\(\int (d+e x)^6 (a+c x^2)^4 \, dx\) [489]

Optimal result

Integrand size = 17, antiderivative size = 276 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {\left (c d^2+a e^2\right )^4 (d+e x)^7}{7 e^9}-\frac {c d \left (c d^2+a e^2\right )^3 (d+e x)^8}{e^9}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^9}{9 e^9}-\frac {4 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^9}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{11}}{11 e^9}-\frac {2 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{12}}{3 e^9}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{13}}{13 e^9}-\frac {4 c^4 d (d+e x)^{14}}{7 e^9}+\frac {c^4 (d+e x)^{15}}{15 e^9} \]

[Out]

1/7*(a*e^2+c*d^2)^4*(e*x+d)^7/e^9-c*d*(a*e^2+c*d^2)^3*(e*x+d)^8/e^9+4/9*c*(a*e^2+c*d^2)^2*(a*e^2+7*c*d^2)*(e*x
+d)^9/e^9-4/5*c^2*d*(a*e^2+c*d^2)*(3*a*e^2+7*c*d^2)*(e*x+d)^10/e^9+2/11*c^2*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d
^4)*(e*x+d)^11/e^9-2/3*c^3*d*(3*a*e^2+7*c*d^2)*(e*x+d)^12/e^9+4/13*c^3*(a*e^2+7*c*d^2)*(e*x+d)^13/e^9-4/7*c^4*
d*(e*x+d)^14/e^9+1/15*c^4*(e*x+d)^15/e^9

Rubi [A] (verified)

Time = 0.31 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {2 c^2 (d+e x)^{11} \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )}{11 e^9}+\frac {4 c^3 (d+e x)^{13} \left (a e^2+7 c d^2\right )}{13 e^9}-\frac {2 c^3 d (d+e x)^{12} \left (3 a e^2+7 c d^2\right )}{3 e^9}-\frac {4 c^2 d (d+e x)^{10} \left (a e^2+c d^2\right ) \left (3 a e^2+7 c d^2\right )}{5 e^9}+\frac {4 c (d+e x)^9 \left (a e^2+c d^2\right )^2 \left (a e^2+7 c d^2\right )}{9 e^9}-\frac {c d (d+e x)^8 \left (a e^2+c d^2\right )^3}{e^9}+\frac {(d+e x)^7 \left (a e^2+c d^2\right )^4}{7 e^9}+\frac {c^4 (d+e x)^{15}}{15 e^9}-\frac {4 c^4 d (d+e x)^{14}}{7 e^9} \]

[In]

Int[(d + e*x)^6*(a + c*x^2)^4,x]

[Out]

((c*d^2 + a*e^2)^4*(d + e*x)^7)/(7*e^9) - (c*d*(c*d^2 + a*e^2)^3*(d + e*x)^8)/e^9 + (4*c*(c*d^2 + a*e^2)^2*(7*
c*d^2 + a*e^2)*(d + e*x)^9)/(9*e^9) - (4*c^2*d*(c*d^2 + a*e^2)*(7*c*d^2 + 3*a*e^2)*(d + e*x)^10)/(5*e^9) + (2*
c^2*(35*c^2*d^4 + 30*a*c*d^2*e^2 + 3*a^2*e^4)*(d + e*x)^11)/(11*e^9) - (2*c^3*d*(7*c*d^2 + 3*a*e^2)*(d + e*x)^
12)/(3*e^9) + (4*c^3*(7*c*d^2 + a*e^2)*(d + e*x)^13)/(13*e^9) - (4*c^4*d*(d + e*x)^14)/(7*e^9) + (c^4*(d + e*x
)^15)/(15*e^9)

Rule 711

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps \begin{align*} \text {integral}& = \int \left (\frac {\left (c d^2+a e^2\right )^4 (d+e x)^6}{e^8}-\frac {8 c d \left (c d^2+a e^2\right )^3 (d+e x)^7}{e^8}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^8}{e^8}+\frac {8 c^2 d \left (-7 c d^2-3 a e^2\right ) \left (c d^2+a e^2\right ) (d+e x)^9}{e^8}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{10}}{e^8}-\frac {8 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{11}}{e^8}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{12}}{e^8}-\frac {8 c^4 d (d+e x)^{13}}{e^8}+\frac {c^4 (d+e x)^{14}}{e^8}\right ) \, dx \\ & = \frac {\left (c d^2+a e^2\right )^4 (d+e x)^7}{7 e^9}-\frac {c d \left (c d^2+a e^2\right )^3 (d+e x)^8}{e^9}+\frac {4 c \left (c d^2+a e^2\right )^2 \left (7 c d^2+a e^2\right ) (d+e x)^9}{9 e^9}-\frac {4 c^2 d \left (c d^2+a e^2\right ) \left (7 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^9}+\frac {2 c^2 \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right ) (d+e x)^{11}}{11 e^9}-\frac {2 c^3 d \left (7 c d^2+3 a e^2\right ) (d+e x)^{12}}{3 e^9}+\frac {4 c^3 \left (7 c d^2+a e^2\right ) (d+e x)^{13}}{13 e^9}-\frac {4 c^4 d (d+e x)^{14}}{7 e^9}+\frac {c^4 (d+e x)^{15}}{15 e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 361, normalized size of antiderivative = 1.31 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {6435 a^4 x \left (7 d^6+21 d^5 e x+35 d^4 e^2 x^2+35 d^3 e^3 x^3+21 d^2 e^4 x^4+7 d e^5 x^5+e^6 x^6\right )+715 a^3 c x^3 \left (84 d^6+378 d^5 e x+756 d^4 e^2 x^2+840 d^3 e^3 x^3+540 d^2 e^4 x^4+189 d e^5 x^5+28 e^6 x^6\right )+117 a^2 c^2 x^5 \left (462 d^6+2310 d^5 e x+4950 d^4 e^2 x^2+5775 d^3 e^3 x^3+3850 d^2 e^4 x^4+1386 d e^5 x^5+210 e^6 x^6\right )+15 a c^3 x^7 \left (1716 d^6+9009 d^5 e x+20020 d^4 e^2 x^2+24024 d^3 e^3 x^3+16380 d^2 e^4 x^4+6006 d e^5 x^5+924 e^6 x^6\right )+c^4 x^9 \left (5005 d^6+27027 d^5 e x+61425 d^4 e^2 x^2+75075 d^3 e^3 x^3+51975 d^2 e^4 x^4+19305 d e^5 x^5+3003 e^6 x^6\right )}{45045} \]

