\(\int (d+e x)^6 (a+c x^2)^3 \, dx\) [472]

Optimal result

Integrand size = 17, antiderivative size = 190 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=\frac {\left (c d^2+a e^2\right )^3 (d+e x)^7}{7 e^7}-\frac {3 c d \left (c d^2+a e^2\right )^2 (d+e x)^8}{4 e^7}+\frac {c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^9}{3 e^7}-\frac {2 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^7}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11}}{11 e^7}-\frac {c^3 d (d+e x)^{12}}{2 e^7}+\frac {c^3 (d+e x)^{13}}{13 e^7} \]

[Out]

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 190, 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} \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=\frac {3 c^2 (d+e x)^{11} \left (a e^2+5 c d^2\right )}{11 e^7}-\frac {2 c^2 d (d+e x)^{10} \left (3 a e^2+5 c d^2\right )}{5 e^7}+\frac {c (d+e x)^9 \left (a e^2+c d^2\right ) \left (a e^2+5 c d^2\right )}{3 e^7}-\frac {3 c d (d+e x)^8 \left (a e^2+c d^2\right )^2}{4 e^7}+\frac {(d+e x)^7 \left (a e^2+c d^2\right )^3}{7 e^7}+\frac {c^3 (d+e x)^{13}}{13 e^7}-\frac {c^3 d (d+e x)^{12}}{2 e^7} \]

[In]

Int[(d + e*x)^6*(a + c*x^2)^3,x]

[Out]

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

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 )^3 (d+e x)^6}{e^6}-\frac {6 c d \left (c d^2+a e^2\right )^2 (d+e x)^7}{e^6}+\frac {3 c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^8}{e^6}-\frac {4 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^9}{e^6}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{10}}{e^6}-\frac {6 c^3 d (d+e x)^{11}}{e^6}+\frac {c^3 (d+e x)^{12}}{e^6}\right ) \, dx \\ & = \frac {\left (c d^2+a e^2\right )^3 (d+e x)^7}{7 e^7}-\frac {3 c d \left (c d^2+a e^2\right )^2 (d+e x)^8}{4 e^7}+\frac {c \left (c d^2+a e^2\right ) \left (5 c d^2+a e^2\right ) (d+e x)^9}{3 e^7}-\frac {2 c^2 d \left (5 c d^2+3 a e^2\right ) (d+e x)^{10}}{5 e^7}+\frac {3 c^2 \left (5 c d^2+a e^2\right ) (d+e x)^{11}}{11 e^7}-\frac {c^3 d (d+e x)^{12}}{2 e^7}+\frac {c^3 (d+e x)^{13}}{13 e^7} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 338, normalized size of antiderivative = 1.78 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=a^3 d^6 x+3 a^3 d^5 e x^2+a^2 d^4 \left (c d^2+5 a e^2\right ) x^3+\frac {1}{2} a^2 d^3 e \left (9 c d^2+10 a e^2\right ) x^4+\frac {3}{5} a d^2 \left (c^2 d^4+15 a c d^2 e^2+5 a^2 e^4\right ) x^5+a d e \left (3 c^2 d^4+10 a c d^2 e^2+a^2 e^4\right ) x^6+\frac {1}{7} \left (c^3 d^6+45 a c^2 d^4 e^2+45 a^2 c d^2 e^4+a^3 e^6\right ) x^7+\frac {3}{4} c d e \left (c^2 d^4+10 a c d^2 e^2+3 a^2 e^4\right ) x^8+\frac {1}{3} c e^2 \left (5 c^2 d^4+15 a c d^2 e^2+a^2 e^4\right ) x^9+\frac {1}{5} c^2 d e^3 \left (10 c d^2+9 a e^2\right ) x^{10}+\frac {3}{11} c^2 e^4 \left (5 c d^2+a e^2\right ) x^{11}+\frac {1}{2} c^3 d e^5 x^{12}+\frac {1}{13} c^3 e^6 x^{13} \]

[In]

Integrate[(d + e*x)^6*(a + c*x^2)^3,x]

