Integrand size = 19, antiderivative size = 206 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {512 d^5 \sqrt {a+b x}}{63 (b c-a d)^6 \sqrt {c+d x}} \]
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Time = 0.04 (sec) , antiderivative size = 206, 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} \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {512 d^5 \sqrt {a+b x}}{63 \sqrt {c+d x} (b c-a d)^6}-\frac {256 d^4}{63 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^5}+\frac {64 d^3}{63 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)^4}-\frac {32 d^2}{63 (a+b x)^{5/2} \sqrt {c+d x} (b c-a d)^3}+\frac {20 d}{63 (a+b x)^{7/2} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{9 (a+b x)^{9/2} \sqrt {c+d x} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}-\frac {(10 d) \int \frac {1}{(a+b x)^{9/2} (c+d x)^{3/2}} \, dx}{9 (b c-a d)} \\ & = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}+\frac {\left (80 d^2\right ) \int \frac {1}{(a+b x)^{7/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^2} \\ & = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}-\frac {\left (32 d^3\right ) \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx}{21 (b c-a d)^3} \\ & = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}+\frac {\left (128 d^4\right ) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^4} \\ & = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {\left (256 d^5\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{63 (b c-a d)^5} \\ & = -\frac {2}{9 (b c-a d) (a+b x)^{9/2} \sqrt {c+d x}}+\frac {20 d}{63 (b c-a d)^2 (a+b x)^{7/2} \sqrt {c+d x}}-\frac {32 d^2}{63 (b c-a d)^3 (a+b x)^{5/2} \sqrt {c+d x}}+\frac {64 d^3}{63 (b c-a d)^4 (a+b x)^{3/2} \sqrt {c+d x}}-\frac {256 d^4}{63 (b c-a d)^5 \sqrt {a+b x} \sqrt {c+d x}}-\frac {512 d^5 \sqrt {a+b x}}{63 (b c-a d)^6 \sqrt {c+d x}} \\ \end{align*}
Time = 0.35 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.11 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2 \left (63 a^5 d^5+315 a^4 b d^4 (c+2 d x)+210 a^3 b^2 d^3 \left (-c^2+4 c d x+8 d^2 x^2\right )+126 a^2 b^3 d^2 \left (c^3-2 c^2 d x+8 c d^2 x^2+16 d^3 x^3\right )+9 a b^4 d \left (-5 c^4+8 c^3 d x-16 c^2 d^2 x^2+64 c d^3 x^3+128 d^4 x^4\right )+b^5 \left (7 c^5-10 c^4 d x+16 c^3 d^2 x^2-32 c^2 d^3 x^3+128 c d^4 x^4+256 d^5 x^5\right )\right )}{63 (b c-a d)^6 (a+b x)^{9/2} \sqrt {c+d x}} \]
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Time = 0.53 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.09
method | result | size |
default | \(-\frac {2}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}} \sqrt {d x +c}}-\frac {10 d \left (-\frac {2}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}} \sqrt {d x +c}}-\frac {8 d \left (-\frac {2}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}} \sqrt {d x +c}}-\frac {6 d \left (-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\) | \(225\) |
gosper | \(-\frac {2 \left (256 x^{5} b^{5} d^{5}+1152 x^{4} a \,b^{4} d^{5}+128 x^{4} b^{5} c \,d^{4}+2016 x^{3} a^{2} b^{3} d^{5}+576 x^{3} a \,b^{4} c \,d^{4}-32 x^{3} b^{5} c^{2} d^{3}+1680 x^{2} a^{3} b^{2} d^{5}+1008 x^{2} a^{2} b^{3} c \,d^{4}-144 x^{2} a \,b^{4} c^{2} d^{3}+16 x^{2} b^{5} c^{3} d^{2}+630 x \,a^{4} b \,d^{5}+840 x \,a^{3} b^{2} c \,d^{4}-252 x \,a^{2} b^{3} c^{2} d^{3}+72 x a \,b^{4} c^{3} d^{2}-10 x \,b^{5} c^{4} d +63 a^{5} d^{5}+315 a^{4} b c \,d^{4}-210 a^{3} b^{2} c^{2} d^{3}+126 a^{2} b^{3} c^{3} d^{2}-45 a \,b^{4} c^{4} d +7 b^{5} c^{5}\right )}{63 \left (b x +a \right )^{\frac {9}{2}} \sqrt {d x +c}\, \left (a^{6} d^{6}-6 a^{5} b c \,d^{5}+15 a^{4} b^{2} c^{2} d^{4}-20 a^{3} b^{3} c^{3} d^{3}+15 a^{2} b^{4} c^{4} d^{2}-6 a \,b^{5} c^{5} d +b^{6} c^{6}\right )}\) | \(356\) |
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Leaf count of result is larger than twice the leaf count of optimal. 955 vs. \(2 (170) = 340\).