[In]

Integrate[(d + e*x)^6*(a + c*x^2)^4,x]

[Out]

(6435*a^4*x*(7*d^6 + 21*d^5*e*x + 35*d^4*e^2*x^2 + 35*d^3*e^3*x^3 + 21*d^2*e^4*x^4 + 7*d*e^5*x^5 + e^6*x^6) +
715*a^3*c*x^3*(84*d^6 + 378*d^5*e*x + 756*d^4*e^2*x^2 + 840*d^3*e^3*x^3 + 540*d^2*e^4*x^4 + 189*d*e^5*x^5 + 28
*e^6*x^6) + 117*a^2*c^2*x^5*(462*d^6 + 2310*d^5*e*x + 4950*d^4*e^2*x^2 + 5775*d^3*e^3*x^3 + 3850*d^2*e^4*x^4 +
1386*d*e^5*x^5 + 210*e^6*x^6) + 15*a*c^3*x^7*(1716*d^6 + 9009*d^5*e*x + 20020*d^4*e^2*x^2 + 24024*d^3*e^3*x^3
+ 16380*d^2*e^4*x^4 + 6006*d*e^5*x^5 + 924*e^6*x^6) + c^4*x^9*(5005*d^6 + 27027*d^5*e*x + 61425*d^4*e^2*x^2 +
75075*d^3*e^3*x^3 + 51975*d^2*e^4*x^4 + 19305*d*e^5*x^5 + 3003*e^6*x^6))/45045

Maple [A] (verified)

Time = 2.15 (sec) , antiderivative size = 435, normalized size of antiderivative = 1.58

