Integrand size = 19, antiderivative size = 200 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=-\frac {2 \left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac {12 c d \left (c d^2+a e^2\right )^2}{e^7 \sqrt {d+e x}}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \sqrt {d+e x}}{e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3/2}}{3 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {12 c^3 d (d+e x)^{7/2}}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \]
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Time = 0.06 (sec) , antiderivative size = 200, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.053, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {6 c^2 (d+e x)^{5/2} \left (a e^2+5 c d^2\right )}{5 e^7}-\frac {8 c^2 d (d+e x)^{3/2} \left (3 a e^2+5 c d^2\right )}{3 e^7}+\frac {6 c \sqrt {d+e x} \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{e^7}+\frac {12 c d \left (a e^2+c d^2\right )^2}{e^7 \sqrt {d+e x}}-\frac {2 \left (a e^2+c d^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7}-\frac {12 c^3 d (d+e x)^{7/2}}{7 e^7} \]
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Rule 711
Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^3}{e^6 (d+e x)^{5/2}}-\frac {6 c d \left (c d^2+a e^2\right )^2}{e^6 (d+e x)^{3/2}}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right )}{e^6 \sqrt {d+e x}}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) \sqrt {d+e x}}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{3/2}}{e^6}-\frac {6 c^3 d (d+e x)^{5/2}}{e^6}+\frac {c^3 (d+e x)^{7/2}}{e^6}\right ) \, dx \\ & = -\frac {2 \left (c d^2+a e^2\right )^3}{3 e^7 (d+e x)^{3/2}}+\frac {12 c d \left (c d^2+a e^2\right )^2}{e^7 \sqrt {d+e x}}+\frac {6 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) \sqrt {d+e x}}{e^7}-\frac {8 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{3/2}}{3 e^7}+\frac {6 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{5/2}}{5 e^7}-\frac {12 c^3 d (d+e x)^{7/2}}{7 e^7}+\frac {2 c^3 (d+e x)^{9/2}}{9 e^7} \\ \end{align*}
Time = 0.14 (sec) , antiderivative size = 171, normalized size of antiderivative = 0.86 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \left (-105 a^3 e^6+315 a^2 c e^4 \left (8 d^2+12 d e x+3 e^2 x^2\right )+63 a c^2 e^2 \left (128 d^4+192 d^3 e x+48 d^2 e^2 x^2-8 d e^3 x^3+3 e^4 x^4\right )+5 c^3 \left (1024 d^6+1536 d^5 e x+384 d^4 e^2 x^2-64 d^3 e^3 x^3+24 d^2 e^4 x^4-12 d e^5 x^5+7 e^6 x^6\right )\right )}{315 e^7 (d+e x)^{3/2}} \]
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Time = 2.40 (sec) , antiderivative size = 163, normalized size of antiderivative = 0.82
method | result | size |
risch | \(\frac {2 c \left (35 c^{2} x^{4} e^{4}-130 x^{3} c^{2} d \,e^{3}+189 x^{2} a c \,e^{4}+345 x^{2} c^{2} d^{2} e^{2}-882 x a c d \,e^{3}-880 x \,c^{2} d^{3} e +945 a^{2} e^{4}+4599 a c \,d^{2} e^{2}+3335 c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{7}}-\frac {2 \left (-18 x c d e +e^{2} a -17 c \,d^{2}\right ) \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{3 e^{7} \left (e x +d \right )^{\frac {3}{2}}}\) | \(163\) |
