Integrand size = 19, antiderivative size = 171 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=-\frac {2}{7 (b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}+\frac {16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {128 d^3}{35 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {256 d^4 \sqrt {a+b x}}{35 (b c-a d)^5 \sqrt {c+d x}} \]
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Time = 0.03 (sec) , antiderivative size = 171, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {256 d^4 \sqrt {a+b x}}{35 \sqrt {c+d x} (b c-a d)^5}+\frac {128 d^3}{35 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^4}-\frac {32 d^2}{35 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^3}+\frac {16 d}{35 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{7 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)} \]
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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} \sqrt {c+d x}}-\frac {(8 d) \int \frac {1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx}{7 (b c-a d)} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}+\frac {16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {\left (48 d^2\right ) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^2} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}+\frac {16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {\left (64 d^3\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^3} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}+\frac {16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {128 d^3}{35 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {\left (128 d^4\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{35 (b c-a d)^4} \\ & = -\frac {2}{7 (b c-a d) (a+b x)^{7/2} \sqrt {c+d x}}+\frac {16 d}{35 (b c-a d)^2 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {32 d^2}{35 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {128 d^3}{35 (b c-a d)^4 \sqrt {a+b x} \sqrt {c+d x}}+\frac {256 d^4 \sqrt {a+b x}}{35 (b c-a d)^5 \sqrt {c+d x}} \\ \end{align*}
Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.97 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {2 \left (35 a^4 d^4+140 a^3 b d^3 (c+2 d x)+70 a^2 b^2 d^2 \left (-c^2+4 c d x+8 d^2 x^2\right )+28 a b^3 d \left (c^3-2 c^2 d x+8 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (-5 c^4+8 c^3 d x-16 c^2 d^2 x^2+64 c d^3 x^3+128 d^4 x^4\right )\right )}{35 (b c-a d)^5 (a+b x)^{7/2} \sqrt {c+d x}} \]
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Time = 0.53 (sec) , antiderivative size = 185, normalized size of antiderivative = 1.08
method | result | size |
default | \(-\frac {2}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}}-\frac {8 d \left (-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}}-\frac {6 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\) | \(185\) |
gosper | \(-\frac {2 \left (128 d^{4} x^{4} b^{4}+448 a \,b^{3} d^{4} x^{3}+64 b^{4} c \,d^{3} x^{3}+560 a^{2} b^{2} d^{4} x^{2}+224 a \,b^{3} c \,d^{3} x^{2}-16 b^{4} c^{2} d^{2} x^{2}+280 a^{3} b \,d^{4} x +280 a^{2} b^{2} c \,d^{3} x -56 a \,b^{3} c^{2} d^{2} x +8 b^{4} c^{3} d x +35 a^{4} d^{4}+140 a^{3} b c \,d^{3}-70 a^{2} b^{2} c^{2} d^{2}+28 a \,b^{3} c^{3} d -5 b^{4} c^{4}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}\, \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(256\) |
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Leaf count of result is larger than twice the leaf count of optimal. 689 vs. \(2 (141) = 282\).
Time = 1.40 (sec) , antiderivative size = 689, normalized size of antiderivative = 4.03 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (128 \, b^{4} d^{4} x^{4} - 5 \, b^{4} c^{4} + 28 \, a b^{3} c^{3} d - 70 \, a^{2} b^{2} c^{2} d^{2} + 140 \, a^{3} b c d^{3} + 35 \, a^{4} d^{4} + 64 \, {\left (b^{4} c d^{3} + 7 \, a b^{3} d^{4}\right )} x^{3} - 16 \, {\left (b^{4} c^{2} d^{2} - 14 \, a b^{3} c d^{3} - 35 \, a^{2} b^{2} d^{4}\right )} x^{2} + 8 \, {\left (b^{4} c^{3} d - 7 \, a b^{3} c^{2} d^{2} + 35 \, a^{2} b^{2} c d^{3} + 35 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{35 \, {\left (a^{4} b^{5} c^{6} - 5 \, a^{5} b^{4} c^{5} d + 10 \, a^{6} b^{3} c^{4} d^{2} - 10 \, a^{7} b^{2} c^{3} d^{3} + 5 \, a^{8} b c^{2} d^{4} - a^{9} c d^{5} + {\left (b^{9} c^{5} d - 5 \, a b^{8} c^{4} d^{2} + 10 \, a^{2} b^{7} c^{3} d^{3} - 10 \, a^{3} b^{6} c^{2} d^{4} + 5 \, a^{4} b^{5} c d^{5} - a^{5} b^{4} d^{6}\right )} x^{5} + {\left (b^{9} c^{6} - a b^{8} c^{5} d - 10 \, a^{2} b^{7} c^{4} d^{2} + 30 \, a^{3} b^{6} c^{3} d^{3} - 35 \, a^{4} b^{5} c^{2} d^{4} + 19 \, a^{5} b^{4} c d^{5} - 4 \, a^{6} b^{3} d^{6}\right )} x^{4} + 2 \, {\left (2 \, a b^{8} c^{6} - 7 \, a^{2} b^{7} c^{5} d + 5 \, a^{3} b^{6} c^{4} d^{2} + 10 \, a^{4} b^{5} c^{3} d^{3} - 20 \, a^{5} b^{4} c^{2} d^{4} + 13 \, a^{6} b^{3} c d^{5} - 3 \, a^{7} b^{2} d^{6}\right )} x^{3} + 2 \, {\left (3 \, a^{2} b^{7} c^{6} - 13 \, a^{3} b^{6} c^{5} d + 20 \, a^{4} b^{5} c^{4} d^{2} - 10 \, a^{5} b^{4} c^{3} d^{3} - 5 \, a^{6} b^{3} c^{2} d^{4} + 7 \, a^{7} b^{2} c d^{5} - 2 \, a^{8} b d^{6}\right )} x^{2} + {\left (4 \, a^{3} b^{6} c^{6} - 19 \, a^{4} b^{5} c^{5} d + 35 \, a^{5} b^{4} c^{4} d^{2} - 30 \, a^{6} b^{3} c^{3} d^{3} + 10 \, a^{7} b^{2} c^{2} d^{4} + a^{8} b c d^{5} - a^{9} d^{6}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 1518 vs. \(2 (141) = 282\).
Time = 0.73 (sec) , antiderivative size = 1518, normalized size of antiderivative = 8.88 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
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Time = 1.52 (sec) , antiderivative size = 337, normalized size of antiderivative = 1.97 \[ \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {256\,b\,d^3\,x^4}{35\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^2\,x^3\,\left (7\,a\,d+b\,c\right )}{35\,{\left (a\,d-b\,c\right )}^5}+\frac {70\,a^4\,d^4+280\,a^3\,b\,c\,d^3-140\,a^2\,b^2\,c^2\,d^2+56\,a\,b^3\,c^3\,d-10\,b^4\,c^4}{35\,b^3\,d\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (560\,a^3\,b\,d^4+560\,a^2\,b^2\,c\,d^3-112\,a\,b^3\,c^2\,d^2+16\,b^4\,c^3\,d\right )}{35\,b^3\,d\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d\,x^2\,\left (35\,a^2\,d^2+14\,a\,b\,c\,d-b^2\,c^2\right )}{35\,b\,{\left (a\,d-b\,c\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^3\,c\,\sqrt {a+b\,x}}{b^3\,d}+\frac {x^3\,\left (3\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {3\,a\,x^2\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d}+\frac {a^2\,x\,\left (a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d}} \]
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