method	result	size
norman	\(d^{6} a^{4} x +3 d^{5} e \,a^{4} x^{2}+\left (5 d^{4} e^{2} a^{4}+\frac {4}{3} d^{6} c \,a^{3}\right ) x^{3}+\left (5 d^{3} e^{3} a^{4}+6 d^{5} e c \,a^{3}\right ) x^{4}+\left (3 d^{2} e^{4} a^{4}+12 d^{4} e^{2} c \,a^{3}+\frac {6}{5} d^{6} a^{2} c^{2}\right ) x^{5}+\left (d \,e^{5} a^{4}+\frac {40}{3} d^{3} e^{3} c \,a^{3}+6 d^{5} e \,a^{2} c^{2}\right ) x^{6}+\left (\frac {1}{7} e^{6} a^{4}+\frac {60}{7} d^{2} e^{4} c \,a^{3}+\frac {90}{7} d^{4} e^{2} a^{2} c^{2}+\frac {4}{7} d^{6} c^{3} a \right ) x^{7}+\left (3 d \,e^{5} c \,a^{3}+15 d^{3} e^{3} a^{2} c^{2}+3 d^{5} e \,c^{3} a \right ) x^{8}+\left (\frac {4}{9} e^{6} c \,a^{3}+10 d^{2} e^{4} a^{2} c^{2}+\frac {20}{3} d^{4} e^{2} c^{3} a +\frac {1}{9} d^{6} c^{4}\right ) x^{9}+\left (\frac {18}{5} d \,e^{5} a^{2} c^{2}+8 d^{3} e^{3} c^{3} a +\frac {3}{5} d^{5} e \,c^{4}\right ) x^{10}+\left (\frac {6}{11} e^{6} a^{2} c^{2}+\frac {60}{11} d^{2} e^{4} c^{3} a +\frac {15}{11} d^{4} e^{2} c^{4}\right ) x^{11}+\left (2 d \,e^{5} c^{3} a +\frac {5}{3} d^{3} e^{3} c^{4}\right ) x^{12}+\left (\frac {4}{13} e^{6} c^{3} a +\frac {15}{13} d^{2} e^{4} c^{4}\right ) x^{13}+\frac {3 d \,e^{5} c^{4} x^{14}}{7}+\frac {e^{6} c^{4} x^{15}}{15}\)	\(435\)
default	\(\frac {e^{6} c^{4} x^{15}}{15}+\frac {3 d \,e^{5} c^{4} x^{14}}{7}+\frac {\left (4 e^{6} c^{3} a +15 d^{2} e^{4} c^{4}\right ) x^{13}}{13}+\frac {\left (24 d \,e^{5} c^{3} a +20 d^{3} e^{3} c^{4}\right ) x^{12}}{12}+\frac {\left (6 e^{6} a^{2} c^{2}+60 d^{2} e^{4} c^{3} a +15 d^{4} e^{2} c^{4}\right ) x^{11}}{11}+\frac {\left (36 d \,e^{5} a^{2} c^{2}+80 d^{3} e^{3} c^{3} a +6 d^{5} e \,c^{4}\right ) x^{10}}{10}+\frac {\left (4 e^{6} c \,a^{3}+90 d^{2} e^{4} a^{2} c^{2}+60 d^{4} e^{2} c^{3} a +d^{6} c^{4}\right ) x^{9}}{9}+\frac {\left (24 d \,e^{5} c \,a^{3}+120 d^{3} e^{3} a^{2} c^{2}+24 d^{5} e \,c^{3} a \right ) x^{8}}{8}+\frac {\left (e^{6} a^{4}+60 d^{2} e^{4} c \,a^{3}+90 d^{4} e^{2} a^{2} c^{2}+4 d^{6} c^{3} a \right ) x^{7}}{7}+\frac {\left (6 d \,e^{5} a^{4}+80 d^{3} e^{3} c \,a^{3}+36 d^{5} e \,a^{2} c^{2}\right ) x^{6}}{6}+\frac {\left (15 d^{2} e^{4} a^{4}+60 d^{4} e^{2} c \,a^{3}+6 d^{6} a^{2} c^{2}\right ) x^{5}}{5}+\frac {\left (20 d^{3} e^{3} a^{4}+24 d^{5} e c \,a^{3}\right ) x^{4}}{4}+\frac {\left (15 d^{4} e^{2} a^{4}+4 d^{6} c \,a^{3}\right ) x^{3}}{3}+3 d^{5} e \,a^{4} x^{2}+d^{6} a^{4} x\)	\(445\)
gosper	\(\frac {3}{5} x^{10} d^{5} e \,c^{4}+\frac {6}{11} x^{11} e^{6} a^{2} c^{2}+\frac {60}{11} x^{11} d^{2} e^{4} c^{3} a +2 x^{12} d \,e^{5} c^{3} a +6 a^{3} c \,d^{5} e \,x^{4}+3 a^{3} c d \,e^{5} x^{8}+15 a^{2} c^{2} d^{3} e^{3} x^{8}+3 a \,c^{3} d^{5} e \,x^{8}+12 x^{5} d^{4} e^{2} c \,a^{3}+\frac {40}{3} x^{6} d^{3} e^{3} c \,a^{3}+6 x^{6} d^{5} e \,a^{2} c^{2}+\frac {60}{7} x^{7} d^{2} e^{4} c \,a^{3}+\frac {90}{7} x^{7} d^{4} e^{2} a^{2} c^{2}+10 x^{9} d^{2} e^{4} a^{2} c^{2}+\frac {20}{3} x^{9} d^{4} e^{2} c^{3} a +\frac {18}{5} x^{10} d \,e^{5} a^{2} c^{2}+8 