\(\int (2 c x+3 d x^2) (a+c x^2+d x^3)^7 \, dx\) [198]

Optimal result

Integrand size = 26, antiderivative size = 18 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+c x^2+d x^3\right )^8 \]

[Out]

1/8*(d*x^3+c*x^2+a)^8

Rubi [A] (verified)

Time = 0.03 (sec) , antiderivative size = 18, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.038, Rules used = {1602} \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \left (a+c x^2+d x^3\right )^8 \]

[In]

Int[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7,x]

[Out]

(a + c*x^2 + d*x^3)^8/8

Rule 1602

Int[(Pp_)*(Qq_)^(m_.), x_Symbol] :> With[{p = Expon[Pp, x], q = Expon[Qq, x]}, Simp[Coeff[Pp, x, p]*x^(p - q +
1)*(Qq^(m + 1)/((p + m*q + 1)*Coeff[Qq, x, q])), x] /; NeQ[p + m*q + 1, 0] && EqQ[(p + m*q + 1)*Coeff[Qq, x,
q]*Pp, Coeff[Pp, x, p]*x^(p - q)*((p - q + 1)*Qq + (m + 1)*x*D[Qq, x])]] /; FreeQ[m, x] && PolyQ[Pp, x] && Pol
yQ[Qq, x] && NeQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {1}{8} \left (a+c x^2+d x^3\right )^8 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(115\) vs. \(2(18)=36\).

Time = 0.04 (sec) , antiderivative size = 115, normalized size of antiderivative = 6.39 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} x^2 (c+d x) \left (8 a^7+28 a^6 x^2 (c+d x)+56 a^5 x^4 (c+d x)^2+70 a^4 x^6 (c+d x)^3+56 a^3 x^8 (c+d x)^4+28 a^2 x^{10} (c+d x)^5+8 a x^{12} (c+d x)^6+x^{14} (c+d x)^7\right ) \]

[In]

Integrate[(2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7,x]

[Out]

(x^2*(c + d*x)*(8*a^7 + 28*a^6*x^2*(c + d*x) + 56*a^5*x^4*(c + d*x)^2 + 70*a^4*x^6*(c + d*x)^3 + 56*a^3*x^8*(c
+ d*x)^4 + 28*a^2*x^10*(c + d*x)^5 + 8*a*x^12*(c + d*x)^6 + x^14*(c + d*x)^7))/8

Maple [A] (verified)

Time = 0.10 (sec) , antiderivative size = 17, normalized size of antiderivative = 0.94

