Integrand size = 19, antiderivative size = 207 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {512 b d^4 \sqrt {a+b x}}{21 (b c-a d)^6 \sqrt {c+d x}} \]
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Time = 0.05 (sec) , antiderivative size = 207, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {512 b d^4 \sqrt {a+b x}}{21 \sqrt {c+d x} (b c-a d)^6}+\frac {256 d^4 \sqrt {a+b x}}{21 (c+d x)^{3/2} (b c-a d)^5}+\frac {64 d^3}{7 \sqrt {a+b x} (c+d x)^{3/2} (b c-a d)^4}-\frac {32 d^2}{21 (a+b x)^{3/2} (c+d x)^{3/2} (b c-a d)^3}+\frac {4 d}{7 (a+b x)^{5/2} (c+d x)^{3/2} (b c-a d)^2}-\frac {2}{7 (a+b x)^{7/2} (c+d x)^{3/2} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}-\frac {(10 d) \int \frac {1}{(a+b x)^{7/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}+\frac {\left (16 d^2\right ) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^2} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}-\frac {\left (32 d^3\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^3} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {\left (128 d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{5/2}} \, dx}{7 (b c-a d)^4} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {\left (256 b d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^5} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} (c+d x)^{3/2}}+\frac {4 d}{7 (b c-a d)^2 (a+b x)^{5/2} (c+d x)^{3/2}}-\frac {32 d^2}{21 (b c-a d)^3 (a+b x)^{3/2} (c+d x)^{3/2}}+\frac {64 d^3}{7 (b c-a d)^4 \sqrt {a+b x} (c+d x)^{3/2}}+\frac {256 d^4 \sqrt {a+b x}}{21 (b c-a d)^5 (c+d x)^{3/2}}+\frac {512 b d^4 \sqrt {a+b x}}{21 (b c-a d)^6 \sqrt {c+d x}} \\ \end{align*}
Time = 0.31 (sec) , antiderivative size = 233, normalized size of antiderivative = 1.13 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {2 \left (-7 a^5 d^5+35 a^4 b d^4 (3 c+2 d x)+70 a^3 b^2 d^3 \left (3 c^2+12 c d x+8 d^2 x^2\right )+70 a^2 b^3 d^2 \left (-c^3+6 c^2 d x+24 c d^2 x^2+16 d^3 x^3\right )+7 a b^4 d \left (3 c^4-8 c^3 d x+48 c^2 d^2 x^2+192 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (-3 c^5+6 c^4 d x-16 c^3 d^2 x^2+96 c^2 d^3 x^3+384 c d^4 x^4+256 d^5 x^5\right )\right )}{21 (b c-a d)^6 (a+b x)^{7/2} (c+d x)^{3/2}} \]
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Time = 0.53 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.04
method | result | size |
default | \(-\frac {2}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {10 d \left (-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {8 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \left (d x +c \right )^{\frac {3}{2}}}-\frac {2 d \left (-\frac {2}{\left (-a d +b c \right ) \left (d x +c \right )^{\frac {3}{2}} \sqrt {b x +a}}-\frac {4 d \left (-\frac {2 \sqrt {b x +a}}{3 \left (a d -b c \right ) \left (d x +c \right )^{\frac {3}{2}}}+\frac {4 b \sqrt {b x +a}}{3 \left (a d -b c \right )^{2} \sqrt {d x +c}}\right )}{-a d +b c}\right )}{-a d +b c}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\) | \(215\) |
gosper | \(-\frac {2 \left (-256 x^{5} b^{5} d^{5}-896 x^{4} a \,b^{4} d^{5}-384 x^{4} b^{5} c \,d^{4}-1120 x^{3} a^{2} b^{3} d^{5}-1344 x^{3} a \,b^{4} c \,d^{4}-96 x^{3} b^{5} c^{2} d^{3}-560 x^{2} a^{3} b^{2} d^{5}-1680 x^{2} a^{2} b^{3} c \,d^{4}-336 x^{2} a \,b^{4} c^{2} d^{3}+16 x^{2} b^{5} c^{3} d^{2}-70 x \,a^{4} b \,d^{5}-840 x \,a^{3} b^{2} c \,d^{4}-420 x \,a^{2} b^{3} c^{2} d^{3}+56 x a \,b^{4} c^{3} d^{2}-6 x \,b^{5} c^{4} d +7 a^{5} d^{5}-105 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+70 a^{2} b^{3} c^{3} d^{2}-21 a \,b^{4} c^{4} d +3 b^{5} c^{5}\right )}{21 \left (b x +a \right )^{\frac {7}{2}} \left (d x +c \right )^{\frac {3}{2}} \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}\) | \(356\) |
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Leaf count of result is larger than twice the leaf count of optimal. 999 vs. \(2 (171) = 342\).
