Integrand size = 19, antiderivative size = 171 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac {256 d^4 \sqrt {c+d x}}{315 (b c-a d)^5 \sqrt {a+b x}} \]
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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} \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {256 d^4 \sqrt {c+d x}}{315 \sqrt {a+b x} (b c-a d)^5}+\frac {128 d^3 \sqrt {c+d x}}{315 (a+b x)^{3/2} (b c-a d)^4}-\frac {32 d^2 \sqrt {c+d x}}{105 (a+b x)^{5/2} (b c-a d)^3}+\frac {16 d \sqrt {c+d x}}{63 (a+b x)^{7/2} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{9 (a+b x)^{9/2} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}-\frac {(8 d) \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx}{9 (b c-a d)} \\ & = -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}+\frac {\left (16 d^2\right ) \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx}{21 (b c-a d)^2} \\ & = -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}-\frac {\left (64 d^3\right ) \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{105 (b c-a d)^3} \\ & = -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}+\frac {\left (128 d^4\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{315 (b c-a d)^4} \\ & = -\frac {2 \sqrt {c+d x}}{9 (b c-a d) (a+b x)^{9/2}}+\frac {16 d \sqrt {c+d x}}{63 (b c-a d)^2 (a+b x)^{7/2}}-\frac {32 d^2 \sqrt {c+d x}}{105 (b c-a d)^3 (a+b x)^{5/2}}+\frac {128 d^3 \sqrt {c+d x}}{315 (b c-a d)^4 (a+b x)^{3/2}}-\frac {256 d^4 \sqrt {c+d x}}{315 (b c-a d)^5 \sqrt {a+b x}} \\ \end{align*}
Time = 0.26 (sec) , antiderivative size = 168, normalized size of antiderivative = 0.98 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x} \left (315 a^4 d^4-420 a^3 b d^3 (c-2 d x)+126 a^2 b^2 d^2 \left (3 c^2-4 c d x+8 d^2 x^2\right )+36 a b^3 d \left (-5 c^3+6 c^2 d x-8 c d^2 x^2+16 d^3 x^3\right )+b^4 \left (35 c^4-40 c^3 d x+48 c^2 d^2 x^2-64 c d^3 x^3+128 d^4 x^4\right )\right )}{315 (b c-a d)^5 (a+b x)^{9/2}} \]
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Time = 0.52 (sec) , antiderivative size = 175, normalized size of antiderivative = 1.02
method | result | size |
default | \(-\frac {2 \sqrt {d x +c}}{9 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {9}{2}}}-\frac {8 d \left (-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\right )}{9 \left (-a d +b c \right )}\) | \(175\) |
gosper | \(\frac {2 \sqrt {d x +c}\, \left (128 d^{4} x^{4} b^{4}+576 a \,b^{3} d^{4} x^{3}-64 b^{4} c \,d^{3} x^{3}+1008 a^{2} b^{2} d^{4} x^{2}-288 a \,b^{3} c \,d^{3} x^{2}+48 b^{4} c^{2} d^{2} x^{2}+840 a^{3} b \,d^{4} x -504 a^{2} b^{2} c \,d^{3} x +216 a \,b^{3} c^{2} d^{2} x -40 b^{4} c^{3} d x +315 a^{4} d^{4}-420 a^{3} b c \,d^{3}+378 a^{2} b^{2} c^{2} d^{2}-180 a \,b^{3} c^{3} d +35 b^{4} c^{4}\right )}{315 \left (b x +a \right )^{\frac {9}{2}} \left (a^{5} d^{5}-5 a^{4} b c \,d^{4}+10 a^{3} b^{2} c^{2} d^{3}-10 a^{2} b^{3} c^{3} d^{2}+5 a \,b^{4} c^{4} d -b^{5} c^{5}\right )}\) | \(256\) |
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Leaf count of result is larger than twice the leaf count of optimal. 638 vs. \(2 (141) = 282\).