method	result	size
default	\(\frac {\left (x^{3} d +c \,x^{2}+a \right )^{8}}{8}\)	\(17\)
norman	\(\frac {7 x^{22} c^{2} d^{6}}{2}+c \,d^{7} x^{23}+\frac {d^{8} x^{24}}{8}+\left (7 a c \,d^{6}+\frac {35}{4} c^{4} d^{4}\right ) x^{20}+\left (d^{7} a +7 c^{3} d^{5}\right ) x^{21}+\left (21 a \,c^{2} d^{5}+7 c^{5} d^{3}\right ) x^{19}+\left (\frac {105}{2} a^{2} c^{2} d^{4}+21 a \,c^{5} d^{2}+\frac {1}{8} c^{8}\right ) x^{16}+\left (21 a^{2} c \,d^{5}+35 a \,c^{4} d^{3}+c^{7} d \right ) x^{17}+\left (\frac {7}{2} a^{2} d^{6}+35 a \,c^{3} d^{4}+\frac {7}{2} c^{6} d^{2}\right ) x^{18}+\left (35 a^{3} c \,d^{4}+\frac {105}{2} a^{2} c^{4} d^{2}+a \,c^{7}\right ) x^{14}+\left (7 a^{3} d^{5}+70 a^{2} c^{3} d^{3}+7 a \,c^{6} d \right ) x^{15}+\left (21 a^{5} c \,d^{2}+\frac {35}{4} a^{4} c^{4}\right ) x^{8}+\left (7 a^{5} d^{3}+35 c^{3} a^{4} d \right ) x^{9}+\left (\frac {105}{2} a^{4} c^{2} d^{2}+7 a^{3} c^{5}\right ) x^{10}+\left (35 a^{4} c \,d^{3}+35 a^{3} c^{4} d \right ) x^{11}+\left (\frac {35}{4} a^{4} d^{4}+70 a^{3} c^{3} d^{2}+\frac {7}{2} a^{2} c^{6}\right ) x^{12}+\left (70 a^{3} c^{2} d^{3}+21 a^{2} c^{5} d \right ) x^{13}+x^{2} c \,a^{7}+a^{7} d \,x^{3}+\frac {7 x^{4} a^{6} c^{2}}{2}+7 a^{6} c d \,x^{5}+\left (\frac {7}{2} a^{6} d^{2}+7 a^{5} c^{3}\right ) x^{6}+21 a^{5} c^{2} d \,x^{7}\)	\(456\)
parallelrisch	\(\frac {7}{2} x^{4} a^{6} c^{2}+\frac {7}{2} x^{6} a^{6} d^{2}+7 x^{6} a^{5} c^{3}+x^{2} c \,a^{7}+35 a^{4} c^{3} d \,x^{9}+35 a^{4} c \,d^{3} x^{11}+35 a^{3} c^{4} d \,x^{11}+c \,d^{7} x^{23}+a \,d^{7} x^{21}+21 a^{2} c \,d^{5} x^{17}+35 a \,c^{4} d^{3} x^{17}+7 c^{3} d^{5} x^{21}+a^{7} d \,x^{3}+7 a^{3} d^{5} x^{15}+\frac {35}{4} x^{20} c^{4} d^{4}+\frac {7}{2} x^{22} c^{2} d^{6}+7 c^{5} d^{3} x^{19}+c^{7} d \,x^{17}+7 a^{5} d^{3} x^{9}+x^{14} a \,c^{7}+35 x^{18} a \,c^{3} d^{4}+70 a^{3} c^{2} d^{3} x^{13}+70 x^{12} a^{3} c^{3} d^{2}+7 a^{6} c d \,x^{5}+21 a^{5} c^{2} d \,x^{7}+70 a^{2} c^{3} d^{3} x^{15}+7 a \,c^{6} d \,x^{15}+\frac {1}{8} d^{8} x^{24}+21 x^{8} a^{5} c \,d^{2}+\frac {105}{2} x^{10} a^{4} c^{2} d^{2}+\frac {7}{2} x^{18} a^{2} d^{6}+\frac {7}{2} x^{18} c^{6} d^{2}+\frac {35}{4} x^{8} a^{4} c^{4}+7 x^{10} a^{3} c^{5}+\frac {105}{2} x^{14} a^{2} c^{4} d^{2}+\frac {105}{2} x^{16} a^{2} c^{2} d^{4}+21 x^{16} a \,c^{5} d^{2}+\frac {35}{4} x^{12} a^{4} d^{4}+\frac {7}{2} x^{12} a^{2} c^{6}+35 x^{14} a^{3} c \,d^{4}+21 a^{2} c^{5} d \,x^{13}+7 x^{20} a c \,d^{6}+21 a \,c^{2} d^{5} x^{19}+\frac {1}{8} x^{16} c^{8}\)	\(489\)