x^{10} d^{3} e^{3} c^{3} a +3 d^{5} e \,a^{4} x^{2}+\frac {3}{7} d \,e^{5} c^{4} x^{14}+\frac {15}{11} x^{11} d^{4} e^{2} c^{4}+\frac {5}{3} x^{12} d^{3} e^{3} c^{4}+\frac {4}{13} x^{13} e^{6} c^{3} a +\frac {15}{13} x^{13} d^{2} e^{4} c^{4}+5 x^{3} d^{4} e^{2} a^{4}+\frac {4}{3} x^{3} d^{6} c \,a^{3}+3 x^{5} d^{2} e^{4} a^{4}+\frac {6}{5} x^{5} d^{6} a^{2} c^{2}+x^{6} d \,e^{5} a^{4}+\frac {1}{15} e^{6} c^{4} x^{15}+\frac {4}{7} x^{7} d^{6} c^{3} a +\frac {4}{9} x^{9} e^{6} c \,a^{3}+5 a^{4} d^{3} e^{3} x^{4}+\frac {1}{7} x^{7} e^{6} a^{4}+\frac {1}{9} x^{9} d^{6} c^{4}+d^{6} a^{4} x\)	\(473\)
risch	\(\frac {3}{5} x^{10} d^{5} e \,c^{4}+\frac {6}{11} x^{11} e^{6} a^{2} c^{2}+\frac {60}{11} x^{11} d^{2} e^{4} c^{3} a +2 x^{12} d \,e^{5} c^{3} a +6 a^{3} c \,d^{5} e \,x^{4}+3 a^{3} c d \,e^{5} x^{8}+15 a^{2} c^{2} d^{3} e^{3} x^{8}+3 a \,c^{3} d^{5} e \,x^{8}+12 x^{5} d^{4} e^{2} c \,a^{3}+\frac {40}{3} x^{6} d^{3} e^{3} c \,a^{3}+6 x^{6} d^{5} e \,a^{2} c^{2}+\frac {60}{7} x^{7} d^{2} e^{4} c \,a^{3}+\frac {90}{7} x^{7} d^{4} e^{2} a^{2} c^{2}+10 x^{9} d^{2} e^{4} a^{2} c^{2}+\frac {20}{3} x^{9} d^{4} e^{2} c^{3} a +\frac {18}{5} x^{10} d \,e^{5} a^{2} c^{2}+8 x^{10} d^{3} e^{3} c^{3} a +3 d^{5} e \,a^{4} x^{2}+\frac {3}{7} d \,e^{5} c^{4} x^{14}+\frac {15}{11} x^{11} d^{4} e^{2} c^{4}+\frac {5}{3} x^{12} d^{3} e^{3} c^{4}+\frac {4}{13} x^{13} e^{6} c^{3} a +\frac {15}{13} x^{13} d^{2} e^{4} c^{4}+5 x^{3} d^{4} e^{2} a^{4}+\frac {4}{3} x^{3} d^{6} c \,a^{3}+3 x^{5} d^{2} e^{4} a^{4}+\frac {6}{5} x^{5} d^{6} a^{2} c^{2}+x^{6} d \,e^{5} a^{4}+\frac {1}{15} e^{6} c^{4} x^{15}+\frac {4}{7} x^{7} d^{6} c^{3} a +\frac {4}{9} x^{9} e^{6} c \,a^{3}+5 a^{4} d^{3} e^{3} x^{4}+\frac {1}{7} x^{7} e^{6} a^{4}+\frac {1}{9} x^{9} d^{6} c^{4}+d^{6} a^{4} x\)	\(473\)
parallelrisch	\(\frac {3}{5} x^{10} d^{5} e \,c^{4}+\frac {6}{11} x^{11} e^{6} a^{2} c^{2}+\frac {60}{11} x^{11} d^{2} e^{4} c^{3} a +2 x^{12} d \,e^{5} c^{3} a +6 a^{3} c \,d^{5} e \,x^{4}+3 a^{3} c d \,e^{5} x^{8}+15 a^{2} c^{2} d^{3} e^{3} x^{8}+3 a \,c^{3} d^{5} e \,x^{8}+12 x^{5} d^{4} e^{2} c \,a^{3}+\frac {40}{3} x^{6} d^{3} e^{3} c \,a^{3}+6 x^{6} d^{5} e \,a^{2} c^{2}+\frac {60}{7} x^{7} d^{2} e^{4} c \,a^{3}+\frac {90}{7} x^{7} d^{4} e^{2} a^{2} c^{2}+10 x^{9} d^{2} e^{4} a^{2} c^{2}+\frac {20}{3} x^{9} d^{4} e^{2} c^{3} a +\frac {18}{5} x^{10} d \,e^{5} a^{2} c^{2}+8 x^{10} d^{3} e^{3} c^{3} a +3 d^{5} e \,a^{4} x^{2}+\frac {3}{7} d \,e^{5} c^{4} x^{14}+\frac {15}{11} x^{11} d^{4} e^{2} c^{4}+\frac {5}{3} x^{12} d^{3} e^{3} c^{4}+\frac {4}{13} x^{13} e^{6} c^{3} a +\frac {15}{13} x^{13} d^{2} e^{4} c^{4}+5 x^{3} d^{4} e^{2} a^{4}+\frac {4}{3} x^{3} d^{6} c \,a^{3}+3 x^{5} d^{2} e^{4} a^{4}+\frac {6}{5} x^{5} d^{6} a^{2} c^{2}+x^{6} d \,e^{5} a^{4}+\frac {1}{15} e^{6} c^{4} x^{15}+\frac {4}{7} x^{7} d^{6} c^{3} a +\frac {4}{9} x^{9} e^{6} c \,a^{3}+5 a^{4} d^{3} e^{3} x^{4}+\frac {1}{7} x^{7} e^{6} a^{4}+\frac {1}{9} x^{9} d^{6} c^{4}+d^{6} a^{4} x\)	\(473\)

