Integrand size = 33, antiderivative size = 212 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {2 (b d-a e)^7 \sqrt {d+e x}}{e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{3/2}}{3 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^8}+\frac {10 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{9/2}}{9 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{11/2}}{11 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \]
[Out]
Time = 0.05 (sec) , antiderivative size = 212, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.061, Rules used = {27, 45} \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=-\frac {14 b^6 (d+e x)^{13/2} (b d-a e)}{13 e^8}+\frac {42 b^5 (d+e x)^{11/2} (b d-a e)^2}{11 e^8}-\frac {70 b^4 (d+e x)^{9/2} (b d-a e)^3}{9 e^8}+\frac {10 b^3 (d+e x)^{7/2} (b d-a e)^4}{e^8}-\frac {42 b^2 (d+e x)^{5/2} (b d-a e)^5}{5 e^8}+\frac {14 b (d+e x)^{3/2} (b d-a e)^6}{3 e^8}-\frac {2 \sqrt {d+e x} (b d-a e)^7}{e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \]
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Rule 27
Rule 45
Rubi steps \begin{align*} \text {integral}& = \int \frac {(a+b x)^7}{\sqrt {d+e x}} \, dx \\ & = \int \left (\frac {(-b d+a e)^7}{e^7 \sqrt {d+e x}}+\frac {7 b (b d-a e)^6 \sqrt {d+e x}}{e^7}-\frac {21 b^2 (b d-a e)^5 (d+e x)^{3/2}}{e^7}+\frac {35 b^3 (b d-a e)^4 (d+e x)^{5/2}}{e^7}-\frac {35 b^4 (b d-a e)^3 (d+e x)^{7/2}}{e^7}+\frac {21 b^5 (b d-a e)^2 (d+e x)^{9/2}}{e^7}-\frac {7 b^6 (b d-a e) (d+e x)^{11/2}}{e^7}+\frac {b^7 (d+e x)^{13/2}}{e^7}\right ) \, dx \\ & = -\frac {2 (b d-a e)^7 \sqrt {d+e x}}{e^8}+\frac {14 b (b d-a e)^6 (d+e x)^{3/2}}{3 e^8}-\frac {42 b^2 (b d-a e)^5 (d+e x)^{5/2}}{5 e^8}+\frac {10 b^3 (b d-a e)^4 (d+e x)^{7/2}}{e^8}-\frac {70 b^4 (b d-a e)^3 (d+e x)^{9/2}}{9 e^8}+\frac {42 b^5 (b d-a e)^2 (d+e x)^{11/2}}{11 e^8}-\frac {14 b^6 (b d-a e) (d+e x)^{13/2}}{13 e^8}+\frac {2 b^7 (d+e x)^{15/2}}{15 e^8} \\ \end{align*}
Time = 0.17 (sec) , antiderivative size = 375, normalized size of antiderivative = 1.77 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \sqrt {d+e x} \left (6435 a^7 e^7+15015 a^6 b e^6 (-2 d+e x)+9009 a^5 b^2 e^5 \left (8 d^2-4 d e x+3 e^2 x^2\right )+6435 a^4 b^3 e^4 \left (-16 d^3+8 d^2 e x-6 d e^2 x^2+5 e^3 x^3\right )+715 a^3 b^4 e^3 \left (128 d^4-64 d^3 e x+48 d^2 e^2 x^2-40 d e^3 x^3+35 e^4 x^4\right )+195 a^2 b^5 e^2 \left (-256 d^5+128 d^4 e x-96 d^3 e^2 x^2+80 d^2 e^3 x^3-70 d e^4 x^4+63 e^5 x^5\right )+15 a b^6 e \left (1024 d^6-512 d^5 e x+384 d^4 e^2 x^2-320 d^3 e^3 x^3+280 d^2 e^4 x^4-252 d e^5 x^5+231 e^6 x^6\right )+b^7 \left (-2048 d^7+1024 d^6 e x-768 d^5 e^2 x^2+640 d^4 e^3 x^3-560 d^3 e^4 x^4+504 d^2 e^5 x^5-462 d e^6 x^6+429 e^7 x^7\right )\right )}{6435 e^8} \]
[In]
[Out]
Time = 0.35 (sec) , antiderivative size = 359, normalized size of antiderivative = 1.69
method | result | size |