gosper	\(\frac {x^{2} \left (d^{8} x^{22}+8 c \,d^{7} x^{21}+28 x^{20} c^{2} d^{6}+8 a \,d^{7} x^{19}+56 c^{3} d^{5} x^{19}+56 x^{18} a c \,d^{6}+70 x^{18} c^{4} d^{4}+168 a \,c^{2} d^{5} x^{17}+56 c^{5} d^{3} x^{17}+28 x^{16} a^{2} d^{6}+280 x^{16} a \,c^{3} d^{4}+28 x^{16} c^{6} d^{2}+168 a^{2} c \,d^{5} x^{15}+280 a \,c^{4} d^{3} x^{15}+8 c^{7} d \,x^{15}+420 x^{14} a^{2} c^{2} d^{4}+168 x^{14} a \,c^{5} d^{2}+x^{14} c^{8}+56 a^{3} d^{5} x^{13}+560 a^{2} c^{3} d^{3} x^{13}+56 a \,c^{6} d \,x^{13}+280 x^{12} a^{3} c \,d^{4}+420 x^{12} a^{2} c^{4} d^{2}+8 x^{12} a \,c^{7}+560 a^{3} c^{2} d^{3} x^{11}+168 a^{2} c^{5} d \,x^{11}+70 x^{10} a^{4} d^{4}+560 x^{10} a^{3} c^{3} d^{2}+28 x^{10} a^{2} c^{6}+280 a^{4} c \,d^{3} x^{9}+280 a^{3} c^{4} d \,x^{9}+420 x^{8} a^{4} c^{2} d^{2}+56 x^{8} a^{3} c^{5}+56 a^{5} d^{3} x^{7}+280 a^{4} c^{3} d \,x^{7}+168 x^{6} a^{5} c \,d^{2}+70 x^{6} a^{4} c^{4}+168 a^{5} c^{2} d \,x^{5}+28 x^{4} a^{6} d^{2}+56 x^{4} a^{5} c^{3}+56 a^{6} c d \,x^{3}+28 x^{2} a^{6} c^{2}+8 a^{7} d x +8 c \,a^{7}\right )}{8}\)	\(493\)
risch	\(\frac {7}{2} x^{4} a^{6} c^{2}+\frac {7}{2} x^{6} a^{6} d^{2}+7 x^{6} a^{5} c^{3}+x^{2} c \,a^{7}+35 a^{4} c^{3} d \,x^{9}+35 a^{4} c \,d^{3} x^{11}+35 a^{3} c^{4} d \,x^{11}+c \,d^{7} x^{23}+a \,d^{7} x^{21}+21 a^{2} c \,d^{5} x^{17}+35 a \,c^{4} d^{3} x^{17}+7 c^{3} d^{5} x^{21}+a^{7} d \,x^{3}+7 a^{3} d^{5} x^{15}+\frac {35}{4} x^{20} c^{4} d^{4}+\frac {7}{2} x^{22} c^{2} d^{6}+7 c^{5} d^{3} x^{19}+c^{7} d \,x^{17}+7 a^{5} d^{3} x^{9}+x^{14} a \,c^{7}+35 x^{18} a \,c^{3} d^{4}+70 a^{3} c^{2} d^{3} x^{13}+70 x^{12} a^{3} c^{3} d^{2}+7 a^{6} c d \,x^{5}+21 a^{5} c^{2} d \,x^{7}+70 a^{2} c^{3} d^{3} x^{15}+7 a \,c^{6} d \,x^{15}+\frac {1}{8} d^{8} x^{24}+21 x^{8} a^{5} c \,d^{2}+\frac {105}{2} x^{10} a^{4} c^{2} d^{2}+\frac {7}{2} x^{18} a^{2} d^{6}+\frac {7}{2} x^{18} c^{6} d^{2}+\frac {35}{4} x^{8} a^{4} c^{4}+7 x^{10} a^{3} c^{5}+\frac {105}{2} x^{14} a^{2} c^{4} d^{2}+\frac {105}{2} x^{16} a^{2} c^{2} d^{4}+21 x^{16} a \,c^{5} d^{2}+\frac {35}{4} x^{12} a^{4} d^{4}+\frac {7}{2} x^{12} a^{2} c^{6}+35 x^{14} a^{3} c \,d^{4}+\frac {1}{8} a^{8}+21 a^{2} c^{5} d \,x^{13}+7 x^{20} a c \,d^{6}+21 a \,c^{2} d^{5} x^{19}+\frac {1}{8} x^{16} c^{8}\)	\(494\)