[In]

int((e*x+d)^6*(c*x^2+a)^4,x,method=_RETURNVERBOSE)

[Out]

d^6*a^4*x+3*d^5*e*a^4*x^2+(5*d^4*e^2*a^4+4/3*d^6*c*a^3)*x^3+(5*a^4*d^3*e^3+6*a^3*c*d^5*e)*x^4+(3*d^2*e^4*a^4+1
2*d^4*e^2*c*a^3+6/5*d^6*a^2*c^2)*x^5+(d*e^5*a^4+40/3*d^3*e^3*c*a^3+6*d^5*e*a^2*c^2)*x^6+(1/7*e^6*a^4+60/7*d^2*
e^4*c*a^3+90/7*d^4*e^2*a^2*c^2+4/7*d^6*c^3*a)*x^7+(3*a^3*c*d*e^5+15*a^2*c^2*d^3*e^3+3*a*c^3*d^5*e)*x^8+(4/9*e^
6*c*a^3+10*d^2*e^4*a^2*c^2+20/3*d^4*e^2*c^3*a+1/9*d^6*c^4)*x^9+(18/5*d*e^5*a^2*c^2+8*d^3*e^3*c^3*a+3/5*d^5*e*c
^4)*x^10+(6/11*e^6*a^2*c^2+60/11*d^2*e^4*c^3*a+15/11*d^4*e^2*c^4)*x^11+(2*d*e^5*c^3*a+5/3*d^3*e^3*c^4)*x^12+(4
/13*e^6*c^3*a+15/13*d^2*e^4*c^4)*x^13+3/7*d*e^5*c^4*x^14+1/15*e^6*c^4*x^15

Fricas [A] (verification not implemented)

none

Time = 0.67 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.60 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {1}{15} \, c^{4} e^{6} x^{15} + \frac {3}{7} \, c^{4} d e^{5} x^{14} + \frac {1}{13} \, {\left (15 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{13} + \frac {1}{3} \, {\left (5 \, c^{4} d^{3} e^{3} + 6 \, a c^{3} d e^{5}\right )} x^{12} + 3 \, a^{4} d^{5} e x^{2} + \frac {3}{11} \, {\left (5 \, c^{4} d^{4} e^{2} + 20 \, a c^{3} d^{2} e^{4} + 2 \, a^{2} c^{2} e^{6}\right )} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, c^{4} d^{5} e + 40 \, a c^{3} d^{3} e^{3} + 18 \, a^{2} c^{2} d e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{6} + 60 \, a c^{3} d^{4} e^{2} + 90 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x^{9} + 3 \, {\left (a c^{3} d^{5} e + 5 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, a c^{3} d^{6} + 90 \, a^{2} c^{2} d^{4} e^{2} + 60 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (18 \, a^{2} c^{2} d^{5} e + 40 \, a^{3} c d^{3} e^{3} + 3 \, a^{4} d e^{5}\right )} x^{6} + \frac {3}{5} \, {\left (2 \, a^{2} c^{2} d^{6} + 20 \, a^{3} c d^{4} e^{2} + 5 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (6 \, a^{3} c d^{5} e + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{6} + 15 \, a^{4} d^{4} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^4,x, algorithm="fricas")

[Out]

1/15*c^4*e^6*x^15 + 3/7*c^4*d*e^5*x^14 + 1/13*(15*c^4*d^2*e^4 + 4*a*c^3*e^6)*x^13 + 1/3*(5*c^4*d^3*e^3 + 6*a*c
^3*d*e^5)*x^12 + 3*a^4*d^5*e*x^2 + 3/11*(5*c^4*d^4*e^2 + 20*a*c^3*d^2*e^4 + 2*a^2*c^2*e^6)*x^11 + a^4*d^6*x +
1/5*(3*c^4*d^5*e + 40*a*c^3*d^3*e^3 + 18*a^2*c^2*d*e^5)*x^10 + 1/9*(c^4*d^6 + 60*a*c^3*d^4*e^2 + 90*a^2*c^2*d^
2*e^4 + 4*a^3*c*e^6)*x^9 + 3*(a*c^3*d^5*e + 5*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*x^8 + 1/7*(4*a*c^3*d^6 + 90*a^2*c
^2*d^4*e^2 + 60*a^3*c*d^2*e^4 + a^4*e^6)*x^7 + 1/3*(18*a^2*c^2*d^5*e + 40*a^3*c*d^3*e^3 + 3*a^4*d*e^5)*x^6 + 3
/5*(2*a^2*c^2*d^6 + 20*a^3*c*d^4*e^2 + 5*a^4*d^2*e^4)*x^5 + (6*a^3*c*d^5*e + 5*a^4*d^3*e^3)*x^4 + 1/3*(4*a^3*c
*d^6 + 15*a^4*d^4*e^2)*x^3

