Integrand size = 17, antiderivative size = 208 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {3 d^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}} \]
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Time = 0.08 (sec) , antiderivative size = 208, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.235, Rules used = {43, 44, 65, 214} \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\frac {3 d^5 \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (a+b x) (b c-a d)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (a+b x)^2 (b c-a d)^2}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (a+b x)^3 (b c-a d)}-\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5} \]
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Rule 43
Rule 44
Rule 65
Rule 214
Rubi steps \begin{align*} \text {integral}& = -\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {(3 d) \int \frac {\sqrt {c+d x}}{(a+b x)^5} \, dx}{10 b} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {\left (3 d^2\right ) \int \frac {1}{(a+b x)^4 \sqrt {c+d x}} \, dx}{80 b^2} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {d^3 \int \frac {1}{(a+b x)^3 \sqrt {c+d x}} \, dx}{32 b^2 (b c-a d)} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {\left (3 d^4\right ) \int \frac {1}{(a+b x)^2 \sqrt {c+d x}} \, dx}{128 b^2 (b c-a d)^2} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {\left (3 d^5\right ) \int \frac {1}{(a+b x) \sqrt {c+d x}} \, dx}{256 b^2 (b c-a d)^3} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}-\frac {\left (3 d^4\right ) \text {Subst}\left (\int \frac {1}{a-\frac {b c}{d}+\frac {b x^2}{d}} \, dx,x,\sqrt {c+d x}\right )}{128 b^2 (b c-a d)^3} \\ & = -\frac {3 d \sqrt {c+d x}}{40 b^2 (a+b x)^4}-\frac {d^2 \sqrt {c+d x}}{80 b^2 (b c-a d) (a+b x)^3}+\frac {d^3 \sqrt {c+d x}}{64 b^2 (b c-a d)^2 (a+b x)^2}-\frac {3 d^4 \sqrt {c+d x}}{128 b^2 (b c-a d)^3 (a+b x)}-\frac {(c+d x)^{3/2}}{5 b (a+b x)^5}+\frac {3 d^5 \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{128 b^{5/2} (b c-a d)^{7/2}} \\ \end{align*}
Time = 1.56 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.07 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\frac {\frac {\sqrt {b} \sqrt {c+d x} \left (-15 a^4 d^4-10 a^3 b d^3 (c+7 d x)+2 a^2 b^2 d^2 \left (124 c^2+233 c d x+64 d^2 x^2\right )-2 a b^3 d \left (168 c^3+256 c^2 d x+23 c d^2 x^2-35 d^3 x^3\right )+b^4 \left (128 c^4+176 c^3 d x+8 c^2 d^2 x^2-10 c d^3 x^3+15 d^4 x^4\right )\right )}{(-b c+a d)^3 (a+b x)^5}+\frac {15 d^5 \arctan \left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {-b c+a d}}\right )}{(-b c+a d)^{7/2}}}{640 b^{5/2}} \]
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Time = 0.37 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.05
method | result | size |
pseudoelliptic | \(-\frac {3 \left (\left (\left (-b^{4} x^{4}-\frac {14}{3} a \,b^{3} x^{3}-\frac {128}{15} a^{2} b^{2} x^{2}+\frac {14}{3} a^{3} b x +a^{4}\right ) d^{4}+\frac {2 \left (b^{3} x^{3}+\frac {23}{5} a \,b^{2} x^{2}-\frac {233}{5} a^{2} b x +a^{3}\right ) b c \,d^{3}}{3}-\frac {248 b^{2} c^{2} \left (\frac {1}{31} b^{2} x^{2}-\frac {64}{31} a b x +a^{2}\right ) d^{2}}{15}+\frac {112 b^{3} \left (-\frac {11 b x}{21}+a \right ) c^{3} d}{5}-\frac {128 b^{4} c^{4}}{15}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {d x +c}-d^{5} \left (b x +a \right )^{5} \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )\right )}{128 \sqrt {\left (a d -b c \right ) b}\, \left (b x +a \right )^{5} b^{2} \left (a d -b c \right )^{3}}\) | \(219\) |
derivativedivides | \(2 d^{5} \left (\frac {\frac {3 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 a d -10 b c}-\frac {7 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{256 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(237\) |
default | \(2 d^{5} \left (\frac {\frac {3 b^{2} \left (d x +c \right )^{\frac {9}{2}}}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right )}+\frac {7 b \left (d x +c \right )^{\frac {7}{2}}}{128 \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right )}+\frac {\left (d x +c \right )^{\frac {5}{2}}}{10 a d -10 b c}-\frac {7 \left (d x +c \right )^{\frac {3}{2}}}{128 b}-\frac {3 \left (a d -b c \right ) \sqrt {d x +c}}{256 b^{2}}}{\left (\left (d x +c \right ) b +a d -b c \right )^{5}}+\frac {3 \arctan \left (\frac {b \sqrt {d x +c}}{\sqrt {\left (a d -b c \right ) b}}\right )}{256 \left (a^{3} d^{3}-3 a^{2} b c \,d^{2}+3 a \,b^{2} c^{2} d -b^{3} c^{3}\right ) b^{2} \sqrt {\left (a d -b c \right ) b}}\right )\) | \(237\) |
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Leaf count of result is larger than twice the leaf count of optimal. 739 vs. \(2 (176) = 352\).
