3.663 \(\int (2+3 x)^6 (1+(2+3 x)^7+(2+3 x)^{14}) \, dx\)

Optimal. Leaf size=34 \[ \frac{1}{63} (3 x+2)^{21}+\frac{1}{42} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7 \]

[Out]

(2 + 3*x)^7/21 + (2 + 3*x)^14/42 + (2 + 3*x)^21/63

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Rubi [A]  time = 0.0356623, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083, Rules used = {1390, 14} \[ \frac{1}{63} (3 x+2)^{21}+\frac{1}{42} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7 \]

Antiderivative was successfully verified.

[In]

Int[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14),x]

[Out]

(2 + 3*x)^7/21 + (2 + 3*x)^14/42 + (2 + 3*x)^21/63

Rule 1390

Int[(u_)^(m_.)*((a_.) + (c_.)*(v_)^(n2_.) + (b_.)*(v_)^(n_))^(p_.), x_Symbol] :> Dist[u^m/(Coefficient[v, x, 1
]*v^m), Subst[Int[x^m*(a + b*x^n + c*x^(2*n))^p, x], x, v], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n]
&& LinearPairQ[u, v, x]

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rubi steps

\begin{align*} \int (2+3 x)^6 \left (1+(2+3 x)^7+(2+3 x)^{14}\right ) \, dx &=\frac{1}{3} \operatorname{Subst}\left (\int x^6 \left (1+x^7+x^{14}\right ) \, dx,x,2+3 x\right )\\ &=\frac{1}{3} \operatorname{Subst}\left (\int \left (x^6+x^{13}+x^{20}\right ) \, dx,x,2+3 x\right )\\ &=\frac{1}{21} (2+3 x)^7+\frac{1}{42} (2+3 x)^{14}+\frac{1}{63} (2+3 x)^{21}\\ \end{align*}

Mathematica [A]  time = 0.0132572, size = 34, normalized size = 1. \[ \frac{1}{63} (3 x+2)^{21}+\frac{1}{42} (3 x+2)^{14}+\frac{1}{21} (3 x+2)^7 \]

Antiderivative was successfully verified.

[In]

Integrate[(2 + 3*x)^6*(1 + (2 + 3*x)^7 + (2 + 3*x)^14),x]

[Out]

(2 + 3*x)^7/21 + (2 + 3*x)^14/42 + (2 + 3*x)^21/63

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Maple [B]  time = 0.002, size = 105, normalized size = 3.1 \begin{align*}{\frac{1162261467\,{x}^{21}}{7}}[phone]\,{x}^{20}[phone]\,{x}^{19}[phone]\,{x}^{18}[phone]\,{x}^{17}[phone]\,{x}^{16}[phone]\,{x}^{15}+{\frac{15819767221203\,{x}^{14}}{14}}[phone]\,{x}^{13}[phone]\,{x}^{12}[phone]\,{x}^{11}[phone]\,{x}^{10}[phone]\,{x}^{9}[phone]\,{x}^{8}[phone]\,{x}^{7}[phone]\,{x}^{6}[phone]\,{x}^{5}[phone]\,{x}^{4}[phone]\,{x}^{3}[phone]\,{x}^{2}+1056832\,x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14),x)

[Out]

1162261467/7*x^21+2324522934*x^20+15496819560*x^19+65431015920*x^18+196293047760*x^17+444930908256*x^16+790988
281344*x^15+15819767221203/14*x^14+1318314865122*x^13+1269491970942*x^12+1015602174288*x^11+677082445416*x^10+
376174427616*x^9+173635132896*x^8+66158154783*x^7+20588764518*x^6+5149786572*x^5+1010576952*x^4+149902032*x^3+
15808800*x^2+1056832*x

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Maxima [B]  time = 0.969965, size = 140, normalized size = 4.12 \begin{align*} \frac{1162261467}{7} \, x^{21} + 2324522934 \, x^{20} + 15496819560 \, x^{19} + 65431015920 \, x^{18} + 196293047760 \, x^{17} + 444930908256 \, x^{16} + 790988281344 \, x^{15} + \frac{15819767221203}{14} \, x^{14} + 1318314865122 \, x^{13} + 1269491970942 \, x^{12} + 1015602174288 \, x^{11} + 677082445416 \, x^{10} + 376174427616 \, x^{9} + 173635132896 \, x^{8} + 66158154783 \, x^{7} + 20588764518 \, x^{6} + 5149786572 \, x^{5} + 1010576952 \, x^{4} + 149902032 \, x^{3} + 15808800 \, x^{2} + 1056832 \, x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14),x, algorithm="maxima")