Time = 3.41 (sec) , antiderivative size = 955, normalized size of antiderivative = 4.64 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {2 \, {\left (256 \, b^{5} d^{5} x^{5} + 7 \, b^{5} c^{5} - 45 \, a b^{4} c^{4} d + 126 \, a^{2} b^{3} c^{3} d^{2} - 210 \, a^{3} b^{2} c^{2} d^{3} + 315 \, a^{4} b c d^{4} + 63 \, a^{5} d^{5} + 128 \, {\left (b^{5} c d^{4} + 9 \, a b^{4} d^{5}\right )} x^{4} - 32 \, {\left (b^{5} c^{2} d^{3} - 18 \, a b^{4} c d^{4} - 63 \, a^{2} b^{3} d^{5}\right )} x^{3} + 16 \, {\left (b^{5} c^{3} d^{2} - 9 \, a b^{4} c^{2} d^{3} + 63 \, a^{2} b^{3} c d^{4} + 105 \, a^{3} b^{2} d^{5}\right )} x^{2} - 2 \, {\left (5 \, b^{5} c^{4} d - 36 \, a b^{4} c^{3} d^{2} + 126 \, a^{2} b^{3} c^{2} d^{3} - 420 \, a^{3} b^{2} c d^{4} - 315 \, a^{4} b d^{5}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{63 \, {\left (a^{5} b^{6} c^{7} - 6 \, a^{6} b^{5} c^{6} d + 15 \, a^{7} b^{4} c^{5} d^{2} - 20 \, a^{8} b^{3} c^{4} d^{3} + 15 \, a^{9} b^{2} c^{3} d^{4} - 6 \, a^{10} b c^{2} d^{5} + a^{11} c d^{6} + {\left (b^{11} c^{6} d - 6 \, a b^{10} c^{5} d^{2} + 15 \, a^{2} b^{9} c^{4} d^{3} - 20 \, a^{3} b^{8} c^{3} d^{4} + 15 \, a^{4} b^{7} c^{2} d^{5} - 6 \, a^{5} b^{6} c d^{6} + a^{6} b^{5} d^{7}\right )} x^{6} + {\left (b^{11} c^{7} - a b^{10} c^{6} d - 15 \, a^{2} b^{9} c^{5} d^{2} + 55 \, a^{3} b^{8} c^{4} d^{3} - 85 \, a^{4} b^{7} c^{3} d^{4} + 69 \, a^{5} b^{6} c^{2} d^{5} - 29 \, a^{6} b^{5} c d^{6} + 5 \, a^{7} b^{4} d^{7}\right )} x^{5} + 5 \, {\left (a b^{10} c^{7} - 4 \, a^{2} b^{9} c^{6} d + 3 \, a^{3} b^{8} c^{5} d^{2} + 10 \, a^{4} b^{7} c^{4} d^{3} - 25 \, a^{5} b^{6} c^{3} d^{4} + 24 \, a^{6} b^{5} c^{2} d^{5} - 11 \, a^{7} b^{4} c d^{6} + 2 \, a^{8} b^{3} d^{7}\right )} x^{4} + 10 \, {\left (a^{2} b^{9} c^{7} - 5 \, a^{3} b^{8} c^{6} d + 9 \, a^{4} b^{7} c^{5} d^{2} - 5 \, a^{5} b^{6} c^{4} d^{3} - 5 \, a^{6} b^{5} c^{3} d^{4} + 9 \, a^{7} b^{4} c^{2} d^{5} - 5 \, a^{8} b^{3} c d^{6} + a^{9} b^{2} d^{7}\right )} x^{3} + 5 \, {\left (2 \, a^{3} b^{8} c^{7} - 11 \, a^{4} b^{7} c^{6} d + 24 \, a^{5} b^{6} c^{5} d^{2} - 