[Out]

a^3*d^6*x + 3*a^3*d^5*e*x^2 + a^2*d^4*(c*d^2 + 5*a*e^2)*x^3 + (a^2*d^3*e*(9*c*d^2 + 10*a*e^2)*x^4)/2 + (3*a*d^
2*(c^2*d^4 + 15*a*c*d^2*e^2 + 5*a^2*e^4)*x^5)/5 + a*d*e*(3*c^2*d^4 + 10*a*c*d^2*e^2 + a^2*e^4)*x^6 + ((c^3*d^6
+ 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)*x^7)/7 + (3*c*d*e*(c^2*d^4 + 10*a*c*d^2*e^2 + 3*a^2*e^4)*x^8
)/4 + (c*e^2*(5*c^2*d^4 + 15*a*c*d^2*e^2 + a^2*e^4)*x^9)/3 + (c^2*d*e^3*(10*c*d^2 + 9*a*e^2)*x^10)/5 + (3*c^2*
e^4*(5*c*d^2 + a*e^2)*x^11)/11 + (c^3*d*e^5*x^12)/2 + (c^3*e^6*x^13)/13

Maple [A] (verified)

Time = 2.51 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.77

method	result	size
norman	\(\frac {e^{6} c^{3} x^{13}}{13}+\frac {d \,e^{5} c^{3} x^{12}}{2}+\left (\frac {3}{11} e^{6} c^{2} a +\frac {15}{11} d^{2} e^{4} c^{3}\right ) x^{11}+\left (\frac {9}{5} d \,e^{5} c^{2} a +2 d^{3} e^{3} c^{3}\right ) x^{10}+\left (\frac {1}{3} e^{6} a^{2} c +5 d^{2} e^{4} c^{2} a +\frac {5}{3} d^{4} e^{2} c^{3}\right ) x^{9}+\left (\frac {9}{4} d \,e^{5} a^{2} c +\frac {15}{2} d^{3} e^{3} c^{2} a +\frac {3}{4} c^{3} d^{5} e \right ) x^{8}+\left (\frac {1}{7} e^{6} a^{3}+\frac {45}{7} d^{2} e^{4} a^{2} c +\frac {45}{7} d^{4} e^{2} c^{2} a +\frac {1}{7} c^{3} d^{6}\right ) x^{7}+\left (d \,e^{5} a^{3}+10 d^{3} e^{3} a^{2} c +3 d^{5} e \,c^{2} a \right ) x^{6}+\left (3 d^{2} e^{4} a^{3}+9 d^{4} e^{2} a^{2} c +\frac {3}{5} d^{6} c^{2} a \right ) x^{5}+\left (5 d^{3} e^{3} a^{3}+\frac {9}{2} d^{5} e \,a^{2} c \right ) x^{4}+\left (5 d^{4} e^{2} a^{3}+d^{6} a^{2} c \right ) x^{3}+3 d^{5} e \,a^{3} x^{2}+d^{6} a^{3} x\)	\(336\)
default	\(\frac {e^{6} c^{3} x^{13}}{13}+\frac {d \,e^{5} c^{3} x^{12}}{2}+\frac {\left (3 e^{6} c^{2} a +15 d^{2} e^{4} c^{3}\right ) x^{11}}{11}+\frac {\left (18 d \,e^{5} c^{2} a +20 d^{3} e^{3} c^{3}\right ) x^{10}}{10}+\frac {\left (3 e^{6} a^{2} c +45 d^{2} e^{4} c^{2} a +15 d^{4} e^{2} c^{3}\right ) x^{9}}{9}+\frac {\left (18 d \,e^{5} a^{2} c +60 d^{3} e^{3} c^{2} a +6 c^{3} d^{5} e \right ) x^{8}}{8}+\frac {\left (e^{6} a^{3}+45 d^{2} e^{4} a^{2} c +45 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) x^{7}}{7}+\frac {\left (6 d \,e^{5} a^{3}+60 d^{3} e^{3} a^{2} c +18 d^{5} e \,c^{2} a \right ) x^{6}}{6}+\frac {\left (15 d^{2} e^{4} a^{3}+45 d^{4} e^{2} a^{2} c +3 d^{6} c^{2} a \right ) x^{5}}{5}+\frac {\left (20 d^{3} e^{3} a^{3}+18 d^{5} e \,a^{2} c \right ) x^{4}}{4}+\frac {\left (15 d^{4} e^{2} a^{3}+3 d^{6} a^{2} c \right ) x^{3}}{3}+3 d^{5} e \,a^{3} x^{2}+d^{6} a^{3} x\)	\(345\)
