Integrand size = 19, antiderivative size = 136 \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=-\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}+\frac {32 d^3 \sqrt {c+d x}}{35 (b c-a d)^4 \sqrt {a+b x}} \]
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Time = 0.02 (sec) , antiderivative size = 136, normalized size of antiderivative = 1.00, number of steps used = 4, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\frac {32 d^3 \sqrt {c+d x}}{35 \sqrt {a+b x} (b c-a d)^4}-\frac {16 d^2 \sqrt {c+d x}}{35 (a+b x)^{3/2} (b c-a d)^3}+\frac {12 d \sqrt {c+d x}}{35 (a+b x)^{5/2} (b c-a d)^2}-\frac {2 \sqrt {c+d x}}{7 (a+b x)^{7/2} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}-\frac {(6 d) \int \frac {1}{(a+b x)^{7/2} \sqrt {c+d x}} \, dx}{7 (b c-a d)} \\ & = -\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}+\frac {\left (24 d^2\right ) \int \frac {1}{(a+b x)^{5/2} \sqrt {c+d x}} \, dx}{35 (b c-a d)^2} \\ & = -\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}-\frac {\left (16 d^3\right ) \int \frac {1}{(a+b x)^{3/2} \sqrt {c+d x}} \, dx}{35 (b c-a d)^3} \\ & = -\frac {2 \sqrt {c+d x}}{7 (b c-a d) (a+b x)^{7/2}}+\frac {12 d \sqrt {c+d x}}{35 (b c-a d)^2 (a+b x)^{5/2}}-\frac {16 d^2 \sqrt {c+d x}}{35 (b c-a d)^3 (a+b x)^{3/2}}+\frac {32 d^3 \sqrt {c+d x}}{35 (b c-a d)^4 \sqrt {a+b x}} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 116, normalized size of antiderivative = 0.85 \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (35 a^3 d^3-35 a^2 b d^2 (c-2 d x)+7 a b^2 d \left (3 c^2-4 c d x+8 d^2 x^2\right )+b^3 \left (-5 c^3+6 c^2 d x-8 c d^2 x^2+16 d^3 x^3\right )\right )}{35 (b c-a d)^4 (a+b x)^{7/2}} \]
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Time = 0.52 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.99
method | result | size |
default | \(-\frac {2 \sqrt {d x +c}}{7 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {7}{2}}}-\frac {6 d \left (-\frac {2 \sqrt {d x +c}}{5 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {5}{2}}}-\frac {4 d \left (-\frac {2 \sqrt {d x +c}}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}}}+\frac {4 d \sqrt {d x +c}}{3 \left (-a d +b c \right )^{2} \sqrt {b x +a}}\right )}{5 \left (-a d +b c \right )}\right )}{7 \left (-a d +b c \right )}\) | \(135\) |
gosper | \(\frac {2 \sqrt {d x +c}\, \left (16 d^{3} x^{3} b^{3}+56 x^{2} a \,b^{2} d^{3}-8 x^{2} b^{3} c \,d^{2}+70 x \,a^{2} b \,d^{3}-28 x a \,b^{2} c \,d^{2}+6 x \,b^{3} c^{2} d +35 a^{3} d^{3}-35 a^{2} b c \,d^{2}+21 a \,b^{2} c^{2} d -5 b^{3} c^{3}\right )}{35 \left (b x +a \right )^{\frac {7}{2}} \left (a^{4} d^{4}-4 a^{3} b c \,d^{3}+6 a^{2} b^{2} c^{2} d^{2}-4 a \,b^{3} c^{3} d +b^{4} c^{4}\right )}\) | \(171\) |
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Leaf count of result is larger than twice the leaf count of optimal. 419 vs. \(2 (112) = 224\).
Time = 0.63 (sec) , antiderivative size = 419, normalized size of antiderivative = 3.08 \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\frac {2 \, {\left (16 \, b^{3} d^{3} x^{3} - 5 \, b^{3} c^{3} + 21 \, a b^{2} c^{2} d - 35 \, a^{2} b c d^{2} + 35 \, a^{3} d^{3} - 8 \, {\left (b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{2} + 2 \, {\left (3 \, b^{3} c^{2} d - 14 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{35 \, {\left (a^{4} b^{4} c^{4} - 4 \, a^{5} b^{3} c^{3} d + 6 \, a^{6} b^{2} c^{2} d^{2} - 4 \, a^{7} b c d^{3} + a^{8} d^{4} + {\left (b^{8} c^{4} - 4 \, a b^{7} c^{3} d + 6 \, a^{2} b^{6} c^{2} d^{2} - 4 \, a^{3} b^{5} c d^{3} + a^{4} b^{4} d^{4}\right )} x^{4} + 4 \, {\left (a b^{7} c^{4} - 4 \, a^{2} b^{6} c^{3} d + 6 \, a^{3} b^{5} c^{2} d^{2} - 4 \, a^{4} b^{4} c d^{3} + a^{5} b^{3} d^{4}\right )} x^{3} + 6 \, {\left (a^{2} b^{6} c^{4} - 4 \, a^{3} b^{5} c^{3} d + 6 \, a^{4} b^{4} c^{2} d^{2} - 4 \, a^{5} b^{3} c d^{3} + a^{6} b^{2} d^{4}\right )} x^{2} + 4 \, {\left (a^{3} b^{5} c^{4} - 4 \, a^{4} b^{4} c^{3} d + 6 \, a^{5} b^{3} c^{2} d^{2} - 4 \, a^{6} b^{2} c d^{3} + a^{7} b d^{4}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {9}{2}} \sqrt {c + d x}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 386 vs. \(2 (112) = 224\).
Time = 0.35 (sec) , antiderivative size = 386, normalized size of antiderivative = 2.84 \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\frac {64 \, {\left (b^{6} c^{3} - 3 \, a b^{5} c^{2} d + 3 \, a^{2} b^{4} c d^{2} - a^{3} b^{3} d^{3} - 7 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c^{2} + 14 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} c d - 7 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a^{2} b^{2} d^{2} + 21 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} c - 21 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} a b d - 35 \, {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{6}\right )} \sqrt {b d} b^{4} d^{3}}{35 \, {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{7} {\left | b \right |}} \]
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Time = 1.26 (sec) , antiderivative size = 209, normalized size of antiderivative = 1.54 \[ \int \frac {1}{(a+b x)^{9/2} \sqrt {c+d x}} \, dx=\frac {\sqrt {c+d\,x}\,\left (\frac {32\,d^3\,x^3}{35\,{\left (a\,d-b\,c\right )}^4}+\frac {70\,a^3\,d^3-70\,a^2\,b\,c\,d^2+42\,a\,b^2\,c^2\,d-10\,b^3\,c^3}{35\,b^3\,{\left (a\,d-b\,c\right )}^4}+\frac {4\,d\,x\,\left (35\,a^2\,d^2-14\,a\,b\,c\,d+3\,b^2\,c^2\right )}{35\,b^2\,{\left (a\,d-b\,c\right )}^4}+\frac {16\,d^2\,x^2\,\left (7\,a\,d-b\,c\right )}{35\,b\,{\left (a\,d-b\,c\right )}^4}\right )}{x^3\,\sqrt {a+b\,x}+\frac {a^3\,\sqrt {a+b\,x}}{b^3}+\frac {3\,a\,x^2\,\sqrt {a+b\,x}}{b}+\frac {3\,a^2\,x\,\sqrt {a+b\,x}}{b^2}} \]
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