pseudoelliptic | \(\frac {2 \left (\left (\frac {1}{15} b^{7} x^{7}+a^{7}+\frac {7}{13} a \,b^{6} x^{6}+\frac {21}{11} a^{2} b^{5} x^{5}+\frac {35}{9} a^{3} b^{4} x^{4}+5 a^{4} b^{3} x^{3}+\frac {21}{5} a^{5} b^{2} x^{2}+\frac {7}{3} a^{6} b x \right ) e^{7}-\frac {14 b \left (\frac {1}{65} b^{6} x^{6}+\frac {18}{143} a \,b^{5} x^{5}+\frac {5}{11} a^{2} b^{4} x^{4}+\frac {20}{21} a^{3} b^{3} x^{3}+\frac {9}{7} a^{4} b^{2} x^{2}+\frac {6}{5} a^{5} b x +a^{6}\right ) d \,e^{6}}{3}+\frac {56 b^{2} \left (\frac {1}{143} b^{5} x^{5}+\frac {25}{429} a \,b^{4} x^{4}+\frac {50}{231} a^{2} b^{3} x^{3}+\frac {10}{21} a^{3} b^{2} x^{2}+\frac {5}{7} a^{4} b x +a^{5}\right ) d^{2} e^{5}}{5}-16 b^{3} d^{3} \left (\frac {7}{1287} x^{4} b^{4}+\frac {20}{429} a \,b^{3} x^{3}+\frac {2}{11} x^{2} b^{2} a^{2}+\frac {4}{9} b \,a^{3} x +a^{4}\right ) e^{4}+\frac {128 b^{4} \left (\frac {1}{143} x^{3} b^{3}+\frac {9}{143} a \,b^{2} x^{2}+\frac {3}{11} b \,a^{2} x +a^{3}\right ) d^{4} e^{3}}{9}-\frac {256 b^{5} \left (\frac {1}{65} b^{2} x^{2}+\frac {2}{13} a b x +a^{2}\right ) d^{5} e^{2}}{33}+\frac {1024 b^{6} \left (\frac {b x}{15}+a \right ) d^{6} e}{429}-\frac {2048 b^{7} d^{7}}{6435}\right ) \sqrt {e x +d}}{e^{8}}\) | \(359\) |
gosper | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
trager | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
risch | \(\frac {2 \left (429 x^{7} b^{7} e^{7}+3465 x^{6} a \,b^{6} e^{7}-462 x^{6} b^{7} d \,e^{6}+12285 x^{5} a^{2} b^{5} e^{7}-3780 x^{5} a \,b^{6} d \,e^{6}+504 x^{5} b^{7} d^{2} e^{5}+25025 x^{4} a^{3} b^{4} e^{7}-13650 x^{4} a^{2} b^{5} d \,e^{6}+4200 x^{4} a \,b^{6} d^{2} e^{5}-560 x^{4} b^{7} d^{3} e^{4}+32175 x^{3} a^{4} b^{3} e^{7}-28600 x^{3} a^{3} b^{4} d \,e^{6}+15600 x^{3} a^{2} b^{5} d^{2} e^{5}-4800 x^{3} a \,b^{6} d^{3} e^{4}+640 x^{3} b^{7} d^{4} e^{3}+27027 x^{2} a^{5} b^{2} e^{7}-38610 x^{2} a^{4} b^{3} d \,e^{6}+34320 x^{2} a^{3} b^{4} d^{2} e^{5}-18720 x^{2} a^{2} b^{5} d^{3} e^{4}+5760 x^{2} a \,b^{6} d^{4} e^{3}-768 x^{2} b^{7} d^{5} e^{2}+15015 x \,a^{6} b \,e^{7}-36036 x \,a^{5} b^{2} d \,e^{6}+51480 x \,a^{4} b^{3} d^{2} e^{5}-45760 x \,a^{3} b^{4} d^{3} e^{4}+24960 x \,a^{2} b^{5} d^{4} e^{3}-7680 x a \,b^{6} d^{5} e^{2}+1024 x \,b^{7} d^{6} e +6435 e^{7} a^{7}-30030 b d \,e^{6} a^{6}+72072 b^{2} d^{2} e^{5} a^{5}-102960 b^{3} d^{3} e^{4} a^{4}+91520 b^{4} d^{4} e^{3} a^{3}-49920 b^{5} d^{5} e^{2} a^{2}+15360 b^{6} d^{6} e a -2048 b^{7} d^{7}\right ) \sqrt {e x +d}}{6435 e^{8}}\) | \(498\) |
derivativedivides | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(935\) |
default | \(\frac {\frac {2 b^{7} \left (e x +d \right )^{\frac {15}{2}}}{15}+\frac {2 \left (\left (a e -b d \right ) b^{6}+3 b^{5} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {13}{2}}}{13}+\frac {2 \left (3 \left (a e -b d \right ) \left (2 a b e -2 b^{2} d \right ) b^{4}+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {11}{2}}}{11}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{4}+2 \left (2 a b e -2 b^{2} d \right )^{2} b^{2}+b^{2} \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )\right ) \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 \left (\left (a e -b d \right ) \left (4 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 a b e -2 b^{2} d \right ) b^{2}+\left (2 a b e -2 b^{2} d \right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )\right )+b \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )\right ) \left (e x +d \right )^{\frac {7}{2}}}{7}+\frac {2 \left (\left (a e -b d \right ) \left (\left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) \left (2 \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right ) b^{2}+\left (2 a b e -2 b^{2} d \right )^{2}\right )+2 \left (2 a b e -2 b^{2} d \right )^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )+b^{2} \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2}\right )+3 b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )\right ) \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {2 \left (3 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{2} \left (2 a b e -2 b^{2} d \right )+b \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3}\right ) \left (e x +d \right )^{\frac {3}{2}}}{3}+2 \left (a e -b d \right ) \left (e^{2} a^{2}-2 a b d e +b^{2} d^{2}\right )^{3} \sqrt {e x +d}}{e^{8}}\) | \(935\) |
[In]
[Out]
Leaf count of result is larger than twice the leaf count of optimal. 463 vs. \(2 (184) = 368\).
Time = 0.28 (sec) , antiderivative size = 463, normalized size of antiderivative = 2.18 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (429 \, b^{7} e^{7} x^{7} - 2048 \, b^{7} d^{7} + 15360 \, a b^{6} d^{6} e - 49920 \, a^{2} b^{5} d^{5} e^{2} + 91520 \, a^{3} b^{4} d^{4} e^{3} - 102960 \, a^{4} b^{3} d^{3} e^{4} + 72072 \, a^{5} b^{2} d^{2} e^{5} - 30030 \, a^{6} b d e^{6} + 6435 \, a^{7} e^{7} - 231 \, {\left (2 \, b^{7} d e^{6} - 15 \, a b^{6} e^{7}\right )} x^{6} + 63 \, {\left (8 \, b^{7} d^{2} e^{5} - 60 \, a b^{6} d e^{6} + 195 \, a^{2} b^{5} e^{7}\right )} x^{5} - 35 \, {\left (16 \, b^{7} d^{3} e^{4} - 120 \, a b^{6} d^{2} e^{5} + 390 \, a^{2} b^{5} d e^{6} - 715 \, a^{3} b^{4} e^{7}\right )} x^{4} + 5 \, {\left (128 \, b^{7} d^{4} e^{3} - 960 \, a b^{6} d^{3} e^{4} + 3120 \, a^{2} b^{5} d^{2} e^{5} - 5720 \, a^{3} b^{4} d e^{6} + 6435 \, a^{4} b^{3} e^{7}\right )} x^{3} - 3 \, {\left (256 \, b^{7} d^{5} e^{2} - 1920 \, a b^{6} d^{4} e^{3} + 6240 \, a^{2} b^{5} d^{3} e^{4} - 11440 \, a^{3} b^{4} d^{2} e^{5} + 12870 \, a^{4} b^{3} d e^{6} - 9009 \, a^{5} b^{2} e^{7}\right )} x^{2} + {\left (1024 \, b^{7} d^{6} e - 7680 \, a b^{6} d^{5} e^{2} + 24960 \, a^{2} b^{5} d^{4} e^{3} - 45760 \, a^{3} b^{4} d^{3} e^{4} + 51480 \, a^{4} b^{3} d^{2} e^{5} - 36036 \, a^{5} b^{2} d e^{6} + 15015 \, a^{6} b e^{7}\right )} x\right )} \sqrt {e x + d}}{6435 \, e^{8}} \]
[In]
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Leaf count of result is larger than twice the leaf count of optimal. 575 vs. \(2 (197) = 394\).