gosper	\(5 x^{9} d^{2} e^{4} c^{2} a +\frac {9}{4} x^{8} d \,e^{5} a^{2} c +\frac {9}{5} x^{10} d \,e^{5} c^{2} a +\frac {45}{7} x^{7} d^{2} e^{4} a^{2} c +10 a^{2} c \,d^{3} e^{3} x^{6}+3 a \,c^{2} d^{5} e \,x^{6}+\frac {3}{5} x^{5} d^{6} c^{2} a +5 x^{4} d^{3} e^{3} a^{3}+\frac {9}{2} x^{4} d^{5} e \,a^{2} c +\frac {45}{7} x^{7} d^{4} e^{2} c^{2} a +9 x^{5} d^{4} e^{2} a^{2} c +\frac {15}{2} x^{8} d^{3} e^{3} c^{2} a +\frac {1}{13} e^{6} c^{3} x^{13}+d^{6} a^{3} x +\frac {5}{3} x^{9} d^{4} e^{2} c^{3}+\frac {3}{4} x^{8} c^{3} d^{5} e +3 x^{5} d^{2} e^{4} a^{3}+a^{3} d \,e^{5} x^{6}+5 a^{3} d^{4} e^{2} x^{3}+a^{2} c \,d^{6} x^{3}+\frac {15}{11} x^{11} d^{2} e^{4} c^{3}+\frac {3}{11} x^{11} e^{6} c^{2} a +\frac {1}{2} d \,e^{5} c^{3} x^{12}+3 d^{5} e \,a^{3} x^{2}+2 x^{10} d^{3} e^{3} c^{3}+\frac {1}{3} x^{9} e^{6} a^{2} c +\frac {1}{7} x^{7} e^{6} a^{3}+\frac {1}{7} x^{7} c^{3} d^{6}\)	\(363\)
risch	\(5 x^{9} d^{2} e^{4} c^{2} a +\frac {9}{4} x^{8} d \,e^{5} a^{2} c +\frac {9}{5} x^{10} d \,e^{5} c^{2} a +\frac {45}{7} x^{7} d^{2} e^{4} a^{2} c +10 a^{2} c \,d^{3} e^{3} x^{6}+3 a \,c^{2} d^{5} e \,x^{6}+\frac {3}{5} x^{5} d^{6} c^{2} a +5 x^{4} d^{3} e^{3} a^{3}+\frac {9}{2} x^{4} d^{5} e \,a^{2} c +\frac {45}{7} x^{7} d^{4} e^{2} c^{2} a +9 x^{5} d^{4} e^{2} a^{2} c +\frac {15}{2} x^{8} d^{3} e^{3} c^{2} a +\frac {1}{13} e^{6} c^{3} x^{13}+d^{6} a^{3} x +\frac {5}{3} x^{9} d^{4} e^{2} c^{3}+\frac {3}{4} x^{8} c^{3} d^{5} e +3 x^{5} d^{2} e^{4} a^{3}+a^{3} d \,e^{5} x^{6}+5 a^{3} d^{4} e^{2} x^{3}+a^{2} c \,d^{6} x^{3}+\frac {15}{11} x^{11} d^{2} e^{4} c^{3}+\frac {3}{11} x^{11} e^{6} c^{2} a +\frac {1}{2} d \,e^{5} c^{3} x^{12}+3 d^{5} e \,a^{3} x^{2}+2 x^{10} d^{3} e^{3} c^{3}+\frac {1}{3} x^{9} e^{6} a^{2} c +\frac {1}{7} x^{7} e^{6} a^{3}+\frac {1}{7} x^{7} c^{3} d^{6}\)	\(363\)
parallelrisch	\(5 x^{9} d^{2} e^{4} c^{2} a +\frac {9}{4} x^{8} d \,e^{5} a^{2} c +\frac {9}{5} x^{10} d \,e^{5} c^{2} a +\frac {45}{7} x^{7} d^{2} e^{4} a^{2} c +10 a^{2} c \,d^{3} e^{3} x^{6}+3 a \,c^{2} d^{5} e \,x^{6}+\frac {3}{5} x^{5} d^{6} c^{2} a +5 x^{4} d^{3} e^{3} a^{3}+\frac {9}{2} x^{4} d^{5} e \,a^{2} c +\frac {45}{7} x^{7} d^{4} e^{2} c^{2} a +9 x^{5} d^{4} e^{2} a^{2} c +\frac {15}{2} x^{8} d^{3} e^{3} c^{2} a +\frac {1}{13} e^{6} c^{3} x^{13}+d^{6} a^{3} x +\frac {5}{3} x^{9} d^{4} e^{2} c^{3}+\frac {3}{4} x^{8} c^{3} d^{5} e +3 x^{5} d^{2} e^{4} a^{3}+a^{3} d \,e^{5} x^{6}+5 a^{3} d^{4} e^{2} x^{3}+a^{2} c \,d^{6} x^{3}+\frac {15}{11} x^{11} d^{2} e^{4} c^{3}+\frac {3}{11} x^{11} e^{6} c^{2} a +\frac {1}{2} d \,e^{5} c^{3} x^{12}+3 d^{5} e \,a^{3} x^{2}+2 x^{10} d^{3} e^{3} c^{3}+\frac {1}{3} x^{9} e^{6} a^{2} c +\frac {1}{7} x^{7} e^{6} a^{3}+\frac {1}{7} x^{7} c^{3} d^{6}\)	\(363\)