Time = 2.05 (sec) , antiderivative size = 638, normalized size of antiderivative = 3.73 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {2 \, {\left (128 \, b^{4} d^{4} x^{4} + 35 \, b^{4} c^{4} - 180 \, a b^{3} c^{3} d + 378 \, a^{2} b^{2} c^{2} d^{2} - 420 \, a^{3} b c d^{3} + 315 \, a^{4} d^{4} - 64 \, {\left (b^{4} c d^{3} - 9 \, a b^{3} d^{4}\right )} x^{3} + 48 \, {\left (b^{4} c^{2} d^{2} - 6 \, a b^{3} c d^{3} + 21 \, a^{2} b^{2} d^{4}\right )} x^{2} - 8 \, {\left (5 \, b^{4} c^{3} d - 27 \, a b^{3} c^{2} d^{2} + 63 \, a^{2} b^{2} c d^{3} - 105 \, a^{3} b d^{4}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{315 \, {\left (a^{5} b^{5} c^{5} - 5 \, a^{6} b^{4} c^{4} d + 10 \, a^{7} b^{3} c^{3} d^{2} - 10 \, a^{8} b^{2} c^{2} d^{3} + 5 \, a^{9} b c d^{4} - a^{10} d^{5} + {\left (b^{10} c^{5} - 5 \, a b^{9} c^{4} d + 10 \, a^{2} b^{8} c^{3} d^{2} - 10 \, a^{3} b^{7} c^{2} d^{3} + 5 \, a^{4} b^{6} c d^{4} - a^{5} b^{5} d^{5}\right )} x^{5} + 5 \, {\left (a b^{9} c^{5} - 5 \, a^{2} b^{8} c^{4} d + 10 \, a^{3} b^{7} c^{3} d^{2} - 10 \, a^{4} b^{6} c^{2} d^{3} + 5 \, a^{5} b^{5} c d^{4} - a^{6} b^{4} d^{5}\right )} x^{4} + 10 \, {\left (a^{2} b^{8} c^{5} - 5 \, a^{3} b^{7} c^{4} d + 10 \, a^{4} b^{6} c^{3} d^{2} - 10 \, a^{5} b^{5} c^{2} d^{3} + 5 \, a^{6} b^{4} c d^{4} - a^{7} b^{3} d^{5}\right )} x^{3} + 10 \, {\left (a^{3} b^{7} c^{5} - 5 \, a^{4} b^{6} c^{4} d + 10 \, a^{5} b^{5} c^{3} d^{2} - 10 \, a^{6} b^{4} c^{2} d^{3} + 5 \, a^{7} b^{3} c d^{4} - a^{8} b^{2} d^{5}\right )} x^{2} + 5 \, {\left (a^{4} b^{6} c^{5} - 5 \, a^{5} b^{5} c^{4} d + 10 \, a^{6} b^{4} c^{3} d^{2} - 10 \, a^{7} b^{3} c^{2} d^{3} + 5 \, a^{8} b^{2} c d^{4} - a^{9} b d^{5}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {11}{2}} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 596 vs. \(2 (141) = 282\).
Time = 0.39 (sec) , antiderivative size = 596, normalized size of antiderivative = 3.49 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=-\frac {512 \, {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4} - 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{6} c^{3} + 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{5} c^{2} d - 27 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{4} c d^{2} + 9 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{3} b^{3} d^{3} + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{4} c^{2} - 72 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b^{3} c d + 36 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a^{2} b^{2} d^{2} - 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} b^{2} c + 84 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6} a b d + 126 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{8}\right )} \sqrt {b d} b^{5} d^{4}}{315 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{9} {\left | b \right |}} \]
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Time = 1.34 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.77 \[ \int \frac {1}{(a+b x)^{11/2} \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {256\,d^4\,x^4}{315\,{\left (a\,d-b\,c\right )}^5}+\frac {630\,a^4\,d^4-840\,a^3\,b\,c\,d^3+756\,a^2\,b^2\,c^2\,d^2-360\,a\,b^3\,c^3\,d+70\,b^4\,c^4}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {x\,\left (1680\,a^3\,b\,d^4-1008\,a^2\,b^2\,c\,d^3+432\,a\,b^3\,c^2\,d^2-80\,b^4\,c^3\,d\right )}{315\,b^4\,{\left (a\,d-b\,c\right )}^5}+\frac {128\,d^3\,x^3\,\left (9\,a\,d-b\,c\right )}{315\,b\,{\left (a\,d-b\,c\right )}^5}+\frac {32\,d^2\,x^2\,\left (21\,a^2\,d^2-6\,a\,b\,c\,d+b^2\,c^2\right )}{105\,b^2\,{\left (a\,d-b\,c\right )}^5}\right )}{x^4\,\sqrt {a+b\,x}+\frac {a^4\,\sqrt {a+b\,x}}{b^4}+\frac {6\,a^2\,x^2\,\sqrt {a+b\,x}}{b^2}+\frac {4\,a\,x^3\,\sqrt {a+b\,x}}{b}+\frac {4\,a^3\,x\,\sqrt {a+b\,x}}{b^3}} \]
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