\(\int x (2 c+3 d x) (c x^2+d x^3)^7 \, dx\) [203]

Optimal result

Integrand size = 23, antiderivative size = 14 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} x^{16} (c+d x)^8 \]

[Out]

1/8*x^16*(d*x+c)^8

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 14, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1598, 75} \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} x^{16} (c+d x)^8 \]

[In]

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

[Out]

(x^16*(c + d*x)^8)/8

Rule 75

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p_.), x_Symbol] :> Simp[b*(c + d*x)^
(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && NeQ[n + p + 2, 0] &
& EqQ[a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)), 0]

Rule 1598

Int[(u_.)*(x_)^(m_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(m + n*p)*(a + b*x^(q -
p))^n, x] /; FreeQ[{a, b, m, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rubi steps \begin{align*} \text {integral}& = \int x^{15} (c+d x)^7 (2 c+3 d x) \, dx \\ & = \frac {1}{8} x^{16} (c+d x)^8 \\ \end{align*}

Mathematica [B] (verified)

Leaf count is larger than twice the leaf count of optimal. \(98\) vs. \(2(14)=28\).

Time = 0.00 (sec) , antiderivative size = 98, normalized size of antiderivative = 7.00 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^8 x^{16}}{8}+c^7 d x^{17}+\frac {7}{2} c^6 d^2 x^{18}+7 c^5 d^3 x^{19}+\frac {35}{4} c^4 d^4 x^{20}+7 c^3 d^5 x^{21}+\frac {7}{2} c^2 d^6 x^{22}+c d^7 x^{23}+\frac {d^8 x^{24}}{8} \]

[In]

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

[Out]

(c^8*x^16)/8 + c^7*d*x^17 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20)/4 + 7*c^3*d^5*x^21 + (7*c^
2*d^6*x^22)/2 + c*d^7*x^23 + (d^8*x^24)/8

Maple [A] (verified)

Time = 0.72 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.93

method	result	size
gosper	\(\frac {x^{16} \left (d x +c \right )^{8}}{8}\)	\(13\)
default	\(\frac {\left (x^{3} d +c \,x^{2}\right )^{8}}{8}\)	\(16\)
norman	\(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\)	\(89\)
risch	\(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\)	\(89\)
parallelrisch	\(\frac {7}{2} x^{22} c^{2} d^{6}+c \,d^{7} x^{23}+\frac {1}{8} d^{8} x^{24}+\frac {1}{8} x^{16} c^{8}+c^{7} d \,x^{17}+\frac {7}{2} x^{18} c^{6} d^{2}+7 c^{5} d^{3} x^{19}+\frac {35}{4} x^{20} c^{4} d^{4}+7 c^{3} d^{5} x^{21}\)	\(89\)

[In]

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

[Out]

1/8*x^16*(d*x+c)^8

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).

Time = 0.24 (sec) , antiderivative size = 88, normalized size of antiderivative = 6.29 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]

[In]

integrate(x*(3*d*x+2*c)*(d*x^3+c*x^2)^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*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 97 vs. \(2 (10) = 20\).

Time = 0.04 (sec) , antiderivative size = 97, normalized size of antiderivative = 6.93 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^{8} x^{16}}{8} + c^{7} d x^{17} + \frac {7 c^{6} d^{2} x^{18}}{2} + 7 c^{5} d^{3} x^{19} + \frac {35 c^{4} d^{4} x^{20}}{4} + 7 c^{3} d^{5} x^{21} + \frac {7 c^{2} d^{6} x^{22}}{2} + c d^{7} x^{23} + \frac {d^{8} x^{24}}{8} \]

[In]

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

[Out]

c**8*x**16/8 + c**7*d*x**17 + 7*c**6*d**2*x**18/2 + 7*c**5*d**3*x**19 + 35*c**4*d**4*x**20/4 + 7*c**3*d**5*x**
21 + 7*c**2*d**6*x**22/2 + c*d**7*x**23 + d**8*x**24/8

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).

Time = 0.19 (sec) , antiderivative size = 88, normalized size of antiderivative = 6.29 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]

[In]

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

[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*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 88 vs. \(2 (12) = 24\).

Time = 0.27 (sec) , antiderivative size = 88, normalized size of antiderivative = 6.29 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac {7}{2} \, c^{2} d^{6} x^{22} + 7 \, c^{3} d^{5} x^{21} + \frac {35}{4} \, c^{4} d^{4} x^{20} + 7 \, c^{5} d^{3} x^{19} + \frac {7}{2} \, c^{6} d^{2} x^{18} + c^{7} d x^{17} + \frac {1}{8} \, c^{8} x^{16} \]

[In]

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

[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*x^21 + 35/4*c^4*d^4*x^20 + 7*c^5*d^3*x^19 + 7/2*c^6*d
^2*x^18 + c^7*d*x^17 + 1/8*c^8*x^16

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 88, normalized size of antiderivative = 6.29 \[ \int x (2 c+3 d x) \left (c x^2+d x^3\right )^7 \, dx=\frac {c^8\,x^{16}}{8}+c^7\,d\,x^{17}+\frac {7\,c^6\,d^2\,x^{18}}{2}+7\,c^5\,d^3\,x^{19}+\frac {35\,c^4\,d^4\,x^{20}}{4}+7\,c^3\,d^5\,x^{21}+\frac {7\,c^2\,d^6\,x^{22}}{2}+c\,d^7\,x^{23}+\frac {d^8\,x^{24}}{8} \]

[In]

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

[Out]

(c^8*x^16)/8 + (d^8*x^24)/8 + c^7*d*x^17 + c*d^7*x^23 + (7*c^6*d^2*x^18)/2 + 7*c^5*d^3*x^19 + (35*c^4*d^4*x^20
)/4 + 7*c^3*d^5*x^21 + (7*c^2*d^6*x^22)/2