Sympy [A] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 486, normalized size of antiderivative = 1.76 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=a^{4} d^{6} x + 3 a^{4} d^{5} e x^{2} + \frac {3 c^{4} d e^{5} x^{14}}{7} + \frac {c^{4} e^{6} x^{15}}{15} + x^{13} \cdot \left (\frac {4 a c^{3} e^{6}}{13} + \frac {15 c^{4} d^{2} e^{4}}{13}\right ) + x^{12} \cdot \left (2 a c^{3} d e^{5} + \frac {5 c^{4} d^{3} e^{3}}{3}\right ) + x^{11} \cdot \left (\frac {6 a^{2} c^{2} e^{6}}{11} + \frac {60 a c^{3} d^{2} e^{4}}{11} + \frac {15 c^{4} d^{4} e^{2}}{11}\right ) + x^{10} \cdot \left (\frac {18 a^{2} c^{2} d e^{5}}{5} + 8 a c^{3} d^{3} e^{3} + \frac {3 c^{4} d^{5} e}{5}\right ) + x^{9} \cdot \left (\frac {4 a^{3} c e^{6}}{9} + 10 a^{2} c^{2} d^{2} e^{4} + \frac {20 a c^{3} d^{4} e^{2}}{3} + \frac {c^{4} d^{6}}{9}\right ) + x^{8} \cdot \left (3 a^{3} c d e^{5} + 15 a^{2} c^{2} d^{3} e^{3} + 3 a c^{3} d^{5} e\right ) + x^{7} \left (\frac {a^{4} e^{6}}{7} + \frac {60 a^{3} c d^{2} e^{4}}{7} + \frac {90 a^{2} c^{2} d^{4} e^{2}}{7} + \frac {4 a c^{3} d^{6}}{7}\right ) + x^{6} \left (a^{4} d e^{5} + \frac {40 a^{3} c d^{3} e^{3}}{3} + 6 a^{2} c^{2} d^{5} e\right ) + x^{5} \cdot \left (3 a^{4} d^{2} e^{4} + 12 a^{3} c d^{4} e^{2} + \frac {6 a^{2} c^{2} d^{6}}{5}\right ) + x^{4} \cdot \left (5 a^{4} d^{3} e^{3} + 6 a^{3} c d^{5} e\right ) + x^{3} \cdot \left (5 a^{4} d^{4} e^{2} + \frac {4 a^{3} c d^{6}}{3}\right ) \]

[In]

integrate((e*x+d)**6*(c*x**2+a)**4,x)

[Out]

a**4*d**6*x + 3*a**4*d**5*e*x**2 + 3*c**4*d*e**5*x**14/7 + c**4*e**6*x**15/15 + x**13*(4*a*c**3*e**6/13 + 15*c
**4*d**2*e**4/13) + x**12*(2*a*c**3*d*e**5 + 5*c**4*d**3*e**3/3) + x**11*(6*a**2*c**2*e**6/11 + 60*a*c**3*d**2
*e**4/11 + 15*c**4*d**4*e**2/11) + x**10*(18*a**2*c**2*d*e**5/5 + 8*a*c**3*d**3*e**3 + 3*c**4*d**5*e/5) + x**9
*(4*a**3*c*e**6/9 + 10*a**2*c**2*d**2*e**4 + 20*a*c**3*d**4*e**2/3 + c**4*d**6/9) + x**8*(3*a**3*c*d*e**5 + 15
*a**2*c**2*d**3*e**3 + 3*a*c**3*d**5*e) + x**7*(a**4*e**6/7 + 60*a**3*c*d**2*e**4/7 + 90*a**2*c**2*d**4*e**2/7
+ 4*a*c**3*d**6/7) + x**6*(a**4*d*e**5 + 40*a**3*c*d**3*e**3/3 + 6*a**2*c**2*d**5*e) + x**5*(3*a**4*d**2*e**4
+ 12*a**3*c*d**4*e**2 + 6*a**2*c**2*d**6/5) + x**4*(5*a**4*d**3*e**3 + 6*a**3*c*d**5*e) + x**3*(5*a**4*d**4*e
**2 + 4*a**3*c*d**6/3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 441, normalized size of antiderivative = 1.60 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {1}{15} \, c^{4} e^{6} x^{15} + \frac {3}{7} \, c^{4} d e^{5} x^{14} + \frac {1}{13} \, {\left (15 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{13} + \frac {1}{3} \, {\left (5 \, c^{4} d^{3} e^{3} + 6 \, a c^{3} d e^{5}\right )} x^{12} + 3 \, a^{4} d^{5} e x^{2} + \frac {3}{11} \, {\left (5 \, c^{4} d^{4} e^{2} + 20 \, a c^{3} d^{2} e^{4} + 2 \, a^{2} c^{2} e^{6}\right )} x^{11} + a^{4} d^{6} x + \frac {1}{5} \, {\left (3 \, c^{4} d^{5} e + 40 \, a c^{3} d^{3} e^{3} + 18 \, a^{2} c^{2} d e^{5}\right )} x^{10} + \frac {1}{9} \, {\left (c^{4} d^{6} + 60 \, a c^{3} d^{4} e^{2} + 90 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x^{9} + 3 \, {\left (a c^{3} d^{5} e + 5 \, a^{2} c^{2} d^{3} e^{3} + a^{3} c d e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (4 \, a c^{3} d^{6} + 90 \, a^{2} c^{2} d^{4} e^{2} + 60 \, a^{3} c d^{2} e^{4} + a^{4} e^{6}\right )} x^{7} + \frac {1}{3} \, {\left (18 \, a^{2} c^{2} d^{5} e + 40 \, a^{3} c d^{3} e^{3} + 3 \, a^{4} d e^{5}\right )} x^{6} + \frac {3}{5} \, {\left (2 \, a^{2} c^{2} d^{6} + 20 \, a^{3} c d^{4} e^{2} + 5 \, a^{4} d^{2} e^{4}\right )} x^{5} + {\left (6 \, a^{3} c d^{5} e + 5 \, a^{4} d^{3} e^{3}\right )} x^{4} + \frac {1}{3} \, {\left (4 \, a^{3} c d^{6} + 15 \, a^{4} d^{4} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^4,x, algorithm="maxima")

