3.193 \(\int (b+2 c x+3 d x^2) (b x+c x^2+d x^3)^7 \, dx\)

Optimal. Leaf size=20 \[ \frac{1}{8} \left (b x+c x^2+d x^3\right )^8 \]

[Out]

(b*x + c*x^2 + d*x^3)^8/8

________________________________________________________________________________________

Rubi [A]  time = 0.0521532, antiderivative size = 20, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 29, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.034, Rules used = {1588} \[ \frac{1}{8} \left (b x+c x^2+d x^3\right )^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(b*x + c*x^2 + d*x^3)^8/8

Rule 1588

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*} \int \left (b+2 c x+3 d x^2\right ) \left (b x+c x^2+d x^3\right )^7 \, dx &=\frac{1}{8} \left (b x+c x^2+d x^3\right )^8\\ \end{align*}

Mathematica [A]  time = 0.0264014, size = 18, normalized size = 0.9 \[ \frac{1}{8} x^8 (b+x (c+d x))^8 \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x^8*(b + x*(c + d*x))^8)/8

________________________________________________________________________________________

Maple [B]  time = 0.003, size = 5596, normalized size = 279.8 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

result too large to display

________________________________________________________________________________________

Maxima [A]  time = 1.00126, size = 24, normalized size = 1.2 \begin{align*} \frac{1}{8} \,{\left (d x^{3} + c x^{2} + b x\right )}^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*(d*x^3 + c*x^2 + b*x)^8