[In]

int((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x,method=_RETURNVERBOSE)

[Out]

1/8*(d*x^3+c*x^2+a)^8

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 458 vs. \(2 (16) = 32\).

Time = 0.25 (sec) , antiderivative size = 458, normalized size of antiderivative = 25.44 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + {\left (7 \, c^{3} d^{5} + a d^{7}\right )} x^{21} + \frac {7}{4} \, {\left (5 \, c^{4} d^{4} + 4 \, a c d^{6}\right )} x^{20} + 7 \, {\left (c^{5} d^{3} + 3 \, a c^{2} d^{5}\right )} x^{19} + \frac {7}{2} \, {\left (c^{6} d^{2} + 10 \, a c^{3} d^{4} + a^{2} d^{6}\right )} x^{18} + {\left (c^{7} d + 35 \, a c^{4} d^{3} + 21 \, a^{2} c d^{5}\right )} x^{17} + \frac {1}{8} \, {\left (c^{8} + 168 \, a c^{5} d^{2} + 420 \, a^{2} c^{2} d^{4}\right )} x^{16} + 7 \, {\left (a c^{6} d + 10 \, a^{2} c^{3} d^{3} + a^{3} d^{5}\right )} x^{15} + 21 \, a^{5} c^{2} d x^{7} + \frac {1}{2} \, {\left (2 \, a c^{7} + 105 \, a^{2} c^{4} d^{2} + 70 \, a^{3} c d^{4}\right )} x^{14} + 7 \, {\left (3 \, a^{2} c^{5} d + 10 \, a^{3} c^{2} d^{3}\right )} x^{13} + 7 \, a^{6} c d x^{5} + \frac {7}{4} \, {\left (2 \, a^{2} c^{6} + 40 \, a^{3} c^{3} d^{2} + 5 \, a^{4} d^{4}\right )} x^{12} + \frac {7}{2} \, a^{6} c^{2} x^{4} + 35 \, {\left (a^{3} c^{4} d + a^{4} c d^{3}\right )} x^{11} + a^{7} d x^{3} + \frac {7}{2} \, {\left (2 \, a^{3} c^{5} + 15 \, a^{4} c^{2} d^{2}\right )} x^{10} + a^{7} c x^{2} + 7 \, {\left (5 \, a^{4} c^{3} d + a^{5} d^{3}\right )} x^{9} + \frac {7}{4} \, {\left (5 \, a^{4} c^{4} + 12 \, a^{5} c d^{2}\right )} x^{8} + \frac {7}{2} \, {\left (2 \, a^{5} c^{3} + a^{6} d^{2}\right )} x^{6} \]

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="fricas")

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + (7*c^3*d^5 + a*d^7)*x^21 + 7/4*(5*c^4*d^4 + 4*a*c*d^6)*x^20 + 7
*(c^5*d^3 + 3*a*c^2*d^5)*x^19 + 7/2*(c^6*d^2 + 10*a*c^3*d^4 + a^2*d^6)*x^18 + (c^7*d + 35*a*c^4*d^3 + 21*a^2*c
*d^5)*x^17 + 1/8*(c^8 + 168*a*c^5*d^2 + 420*a^2*c^2*d^4)*x^16 + 7*(a*c^6*d + 10*a^2*c^3*d^3 + a^3*d^5)*x^15 +
21*a^5*c^2*d*x^7 + 1/2*(2*a*c^7 + 105*a^2*c^4*d^2 + 70*a^3*c*d^4)*x^14 + 7*(3*a^2*c^5*d + 10*a^3*c^2*d^3)*x^13
+ 7*a^6*c*d*x^5 + 7/4*(2*a^2*c^6 + 40*a^3*c^3*d^2 + 5*a^4*d^4)*x^12 + 7/2*a^6*c^2*x^4 + 35*(a^3*c^4*d + a^4*c
*d^3)*x^11 + a^7*d*x^3 + 7/2*(2*a^3*c^5 + 15*a^4*c^2*d^2)*x^10 + a^7*c*x^2 + 7*(5*a^4*c^3*d + a^5*d^3)*x^9 + 7
/4*(5*a^4*c^4 + 12*a^5*c*d^2)*x^8 + 7/2*(2*a^5*c^3 + a^6*d^2)*x^6

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 484 vs. \(2 (14) = 28\).

Time = 0.08 (sec) , antiderivative size = 484, normalized size of antiderivative = 26.89 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=a^{7} c x^{2} + a^{7} d x^{3} + \frac {7 a^{6} c^{2} x^{4}}{2} + 7 a^{6} c d x^{5} + 21 a^{5} c^{2} d x^{7} + \frac {7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} + x^{21} \left (a d^{7} + 7 c^{3} d^{5}\right ) + x^{20} \cdot \left (7 a c d^{6} + \frac {35 c^{4} d^{4}}{4}\right ) + x^{19} \cdot \left (21 a c^{2} d^{5} + 7 c^{5} d^{3}\right ) + x^{18} \cdot \left (\frac {7 a^{2} d^{6}}{2} + 35 a c^{3} d^{4} + \frac {7 c^{6} d^{2}}{2}\right ) + x^{17} \cdot \left (21 a^{2} c d^{5} + 35 a c^{4} d^{3} + c^{7} d\right ) + x^{16} \cdot \left (\frac {105 a^{2} c^{2} d^{4}}{2} + 21 a c^{5} d^{2} + \frac {c^{8}}{8}\right ) + x^{15} \cdot \left (7 a^{3} d^{5} + 70 a^{2} c^{3} d^{3} + 7 a c^{6} d\right ) + x^{14} \cdot \left (35 a^{3} c d^{4} + \frac {105 a^{2} c^{4} d^{2}}{2} + a c^{7}\right ) + x^{13} \cdot \left (70 a^{3} c^{2} d^{3} + 21 a^{2} c^{5} d\right ) + x^{12} \cdot \left (\frac {35 a^{4} d^{4}}{4} + 70 a^{3} c^{3} d^{2} + \frac {7 a^{2} c^{6}}{2}\right ) + x^{11} \cdot \left (35 a^{4} c d^{3} + 35 a^{3} c^{4} d\right ) + x^{10} \cdot \left (\frac {105 a^{4} c^{2} d^{2}}{2} + 7 a^{3} c^{5}\right ) + x^{9} \cdot \left (7 a^{5} d^{3} + 35 a^{4} c^{3} d\right ) + x^{8} \cdot \left (21 a^{5} c d^{2} + \frac {35 a^{4} c^{4}}{4}\right ) + x^{6} \cdot \left (\frac {7 a^{6} d^{2}}{2} + 7 a^{5} c^{3}\right ) \]

