### 3.101 $$\int (b+2 c x) (b x+c x^2)^{13} \, dx$$

Optimal. Leaf size=15 $\frac{1}{14} \left (b x+c x^2\right )^{14}$

[Out]

(b*x + c*x^2)^14/14

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Rubi [A]  time = 0.0138468, antiderivative size = 15, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 18, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.056, Rules used = {629} $\frac{1}{14} \left (b x+c x^2\right )^{14}$

Antiderivative was successfully veriﬁed.

[In]

Int[(b + 2*c*x)*(b*x + c*x^2)^13,x]

[Out]

(b*x + c*x^2)^14/14

Rule 629

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

Rubi steps

\begin{align*} \int (b+2 c x) \left (b x+c x^2\right )^{13} \, dx &=\frac{1}{14} \left (b x+c x^2\right )^{14}\\ \end{align*}

Mathematica [B]  time = 0.0052051, size = 172, normalized size = 11.47 $\frac{13}{2} b^2 c^{12} x^{26}+26 b^3 c^{11} x^{25}+\frac{143}{2} b^4 c^{10} x^{24}+143 b^5 c^9 x^{23}+\frac{429}{2} b^6 c^8 x^{22}+\frac{1716}{7} b^7 c^7 x^{21}+\frac{429}{2} b^8 c^6 x^{20}+143 b^9 c^5 x^{19}+\frac{143}{2} b^{10} c^4 x^{18}+26 b^{11} c^3 x^{17}+\frac{13}{2} b^{12} c^2 x^{16}+b^{13} c x^{15}+\frac{b^{14} x^{14}}{14}+b c^{13} x^{27}+\frac{c^{14} x^{28}}{14}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(b + 2*c*x)*(b*x + c*x^2)^13,x]

[Out]

(b^14*x^14)/14 + b^13*c*x^15 + (13*b^12*c^2*x^16)/2 + 26*b^11*c^3*x^17 + (143*b^10*c^4*x^18)/2 + 143*b^9*c^5*x
^19 + (429*b^8*c^6*x^20)/2 + (1716*b^7*c^7*x^21)/7 + (429*b^6*c^8*x^22)/2 + 143*b^5*c^9*x^23 + (143*b^4*c^10*x
^24)/2 + 26*b^3*c^11*x^25 + (13*b^2*c^12*x^26)/2 + b*c^13*x^27 + (c^14*x^28)/14

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Maple [B]  time = 0.003, size = 155, normalized size = 10.3 \begin{align*}{\frac{{c}^{14}{x}^{28}}{14}}+b{c}^{13}{x}^{27}+{\frac{13\,{b}^{2}{c}^{12}{x}^{26}}{2}}+26\,{b}^{3}{c}^{11}{x}^{25}+{\frac{143\,{b}^{4}{c}^{10}{x}^{24}}{2}}+143\,{b}^{5}{c}^{9}{x}^{23}+{\frac{429\,{b}^{6}{c}^{8}{x}^{22}}{2}}+{\frac{1716\,{b}^{7}{c}^{7}{x}^{21}}{7}}+{\frac{429\,{b}^{8}{c}^{6}{x}^{20}}{2}}+143\,{b}^{9}{c}^{5}{x}^{19}+{\frac{143\,{b}^{10}{c}^{4}{x}^{18}}{2}}+26\,{b}^{11}{c}^{3}{x}^{17}+{\frac{13\,{b}^{12}{c}^{2}{x}^{16}}{2}}+{b}^{13}c{x}^{15}+{\frac{{b}^{14}{x}^{14}}{14}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((2*c*x+b)*(c*x^2+b*x)^13,x)

[Out]

1/14*c^14*x^28+b*c^13*x^27+13/2*b^2*c^12*x^26+26*b^3*c^11*x^25+143/2*b^4*c^10*x^24+143*b^5*c^9*x^23+429/2*b^6*
c^8*x^22+1716/7*b^7*c^7*x^21+429/2*b^8*c^6*x^20+143*b^9*c^5*x^19+143/2*b^10*c^4*x^18+26*b^11*c^3*x^17+13/2*b^1
2*c^2*x^16+b^13*c*x^15+1/14*b^14*x^14

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Maxima [A]  time = 1.00223, size = 18, normalized size = 1.2 \begin{align*} \frac{1}{14} \,{\left (c x^{2} + b x\right )}^{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x)^13,x, algorithm="maxima")

[Out]