pseudoelliptic | \(\frac {\left (70 e^{6} x^{6}-120 d \,e^{5} x^{5}+240 d^{2} e^{4} x^{4}-640 x^{3} d^{3} e^{3}+3840 d^{4} e^{2} x^{2}+15360 d^{5} e x +10240 d^{6}\right ) c^{3}+16128 \left (\frac {3}{128} e^{4} x^{4}-\frac {1}{16} d \,e^{3} x^{3}+\frac {3}{8} d^{2} e^{2} x^{2}+\frac {3}{2} d^{3} e x +d^{4}\right ) e^{2} a \,c^{2}+5040 \left (\frac {3}{8} x^{2} e^{2}+\frac {3}{2} d e x +d^{2}\right ) e^{4} a^{2} c -210 e^{6} a^{3}}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(163\) |
gosper | \(-\frac {2 \left (-35 x^{6} c^{3} e^{6}+60 x^{5} c^{3} d \,e^{5}-189 x^{4} a \,c^{2} e^{6}-120 x^{4} c^{3} d^{2} e^{4}+504 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}-945 x^{2} a^{2} c \,e^{6}-3024 x^{2} a \,c^{2} d^{2} e^{4}-1920 x^{2} c^{3} d^{4} e^{2}-3780 x \,a^{2} c d \,e^{5}-12096 x a \,c^{2} d^{3} e^{3}-7680 x \,c^{3} d^{5} e +105 e^{6} a^{3}-2520 d^{2} e^{4} a^{2} c -8064 d^{4} e^{2} c^{2} a -5120 c^{3} d^{6}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(205\) |
trager | \(-\frac {2 \left (-35 x^{6} c^{3} e^{6}+60 x^{5} c^{3} d \,e^{5}-189 x^{4} a \,c^{2} e^{6}-120 x^{4} c^{3} d^{2} e^{4}+504 x^{3} a \,c^{2} d \,e^{5}+320 x^{3} c^{3} d^{3} e^{3}-945 x^{2} a^{2} c \,e^{6}-3024 x^{2} a \,c^{2} d^{2} e^{4}-1920 x^{2} c^{3} d^{4} e^{2}-3780 x \,a^{2} c d \,e^{5}-12096 x a \,c^{2} d^{3} e^{3}-7680 x \,c^{3} d^{5} e +105 e^{6} a^{3}-2520 d^{2} e^{4} a^{2} c -8064 d^{4} e^{2} c^{2} a -5120 c^{3} d^{6}\right )}{315 \left (e x +d \right )^{\frac {3}{2}} e^{7}}\) | \(205\) |
derivativedivides | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}-8 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a^{2} c \,e^{4} \sqrt {e x +d}+36 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}+\frac {12 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(229\) |
default | \(\frac {\frac {2 c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}-\frac {12 c^{3} d \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {6 a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+6 c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}-8 a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {40 c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}+6 a^{2} c \,e^{4} \sqrt {e x +d}+36 a \,c^{2} d^{2} e^{2} \sqrt {e x +d}+30 c^{3} d^{4} \sqrt {e x +d}+\frac {12 c d \left (a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}\right )}{\sqrt {e x +d}}-\frac {2 \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{7}}\) | \(229\) |
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Time = 0.45 (sec) , antiderivative size = 223, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (35 \, c^{3} e^{6} x^{6} - 60 \, c^{3} d e^{5} x^{5} + 5120 \, c^{3} d^{6} + 8064 \, a c^{2} d^{4} e^{2} + 2520 \, a^{2} c d^{2} e^{4} - 105 \, a^{3} e^{6} + 3 \, {\left (40 \, c^{3} d^{2} e^{4} + 63 \, a c^{2} e^{6}\right )} x^{4} - 8 \, {\left (40 \, c^{3} d^{3} e^{3} + 63 \, a c^{2} d e^{5}\right )} x^{3} + 3 \, {\left (640 \, c^{3} d^{4} e^{2} + 1008 \, a c^{2} d^{2} e^{4} + 315 \, a^{2} c e^{6}\right )} x^{2} + 12 \, {\left (640 \, c^{3} d^{5} e + 1008 \, a c^{2} d^{3} e^{3} + 315 \, a^{2} c d e^{5}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{9} x^{2} + 2 \, d e^{8} x + d^{2} e^{7}\right )}} \]
