3.336 \(\int \left (a x^m+b x^{1+6 m}\right )^5 \, dx\)

Optimal. Leaf size=27 \[ \frac{\left (a+b x^{5 m+1}\right )^6}{6 b (5 m+1)} \]

[Out]

(a + b*x^(1 + 5*m))^6/(6*b*(1 + 5*m))

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Rubi [A]  time = 0.0262613, antiderivative size = 27, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{\left (a+b x^{5 m+1}\right )^6}{6 b (5 m+1)} \]

Antiderivative was successfully verified.

[In]  Int[(a*x^m + b*x^(1 + 6*m))^5,x]

[Out]

(a + b*x^(1 + 5*m))^6/(6*b*(1 + 5*m))

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Rubi in Sympy [A]  time = 3.56607, size = 19, normalized size = 0.7 \[ \frac{\left (a + b x^{5 m + 1}\right )^{6}}{6 b \left (5 m + 1\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((a*x**m+b*x**(1+6*m))**5,x)

[Out]

(a + b*x**(5*m + 1))**6/(6*b*(5*m + 1))

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Mathematica [B]  time = 0.0725136, size = 88, normalized size = 3.26 \[ \frac{x^{5 m+1} \left (6 a^5+15 a^4 b x^{5 m+1}+20 a^3 b^2 x^{10 m+2}+15 a^2 b^3 x^{15 m+3}+6 a b^4 x^{20 m+4}+b^5 x^{25 m+5}\right )}{30 m+6} \]

Antiderivative was successfully verified.

[In]  Integrate[(a*x^m + b*x^(1 + 6*m))^5,x]

[Out]

(x^(1 + 5*m)*(6*a^5 + 15*a^4*b*x^(1 + 5*m) + 20*a^3*b^2*x^(2 + 10*m) + 15*a^2*b^
3*x^(3 + 15*m) + 6*a*b^4*x^(4 + 20*m) + b^5*x^(5 + 25*m)))/(6 + 30*m)

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Maple [B]  time = 0.035, size = 126, normalized size = 4.7 \[{\frac{{b}^{5}{x}^{6} \left ({x}^{m} \right ) ^{30}}{6+30\,m}}+{\frac{a{b}^{4}{x}^{5} \left ({x}^{m} \right ) ^{25}}{1+5\,m}}+{\frac{5\,{a}^{2}{b}^{3}{x}^{4} \left ({x}^{m} \right ) ^{20}}{2+10\,m}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{3} \left ({x}^{m} \right ) ^{15}}{3+15\,m}}+{\frac{5\,{a}^{4}b{x}^{2} \left ({x}^{m} \right ) ^{10}}{2+10\,m}}+{\frac{{a}^{5}x \left ({x}^{m} \right ) ^{5}}{1+5\,m}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((a*x^m+b*x^(1+6*m))^5,x)

[Out]

1/6*b^5*x^6/(1+5*m)*(x^m)^30+a*b^4*x^5/(1+5*m)*(x^m)^25+5/2*a^2*b^3*x^4/(1+5*m)*
(x^m)^20+10/3*a^3*b^2*x^3/(1+5*m)*(x^m)^15+5/2*a^4*b*x^2/(1+5*m)*(x^m)^10+a^5/(1
+5*m)*x*(x^m)^5

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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((b*x^(6*m + 1) + a*x^m)^5,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.237145, size = 126, normalized size = 4.67 \[ \frac{b^{5} x^{6} x^{30 \, m} + 6 \, a b^{4} x^{5} x^{25 \, m} + 15 \, a^{2} b^{3} x^{4} x^{20 \, m} + 20 \, a^{3} b^{2} x^{3} x^{15 \, m} + 15 \, a^{4} b x^{2} x^{10 \, m} + 6 \, a^{5} x x^{5 \, m}}{6 \,{\left (5 \, m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(6*m + 1) + a*x^m)^5,x, algorithm="fricas")

[Out]

1/6*(b^5*x^6*x^(30*m) + 6*a*b^4*x^5*x^(25*m) + 15*a^2*b^3*x^4*x^(20*m) + 20*a^3*
b^2*x^3*x^(15*m) + 15*a^4*b*x^2*x^(10*m) + 6*a^5*x*x^(5*m))/(5*m + 1)

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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((a*x**m+b*x**(1+6*m))**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.224648, size = 134, normalized size = 4.96 \[ \frac{b^{5} x^{6} e^{\left (30 \, m{\rm ln}\left (x\right )\right )} + 6 \, a b^{4} x^{5} e^{\left (25 \, m{\rm ln}\left (x\right )\right )} + 15 \, a^{2} b^{3} x^{4} e^{\left (20 \, m{\rm ln}\left (x\right )\right )} + 20 \, a^{3} b^{2} x^{3} e^{\left (15 \, m{\rm ln}\left (x\right )\right )} + 15 \, a^{4} b x^{2} e^{\left (10 \, m{\rm ln}\left (x\right )\right )} + 6 \, a^{5} x e^{\left (5 \, m{\rm ln}\left (x\right )\right )}}{6 \,{\left (5 \, m + 1\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^(6*m + 1) + a*x^m)^5,x, algorithm="giac")

[Out]

1/6*(b^5*x^6*e^(30*m*ln(x)) + 6*a*b^4*x^5*e^(25*m*ln(x)) + 15*a^2*b^3*x^4*e^(20*
m*ln(x)) + 20*a^3*b^2*x^3*e^(15*m*ln(x)) + 15*a^4*b*x^2*e^(10*m*ln(x)) + 6*a^5*x
*e^(5*m*ln(x)))/(5*m + 1)