[Out]

1/15*c^4*e^6*x^15 + 3/7*c^4*d*e^5*x^14 + 1/13*(15*c^4*d^2*e^4 + 4*a*c^3*e^6)*x^13 + 1/3*(5*c^4*d^3*e^3 + 6*a*c
^3*d*e^5)*x^12 + 3*a^4*d^5*e*x^2 + 3/11*(5*c^4*d^4*e^2 + 20*a*c^3*d^2*e^4 + 2*a^2*c^2*e^6)*x^11 + a^4*d^6*x +
1/5*(3*c^4*d^5*e + 40*a*c^3*d^3*e^3 + 18*a^2*c^2*d*e^5)*x^10 + 1/9*(c^4*d^6 + 60*a*c^3*d^4*e^2 + 90*a^2*c^2*d^
2*e^4 + 4*a^3*c*e^6)*x^9 + 3*(a*c^3*d^5*e + 5*a^2*c^2*d^3*e^3 + a^3*c*d*e^5)*x^8 + 1/7*(4*a*c^3*d^6 + 90*a^2*c
^2*d^4*e^2 + 60*a^3*c*d^2*e^4 + a^4*e^6)*x^7 + 1/3*(18*a^2*c^2*d^5*e + 40*a^3*c*d^3*e^3 + 3*a^4*d*e^5)*x^6 + 3
/5*(2*a^2*c^2*d^6 + 20*a^3*c*d^4*e^2 + 5*a^4*d^2*e^4)*x^5 + (6*a^3*c*d^5*e + 5*a^4*d^3*e^3)*x^4 + 1/3*(4*a^3*c
*d^6 + 15*a^4*d^4*e^2)*x^3

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 472, normalized size of antiderivative = 1.71 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=\frac {1}{15} \, c^{4} e^{6} x^{15} + \frac {3}{7} \, c^{4} d e^{5} x^{14} + \frac {15}{13} \, c^{4} d^{2} e^{4} x^{13} + \frac {4}{13} \, a c^{3} e^{6} x^{13} + \frac {5}{3} \, c^{4} d^{3} e^{3} x^{12} + 2 \, a c^{3} d e^{5} x^{12} + \frac {15}{11} \, c^{4} d^{4} e^{2} x^{11} + \frac {60}{11} \, a c^{3} d^{2} e^{4} x^{11} + \frac {6}{11} \, a^{2} c^{2} e^{6} x^{11} + \frac {3}{5} \, c^{4} d^{5} e x^{10} + 8 \, a c^{3} d^{3} e^{3} x^{10} + \frac {18}{5} \, a^{2} c^{2} d e^{5} x^{10} + \frac {1}{9} \, c^{4} d^{6} x^{9} + \frac {20}{3} \, a c^{3} d^{4} e^{2} x^{9} + 10 \, a^{2} c^{2} d^{2} e^{4} x^{9} + \frac {4}{9} \, a^{3} c e^{6} x^{9} + 3 \, a c^{3} d^{5} e x^{8} + 15 \, a^{2} c^{2} d^{3} e^{3} x^{8} + 3 \, a^{3} c d e^{5} x^{8} + \frac {4}{7} \, a c^{3} d^{6} x^{7} + \frac {90}{7} \, a^{2} c^{2} d^{4} e^{2} x^{7} + \frac {60}{7} \, a^{3} c d^{2} e^{4} x^{7} + \frac {1}{7} \, a^{4} e^{6} x^{7} + 6 \, a^{2} c^{2} d^{5} e x^{6} + \frac {40}{3} \, a^{3} c d^{3} e^{3} x^{6} + a^{4} d e^{5} x^{6} + \frac {6}{5} \, a^{2} c^{2} d^{6} x^{5} + 12 \, a^{3} c d^{4} e^{2} x^{5} + 3 \, a^{4} d^{2} e^{4} x^{5} + 6 \, a^{3} c d^{5} e x^{4} + 5 \, a^{4} d^{3} e^{3} x^{4} + \frac {4}{3} \, a^{3} c d^{6} x^{3} + 5 \, a^{4} d^{4} e^{2} x^{3} + 3 \, a^{4} d^{5} e x^{2} + a^{4} d^{6} x \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^4,x, algorithm="giac")

[Out]