[In]

int((e*x+d)^6*(c*x^2+a)^3,x,method=_RETURNVERBOSE)

[Out]

1/13*e^6*c^3*x^13+1/2*d*e^5*c^3*x^12+(3/11*e^6*c^2*a+15/11*d^2*e^4*c^3)*x^11+(9/5*d*e^5*c^2*a+2*d^3*e^3*c^3)*x
^10+(1/3*e^6*a^2*c+5*d^2*e^4*c^2*a+5/3*d^4*e^2*c^3)*x^9+(9/4*d*e^5*a^2*c+15/2*d^3*e^3*c^2*a+3/4*c^3*d^5*e)*x^8
+(1/7*e^6*a^3+45/7*d^2*e^4*a^2*c+45/7*d^4*e^2*c^2*a+1/7*c^3*d^6)*x^7+(a^3*d*e^5+10*a^2*c*d^3*e^3+3*a*c^2*d^5*e
)*x^6+(3*d^2*e^4*a^3+9*d^4*e^2*a^2*c+3/5*d^6*c^2*a)*x^5+(5*d^3*e^3*a^3+9/2*d^5*e*a^2*c)*x^4+(5*a^3*d^4*e^2+a^2
*c*d^6)*x^3+3*d^5*e*a^3*x^2+d^6*a^3*x

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.77 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, c^{3} e^{6} x^{13} + \frac {1}{2} \, c^{3} d e^{5} x^{12} + \frac {3}{11} \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{11} + 3 \, a^{3} d^{5} e x^{2} + \frac {1}{5} \, {\left (10 \, c^{3} d^{3} e^{3} + 9 \, a c^{2} d e^{5}\right )} x^{10} + a^{3} d^{6} x + \frac {1}{3} \, {\left (5 \, c^{3} d^{4} e^{2} + 15 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{6} + 45 \, a c^{2} d^{4} e^{2} + 45 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{7} + {\left (3 \, a c^{2} d^{5} e + 10 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{6} + \frac {3}{5} \, {\left (a c^{2} d^{6} + 15 \, a^{2} c d^{4} e^{2} + 5 \, a^{3} d^{2} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (9 \, a^{2} c d^{5} e + 10 \, a^{3} d^{3} e^{3}\right )} x^{4} + {\left (a^{2} c d^{6} + 5 \, a^{3} d^{4} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="fricas")

[Out]

1/13*c^3*e^6*x^13 + 1/2*c^3*d*e^5*x^12 + 3/11*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^11 + 3*a^3*d^5*e*x^2 + 1/5*(10*c^3
*d^3*e^3 + 9*a*c^2*d*e^5)*x^10 + a^3*d^6*x + 1/3*(5*c^3*d^4*e^2 + 15*a*c^2*d^2*e^4 + a^2*c*e^6)*x^9 + 3/4*(c^3
*d^5*e + 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x^8 + 1/7*(c^3*d^6 + 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)
*x^7 + (3*a*c^2*d^5*e + 10*a^2*c*d^3*e^3 + a^3*d*e^5)*x^6 + 3/5*(a*c^2*d^6 + 15*a^2*c*d^4*e^2 + 5*a^3*d^2*e^4)
*x^5 + 1/2*(9*a^2*c*d^5*e + 10*a^3*d^3*e^3)*x^4 + (a^2*c*d^6 + 5*a^3*d^4*e^2)*x^3

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 371 vs. \(2 (180) = 360\).