25 \, a^{6} b^{5} c^{4} d^{3} + 10 \, a^{7} b^{4} c^{3} d^{4} + 3 \, a^{8} b^{3} c^{2} d^{5} - 4 \, a^{9} b^{2} c d^{6} + a^{10} b d^{7}\right )} x^{2} + {\left (5 \, a^{4} b^{7} c^{7} - 29 \, a^{5} b^{6} c^{6} d + 69 \, a^{6} b^{5} c^{5} d^{2} - 85 \, a^{7} b^{4} c^{4} d^{3} + 55 \, a^{8} b^{3} c^{3} d^{4} - 15 \, a^{9} b^{2} c^{2} d^{5} - a^{10} b c d^{6} + a^{11} d^{7}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {11}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 2438 vs. \(2 (170) = 340\).
Time = 1.13 (sec) , antiderivative size = 2438, normalized size of antiderivative = 11.83 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=\text {Too large to display} \]
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Time = 1.83 (sec) , antiderivative size = 454, normalized size of antiderivative = 2.20 \[ \int \frac {1}{(a+b x)^{11/2} (c+d x)^{3/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {126\,a^5\,d^5+630\,a^4\,b\,c\,d^4-420\,a^3\,b^2\,c^2\,d^3+252\,a^2\,b^3\,c^3\,d^2-90\,a\,b^4\,c^4\,d+14\,b^5\,c^5}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {512\,b\,d^4\,x^5}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {256\,d^3\,x^4\,\left (9\,a\,d+b\,c\right )}{63\,{\left (a\,d-b\,c\right )}^6}+\frac {x\,\left (1260\,a^4\,b\,d^5+1680\,a^3\,b^2\,c\,d^4-504\,a^2\,b^3\,c^2\,d^3+144\,a\,b^4\,c^3\,d^2-20\,b^5\,c^4\,d\right )}{63\,b^4\,d\,{\left (a\,d-b\,c\right )}^6}+\frac {64\,d^2\,x^3\,\left (63\,a^2\,d^2+18\,a\,b\,c\,d-b^2\,c^2\right )}{63\,b\,{\left (a\,d-b\,c\right )}^6}+\frac {32\,d\,x^2\,\left (105\,a^3\,d^3+63\,a^2\,b\,c\,d^2-9\,a\,b^2\,c^2\,d+b^3\,c^3\right )}{63\,b^2\,{\left (a\,d-b\,c\right )}^6}\right )}{x^5\,\sqrt {a+b\,x}+\frac {a^4\,c\,\sqrt {a+b\,x}}{b^4\,d}+\frac {x^4\,\left (4\,a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}+\frac {2\,a\,x^3\,\left (3\,a\,d+2\,b\,c\right )\,\sqrt {a+b\,x}}{b^2\,d}+\frac {a^3\,x\,\left (a\,d+4\,b\,c\right )\,\sqrt {a+b\,x}}{b^4\,d}+\frac {2\,a^2\,x^2\,\left (2\,a\,d+3\,b\,c\right )\,\sqrt {a+b\,x}}{b^3\,d}} \]
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