3.71 \(\int x^6 \left (a+b x^2\right )^5 \, dx\)

Optimal. Leaf size=69 \[ \frac{a^5 x^7}{7}+\frac{5}{9} a^4 b x^9+\frac{10}{11} a^3 b^2 x^{11}+\frac{10}{13} a^2 b^3 x^{13}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{17}}{17} \]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a
*b^4*x^15)/3 + (b^5*x^17)/17

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Rubi [A]  time = 0.0709719, antiderivative size = 69, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077 \[ \frac{a^5 x^7}{7}+\frac{5}{9} a^4 b x^9+\frac{10}{11} a^3 b^2 x^{11}+\frac{10}{13} a^2 b^3 x^{13}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]  Int[x^6*(a + b*x^2)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a
*b^4*x^15)/3 + (b^5*x^17)/17

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Rubi in Sympy [A]  time = 11.9187, size = 65, normalized size = 0.94 \[ \frac{a^{5} x^{7}}{7} + \frac{5 a^{4} b x^{9}}{9} + \frac{10 a^{3} b^{2} x^{11}}{11} + \frac{10 a^{2} b^{3} x^{13}}{13} + \frac{a b^{4} x^{15}}{3} + \frac{b^{5} x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**6*(b*x**2+a)**5,x)

[Out]

a**5*x**7/7 + 5*a**4*b*x**9/9 + 10*a**3*b**2*x**11/11 + 10*a**2*b**3*x**13/13 +
a*b**4*x**15/3 + b**5*x**17/17

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Mathematica [A]  time = 0.00366061, size = 69, normalized size = 1. \[ \frac{a^5 x^7}{7}+\frac{5}{9} a^4 b x^9+\frac{10}{11} a^3 b^2 x^{11}+\frac{10}{13} a^2 b^3 x^{13}+\frac{1}{3} a b^4 x^{15}+\frac{b^5 x^{17}}{17} \]

Antiderivative was successfully verified.

[In]  Integrate[x^6*(a + b*x^2)^5,x]

[Out]

(a^5*x^7)/7 + (5*a^4*b*x^9)/9 + (10*a^3*b^2*x^11)/11 + (10*a^2*b^3*x^13)/13 + (a
*b^4*x^15)/3 + (b^5*x^17)/17

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Maple [A]  time = 0.002, size = 58, normalized size = 0.8 \[{\frac{{a}^{5}{x}^{7}}{7}}+{\frac{5\,{a}^{4}b{x}^{9}}{9}}+{\frac{10\,{a}^{3}{b}^{2}{x}^{11}}{11}}+{\frac{10\,{a}^{2}{b}^{3}{x}^{13}}{13}}+{\frac{a{b}^{4}{x}^{15}}{3}}+{\frac{{b}^{5}{x}^{17}}{17}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^6*(b*x^2+a)^5,x)

[Out]

1/7*a^5*x^7+5/9*a^4*b*x^9+10/11*a^3*b^2*x^11+10/13*a^2*b^3*x^13+1/3*a*b^4*x^15+1
/17*b^5*x^17

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Maxima [A]  time = 1.34668, size = 77, normalized size = 1.12 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{1}{3} \, a b^{4} x^{15} + \frac{10}{13} \, a^{2} b^{3} x^{13} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^6,x, algorithm="maxima")

[Out]

1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a
^4*b*x^9 + 1/7*a^5*x^7

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Fricas [A]  time = 0.174745, size = 1, normalized size = 0.01 \[ \frac{1}{17} x^{17} b^{5} + \frac{1}{3} x^{15} b^{4} a + \frac{10}{13} x^{13} b^{3} a^{2} + \frac{10}{11} x^{11} b^{2} a^{3} + \frac{5}{9} x^{9} b a^{4} + \frac{1}{7} x^{7} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^6,x, algorithm="fricas")

[Out]

1/17*x^17*b^5 + 1/3*x^15*b^4*a + 10/13*x^13*b^3*a^2 + 10/11*x^11*b^2*a^3 + 5/9*x
^9*b*a^4 + 1/7*x^7*a^5

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Sympy [A]  time = 0.127308, size = 65, normalized size = 0.94 \[ \frac{a^{5} x^{7}}{7} + \frac{5 a^{4} b x^{9}}{9} + \frac{10 a^{3} b^{2} x^{11}}{11} + \frac{10 a^{2} b^{3} x^{13}}{13} + \frac{a b^{4} x^{15}}{3} + \frac{b^{5} x^{17}}{17} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**6*(b*x**2+a)**5,x)

[Out]

a**5*x**7/7 + 5*a**4*b*x**9/9 + 10*a**3*b**2*x**11/11 + 10*a**2*b**3*x**13/13 +
a*b**4*x**15/3 + b**5*x**17/17

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GIAC/XCAS [A]  time = 0.20759, size = 77, normalized size = 1.12 \[ \frac{1}{17} \, b^{5} x^{17} + \frac{1}{3} \, a b^{4} x^{15} + \frac{10}{13} \, a^{2} b^{3} x^{13} + \frac{10}{11} \, a^{3} b^{2} x^{11} + \frac{5}{9} \, a^{4} b x^{9} + \frac{1}{7} \, a^{5} x^{7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^6,x, algorithm="giac")

[Out]

1/17*b^5*x^17 + 1/3*a*b^4*x^15 + 10/13*a^2*b^3*x^13 + 10/11*a^3*b^2*x^11 + 5/9*a
^4*b*x^9 + 1/7*a^5*x^7