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Time = 3.29 (sec) , antiderivative size = 252, normalized size of antiderivative = 1.26 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {6 c^{3} d \left (d + e x\right )^{\frac {7}{2}}}{7 e^{6}} + \frac {c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{6}} + \frac {6 c d \left (a e^{2} + c d^{2}\right )^{2}}{e^{6} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (3 a c^{2} e^{2} + 15 c^{3} d^{2}\right )}{5 e^{6}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \left (- 12 a c^{2} d e^{2} - 20 c^{3} d^{3}\right )}{3 e^{6}} + \frac {\sqrt {d + e x} \left (3 a^{2} c e^{4} + 18 a c^{2} d^{2} e^{2} + 15 c^{3} d^{4}\right )}{e^{6}} - \frac {\left (a e^{2} + c d^{2}\right )^{3}}{3 e^{6} \left (d + e x\right )^{\frac {3}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {a^{3} x + a^{2} c x^{3} + \frac {3 a c^{2} x^{5}}{5} + \frac {c^{3} x^{7}}{7}}{d^{\frac {5}{2}}} & \text {otherwise} \end {cases} \]
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Time = 0.22 (sec) , antiderivative size = 215, normalized size of antiderivative = 1.08 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} - 270 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d + 189 \, {\left (5 \, c^{3} d^{2} + a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 420 \, {\left (5 \, c^{3} d^{3} + 3 \, a c^{2} d e^{2}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 945 \, {\left (5 \, c^{3} d^{4} + 6 \, a c^{2} d^{2} e^{2} + a^{2} c e^{4}\right )} \sqrt {e x + d}}{e^{6}} - \frac {105 \, {\left (c^{3} d^{6} + 3 \, a c^{2} d^{4} e^{2} + 3 \, a^{2} c d^{2} e^{4} + a^{3} e^{6} - 18 \, {\left (c^{3} d^{5} + 2 \, a c^{2} d^{3} e^{2} + a^{2} c d e^{4}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {3}{2}} e^{6}}\right )}}{315 \, e} \]
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Time = 0.28 (sec) , antiderivative size = 258, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {2 \, {\left (18 \, {\left (e x + d\right )} c^{3} d^{5} - c^{3} d^{6} + 36 \, {\left (e x + d\right )} a c^{2} d^{3} e^{2} - 3 \, a c^{2} d^{4} e^{2} + 18 \, {\left (e x + d\right )} a^{2} c d e^{4} - 3 \, a^{2} c d^{2} e^{4} - a^{3} e^{6}\right )}}{3 \, {\left (e x + d\right )}^{\frac {3}{2}} e^{7}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} c^{3} e^{56} - 270 \, {\left (e x + d\right )}^{\frac {7}{2}} c^{3} d e^{56} + 945 \, {\left (e x + d\right )}^{\frac {5}{2}} c^{3} d^{2} e^{56} - 2100 \, {\left (e x + d\right )}^{\frac {3}{2}} c^{3} d^{3} e^{56} + 4725 \, \sqrt {e x + d} c^{3} d^{4} e^{56} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} a c^{2} e^{58} - 1260 \, {\left (e x + d\right )}^{\frac {3}{2}} a c^{2} d e^{58} + 5670 \, \sqrt {e x + d} a c^{2} d^{2} e^{58} + 945 \, \sqrt {e x + d} a^{2} c e^{60}\right )}}{315 \, e^{63}} \]
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Time = 9.47 (sec) , antiderivative size = 225, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+c x^2\right )^3}{(d+e x)^{5/2}} \, dx=\frac {\left (30\,c^3\,d^2+6\,a\,c^2\,e^2\right )\,{\left (d+e\,x\right )}^{5/2}}{5\,e^7}+\frac {\sqrt {d+e\,x}\,\left (6\,a^2\,c\,e^4+36\,a\,c^2\,d^2\,e^2+30\,c^3\,d^4\right )}{e^7}-\frac {\frac {2\,a^3\,e^6}{3}-\left (d+e\,x\right )\,\left (12\,a^2\,c\,d\,e^4+24\,a\,c^2\,d^3\,e^2+12\,c^3\,d^5\right )+\frac {2\,c^3\,d^6}{3}+2\,a\,c^2\,d^4\,e^2+2\,a^2\,c\,d^2\,e^4}{e^7\,{\left (d+e\,x\right )}^{3/2}}+\frac {2\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^7}-\frac {\left (40\,c^3\,d^3+24\,a\,c^2\,d\,e^2\right )\,{\left (d+e\,x\right )}^{3/2}}{3\,e^7}-\frac {12\,c^3\,d\,{\left (d+e\,x\right )}^{7/2}}{7\,e^7} \]
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