1/14*(c*x^2 + b*x)^14

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Fricas [B]  time = 0.936906, size = 387, normalized size = 25.8 \begin{align*} \frac{1}{14} x^{28} c^{14} + x^{27} c^{13} b + \frac{13}{2} x^{26} c^{12} b^{2} + 26 x^{25} c^{11} b^{3} + \frac{143}{2} x^{24} c^{10} b^{4} + 143 x^{23} c^{9} b^{5} + \frac{429}{2} x^{22} c^{8} b^{6} + \frac{1716}{7} x^{21} c^{7} b^{7} + \frac{429}{2} x^{20} c^{6} b^{8} + 143 x^{19} c^{5} b^{9} + \frac{143}{2} x^{18} c^{4} b^{10} + 26 x^{17} c^{3} b^{11} + \frac{13}{2} x^{16} c^{2} b^{12} + x^{15} c b^{13} + \frac{1}{14} x^{14} b^{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x)^13,x, algorithm="fricas")

[Out]

1/14*x^28*c^14 + x^27*c^13*b + 13/2*x^26*c^12*b^2 + 26*x^25*c^11*b^3 + 143/2*x^24*c^10*b^4 + 143*x^23*c^9*b^5
+ 429/2*x^22*c^8*b^6 + 1716/7*x^21*c^7*b^7 + 429/2*x^20*c^6*b^8 + 143*x^19*c^5*b^9 + 143/2*x^18*c^4*b^10 + 26*
x^17*c^3*b^11 + 13/2*x^16*c^2*b^12 + x^15*c*b^13 + 1/14*x^14*b^14

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Sympy [B]  time = 0.117751, size = 175, normalized size = 11.67 \begin{align*} \frac{b^{14} x^{14}}{14} + b^{13} c x^{15} + \frac{13 b^{12} c^{2} x^{16}}{2} + 26 b^{11} c^{3} x^{17} + \frac{143 b^{10} c^{4} x^{18}}{2} + 143 b^{9} c^{5} x^{19} + \frac{429 b^{8} c^{6} x^{20}}{2} + \frac{1716 b^{7} c^{7} x^{21}}{7} + \frac{429 b^{6} c^{8} x^{22}}{2} + 143 b^{5} c^{9} x^{23} + \frac{143 b^{4} c^{10} x^{24}}{2} + 26 b^{3} c^{11} x^{25} + \frac{13 b^{2} c^{12} x^{26}}{2} + b c^{13} x^{27} + \frac{c^{14} x^{28}}{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x**2+b*x)**13,x)

[Out]

b**14*x**14/14 + b**13*c*x**15 + 13*b**12*c**2*x**16/2 + 26*b**11*c**3*x**17 + 143*b**10*c**4*x**18/2 + 143*b*
*9*c**5*x**19 + 429*b**8*c**6*x**20/2 + 1716*b**7*c**7*x**21/7 + 429*b**6*c**8*x**22/2 + 143*b**5*c**9*x**23 +
143*b**4*c**10*x**24/2 + 26*b**3*c**11*x**25 + 13*b**2*c**12*x**26/2 + b*c**13*x**27 + c**14*x**28/14

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Giac [B]  time = 1.08786, size = 208, normalized size = 13.87 \begin{align*} \frac{1}{14} \, c^{14} x^{28} + b c^{13} x^{27} + \frac{13}{2} \, b^{2} c^{12} x^{26} + 26 \, b^{3} c^{11} x^{25} + \frac{143}{2} \, b^{4} c^{10} x^{24} + 143 \, b^{5} c^{9} x^{23} + \frac{429}{2} \, b^{6} c^{8} x^{22} + \frac{1716}{7} \, b^{7} c^{7} x^{21} + \frac{429}{2} \, b^{8} c^{6} x^{20} + 143 \, b^{9} c^{5} x^{19} + \frac{143}{2} \, b^{10} c^{4} x^{18} + 26 \, b^{11} c^{3} x^{17} + \frac{13}{2} \, b^{12} c^{2} x^{16} + b^{13} c x^{15} + \frac{1}{14} \, b^{14} x^{14} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*c*x+b)*(c*x^2+b*x)^13,x, algorithm="giac")

[Out]

1/14*c^14*x^28 + b*c^13*x^27 + 13/2*b^2*c^12*x^26 + 26*b^3*c^11*x^25 + 143/2*b^4*c^10*x^24 + 143*b^5*c^9*x^23
+ 429/2*b^6*c^8*x^22 + 1716/7*b^7*c^7*x^21 + 429/2*b^8*c^6*x^20 + 143*b^9*c^5*x^19 + 143/2*b^10*c^4*x^18 + 26*
b^11*c^3*x^17 + 13/2*b^12*c^2*x^16 + b^13*c*x^15 + 1/14*b^14*x^14