[In]

integrate((3*d*x**2+2*c*x)*(d*x**3+c*x**2+a)**7,x)

[Out]

a**7*c*x**2 + a**7*d*x**3 + 7*a**6*c**2*x**4/2 + 7*a**6*c*d*x**5 + 21*a**5*c**2*d*x**7 + 7*c**2*d**6*x**22/2 +
c*d**7*x**23 + d**8*x**24/8 + x**21*(a*d**7 + 7*c**3*d**5) + x**20*(7*a*c*d**6 + 35*c**4*d**4/4) + x**19*(21*
a*c**2*d**5 + 7*c**5*d**3) + x**18*(7*a**2*d**6/2 + 35*a*c**3*d**4 + 7*c**6*d**2/2) + x**17*(21*a**2*c*d**5 +
35*a*c**4*d**3 + c**7*d) + x**16*(105*a**2*c**2*d**4/2 + 21*a*c**5*d**2 + c**8/8) + x**15*(7*a**3*d**5 + 70*a*
*2*c**3*d**3 + 7*a*c**6*d) + x**14*(35*a**3*c*d**4 + 105*a**2*c**4*d**2/2 + a*c**7) + x**13*(70*a**3*c**2*d**3
+ 21*a**2*c**5*d) + x**12*(35*a**4*d**4/4 + 70*a**3*c**3*d**2 + 7*a**2*c**6/2) + x**11*(35*a**4*c*d**3 + 35*a
**3*c**4*d) + x**10*(105*a**4*c**2*d**2/2 + 7*a**3*c**5) + x**9*(7*a**5*d**3 + 35*a**4*c**3*d) + x**8*(21*a**5
*c*d**2 + 35*a**4*c**4/4) + x**6*(7*a**6*d**2/2 + 7*a**5*c**3)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 16, normalized size of antiderivative = 0.89 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2} + a\right )}^{8} \]

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="maxima")

[Out]

1/8*(d*x^3 + c*x^2 + a)^8

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 136 vs. \(2 (16) = 32\).

Time = 0.29 (sec) , antiderivative size = 136, normalized size of antiderivative = 7.56 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, {\left (d x^{3} + c x^{2}\right )}^{8} + {\left (d x^{3} + c x^{2}\right )}^{7} a + \frac {7}{2} \, {\left (d x^{3} + c x^{2}\right )}^{6} a^{2} + 7 \, {\left (d x^{3} + c x^{2}\right )}^{5} a^{3} + \frac {35}{4} \, {\left (d x^{3} + c x^{2}\right )}^{4} a^{4} + 7 \, {\left (d x^{3} + c x^{2}\right )}^{3} a^{5} + \frac {7}{2} \, {\left (d x^{3} + c x^{2}\right )}^{2} a^{6} + {\left (d x^{3} + c x^{2}\right )} a^{7} \]

[In]

integrate((3*d*x^2+2*c*x)*(d*x^3+c*x^2+a)^7,x, algorithm="giac")

[Out]

1/8*(d*x^3 + c*x^2)^8 + (d*x^3 + c*x^2)^7*a + 7/2*(d*x^3 + c*x^2)^6*a^2 + 7*(d*x^3 + c*x^2)^5*a^3 + 35/4*(d*x^
3 + c*x^2)^4*a^4 + 7*(d*x^3 + c*x^2)^3*a^5 + 7/2*(d*x^3 + c*x^2)^2*a^6 + (d*x^3 + c*x^2)*a^7

