Integrand size = 19, antiderivative size = 101 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=-\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac {32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}-\frac {128 d^2 (c+d x)^{9/4}}{1989 (b c-a d)^3 (a+b x)^{9/4}} \]
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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} \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=-\frac {128 d^2 (c+d x)^{9/4}}{1989 (a+b x)^{9/4} (b c-a d)^3}+\frac {32 d (c+d x)^{9/4}}{221 (a+b x)^{13/4} (b c-a d)^2}-\frac {4 (c+d x)^{9/4}}{17 (a+b x)^{17/4} (b c-a d)} \]
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Rule 37
Rule 47
Rubi steps \begin{align*} \text {integral}& = -\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}-\frac {(8 d) \int \frac {(c+d x)^{5/4}}{(a+b x)^{17/4}} \, dx}{17 (b c-a d)} \\ & = -\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac {32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}+\frac {\left (32 d^2\right ) \int \frac {(c+d x)^{5/4}}{(a+b x)^{13/4}} \, dx}{221 (b c-a d)^2} \\ & = -\frac {4 (c+d x)^{9/4}}{17 (b c-a d) (a+b x)^{17/4}}+\frac {32 d (c+d x)^{9/4}}{221 (b c-a d)^2 (a+b x)^{13/4}}-\frac {128 d^2 (c+d x)^{9/4}}{1989 (b c-a d)^3 (a+b x)^{9/4}} \\ \end{align*}
Time = 0.87 (sec) , antiderivative size = 77, normalized size of antiderivative = 0.76 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=-\frac {4 (c+d x)^{9/4} \left (221 a^2 d^2+34 a b d (-9 c+4 d x)+b^2 \left (117 c^2-72 c d x+32 d^2 x^2\right )\right )}{1989 (b c-a d)^3 (a+b x)^{17/4}} \]
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Time = 0.33 (sec) , antiderivative size = 105, normalized size of antiderivative = 1.04
method | result | size |
gosper | \(\frac {4 \left (d x +c \right )^{\frac {9}{4}} \left (32 d^{2} x^{2} b^{2}+136 x a b \,d^{2}-72 x \,b^{2} c d +221 a^{2} d^{2}-306 a b c d +117 b^{2} c^{2}\right )}{1989 \left (b x +a \right )^{\frac {17}{4}} \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}\) | \(105\) |
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Leaf count of result is larger than twice the leaf count of optimal. 426 vs. \(2 (83) = 166\).
Time = 0.30 (sec) , antiderivative size = 426, normalized size of antiderivative = 4.22 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=-\frac {4 \, {\left (32 \, b^{2} d^{4} x^{4} + 117 \, b^{2} c^{4} - 306 \, a b c^{3} d + 221 \, a^{2} c^{2} d^{2} - 8 \, {\left (b^{2} c d^{3} - 17 \, a b d^{4}\right )} x^{3} + {\left (5 \, b^{2} c^{2} d^{2} - 34 \, a b c d^{3} + 221 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (81 \, b^{2} c^{3} d - 238 \, a b c^{2} d^{2} + 221 \, a^{2} c d^{3}\right )} x\right )} {\left (b x + a\right )}^{\frac {3}{4}} {\left (d x + c\right )}^{\frac {1}{4}}}{1989 \, {\left (a^{5} b^{3} c^{3} - 3 \, a^{6} b^{2} c^{2} d + 3 \, a^{7} b c d^{2} - a^{8} d^{3} + {\left (b^{8} c^{3} - 3 \, a b^{7} c^{2} d + 3 \, a^{2} b^{6} c d^{2} - a^{3} b^{5} d^{3}\right )} x^{5} + 5 \, {\left (a b^{7} c^{3} - 3 \, a^{2} b^{6} c^{2} d + 3 \, a^{3} b^{5} c d^{2} - a^{4} b^{4} d^{3}\right )} x^{4} + 10 \, {\left (a^{2} b^{6} c^{3} - 3 \, a^{3} b^{5} c^{2} d + 3 \, a^{4} b^{4} c d^{2} - a^{5} b^{3} d^{3}\right )} x^{3} + 10 \, {\left (a^{3} b^{5} c^{3} - 3 \, a^{4} b^{4} c^{2} d + 3 \, a^{5} b^{3} c d^{2} - a^{6} b^{2} d^{3}\right )} x^{2} + 5 \, {\left (a^{4} b^{4} c^{3} - 3 \, a^{5} b^{3} c^{2} d + 3 \, a^{6} b^{2} c d^{2} - a^{7} b d^{3}\right )} x\right )}} \]
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Timed out. \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=\text {Timed out} \]
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\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {21}{4}}} \,d x } \]
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\[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=\int { \frac {{\left (d x + c\right )}^{\frac {5}{4}}}{{\left (b x + a\right )}^{\frac {21}{4}}} \,d x } \]
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Time = 1.06 (sec) , antiderivative size = 268, normalized size of antiderivative = 2.65 \[ \int \frac {(c+d x)^{5/4}}{(a+b x)^{21/4}} \, dx=\frac {{\left (c+d\,x\right )}^{1/4}\,\left (\frac {884\,a^2\,c^2\,d^2-1224\,a\,b\,c^3\,d+468\,b^2\,c^4}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {x^2\,\left (884\,a^2\,d^4-136\,a\,b\,c\,d^3+20\,b^2\,c^2\,d^2\right )}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}+\frac {128\,d^4\,x^4}{1989\,b^2\,{\left (a\,d-b\,c\right )}^3}+\frac {32\,d^3\,x^3\,\left (17\,a\,d-b\,c\right )}{1989\,b^3\,{\left (a\,d-b\,c\right )}^3}+\frac {8\,c\,d\,x\,\left (221\,a^2\,d^2-238\,a\,b\,c\,d+81\,b^2\,c^2\right )}{1989\,b^4\,{\left (a\,d-b\,c\right )}^3}\right )}{x^4\,{\left (a+b\,x\right )}^{1/4}+\frac {a^4\,{\left (a+b\,x\right )}^{1/4}}{b^4}+\frac {6\,a^2\,x^2\,{\left (a+b\,x\right )}^{1/4}}{b^2}+\frac {4\,a\,x^3\,{\left (a+b\,x\right )}^{1/4}}{b}+\frac {4\,a^3\,x\,{\left (a+b\,x\right )}^{1/4}}{b^3}} \]
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