Time = 0.05 (sec) , antiderivative size = 371, normalized size of antiderivative = 1.95 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=a^{3} d^{6} x + 3 a^{3} d^{5} e x^{2} + \frac {c^{3} d e^{5} x^{12}}{2} + \frac {c^{3} e^{6} x^{13}}{13} + x^{11} \cdot \left (\frac {3 a c^{2} e^{6}}{11} + \frac {15 c^{3} d^{2} e^{4}}{11}\right ) + x^{10} \cdot \left (\frac {9 a c^{2} d e^{5}}{5} + 2 c^{3} d^{3} e^{3}\right ) + x^{9} \left (\frac {a^{2} c e^{6}}{3} + 5 a c^{2} d^{2} e^{4} + \frac {5 c^{3} d^{4} e^{2}}{3}\right ) + x^{8} \cdot \left (\frac {9 a^{2} c d e^{5}}{4} + \frac {15 a c^{2} d^{3} e^{3}}{2} + \frac {3 c^{3} d^{5} e}{4}\right ) + x^{7} \left (\frac {a^{3} e^{6}}{7} + \frac {45 a^{2} c d^{2} e^{4}}{7} + \frac {45 a c^{2} d^{4} e^{2}}{7} + \frac {c^{3} d^{6}}{7}\right ) + x^{6} \left (a^{3} d e^{5} + 10 a^{2} c d^{3} e^{3} + 3 a c^{2} d^{5} e\right ) + x^{5} \cdot \left (3 a^{3} d^{2} e^{4} + 9 a^{2} c d^{4} e^{2} + \frac {3 a c^{2} d^{6}}{5}\right ) + x^{4} \cdot \left (5 a^{3} d^{3} e^{3} + \frac {9 a^{2} c d^{5} e}{2}\right ) + x^{3} \cdot \left (5 a^{3} d^{4} e^{2} + a^{2} c d^{6}\right ) \]

[In]

integrate((e*x+d)**6*(c*x**2+a)**3,x)

[Out]

a**3*d**6*x + 3*a**3*d**5*e*x**2 + c**3*d*e**5*x**12/2 + c**3*e**6*x**13/13 + x**11*(3*a*c**2*e**6/11 + 15*c**
3*d**2*e**4/11) + x**10*(9*a*c**2*d*e**5/5 + 2*c**3*d**3*e**3) + x**9*(a**2*c*e**6/3 + 5*a*c**2*d**2*e**4 + 5*
c**3*d**4*e**2/3) + x**8*(9*a**2*c*d*e**5/4 + 15*a*c**2*d**3*e**3/2 + 3*c**3*d**5*e/4) + x**7*(a**3*e**6/7 + 4
5*a**2*c*d**2*e**4/7 + 45*a*c**2*d**4*e**2/7 + c**3*d**6/7) + x**6*(a**3*d*e**5 + 10*a**2*c*d**3*e**3 + 3*a*c*
*2*d**5*e) + x**5*(3*a**3*d**2*e**4 + 9*a**2*c*d**4*e**2 + 3*a*c**2*d**6/5) + x**4*(5*a**3*d**3*e**3 + 9*a**2*
c*d**5*e/2) + x**3*(5*a**3*d**4*e**2 + a**2*c*d**6)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 336, normalized size of antiderivative = 1.77 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, c^{3} e^{6} x^{13} + \frac {1}{2} \, c^{3} d e^{5} x^{12} + \frac {3}{11} \, {\left (5 \, c^{3} d^{2} e^{4} + a c^{2} e^{6}\right )} x^{11} + 3 \, a^{3} d^{5} e x^{2} + \frac {1}{5} \, {\left (10 \, c^{3} d^{3} e^{3} + 9 \, a c^{2} d e^{5}\right )} x^{10} + a^{3} d^{6} x + \frac {1}{3} \, {\left (5 \, c^{3} d^{4} e^{2} + 15 \, a c^{2} d^{2} e^{4} + a^{2} c e^{6}\right )} x^{9} + \frac {3}{4} \, {\left (c^{3} d^{5} e + 10 \, a c^{2} d^{3} e^{3} + 3 \, a^{2} c d e^{5}\right )} x^{8} + \frac {1}{7} \, {\left (c^{3} d^{6} + 45 \, a c^{2} d^{4} e^{2} + 45 \, a^{2} c d^{2} e^{4} + a^{3} e^{6}\right )} x^{7} + {\left (3 \, a c^{2} d^{5} e + 10 \, a^{2} c d^{3} e^{3} + a^{3} d e^{5}\right )} x^{6} + \frac {3}{5} \, {\left (a c^{2} d^{6} + 15 \, a^{2} c d^{4} e^{2} + 5 \, a^{3} d^{2} e^{4}\right )} x^{5} + \frac {1}{2} \, {\left (9 \, a^{2} c d^{5} e + 10 \, a^{3} d^{3} e^{3}\right )} x^{4} + {\left (a^{2} c d^{6} + 5 \, a^{3} d^{4} e^{2}\right )} x^{3} \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="maxima")

