Integrand size = 19, antiderivative size = 101 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=-\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 \sqrt {c+d x}} \]
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Time = 0.01 (sec) , antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.105, Rules used = {47, 37} \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\frac {16 d^2 \sqrt {a+b x}}{3 \sqrt {c+d x} (b c-a d)^3}+\frac {8 d}{3 \sqrt {a+b x} \sqrt {c+d x} (b c-a d)^2}-\frac {2}{3 (a+b x)^{3/2} \sqrt {c+d x} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}-\frac {(4 d) \int \frac {1}{(a+b x)^{3/2} (c+d x)^{3/2}} \, dx}{3 (b c-a d)} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {\left (8 d^2\right ) \int \frac {1}{\sqrt {a+b x} (c+d x)^{3/2}} \, dx}{3 (b c-a d)^2} \\ & = -\frac {2}{3 (b c-a d) (a+b x)^{3/2} \sqrt {c+d x}}+\frac {8 d}{3 (b c-a d)^2 \sqrt {a+b x} \sqrt {c+d x}}+\frac {16 d^2 \sqrt {a+b x}}{3 (b c-a d)^3 \sqrt {c+d x}} \\ \end{align*}
Time = 0.10 (sec) , antiderivative size = 75, normalized size of antiderivative = 0.74 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\frac {2 \left (3 a^2 d^2+6 a b d (c+2 d x)+b^2 \left (-c^2+4 c d x+8 d^2 x^2\right )\right )}{3 (b c-a d)^3 (a+b x)^{3/2} \sqrt {c+d x}} \]
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Time = 0.79 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(-\frac {2 \left (8 d^{2} x^{2} b^{2}+12 x a b \,d^{2}+4 x \,b^{2} c d +3 a^{2} d^{2}+6 a b c d -b^{2} c^{2}\right )}{3 \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}\, \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
default | \(-\frac {2}{3 \left (-a d +b c \right ) \left (b x +a \right )^{\frac {3}{2}} \sqrt {d x +c}}-\frac {4 d \left (-\frac {2}{\left (-a d +b c \right ) \sqrt {b x +a}\, \sqrt {d x +c}}+\frac {4 d \sqrt {b x +a}}{\left (-a d +b c \right ) \sqrt {d x +c}\, \left (a d -b c \right )}\right )}{3 \left (-a d +b c \right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 273 vs. \(2 (83) = 166\).
Time = 0.34 (sec) , antiderivative size = 273, normalized size of antiderivative = 2.70 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\frac {2 \, {\left (8 \, b^{2} d^{2} x^{2} - b^{2} c^{2} + 6 \, a b c d + 3 \, a^{2} d^{2} + 4 \, {\left (b^{2} c d + 3 \, a b d^{2}\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c}}{3 \, {\left (a^{2} b^{3} c^{4} - 3 \, a^{3} b^{2} c^{3} d + 3 \, a^{4} b c^{2} d^{2} - a^{5} c d^{3} + {\left (b^{5} c^{3} d - 3 \, a b^{4} c^{2} d^{2} + 3 \, a^{2} b^{3} c d^{3} - a^{3} b^{2} d^{4}\right )} x^{3} + {\left (b^{5} c^{4} - a b^{4} c^{3} d - 3 \, a^{2} b^{3} c^{2} d^{2} + 5 \, a^{3} b^{2} c d^{3} - 2 \, a^{4} b d^{4}\right )} x^{2} + {\left (2 \, a b^{4} c^{4} - 5 \, a^{2} b^{3} c^{3} d + 3 \, a^{3} b^{2} c^{2} d^{2} + a^{4} b c d^{3} - a^{5} d^{4}\right )} x\right )}} \]
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\[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\int \frac {1}{\left (a + b x\right )^{\frac {5}{2}} \left (c + d x\right )^{\frac {3}{2}}}\, dx \]
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Exception generated. \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 368 vs. \(2 (83) = 166\).
Time = 0.38 (sec) , antiderivative size = 368, normalized size of antiderivative = 3.64 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=\frac {2 \, \sqrt {b x + a} b^{2} d^{2}}{{\left (b^{3} c^{3} {\left | b \right |} - 3 \, a b^{2} c^{2} d {\left | b \right |} + 3 \, a^{2} b c d^{2} {\left | b \right |} - a^{3} d^{3} {\left | b \right |}\right )} \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}} + \frac {4 \, {\left (5 \, \sqrt {b d} b^{6} c^{2} d - 10 \, \sqrt {b d} a b^{5} c d^{2} + 5 \, \sqrt {b d} a^{2} b^{4} d^{3} - 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} b^{4} c d + 12 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2} a b^{3} d^{2} + 3 \, \sqrt {b d} {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{4} b^{2} d\right )}}{3 \, {\left (b^{2} c^{2} {\left | b \right |} - 2 \, a b c d {\left | b \right |} + a^{2} d^{2} {\left | b \right |}\right )} {\left (b^{2} c - a b d - {\left (\sqrt {b d} \sqrt {b x + a} - \sqrt {b^{2} c + {\left (b x + a\right )} b d - a b d}\right )}^{2}\right )}^{3}} \]
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Time = 1.26 (sec) , antiderivative size = 141, normalized size of antiderivative = 1.40 \[ \int \frac {1}{(a+b x)^{5/2} (c+d x)^{3/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {8\,x\,\left (3\,a\,d+b\,c\right )}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {16\,b\,d\,x^2}{3\,{\left (a\,d-b\,c\right )}^3}+\frac {6\,a^2\,d^2+12\,a\,b\,c\,d-2\,b^2\,c^2}{3\,b\,d\,{\left (a\,d-b\,c\right )}^3}\right )}{x^2\,\sqrt {a+b\,x}+\frac {a\,c\,\sqrt {a+b\,x}}{b\,d}+\frac {x\,\left (a\,d+b\,c\right )\,\sqrt {a+b\,x}}{b\,d}} \]
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