Time = 0.27 (sec) , antiderivative size = 1492, normalized size of antiderivative = 7.17 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\text {Too large to display} \]
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Timed out. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\text {Timed out} \]
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Exception generated. \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\text {Exception raised: ValueError} \]
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Leaf count of result is larger than twice the leaf count of optimal. 410 vs. \(2 (176) = 352\).
Time = 0.32 (sec) , antiderivative size = 410, normalized size of antiderivative = 1.97 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=-\frac {3 \, d^{5} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{128 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} \sqrt {-b^{2} c + a b d}} - \frac {15 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{4} d^{5} - 70 \, {\left (d x + c\right )}^{\frac {7}{2}} b^{4} c d^{5} + 128 \, {\left (d x + c\right )}^{\frac {5}{2}} b^{4} c^{2} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {3}{2}} b^{4} c^{3} d^{5} - 15 \, \sqrt {d x + c} b^{4} c^{4} d^{5} + 70 \, {\left (d x + c\right )}^{\frac {7}{2}} a b^{3} d^{6} - 256 \, {\left (d x + c\right )}^{\frac {5}{2}} a b^{3} c d^{6} - 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a b^{3} c^{2} d^{6} + 60 \, \sqrt {d x + c} a b^{3} c^{3} d^{6} + 128 \, {\left (d x + c\right )}^{\frac {5}{2}} a^{2} b^{2} d^{7} + 210 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{2} b^{2} c d^{7} - 90 \, \sqrt {d x + c} a^{2} b^{2} c^{2} d^{7} - 70 \, {\left (d x + c\right )}^{\frac {3}{2}} a^{3} b d^{8} + 60 \, \sqrt {d x + c} a^{3} b c d^{8} - 15 \, \sqrt {d x + c} a^{4} d^{9}}{640 \, {\left (b^{5} c^{3} - 3 \, a b^{4} c^{2} d + 3 \, a^{2} b^{3} c d^{2} - a^{3} b^{2} d^{3}\right )} {\left ({\left (d x + c\right )} b - b c + a d\right )}^{5}} \]
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Time = 0.51 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.91 \[ \int \frac {(c+d x)^{3/2}}{(a+b x)^6} \, dx=\frac {\frac {d^5\,{\left (c+d\,x\right )}^{5/2}}{5\,\left (a\,d-b\,c\right )}-\frac {7\,d^5\,{\left (c+d\,x\right )}^{3/2}}{64\,b}+\frac {3\,b^2\,d^5\,{\left (c+d\,x\right )}^{9/2}}{128\,{\left (a\,d-b\,c\right )}^3}-\frac {3\,d^5\,\left (a\,d-b\,c\right )\,\sqrt {c+d\,x}}{128\,b^2}+\frac {7\,b\,d^5\,{\left (c+d\,x\right )}^{7/2}}{64\,{\left (a\,d-b\,c\right )}^2}}{b^5\,{\left (c+d\,x\right )}^5-{\left (c+d\,x\right )}^2\,\left (-10\,a^3\,b^2\,d^3+30\,a^2\,b^3\,c\,d^2-30\,a\,b^4\,c^2\,d+10\,b^5\,c^3\right )-\left (5\,b^5\,c-5\,a\,b^4\,d\right )\,{\left (c+d\,x\right )}^4+a^5\,d^5-b^5\,c^5+{\left (c+d\,x\right )}^3\,\left (10\,a^2\,b^3\,d^2-20\,a\,b^4\,c\,d+10\,b^5\,c^2\right )+\left (c+d\,x\right )\,\left (5\,a^4\,b\,d^4-20\,a^3\,b^2\,c\,d^3+30\,a^2\,b^3\,c^2\,d^2-20\,a\,b^4\,c^3\,d+5\,b^5\,c^4\right )-10\,a^2\,b^3\,c^3\,d^2+10\,a^3\,b^2\,c^2\,d^3+5\,a\,b^4\,c^4\,d-5\,a^4\,b\,c\,d^4}+\frac {3\,d^5\,\mathrm {atan}\left (\frac {\sqrt {b}\,\sqrt {c+d\,x}}{\sqrt {a\,d-b\,c}}\right )}{128\,b^{5/2}\,{\left (a\,d-b\,c\right )}^{7/2}} \]
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