Time = 1.67 (sec) , antiderivative size = 575, normalized size of antiderivative = 2.71 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\begin {cases} \frac {2 \left (\frac {b^{7} \left (d + e x\right )^{\frac {15}{2}}}{15 e^{7}} + \frac {\left (d + e x\right )^{\frac {13}{2}} \cdot \left (7 a b^{6} e - 7 b^{7} d\right )}{13 e^{7}} + \frac {\left (d + e x\right )^{\frac {11}{2}} \cdot \left (21 a^{2} b^{5} e^{2} - 42 a b^{6} d e + 21 b^{7} d^{2}\right )}{11 e^{7}} + \frac {\left (d + e x\right )^{\frac {9}{2}} \cdot \left (35 a^{3} b^{4} e^{3} - 105 a^{2} b^{5} d e^{2} + 105 a b^{6} d^{2} e - 35 b^{7} d^{3}\right )}{9 e^{7}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \cdot \left (35 a^{4} b^{3} e^{4} - 140 a^{3} b^{4} d e^{3} + 210 a^{2} b^{5} d^{2} e^{2} - 140 a b^{6} d^{3} e + 35 b^{7} d^{4}\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \cdot \left (21 a^{5} b^{2} e^{5} - 105 a^{4} b^{3} d e^{4} + 210 a^{3} b^{4} d^{2} e^{3} - 210 a^{2} b^{5} d^{3} e^{2} + 105 a b^{6} d^{4} e - 21 b^{7} d^{5}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (7 a^{6} b e^{6} - 42 a^{5} b^{2} d e^{5} + 105 a^{4} b^{3} d^{2} e^{4} - 140 a^{3} b^{4} d^{3} e^{3} + 105 a^{2} b^{5} d^{4} e^{2} - 42 a b^{6} d^{5} e + 7 b^{7} d^{6}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (a^{7} e^{7} - 7 a^{6} b d e^{6} + 21 a^{5} b^{2} d^{2} e^{5} - 35 a^{4} b^{3} d^{3} e^{4} + 35 a^{3} b^{4} d^{4} e^{3} - 21 a^{2} b^{5} d^{5} e^{2} + 7 a b^{6} d^{6} e - b^{7} d^{7}\right )}{e^{7}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {\begin {cases} a^{7} x & \text {for}\: b = 0 \\\frac {\left (a^{2} + 2 a b x + b^{2} x^{2}\right )^{4}}{8 b} & \text {otherwise} \end {cases}}{\sqrt {d}} & \text {otherwise} \end {cases} \]
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Leaf count of result is larger than twice the leaf count of optimal. 456 vs. \(2 (184) = 368\).
Time = 0.21 (sec) , antiderivative size = 456, normalized size of antiderivative = 2.15 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} b^{7} - 3465 \, {\left (b^{7} d - a b^{6} e\right )} {\left (e x + d\right )}^{\frac {13}{2}} + 12285 \, {\left (b^{7} d^{2} - 2 \, a b^{6} d e + a^{2} b^{5} e^{2}\right )} {\left (e x + d\right )}^{\frac {11}{2}} - 25025 \, {\left (b^{7} d^{3} - 3 \, a b^{6} d^{2} e + 3 \, a^{2} b^{5} d e^{2} - a^{3} b^{4} e^{3}\right )} {\left (e x + d\right )}^{\frac {9}{2}} + 32175 \, {\left (b^{7} d^{4} - 4 \, a b^{6} d^{3} e + 6 \, a^{2} b^{5} d^{2} e^{2} - 4 \, a^{3} b^{4} d e^{3} + a^{4} b^{3} e^{4}\right )} {\left (e x + d\right )}^{\frac {7}{2}} - 27027 \, {\left (b^{7} d^{5} - 5 \, a b^{6} d^{4} e + 10 \, a^{2} b^{5} d^{3} e^{2} - 10 \, a^{3} b^{4} d^{2} e^{3} + 5 \, a^{4} b^{3} d e^{4} - a^{5} b^{2} e^{5}\right )} {\left (e x + d\right )}^{\frac {5}{2}} + 15015 \, {\left (b^{7} d^{6} - 6 \, a b^{6} d^{5} e + 15 \, a^{2} b^{5} d^{4} e^{2} - 20 \, a^{3} b^{4} d^{3} e^{3} + 15 \, a^{4} b^{3} d^{2} e^{4} - 6 \, a^{5} b^{2} d e^{5} + a^{6} b e^{6}\right )} {\left (e x + d\right )}^{\frac {3}{2}} - 6435 \, {\left (b^{7} d^{7} - 7 \, a b^{6} d^{6} e + 21 \, a^{2} b^{5} d^{5} e^{2} - 35 \, a^{3} b^{4} d^{4} e^{3} + 35 \, a^{4} b^{3} d^{3} e^{4} - 21 \, a^{5} b^{2} d^{2} e^{5} + 7 \, a^{6} b d e^{6} - a^{7} e^{7}\right )} \sqrt {e x + d}\right )}}{6435 \, e^{8}} \]
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Leaf count of result is larger than twice the leaf count of optimal. 477 vs. \(2 (184) = 368\).