1/15*c^4*e^6*x^15 + 3/7*c^4*d*e^5*x^14 + 15/13*c^4*d^2*e^4*x^13 + 4/13*a*c^3*e^6*x^13 + 5/3*c^4*d^3*e^3*x^12 +
2*a*c^3*d*e^5*x^12 + 15/11*c^4*d^4*e^2*x^11 + 60/11*a*c^3*d^2*e^4*x^11 + 6/11*a^2*c^2*e^6*x^11 + 3/5*c^4*d^5*
e*x^10 + 8*a*c^3*d^3*e^3*x^10 + 18/5*a^2*c^2*d*e^5*x^10 + 1/9*c^4*d^6*x^9 + 20/3*a*c^3*d^4*e^2*x^9 + 10*a^2*c^
2*d^2*e^4*x^9 + 4/9*a^3*c*e^6*x^9 + 3*a*c^3*d^5*e*x^8 + 15*a^2*c^2*d^3*e^3*x^8 + 3*a^3*c*d*e^5*x^8 + 4/7*a*c^3
*d^6*x^7 + 90/7*a^2*c^2*d^4*e^2*x^7 + 60/7*a^3*c*d^2*e^4*x^7 + 1/7*a^4*e^6*x^7 + 6*a^2*c^2*d^5*e*x^6 + 40/3*a^
3*c*d^3*e^3*x^6 + a^4*d*e^5*x^6 + 6/5*a^2*c^2*d^6*x^5 + 12*a^3*c*d^4*e^2*x^5 + 3*a^4*d^2*e^4*x^5 + 6*a^3*c*d^5
*e*x^4 + 5*a^4*d^3*e^3*x^4 + 4/3*a^3*c*d^6*x^3 + 5*a^4*d^4*e^2*x^3 + 3*a^4*d^5*e*x^2 + a^4*d^6*x

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.54 \[ \int (d+e x)^6 \left (a+c x^2\right )^4 \, dx=x^7\,\left (\frac {a^4\,e^6}{7}+\frac {60\,a^3\,c\,d^2\,e^4}{7}+\frac {90\,a^2\,c^2\,d^4\,e^2}{7}+\frac {4\,a\,c^3\,d^6}{7}\right )+x^9\,\left (\frac {4\,a^3\,c\,e^6}{9}+10\,a^2\,c^2\,d^2\,e^4+\frac {20\,a\,c^3\,d^4\,e^2}{3}+\frac {c^4\,d^6}{9}\right )+x^3\,\left (5\,a^4\,d^4\,e^2+\frac {4\,c\,a^3\,d^6}{3}\right )+x^{13}\,\left (\frac {15\,c^4\,d^2\,e^4}{13}+\frac {4\,a\,c^3\,e^6}{13}\right )+x^5\,\left (3\,a^4\,d^2\,e^4+12\,a^3\,c\,d^4\,e^2+\frac {6\,a^2\,c^2\,d^6}{5}\right )+x^{11}\,\left (\frac {6\,a^2\,c^2\,e^6}{11}+\frac {60\,a\,c^3\,d^2\,e^4}{11}+\frac {15\,c^4\,d^4\,e^2}{11}\right )+a^4\,d^6\,x+\frac {c^4\,e^6\,x^{15}}{15}+3\,a^4\,d^5\,e\,x^2+\frac {3\,c^4\,d\,e^5\,x^{14}}{7}+a^3\,d^3\,e\,x^4\,\left (6\,c\,d^2+5\,a\,e^2\right )+\frac {c^3\,d\,e^3\,x^{12}\,\left (5\,c\,d^2+6\,a\,e^2\right )}{3}+\frac {a^2\,d\,e\,x^6\,\left (3\,a^2\,e^4+40\,a\,c\,d^2\,e^2+18\,c^2\,d^4\right )}{3}+\frac {c^2\,d\,e\,x^{10}\,\left (18\,a^2\,e^4+40\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )}{5}+3\,a\,c\,d\,e\,x^8\,\left (a^2\,e^4+5\,a\,c\,d^2\,e^2+c^2\,d^4\right ) \]

[In]

int((a + c*x^2)^4*(d + e*x)^6,x)

[Out]

x^7*((a^4*e^6)/7 + (4*a*c^3*d^6)/7 + (60*a^3*c*d^2*e^4)/7 + (90*a^2*c^2*d^4*e^2)/7) + x^9*((c^4*d^6)/9 + (4*a^
3*c*e^6)/9 + (20*a*c^3*d^4*e^2)/3 + 10*a^2*c^2*d^2*e^4) + x^3*((4*a^3*c*d^6)/3 + 5*a^4*d^4*e^2) + x^13*((4*a*c
^3*e^6)/13 + (15*c^4*d^2*e^4)/13) + x^5*((6*a^2*c^2*d^6)/5 + 3*a^4*d^2*e^4 + 12*a^3*c*d^4*e^2) + x^11*((6*a^2*
c^2*e^6)/11 + (15*c^4*d^4*e^2)/11 + (60*a*c^3*d^2*e^4)/11) + a^4*d^6*x + (c^4*e^6*x^15)/15 + 3*a^4*d^5*e*x^2 +
(3*c^4*d*e^5*x^14)/7 + a^3*d^3*e*x^4*(5*a*e^2 + 6*c*d^2) + (c^3*d*e^3*x^12*(6*a*e^2 + 5*c*d^2))/3 + (a^2*d*e*
x^6*(3*a^2*e^4 + 18*c^2*d^4 + 40*a*c*d^2*e^2))/3 + (c^2*d*e*x^10*(18*a^2*e^4 + 3*c^2*d^4 + 40*a*c*d^2*e^2))/5
+ 3*a*c*d*e*x^8*(a^2*e^4 + c^2*d^4 + 5*a*c*d^2*e^2)