3.72 \(\int x^4 (a+b x^2)^5 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.02, antiderivative size = 69, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {270} \[ \frac {10}{11} a^2 b^3 x^{11}+\frac {10}{9} a^3 b^2 x^9+\frac {5}{7} a^4 b x^7+\frac {a^5 x^5}{5}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Int[x^4*(a + b*x^2)^5,x]

[Out]

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

Rule 270

Int[((c_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*(a + b*x^n)^p,
x], x] /; FreeQ[{a, b, c, m, n}, x] && IGtQ[p, 0]

Rubi steps

\begin {align*} \int x^4 \left (a+b x^2\right )^5 \, dx &=\int \left (a^5 x^4+5 a^4 b x^6+10 a^3 b^2 x^8+10 a^2 b^3 x^{10}+5 a b^4 x^{12}+b^5 x^{14}\right ) \, dx\\ &=\frac {a^5 x^5}{5}+\frac {5}{7} a^4 b x^7+\frac {10}{9} a^3 b^2 x^9+\frac {10}{11} a^2 b^3 x^{11}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{15}}{15}\\ \end {align*}

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Mathematica [A]  time = 0.00, size = 69, normalized size = 1.00 \[ \frac {a^5 x^5}{5}+\frac {5}{7} a^4 b x^7+\frac {10}{9} a^3 b^2 x^9+\frac {10}{11} a^2 b^3 x^{11}+\frac {5}{13} a b^4 x^{13}+\frac {b^5 x^{15}}{15} \]

Antiderivative was successfully verified.

[In]

Integrate[x^4*(a + b*x^2)^5,x]

[Out]

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

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fricas [A]  time = 0.74, size = 57, normalized size = 0.83 \[ \frac {1}{15} x^{15} b^{5} + \frac {5}{13} x^{13} b^{4} a + \frac {10}{11} x^{11} b^{3} a^{2} + \frac {10}{9} x^{9} b^{2} a^{3} + \frac {5}{7} x^{7} b a^{4} + \frac {1}{5} x^{5} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 1.01, size = 57, normalized size = 0.83 \[ \frac {1}{15} \, b^{5} x^{15} + \frac {5}{13} \, a b^{4} x^{13} + \frac {10}{11} \, a^{2} b^{3} x^{11} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5,x, algorithm="giac")

[Out]

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

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maple [A]  time = 0.00, size = 58, normalized size = 0.84 \[ \frac {1}{15} b^{5} x^{15}+\frac {5}{13} a \,b^{4} x^{13}+\frac {10}{11} a^{2} b^{3} x^{11}+\frac {10}{9} a^{3} b^{2} x^{9}+\frac {5}{7} a^{4} b \,x^{7}+\frac {1}{5} a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(b*x^2+a)^5,x)

[Out]

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

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maxima [A]  time = 1.31, size = 57, normalized size = 0.83 \[ \frac {1}{15} \, b^{5} x^{15} + \frac {5}{13} \, a b^{4} x^{13} + \frac {10}{11} \, a^{2} b^{3} x^{11} + \frac {10}{9} \, a^{3} b^{2} x^{9} + \frac {5}{7} \, a^{4} b x^{7} + \frac {1}{5} \, a^{5} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(b*x^2+a)^5,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 0.02, size = 57, normalized size = 0.83 \[ \frac {a^5\,x^5}{5}+\frac {5\,a^4\,b\,x^7}{7}+\frac {10\,a^3\,b^2\,x^9}{9}+\frac {10\,a^2\,b^3\,x^{11}}{11}+\frac {5\,a\,b^4\,x^{13}}{13}+\frac {b^5\,x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(a + b*x^2)^5,x)

[Out]

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

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sympy [A]  time = 0.08, size = 66, normalized size = 0.96 \[ \frac {a^{5} x^{5}}{5} + \frac {5 a^{4} b x^{7}}{7} + \frac {10 a^{3} b^{2} x^{9}}{9} + \frac {10 a^{2} b^{3} x^{11}}{11} + \frac {5 a b^{4} x^{13}}{13} + \frac {b^{5} x^{15}}{15} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(b*x**2+a)**5,x)

[Out]

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

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