Time = 4.12 (sec) , antiderivative size = 999, normalized size of antiderivative = 4.83 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {2 \, {\left (256 \, b^{5} d^{5} x^{5} - 3 \, b^{5} c^{5} + 21 \, a b^{4} c^{4} d - 70 \, a^{2} b^{3} c^{3} d^{2} + 210 \, a^{3} b^{2} c^{2} d^{3} + 105 \, a^{4} b c d^{4} - 7 \, a^{5} d^{5} + 128 \, {\left (3 \, b^{5} c d^{4} + 7 \, a b^{4} d^{5}\right )} x^{4} + 32 \, {\left (3 \, b^{5} c^{2} d^{3} + 42 \, a b^{4} c d^{4} + 35 \, a^{2} b^{3} d^{5}\right )} x^{3} - 16 \, {\left (b^{5} c^{3} d^{2} - 21 \, a b^{4} c^{2} d^{3} - 105 \, a^{2} b^{3} c d^{4} - 35 \, a^{3} b^{2} d^{5}\right )} x^{2} + 2 \, {\left (3 \, b^{5} c^{4} d - 28 \, a b^{4} c^{3} d^{2} + 210 \, a^{2} b^{3} c^{2} d^{3} + 420 \, a^{3} b^{2} c d^{4} + 35 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{21 \, {\left (a^{4} b^{6} c^{8} - 6 \, a^{5} b^{5} c^{7} d + 15 \, a^{6} b^{4} c^{6} d^{2} - 20 \, a^{7} b^{3} c^{5} d^{3} + 15 \, a^{8} b^{2} c^{4} d^{4} - 6 \, a^{9} b c^{3} d^{5} + a^{10} c^{2} d^{6} + {\left (b^{10} c^{6} d^{2} - 6 \, a b^{9} c^{5} d^{3} + 15 \, a^{2} b^{8} c^{4} d^{4} - 20 \, a^{3} b^{7} c^{3} d^{5} + 15 \, a^{4} b^{6} c^{2} d^{6} - 6 \, a^{5} b^{5} c d^{7} + a^{6} b^{4} d^{8}\right )} x^{6} + 2 \, {\left (b^{10} c^{7} d - 4 \, a b^{9} c^{6} d^{2} + 3 \, a^{2} b^{8} c^{5} d^{3} + 10 \, a^{3} b^{7} c^{4} d^{4} - 25 \, a^{4} b^{6} c^{3} d^{5} + 24 \, a^{5} b^{5} c^{2} d^{6} - 11 \, a^{6} b^{4} c d^{7} + 2 \, a^{7} b^{3} d^{8}\right )} x^{5} + {\left (b^{10} c^{8} + 2 \, a b^{9} c^{7} d - 27 \, a^{2} b^{8} c^{6} d^{2} + 64 \, a^{3} b^{7} c^{5} d^{3} - 55 \, a^{4} b^{6} c^{4} d^{4} - 6 \, a^{5} b^{5} c^{3} d^{5} + 43 \, a^{6} b^{4} c^{2} d^{6} - 28 \, a^{7} b^{3} c d^{7} + 6 \, a^{8} b^{2} d^{8}\right )} x^{4} + 4 \, {\left (a b^{9} c^{8} - 3 \, a^{2} b^{8} c^{7} d - 2 \, a^{3} b^{7} c^{6} d^{2} + 19 \, a^{4} b^{6} c^{5} d^{3} - 30 \, a^{5} b^{5} c^{4} d^{4} + 19 \, a^{6} b^{4} c^{3} d^{5} - 2 \, a^{7} b^{3} c^{2} d^{6} - 3 \, a^{8} b^{2} c d^{7} + a^{9} b d^{8}\right )} x^{3} + {\left (6 \, a^{2} b^{8} c^{8} - 28 \, a^{3} b^{7} c^{7} d + 43 \, a^{4} b^{6} c^{6} d^{2} - 6 \, a^{5} b^{5} c^{5} d^{3} - 55 \, a^{6} b^{4} c^{4} d^{4} + 64 \, a^{7} b^{3} c^{3} d^{5} - 27 \, a^{8} b^{2} c^{2} d^{6} + 2 \, a^{9} b c d^{7} + a^{10} d^{8}\right )} x^{2} + 2 \, {\left (2 \, a^{3} b^{7} c^{8} - 11 \, a^{4} b^{6} c^{7} d + 24 \, a^{5} b^{5} c^{6} d^{2} - 25 \, a^{6} b^{4} c^{5} d^{3} + 10 \, a^{7} b^{3} c^{4} d^{4} + 3 \, a^{8} b^{2} c^{3} d^{5} - 4 \, a^{9} b c^{2} d^{6} + a^{10} c d^{7}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{2}} \left (c + d x\right )^{\frac {5}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1964 vs. \(2 (171) = 342\).
Time = 1.06 (sec) , antiderivative size = 1964, normalized size of antiderivative = 9.49 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\text {Too large to display} \]
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Time = 1.78 (sec) , antiderivative size = 478, normalized size of antiderivative = 2.31 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{5/2}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {32\,x^2\,\left (35\,a^3\,d^3+105\,a^2\,b\,c\,d^2+21\,a\,b^2\,c^2\,d-b^3\,c^3\right )}{21\,b\,{\left (a\,d-b\,c\right )}^6}-\frac {14\,a^5\,d^5-210\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+140\,a^2\,b^3\,c^3\,d^2-42\,a\,b^4\,c^4\,d+6\,b^5\,c^5}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d\,x^3\,\left (35\,a^2\,d^2+42\,a\,b\,c\,d+3\,b^2\,c^2\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b^2\,d^3\,x^5}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,b\,d^2\,x^4\,\left (7\,a\,d+3\,b\,c\right )}{21\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (140\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4+840\,a^2\,b^3\,c^2\,d^3-112\,a\,b^4\,c^3\,d^2+12\,b^5\,c^4\,d\right )}{21\,b^3\,d^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {x^3\,\sqrt {a+b\,x}\,\left (3\,a^2\,d^2+6\,a\,b\,c\,d+b^2\,c^2\right )}{b^2\,d^2}+\frac {x^4\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {a^3\,c^2\,\sqrt {a+b\,x}}{b^3\,d^2}+\frac {a\,x^2\,\sqrt {a+b\,x}\,\left (a^2\,d^2+6\,a\,b\,c\,d+3\,b^2\,c^2\right )}{b^3\,d^2}+\frac {a^2\,c\,x\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d^2}} \]
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