Time = 0.28 (sec) , antiderivative size = 477, normalized size of antiderivative = 2.25 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2 \, {\left (6435 \, \sqrt {e x + d} a^{7} + \frac {15015 \, {\left ({\left (e x + d\right )}^{\frac {3}{2}} - 3 \, \sqrt {e x + d} d\right )} a^{6} b}{e} + \frac {9009 \, {\left (3 \, {\left (e x + d\right )}^{\frac {5}{2}} - 10 \, {\left (e x + d\right )}^{\frac {3}{2}} d + 15 \, \sqrt {e x + d} d^{2}\right )} a^{5} b^{2}}{e^{2}} + \frac {6435 \, {\left (5 \, {\left (e x + d\right )}^{\frac {7}{2}} - 21 \, {\left (e x + d\right )}^{\frac {5}{2}} d + 35 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{2} - 35 \, \sqrt {e x + d} d^{3}\right )} a^{4} b^{3}}{e^{3}} + \frac {715 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} - 180 \, {\left (e x + d\right )}^{\frac {7}{2}} d + 378 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{2} - 420 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{3} + 315 \, \sqrt {e x + d} d^{4}\right )} a^{3} b^{4}}{e^{4}} + \frac {195 \, {\left (63 \, {\left (e x + d\right )}^{\frac {11}{2}} - 385 \, {\left (e x + d\right )}^{\frac {9}{2}} d + 990 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{2} - 1386 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{3} + 1155 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{4} - 693 \, \sqrt {e x + d} d^{5}\right )} a^{2} b^{5}}{e^{5}} + \frac {15 \, {\left (231 \, {\left (e x + d\right )}^{\frac {13}{2}} - 1638 \, {\left (e x + d\right )}^{\frac {11}{2}} d + 5005 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{2} - 8580 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{3} + 9009 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{4} - 6006 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{5} + 3003 \, \sqrt {e x + d} d^{6}\right )} a b^{6}}{e^{6}} + \frac {{\left (429 \, {\left (e x + d\right )}^{\frac {15}{2}} - 3465 \, {\left (e x + d\right )}^{\frac {13}{2}} d + 12285 \, {\left (e x + d\right )}^{\frac {11}{2}} d^{2} - 25025 \, {\left (e x + d\right )}^{\frac {9}{2}} d^{3} + 32175 \, {\left (e x + d\right )}^{\frac {7}{2}} d^{4} - 27027 \, {\left (e x + d\right )}^{\frac {5}{2}} d^{5} + 15015 \, {\left (e x + d\right )}^{\frac {3}{2}} d^{6} - 6435 \, \sqrt {e x + d} d^{7}\right )} b^{7}}{e^{7}}\right )}}{6435 \, e} \]
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Time = 11.03 (sec) , antiderivative size = 187, normalized size of antiderivative = 0.88 \[ \int \frac {(a+b x) \left (a^2+2 a b x+b^2 x^2\right )^3}{\sqrt {d+e x}} \, dx=\frac {2\,b^7\,{\left (d+e\,x\right )}^{15/2}}{15\,e^8}-\frac {\left (14\,b^7\,d-14\,a\,b^6\,e\right )\,{\left (d+e\,x\right )}^{13/2}}{13\,e^8}+\frac {2\,{\left (a\,e-b\,d\right )}^7\,\sqrt {d+e\,x}}{e^8}+\frac {42\,b^2\,{\left (a\,e-b\,d\right )}^5\,{\left (d+e\,x\right )}^{5/2}}{5\,e^8}+\frac {10\,b^3\,{\left (a\,e-b\,d\right )}^4\,{\left (d+e\,x\right )}^{7/2}}{e^8}+\frac {70\,b^4\,{\left (a\,e-b\,d\right )}^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {42\,b^5\,{\left (a\,e-b\,d\right )}^2\,{\left (d+e\,x\right )}^{11/2}}{11\,e^8}+\frac {14\,b\,{\left (a\,e-b\,d\right )}^6\,{\left (d+e\,x\right )}^{3/2}}{3\,e^8} \]
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