left (-c^2 d+\frac {3 b c^2 (c+d x)}{a+b x}+\frac {6 a c d (c+d x)}{a+b x}+\frac {6 a b c (c+d x)^2}{(a+b x)^2}+\frac {3 a^2 d (c+d x)^2}{(a+b x)^2}-\frac {a^2 b (c+d x)^3}{(a+b x)^3}\right )}{3 (b c-a d)^4 (c+d x)^{3/2}} \]
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Time = 1.74 (sec) , antiderivative size = 162, normalized size of antiderivative = 0.87
method | result | size |
default | \(\frac {\frac {4}{3} a^{2} b \,d^{3} x^{3}+8 a \,b^{2} c \,d^{2} x^{3}+\frac {4}{3} b^{3} c^{2} d \,x^{3}+2 a^{3} d^{3} x^{2}+14 a^{2} b c \,d^{2} x^{2}+14 a \,b^{2} c^{2} d \,x^{2}+2 b^{3} c^{3} x^{2}+8 x \,a^{3} c \,d^{2}+16 x \,a^{2} b \,c^{2} d +8 x a \,b^{2} c^{3}+\frac {16}{3} a^{3} c^{2} d +\frac {16}{3} a^{2} b \,c^{3}}{\left (a d -b c \right )^{4} \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}\) | \(162\) |
gosper | \(\frac {\frac {4}{3} a^{2} b \,d^{3} x^{3}+8 a \,b^{2} c \,d^{2} x^{3}+\frac {4}{3} b^{3} c^{2} d \,x^{3}+2 a^{3} d^{3} x^{2}+14 a^{2} b c \,d^{2} x^{2}+14 a \,b^{2} c^{2} d \,x^{2}+2 b^{3} c^{3} x^{2}+8 x \,a^{3} c \,d^{2}+16 x \,a^{2} b \,c^{2} d +8 x a \,b^{2} c^{3}+\frac {16}{3} a^{3} c^{2} d +\frac {16}{3} a^{2} b \,c^{3}}{\left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(203\) |
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Leaf count of result is larger than twice the leaf count of optimal. 468 vs. \(2 (164) = 328\).
Time = 0.60 (sec) , antiderivative size = 468, normalized size of antiderivative = 2.52 \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (8 \, a^{2} b c^{3} + 8 \, a^{3} c^{2} d + 2 \, {\left (b^{3} c^{2} d + 6 \, a b^{2} c d^{2} + a^{2} b d^{3}\right )} x^{3} + 3 \, {\left (b^{3} c^{3} + 7 \, a b^{2} c^{2} d + 7 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} x^{2} + 12 \, {\left (a b^{2} c^{3} + 2 \, a^{2} b c^{2} d + a^{3} c d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{4} c^{6} - 4 \, a^{3} b^{3} c^{5} d + 6 \, a^{4} b^{2} c^{4} d^{2} - 4 \, a^{5} b c^{3} d^{3} + a^{6} c^{2} d^{4} + {\left (b^{6} c^{4} d^{2} - 4 \, a b^{5} c^{3} d^{3} + 6 \, a^{2} b^{4} c^{2} d^{4} - 4 \, a^{3} b^{3} c d^{5} + a^{4} b^{2} d^{6}\right )} x^{4} + 2 \, {\left (b^{6} c^{5} d - 3 \, a b^{5} c^{4} d^{2} + 2 \, a^{2} b^{4} c^{3} d^{3} + 2 \, a^{3} b^{3} c^{2} d^{4} - 3 \, a^{4} b^{2} c d^{5} + a^{5} b d^{6}\right )} x^{3} + {\left (b^{6} c^{6} - 9 \, a^{2} b^{4} c^{4} d^{2} + 16 \, a^{3} b^{3} c^{3} d^{3} - 9 \, a^{4} b^{2} c^{2} d^{4} + a^{6} d^{6}\right )} x^{2} + 2 \, {\left (a b^{5} c^{6} - 3 \, a^{2} b^{4} c^{5} d + 2 \, a^{3} b^{3} c^{4} d^{2} + 2 \, a^{4} b^{2} c^{3} d^{3} - 3 \, a^{5} b c^{2} d^{4} + a^{6} c d^{5}\right )} x\right )}} \]
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\[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\int \frac {x^{2}}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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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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Leaf count of result is larger than twice the leaf count of optimal. 769 vs. \(2 (164) = 328\).
Time = 0.55 (sec) , antiderivative size = 769, normalized size of antiderivative = 4.13 \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {2 \, \sqrt {b x + a} {\left (\frac {2 \, {\left (b^{7} c^{5} d^{2} {\left | b \right |} - 6 \, a^{2} b^{5} c^{3} d^{4} {\left | b \right |} + 8 \, a^{3} b^{4} c^{2} d^{5} {\left | b \right |} - 3 \, a^{4} b^{3} c d^{6} {\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 {3 \, {\left (b^{8} c^{6} d {\left | b \right |} - 2 \, a b^{7} c^{5} d^{2} {\left | b \right |} - 2 \, a^{2} b^{6} c^{4} d^{3} {\left | b \right |} + 8 \, a^{3} b^{5} c^{3} d^{4} {\left | b \right |} - 7 \, a^{4} b^{4} c^{2} d^{5} {\left | b \right |} + 2 \, a^{5} b^{3} c d^{6} {\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 {8 \, {\left (3 \, \sqrt {b d} a b^{6} c^{3} - 5 \, \sqrt {b d} a^{2} b^{5} c^{2} d + \sqrt {b d} a^{3} b^{4} c d^{2} + \sqrt {b d} a^{4} b^{3} d^{3} - 6 \, \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 b^{4} c^{2} + 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 )}^{2} a^{2} b^{3} c d + 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 )}^{2} a^{3} b^{2} d^{2} + 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 b^{2} c\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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Time = 2.11 (sec) , antiderivative size = 248, normalized size of antiderivative = 1.33 \[ \int \frac {x^2}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {4\,x^3\,\left (a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{3\,d\,{\left (a\,d-b\,c\right )}^4}+\frac {x^2\,\left (6\,a^3\,d^3+42\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d+6\,b^3\,c^3\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,a^2\,c^2\,\left (a\,d+b\,c\right )}{3\,b\,d^2\,{\left (a\,d-b\,c\right )}^4}+\frac {8\,a\,c\,x\,{\left (a\,d+b\,c\right )}^2}{b\,d^2\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a\,c^2\,\sqrt {a+b\,x}}{b\,d^2}+\frac {x^2\,\left (a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {c\,x\,\left (2\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d^2}} \]
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