3.885 \(\int x^{-1+14 n} \left (b+c x^n\right )^{13} \left (b+2 c x^n\right ) \, dx\)

Optimal. Leaf size=21 \[ \frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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Rubi [A]  time = 0.0509615, antiderivative size = 21, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 25, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.08 \[ \frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

Antiderivative was successfully verified.

[In]  Int[x^(-1 + 14*n)*(b + c*x^n)^13*(b + 2*c*x^n),x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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Rubi in Sympy [A]  time = 9.39983, size = 15, normalized size = 0.71 \[ \frac{x^{14 n} \left (b + c x^{n}\right )^{14}}{14 n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**(-1+14*n)*(b+c*x**n)**13*(b+2*c*x**n),x)

[Out]

x**(14*n)*(b + c*x**n)**14/(14*n)

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Mathematica [A]  time = 0.0613228, size = 21, normalized size = 1. \[ \frac{x^{14 n} \left (b+c x^n\right )^{14}}{14 n} \]

Antiderivative was successfully verified.

[In]  Integrate[x^(-1 + 14*n)*(b + c*x^n)^13*(b + 2*c*x^n),x]

[Out]

(x^(14*n)*(b + c*x^n)^14)/(14*n)

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Maple [B]  time = 0.052, size = 230, normalized size = 11. \[{\frac{{c}^{14} \left ({x}^{n} \right ) ^{28}}{14\,n}}+{\frac{b{c}^{13} \left ({x}^{n} \right ) ^{27}}{n}}+{\frac{13\,{b}^{2}{c}^{12} \left ({x}^{n} \right ) ^{26}}{2\,n}}+26\,{\frac{{b}^{3}{c}^{11} \left ({x}^{n} \right ) ^{25}}{n}}+{\frac{143\,{b}^{4}{c}^{10} \left ({x}^{n} \right ) ^{24}}{2\,n}}+143\,{\frac{{b}^{5}{c}^{9} \left ({x}^{n} \right ) ^{23}}{n}}+{\frac{429\,{b}^{6}{c}^{8} \left ({x}^{n} \right ) ^{22}}{2\,n}}+{\frac{1716\,{b}^{7}{c}^{7} \left ({x}^{n} \right ) ^{21}}{7\,n}}+{\frac{429\,{b}^{8}{c}^{6} \left ({x}^{n} \right ) ^{20}}{2\,n}}+143\,{\frac{{b}^{9}{c}^{5} \left ({x}^{n} \right ) ^{19}}{n}}+{\frac{143\,{b}^{10}{c}^{4} \left ({x}^{n} \right ) ^{18}}{2\,n}}+26\,{\frac{{b}^{11}{c}^{3} \left ({x}^{n} \right ) ^{17}}{n}}+{\frac{13\,{b}^{12}{c}^{2} \left ({x}^{n} \right ) ^{16}}{2\,n}}+{\frac{{b}^{13}c \left ({x}^{n} \right ) ^{15}}{n}}+{\frac{{b}^{14} \left ({x}^{n} \right ) ^{14}}{14\,n}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^(-1+14*n)*(b+c*x^n)^13*(b+2*c*x^n),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*(c*x^n + b)^13*x^(14*n - 1),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.234059, size = 255, normalized size = 12.14 \[ \frac{c^{14} x^{28 \, n} + 14 \, b c^{13} x^{27 \, n} + 91 \, b^{2} c^{12} x^{26 \, n} + 364 \, b^{3} c^{11} x^{25 \, n} + 1001 \, b^{4} c^{10} x^{24 \, n} + 2002 \, b^{5} c^{9} x^{23 \, n} + 3003 \, b^{6} c^{8} x^{22 \, n} + 3432 \, b^{7} c^{7} x^{21 \, n} + 3003 \, b^{8} c^{6} x^{20 \, n} + 2002 \, b^{9} c^{5} x^{19 \, n} + 1001 \, b^{10} c^{4} x^{18 \, n} + 364 \, b^{11} c^{3} x^{17 \, n} + 91 \, b^{12} c^{2} x^{16 \, n} + 14 \, b^{13} c x^{15 \, n} + b^{14} x^{14 \, n}}{14 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*(c*x^n + b)^13*x^(14*n - 1),x, algorithm="fricas")