Mupad [B] (verification not implemented)

Time = 9.58 (sec) , antiderivative size = 440, normalized size of antiderivative = 24.44 \[ \int \left (2 c x+3 d x^2\right ) \left (a+c x^2+d x^3\right )^7 \, dx=x^{12}\,\left (\frac {35\,a^4\,d^4}{4}+70\,a^3\,c^3\,d^2+\frac {7\,a^2\,c^6}{2}\right )+x^6\,\left (\frac {7\,a^6\,d^2}{2}+7\,a^5\,c^3\right )+x^{20}\,\left (\frac {35\,c^4\,d^4}{4}+7\,a\,c\,d^6\right )+x^{16}\,\left (\frac {105\,a^2\,c^2\,d^4}{2}+21\,a\,c^5\,d^2+\frac {c^8}{8}\right )+x^{18}\,\left (\frac {7\,a^2\,d^6}{2}+35\,a\,c^3\,d^4+\frac {7\,c^6\,d^2}{2}\right )+\frac {d^8\,x^{24}}{8}+x^{21}\,\left (7\,c^3\,d^5+a\,d^7\right )+a^7\,c\,x^2+a^7\,d\,x^3+c\,d^7\,x^{23}+\frac {7\,a^6\,c^2\,x^4}{2}+\frac {7\,c^2\,d^6\,x^{22}}{2}+21\,a^5\,c^2\,d\,x^7+7\,a\,d\,x^{15}\,\left (a^2\,d^4+10\,a\,c^3\,d^2+c^6\right )+c\,d\,x^{17}\,\left (21\,a^2\,d^4+35\,a\,c^3\,d^2+c^6\right )+\frac {7\,a^4\,c\,x^8\,\left (5\,c^3+12\,a\,d^2\right )}{4}+7\,a^4\,d\,x^9\,\left (5\,c^3+a\,d^2\right )+7\,c^2\,d^3\,x^{19}\,\left (c^3+3\,a\,d^2\right )+\frac {a\,c\,x^{14}\,\left (70\,a^2\,d^4+105\,a\,c^3\,d^2+2\,c^6\right )}{2}+7\,a^6\,c\,d\,x^5+\frac {7\,a^3\,c^2\,x^{10}\,\left (2\,c^3+15\,a\,d^2\right )}{2}+7\,a^2\,c^2\,d\,x^{13}\,\left (3\,c^3+10\,a\,d^2\right )+35\,a^3\,c\,d\,x^{11}\,\left (c^3+a\,d^2\right ) \]

[In]

int((2*c*x + 3*d*x^2)*(a + c*x^2 + d*x^3)^7,x)

[Out]

x^12*((7*a^2*c^6)/2 + (35*a^4*d^4)/4 + 70*a^3*c^3*d^2) + x^6*(7*a^5*c^3 + (7*a^6*d^2)/2) + x^20*((35*c^4*d^4)/
4 + 7*a*c*d^6) + x^16*(c^8/8 + 21*a*c^5*d^2 + (105*a^2*c^2*d^4)/2) + x^18*((7*a^2*d^6)/2 + (7*c^6*d^2)/2 + 35*
a*c^3*d^4) + (d^8*x^24)/8 + x^21*(a*d^7 + 7*c^3*d^5) + a^7*c*x^2 + a^7*d*x^3 + c*d^7*x^23 + (7*a^6*c^2*x^4)/2
+ (7*c^2*d^6*x^22)/2 + 21*a^5*c^2*d*x^7 + 7*a*d*x^15*(c^6 + a^2*d^4 + 10*a*c^3*d^2) + c*d*x^17*(c^6 + 21*a^2*d
^4 + 35*a*c^3*d^2) + (7*a^4*c*x^8*(12*a*d^2 + 5*c^3))/4 + 7*a^4*d*x^9*(a*d^2 + 5*c^3) + 7*c^2*d^3*x^19*(3*a*d^
2 + c^3) + (a*c*x^14*(2*c^6 + 70*a^2*d^4 + 105*a*c^3*d^2))/2 + 7*a^6*c*d*x^5 + (7*a^3*c^2*x^10*(15*a*d^2 + 2*c
^3))/2 + 7*a^2*c^2*d*x^13*(10*a*d^2 + 3*c^3) + 35*a^3*c*d*x^11*(a*d^2 + c^3)