[Out]

1162261467/7*x^21 + 2324522934*x^20 + 15496819560*x^19 + 65431015920*x^18 + 196293047760*x^17 + 444930908256*x
^16 + 790988281344*x^15 + 15819767221203/14*x^14 + 1318314865122*x^13 + 1269491970942*x^12 + 1015602174288*x^1
1 + 677082445416*x^10 + 376174427616*x^9 + 173635132896*x^8 + 66158154783*x^7 + 20588764518*x^6 + 5149786572*x
^5 + 1010576952*x^4 + 149902032*x^3 + 15808800*x^2 + 1056832*x

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Fricas [B]  time = 1.07722, size = 532, normalized size = 15.65 \begin{align*} \frac{1162261467}{7} x^{21} + 2324522934 x^{20} + 15496819560 x^{19} + 65431015920 x^{18} + 196293047760 x^{17} + 444930908256 x^{16} + 790988281344 x^{15} + \frac{15819767221203}{14} x^{14} + 1318314865122 x^{13} + 1269491970942 x^{12} + 1015602174288 x^{11} + 677082445416 x^{10} + 376174427616 x^{9} + 173635132896 x^{8} + 66158154783 x^{7} + 20588764518 x^{6} + 5149786572 x^{5} + 1010576952 x^{4} + 149902032 x^{3} + 15808800 x^{2} + 1056832 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14),x, algorithm="fricas")

[Out]

1162261467/7*x^21 + 2324522934*x^20 + 15496819560*x^19 + 65431015920*x^18 + 196293047760*x^17 + 444930908256*x
^16 + 790988281344*x^15 + 15819767221203/14*x^14 + 1318314865122*x^13 + 1269491970942*x^12 + 1015602174288*x^1
1 + 677082445416*x^10 + 376174427616*x^9 + 173635132896*x^8 + 66158154783*x^7 + 20588764518*x^6 + 5149786572*x
^5 + 1010576952*x^4 + 149902032*x^3 + 15808800*x^2 + 1056832*x

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Sympy [B]  time = 0.092468, size = 107, normalized size = 3.15 \begin{align*} \frac{1162261467 x^{21}}{7} + 2324522934 x^{20} + 15496819560 x^{19} + 65431015920 x^{18} + 196293047760 x^{17} + 444930908256 x^{16} + 790988281344 x^{15} + \frac{15819767221203 x^{14}}{14} + 1318314865122 x^{13} + 1269491970942 x^{12} + 1015602174288 x^{11} + 677082445416 x^{10} + 376174427616 x^{9} + 173635132896 x^{8} + 66158154783 x^{7} + 20588764518 x^{6} + 5149786572 x^{5} + 1010576952 x^{4} + 149902032 x^{3} + 15808800 x^{2} + 1056832 x \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)**6*(1+(2+3*x)**7+(2+3*x)**14),x)

[Out]

1162261467*x**21/7 + 2324522934*x**20 + 15496819560*x**19 + 65431015920*x**18 + 196293047760*x**17 + 444930908
256*x**16 + 790988281344*x**15 + 15819767221203*x**14/14 + 1318314865122*x**13 + 1269491970942*x**12 + 1015602
174288*x**11 + 677082445416*x**10 + 376174427616*x**9 + 173635132896*x**8 + 66158154783*x**7 + 20588764518*x**
6 + 5149786572*x**5 + 1010576952*x**4 + 149902032*x**3 + 15808800*x**2 + 1056832*x

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Giac [A]  time = 1.08124, size = 38, normalized size = 1.12 \begin{align*} \frac{1}{63} \,{\left (3 \, x + 2\right )}^{21} + \frac{1}{42} \,{\left (3 \, x + 2\right )}^{14} + \frac{1}{21} \,{\left (3 \, x + 2\right )}^{7} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2+3*x)^6*(1+(2+3*x)^7+(2+3*x)^14),x, algorithm="giac")

[Out]

1/63*(3*x + 2)^21 + 1/42*(3*x + 2)^14 + 1/21*(3*x + 2)^7