________________________________________________________________________________________

Fricas [B]  time = 1.20495, size = 1118, normalized size = 55.9 \begin{align*} \frac{1}{8} x^{24} d^{8} + x^{23} d^{7} c + \frac{7}{2} x^{22} d^{6} c^{2} + x^{22} d^{7} b + 7 x^{21} d^{5} c^{3} + 7 x^{21} d^{6} c b + \frac{35}{4} x^{20} d^{4} c^{4} + 21 x^{20} d^{5} c^{2} b + \frac{7}{2} x^{20} d^{6} b^{2} + 7 x^{19} d^{3} c^{5} + 35 x^{19} d^{4} c^{3} b + 21 x^{19} d^{5} c b^{2} + \frac{7}{2} x^{18} d^{2} c^{6} + 35 x^{18} d^{3} c^{4} b + \frac{105}{2} x^{18} d^{4} c^{2} b^{2} + 7 x^{18} d^{5} b^{3} + x^{17} d c^{7} + 21 x^{17} d^{2} c^{5} b + 70 x^{17} d^{3} c^{3} b^{2} + 35 x^{17} d^{4} c b^{3} + \frac{1}{8} x^{16} c^{8} + 7 x^{16} d c^{6} b + \frac{105}{2} x^{16} d^{2} c^{4} b^{2} + 70 x^{16} d^{3} c^{2} b^{3} + \frac{35}{4} x^{16} d^{4} b^{4} + x^{15} c^{7} b + 21 x^{15} d c^{5} b^{2} + 70 x^{15} d^{2} c^{3} b^{3} + 35 x^{15} d^{3} c b^{4} + \frac{7}{2} x^{14} c^{6} b^{2} + 35 x^{14} d c^{4} b^{3} + \frac{105}{2} x^{14} d^{2} c^{2} b^{4} + 7 x^{14} d^{3} b^{5} + 7 x^{13} c^{5} b^{3} + 35 x^{13} d c^{3} b^{4} + 21 x^{13} d^{2} c b^{5} + \frac{35}{4} x^{12} c^{4} b^{4} + 21 x^{12} d c^{2} b^{5} + \frac{7}{2} x^{12} d^{2} b^{6} + 7 x^{11} c^{3} b^{5} + 7 x^{11} d c b^{6} + \frac{7}{2} x^{10} c^{2} b^{6} + x^{10} d b^{7} + x^{9} c b^{7} + \frac{1}{8} x^{8} b^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Sympy [B]  time = 0.164026, size = 469, normalized size = 23.45 \begin{align*} \frac{b^{8} x^{8}}{8} + b^{7} c x^{9} + c d^{7} x^{23} + \frac{d^{8} x^{24}}{8} + x^{22} \left (b d^{7} + \frac{7 c^{2} d^{6}}{2}\right ) + x^{21} \left (7 b c d^{6} + 7 c^{3} d^{5}\right ) + x^{20} \left (\frac{7 b^{2} d^{6}}{2} + 21 b c^{2} d^{5} + \frac{35 c^{4} d^{4}}{4}\right ) + x^{19} \left (21 b^{2} c d^{5} + 35 b c^{3} d^{4} + 7 c^{5} d^{3}\right ) + x^{18} \left (7 b^{3} d^{5} + \frac{105 b^{2} c^{2} d^{4}}{2} + 35 b c^{4} d^{3} + \frac{7 c^{6} d^{2}}{2}\right ) + x^{17} \left (35 b^{3} c d^{4} + 70 b^{2} c^{3} d^{3} + 21 b c^{5} d^{2} + c^{7} d\right ) + x^{16} \left (\frac{35 b^{4} d^{4}}{4} + 70 b^{3} c^{2} d^{3} + \frac{105 b^{2} c^{4} d^{2}}{2} + 7 b c^{6} d + \frac{c^{8}}{8}\right ) + x^{15} \left (35 b^{4} c d^{3} + 70 b^{3} c^{3} d^{2} + 21 b^{2} c^{5} d + b c^{7}\right ) + x^{14} \left (7 b^{5} d^{3} + \frac{105 b^{4} c^{2} d^{2}}{2} + 35 b^{3} c^{4} d + \frac{7 b^{2} c^{6}}{2}\right ) + x^{13} \left (21 b^{5} c d^{2} + 35 b^{4} c^{3} d + 7 b^{3} c^{5}\right ) + x^{12} \left (\frac{7 b^{6} d^{2}}{2} + 21 b^{5} c^{2} d + \frac{35 b^{4} c^{4}}{4}\right ) + x^{11} \left (7 b^{6} c d + 7 b^{5} c^{3}\right ) + x^{10} \left (b^{7} d + \frac{7 b^{6} c^{2}}{2}\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Giac [B]  time = 1.17305, size = 670, normalized size = 33.5 \begin{align*} \frac{1}{8} \, d^{8} x^{24} + c d^{7} x^{23} + \frac{7}{2} \, c^{2} d^{6} x^{22} + b d^{7} x^{22} + 7 \, c^{3} d^{5} x^{21} + 7 \, b c d^{6} x^{21} + \frac{35}{4} \, c^{4} d^{4} x^{20} + 21 \, b c^{2} d^{5} x^{20} + \frac{7}{2} \, b^{2} d^{6} x^{20} + 7 \, c^{5} d^{3} x^{19} + 35 \, b c^{3} d^{4} x^{19} + 21 \, b^{2} c d^{5} x^{19} + \frac{7}{2} \, c^{6} d^{2} x^{18} + 35 \, b c^{4} d^{3} x^{18} + \frac{105}{2} \, b^{2} c^{2} d^{4} x^{18} + 7 \, b^{3} d^{5} x^{18} + c^{7} d x^{17} + 21 \, b c^{5} d^{2} x^{17} + 70 \, b^{2} c^{3} d^{3} x^{17} + 35 \, b^{3} c d^{4} x^{17} + \frac{1}{8} \, c^{8} x^{16} + 7 \, b c^{6} d x^{16} + \frac{105}{2} \, b^{2} c^{4} d^{2} x^{16} + 70 \, b^{3} c^{2} d^{3} x^{16} + \frac{35}{4} \, b^{4} d^{4} x^{16} + b c^{7} x^{15} + 21 \, b^{2} c^{5} d x^{15} + 70 \, b^{3} c^{3} d^{2} x^{15} + 35 \, b^{4} c d^{3} x^{15} + \frac{7}{2} \, b^{2} c^{6} x^{14} + 35 \, b^{3} c^{4} d x^{14} + \frac{105}{2} \, b^{4} c^{2} d^{2} x^{14} + 7 \, b^{5} d^{3} x^{14} + 7 \, b^{3} c^{5} x^{13} + 35 \, b^{4} c^{3} d x^{13} + 21 \, b^{5} c d^{2} x^{13} + \frac{35}{4} \, b^{4} c^{4} x^{12} + 21 \, b^{5} c^{2} d x^{12} + \frac{7}{2} \, b^{6} d^{2} x^{12} + 7 \, b^{5} c^{3} x^{11} + 7 \, b^{6} c d x^{11} + \frac{7}{2} \, b^{6} c^{2} x^{10} + b^{7} d x^{10} + b^{7} c x^{9} + \frac{1}{8} \, b^{8} x^{8} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/8*d^8*x^24 + c*d^7*x^23 + 7/2*c^2*d^6*x^22 + b*d^7*x^22 + 7*c^3*d^5*x^21 + 7*b*c*d^6*x^21 + 35/4*c^4*d^4*x^2
0 + 21*b*c^2*d^5*x^20 + 7/2*b^2*d^6*x^20 + 7*c^5*d^3*x^19 + 35*b*c^3*d^4*x^19 + 21*b^2*c*d^5*x^19 + 7/2*c^6*d^
2*x^18 + 35*b*c^4*d^3*x^18 + 105/2*b^2*c^2*d^4*x^18 + 7*b^3*d^5*x^18 + c^7*d*x^17 + 21*b*c^5*d^2*x^17 + 70*b^2
*c^3*d^3*x^17 + 35*b^3*c*d^4*x^17 + 1/8*c^8*x^16 + 7*b*c^6*d*x^16 + 105/2*b^2*c^4*d^2*x^16 + 70*b^3*c^2*d^3*x^
16 + 35/4*b^4*d^4*x^16 + b*c^7*x^15 + 21*b^2*c^5*d*x^15 + 70*b^3*c^3*d^2*x^15 + 35*b^4*c*d^3*x^15 + 7/2*b^2*c^
6*x^14 + 35*b^3*c^4*d*x^14 + 105/2*b^4*c^2*d^2*x^14 + 7*b^5*d^3*x^14 + 7*b^3*c^5*x^13 + 35*b^4*c^3*d*x^13 + 21
*b^5*c*d^2*x^13 + 35/4*b^4*c^4*x^12 + 21*b^5*c^2*d*x^12 + 7/2*b^6*d^2*x^12 + 7*b^5*c^3*x^11 + 7*b^6*c*d*x^11 +
7/2*b^6*c^2*x^10 + b^7*d*x^10 + b^7*c*x^9 + 1/8*b^8*x^8