[Out]

1/13*c^3*e^6*x^13 + 1/2*c^3*d*e^5*x^12 + 3/11*(5*c^3*d^2*e^4 + a*c^2*e^6)*x^11 + 3*a^3*d^5*e*x^2 + 1/5*(10*c^3
*d^3*e^3 + 9*a*c^2*d*e^5)*x^10 + a^3*d^6*x + 1/3*(5*c^3*d^4*e^2 + 15*a*c^2*d^2*e^4 + a^2*c*e^6)*x^9 + 3/4*(c^3
*d^5*e + 10*a*c^2*d^3*e^3 + 3*a^2*c*d*e^5)*x^8 + 1/7*(c^3*d^6 + 45*a*c^2*d^4*e^2 + 45*a^2*c*d^2*e^4 + a^3*e^6)
*x^7 + (3*a*c^2*d^5*e + 10*a^2*c*d^3*e^3 + a^3*d*e^5)*x^6 + 3/5*(a*c^2*d^6 + 15*a^2*c*d^4*e^2 + 5*a^3*d^2*e^4)
*x^5 + 1/2*(9*a^2*c*d^5*e + 10*a^3*d^3*e^3)*x^4 + (a^2*c*d^6 + 5*a^3*d^4*e^2)*x^3

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 362 vs. \(2 (176) = 352\).

Time = 0.28 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.91 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=\frac {1}{13} \, c^{3} e^{6} x^{13} + \frac {1}{2} \, c^{3} d e^{5} x^{12} + \frac {15}{11} \, c^{3} d^{2} e^{4} x^{11} + \frac {3}{11} \, a c^{2} e^{6} x^{11} + 2 \, c^{3} d^{3} e^{3} x^{10} + \frac {9}{5} \, a c^{2} d e^{5} x^{10} + \frac {5}{3} \, c^{3} d^{4} e^{2} x^{9} + 5 \, a c^{2} d^{2} e^{4} x^{9} + \frac {1}{3} \, a^{2} c e^{6} x^{9} + \frac {3}{4} \, c^{3} d^{5} e x^{8} + \frac {15}{2} \, a c^{2} d^{3} e^{3} x^{8} + \frac {9}{4} \, a^{2} c d e^{5} x^{8} + \frac {1}{7} \, c^{3} d^{6} x^{7} + \frac {45}{7} \, a c^{2} d^{4} e^{2} x^{7} + \frac {45}{7} \, a^{2} c d^{2} e^{4} x^{7} + \frac {1}{7} \, a^{3} e^{6} x^{7} + 3 \, a c^{2} d^{5} e x^{6} + 10 \, a^{2} c d^{3} e^{3} x^{6} + a^{3} d e^{5} x^{6} + \frac {3}{5} \, a c^{2} d^{6} x^{5} + 9 \, a^{2} c d^{4} e^{2} x^{5} + 3 \, a^{3} d^{2} e^{4} x^{5} + \frac {9}{2} \, a^{2} c d^{5} e x^{4} + 5 \, a^{3} d^{3} e^{3} x^{4} + a^{2} c d^{6} x^{3} + 5 \, a^{3} d^{4} e^{2} x^{3} + 3 \, a^{3} d^{5} e x^{2} + a^{3} d^{6} x \]

[In]

integrate((e*x+d)^6*(c*x^2+a)^3,x, algorithm="giac")

[Out]