[Out]

1/14*(c^14*x^(28*n) + 14*b*c^13*x^(27*n) + 91*b^2*c^12*x^(26*n) + 364*b^3*c^11*x
^(25*n) + 1001*b^4*c^10*x^(24*n) + 2002*b^5*c^9*x^(23*n) + 3003*b^6*c^8*x^(22*n)
+ 3432*b^7*c^7*x^(21*n) + 3003*b^8*c^6*x^(20*n) + 2002*b^9*c^5*x^(19*n) + 1001*
b^10*c^4*x^(18*n) + 364*b^11*c^3*x^(17*n) + 91*b^12*c^2*x^(16*n) + 14*b^13*c*x^(
15*n) + b^14*x^(14*n))/n

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**(-1+14*n)*(b+c*x**n)**13*(b+2*c*x**n),x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.288667, size = 275, normalized size = 13.1 \[ \frac{c^{14} e^{\left (28 \, n{\rm ln}\left (x\right )\right )} + 14 \, b c^{13} e^{\left (27 \, n{\rm ln}\left (x\right )\right )} + 91 \, b^{2} c^{12} e^{\left (26 \, n{\rm ln}\left (x\right )\right )} + 364 \, b^{3} c^{11} e^{\left (25 \, n{\rm ln}\left (x\right )\right )} + 1001 \, b^{4} c^{10} e^{\left (24 \, n{\rm ln}\left (x\right )\right )} + 2002 \, b^{5} c^{9} e^{\left (23 \, n{\rm ln}\left (x\right )\right )} + 3003 \, b^{6} c^{8} e^{\left (22 \, n{\rm ln}\left (x\right )\right )} + 3432 \, b^{7} c^{7} e^{\left (21 \, n{\rm ln}\left (x\right )\right )} + 3003 \, b^{8} c^{6} e^{\left (20 \, n{\rm ln}\left (x\right )\right )} + 2002 \, b^{9} c^{5} e^{\left (19 \, n{\rm ln}\left (x\right )\right )} + 1001 \, b^{10} c^{4} e^{\left (18 \, n{\rm ln}\left (x\right )\right )} + 364 \, b^{11} c^{3} e^{\left (17 \, n{\rm ln}\left (x\right )\right )} + 91 \, b^{12} c^{2} e^{\left (16 \, n{\rm ln}\left (x\right )\right )} + 14 \, b^{13} c e^{\left (15 \, n{\rm ln}\left (x\right )\right )} + b^{14} e^{\left (14 \, n{\rm ln}\left (x\right )\right )}}{14 \, n} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((2*c*x^n + b)*(c*x^n + b)^13*x^(14*n - 1),x, algorithm="giac")

[Out]

1/14*(c^14*e^(28*n*ln(x)) + 14*b*c^13*e^(27*n*ln(x)) + 91*b^2*c^12*e^(26*n*ln(x)
) + 364*b^3*c^11*e^(25*n*ln(x)) + 1001*b^4*c^10*e^(24*n*ln(x)) + 2002*b^5*c^9*e^
(23*n*ln(x)) + 3003*b^6*c^8*e^(22*n*ln(x)) + 3432*b^7*c^7*e^(21*n*ln(x)) + 3003*
b^8*c^6*e^(20*n*ln(x)) + 2002*b^9*c^5*e^(19*n*ln(x)) + 1001*b^10*c^4*e^(18*n*ln(
x)) + 364*b^11*c^3*e^(17*n*ln(x)) + 91*b^12*c^2*e^(16*n*ln(x)) + 14*b^13*c*e^(15
*n*ln(x)) + b^14*e^(14*n*ln(x)))/n