3.2.1 \(\int (b+2 c x) (b x+c x^2)^{13} \, dx\) [101]

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

[Out]

1/14*(c*x^2+b*x)^14

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Rubi [A]
time = 0.01, antiderivative size = 15, normalized size of antiderivative = 1.00, 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 = {643} \begin {gather*} \frac {1}{14} \left (b x+c x^2\right )^{14} \end {gather*}

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 643

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*}

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(172\) vs. \(2(15)=30\).
time = 0.00, size = 172, normalized size = 11.47 \begin {gather*} \frac {b^{14} x^{14}}{14}+b^{13} c x^{15}+\frac {13}{2} b^{12} c^2 x^{16}+26 b^{11} c^3 x^{17}+\frac {143}{2} b^{10} c^4 x^{18}+143 b^9 c^5 x^{19}+\frac {429}{2} b^8 c^6 x^{20}+\frac {1716}{7} b^7 c^7 x^{21}+\frac {429}{2} b^6 c^8 x^{22}+143 b^5 c^9 x^{23}+\frac {143}{2} b^4 c^{10} x^{24}+26 b^3 c^{11} x^{25}+\frac {13}{2} b^2 c^{12} x^{26}+b c^{13} x^{27}+\frac {c^{14} x^{28}}{14} \end {gather*}

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 [A]
time = 0.20, size = 14, normalized size = 0.93

method	result	size
gosper	\(\frac {\left (c x +b \right )^{14} x^{14}}{14}\)	\(13\)
default	\(\frac {\left (c \,x^{2}+b x \right )^{14}}{14}\)	\(14\)
norman	\(b \,c^{13} x^{27}+\frac {1}{14} c^{14} x^{28}+\frac {1}{14} x^{14} b^{14}+b^{13} c \,x^{15}+\frac {13}{2} x^{16} b^{12} c^{2}+26 b^{11} c^{3} x^{17}+\frac {143}{2} x^{18} b^{10} c^{4}+143 b^{9} c^{5} x^{19}+\frac {429}{2} x^{20} b^{8} c^{6}+\frac {1716}{7} x^{21} b^{7} c^{7}+\frac {429}{2} x^{22} b^{6} c^{8}+143 b^{5} c^{9} x^{23}+\frac {143}{2} x^{24} b^{4} c^{10}+26 b^{3} c^{11} x^{25}+\frac {13}{2} x^{26} b^{2} c^{12}\)	\(155\)
risch	\(b \,c^{13} x^{27}+\frac {1}{14} c^{14} x^{28}+\frac {1}{14} x^{14} b^{14}+b^{13} c \,x^{15}+\frac {13}{2} x^{16} b^{12} c^{2}+26 b^{11} c^{3} x^{17}+\frac {143}{2} x^{18} b^{10} c^{4}+143 b^{9} c^{5} x^{19}+\frac {429}{2} x^{20} b^{8} c^{6}+\frac {1716}{7} x^{21} b^{7} c^{7}+\frac {429}{2} x^{22} b^{6} c^{8}+143 b^{5} c^{9} x^{23}+\frac {143}{2} x^{24} b^{4} c^{10}+26 b^{3} c^{11} x^{25}+\frac {13}{2} x^{26} b^{2} c^{12}\)	\(155\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/14*(c*x^2+b*x)^14

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Maxima [A]
time = 0.29, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{14} \, {\left (c x^{2} + b x\right )}^{14} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 154 vs. \(2 (13) = 26\).
time = 0.33, size = 154, normalized size = 10.27 \begin {gather*} \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 {gather*}

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*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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 175 vs. \(2 (10) = 20\).
time = 0.05, size = 175, normalized size = 11.67 \begin {gather*} \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 {gather*}

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 [A]
time = 2.89, size = 13, normalized size = 0.87 \begin {gather*} \frac {1}{14} \, {\left (c x^{2} + b x\right )}^{14} \end {gather*}

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*x^2 + b*x)^14

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Mupad [B]
time = 2.09, size = 154, normalized size = 10.27 \begin {gather*} \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 {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x + c*x^2)^13*(b + 2*c*x),x)

[Out]

(b^14*x^14)/14 + (c^14*x^28)/14 + b^13*c*x^15 + b*c^13*x^27 + (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

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