1/13*c^3*e^6*x^13 + 1/2*c^3*d*e^5*x^12 + 15/11*c^3*d^2*e^4*x^11 + 3/11*a*c^2*e^6*x^11 + 2*c^3*d^3*e^3*x^10 + 9
/5*a*c^2*d*e^5*x^10 + 5/3*c^3*d^4*e^2*x^9 + 5*a*c^2*d^2*e^4*x^9 + 1/3*a^2*c*e^6*x^9 + 3/4*c^3*d^5*e*x^8 + 15/2
*a*c^2*d^3*e^3*x^8 + 9/4*a^2*c*d*e^5*x^8 + 1/7*c^3*d^6*x^7 + 45/7*a*c^2*d^4*e^2*x^7 + 45/7*a^2*c*d^2*e^4*x^7 +
1/7*a^3*e^6*x^7 + 3*a*c^2*d^5*e*x^6 + 10*a^2*c*d^3*e^3*x^6 + a^3*d*e^5*x^6 + 3/5*a*c^2*d^6*x^5 + 9*a^2*c*d^4*
e^2*x^5 + 3*a^3*d^2*e^4*x^5 + 9/2*a^2*c*d^5*e*x^4 + 5*a^3*d^3*e^3*x^4 + a^2*c*d^6*x^3 + 5*a^3*d^4*e^2*x^3 + 3*
a^3*d^5*e*x^2 + a^3*d^6*x

Mupad [B] (verification not implemented)

Time = 0.18 (sec) , antiderivative size = 329, normalized size of antiderivative = 1.73 \[ \int (d+e x)^6 \left (a+c x^2\right )^3 \, dx=x^7\,\left (\frac {a^3\,e^6}{7}+\frac {45\,a^2\,c\,d^2\,e^4}{7}+\frac {45\,a\,c^2\,d^4\,e^2}{7}+\frac {c^3\,d^6}{7}\right )+x^3\,\left (5\,a^3\,d^4\,e^2+c\,a^2\,d^6\right )+x^{11}\,\left (\frac {15\,c^3\,d^2\,e^4}{11}+\frac {3\,a\,c^2\,e^6}{11}\right )+x^5\,\left (3\,a^3\,d^2\,e^4+9\,a^2\,c\,d^4\,e^2+\frac {3\,a\,c^2\,d^6}{5}\right )+x^9\,\left (\frac {a^2\,c\,e^6}{3}+5\,a\,c^2\,d^2\,e^4+\frac {5\,c^3\,d^4\,e^2}{3}\right )+a^3\,d^6\,x+\frac {c^3\,e^6\,x^{13}}{13}+3\,a^3\,d^5\,e\,x^2+\frac {c^3\,d\,e^5\,x^{12}}{2}+a\,d\,e\,x^6\,\left (a^2\,e^4+10\,a\,c\,d^2\,e^2+3\,c^2\,d^4\right )+\frac {3\,c\,d\,e\,x^8\,\left (3\,a^2\,e^4+10\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{4}+\frac {a^2\,d^3\,e\,x^4\,\left (9\,c\,d^2+10\,a\,e^2\right )}{2}+\frac {c^2\,d\,e^3\,x^{10}\,\left (10\,c\,d^2+9\,a\,e^2\right )}{5} \]

[In]

int((a + c*x^2)^3*(d + e*x)^6,x)

[Out]

x^7*((a^3*e^6)/7 + (c^3*d^6)/7 + (45*a*c^2*d^4*e^2)/7 + (45*a^2*c*d^2*e^4)/7) + x^3*(a^2*c*d^6 + 5*a^3*d^4*e^2
) + x^11*((3*a*c^2*e^6)/11 + (15*c^3*d^2*e^4)/11) + x^5*((3*a*c^2*d^6)/5 + 3*a^3*d^2*e^4 + 9*a^2*c*d^4*e^2) +
x^9*((a^2*c*e^6)/3 + (5*c^3*d^4*e^2)/3 + 5*a*c^2*d^2*e^4) + a^3*d^6*x + (c^3*e^6*x^13)/13 + 3*a^3*d^5*e*x^2 +
(c^3*d*e^5*x^12)/2 + a*d*e*x^6*(a^2*e^4 + 3*c^2*d^4 + 10*a*c*d^2*e^2) + (3*c*d*e*x^8*(3*a^2*e^4 + c^2*d^4 + 10
*a*c*d^2*e^2))/4 + (a^2*d^3*e*x^4*(10*a*e^2 + 9*c*d^2))/2 + (c^2*d*e^3*x^10*(9*a*e^2 + 10*c*d^2))/5