3.1.90 \(\int x^3 (a+b x^2)^8 \, dx\)

Optimal. Leaf size=34 \[ \frac {\left (a+b x^2\right )^{10}}{20 b^2}-\frac {a \left (a+b x^2\right )^9}{18 b^2} \]

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Rubi [A]  time = 0.05, antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.154, Rules used = {266, 43} \begin {gather*} \frac {\left (a+b x^2\right )^{10}}{20 b^2}-\frac {a \left (a+b x^2\right )^9}{18 b^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^3*(a + b*x^2)^8,x]

[Out]

-(a*(a + b*x^2)^9)/(18*b^2) + (a + b*x^2)^10/(20*b^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

\begin {align*} \int x^3 \left (a+b x^2\right )^8 \, dx &=\frac {1}{2} \operatorname {Subst}\left (\int x (a+b x)^8 \, dx,x,x^2\right )\\ &=\frac {1}{2} \operatorname {Subst}\left (\int \left (-\frac {a (a+b x)^8}{b}+\frac {(a+b x)^9}{b}\right ) \, dx,x,x^2\right )\\ &=-\frac {a \left (a+b x^2\right )^9}{18 b^2}+\frac {\left (a+b x^2\right )^{10}}{20 b^2}\\ \end {align*}

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Mathematica [B]  time = 0.00, size = 106, normalized size = 3.12 \begin {gather*} \frac {a^8 x^4}{4}+\frac {4}{3} a^7 b x^6+\frac {7}{2} a^6 b^2 x^8+\frac {28}{5} a^5 b^3 x^{10}+\frac {35}{6} a^4 b^4 x^{12}+4 a^3 b^5 x^{14}+\frac {7}{4} a^2 b^6 x^{16}+\frac {4}{9} a b^7 x^{18}+\frac {b^8 x^{20}}{20} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^3*(a + b*x^2)^8,x]

[Out]

(a^8*x^4)/4 + (4*a^7*b*x^6)/3 + (7*a^6*b^2*x^8)/2 + (28*a^5*b^3*x^10)/5 + (35*a^4*b^4*x^12)/6 + 4*a^3*b^5*x^14
+ (7*a^2*b^6*x^16)/4 + (4*a*b^7*x^18)/9 + (b^8*x^20)/20

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int x^3 \left (a+b x^2\right )^8 \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[x^3*(a + b*x^2)^8,x]

[Out]

IntegrateAlgebraic[x^3*(a + b*x^2)^8, x]

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fricas [B]  time = 0.94, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{20} x^{20} b^{8} + \frac {4}{9} x^{18} b^{7} a + \frac {7}{4} x^{16} b^{6} a^{2} + 4 x^{14} b^{5} a^{3} + \frac {35}{6} x^{12} b^{4} a^{4} + \frac {28}{5} x^{10} b^{3} a^{5} + \frac {7}{2} x^{8} b^{2} a^{6} + \frac {4}{3} x^{6} b a^{7} + \frac {1}{4} x^{4} a^{8} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="fricas")

[Out]

1/20*x^20*b^8 + 4/9*x^18*b^7*a + 7/4*x^16*b^6*a^2 + 4*x^14*b^5*a^3 + 35/6*x^12*b^4*a^4 + 28/5*x^10*b^3*a^5 + 7
/2*x^8*b^2*a^6 + 4/3*x^6*b*a^7 + 1/4*x^4*a^8

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giac [B]  time = 1.18, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{20} \, b^{8} x^{20} + \frac {4}{9} \, a b^{7} x^{18} + \frac {7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {4}{3} \, a^{7} b x^{6} + \frac {1}{4} \, a^{8} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="giac")

[Out]

1/20*b^8*x^20 + 4/9*a*b^7*x^18 + 7/4*a^2*b^6*x^16 + 4*a^3*b^5*x^14 + 35/6*a^4*b^4*x^12 + 28/5*a^5*b^3*x^10 + 7
/2*a^6*b^2*x^8 + 4/3*a^7*b*x^6 + 1/4*a^8*x^4

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maple [B]  time = 0.00, size = 91, normalized size = 2.68 \begin {gather*} \frac {1}{20} b^{8} x^{20}+\frac {4}{9} a \,b^{7} x^{18}+\frac {7}{4} a^{2} b^{6} x^{16}+4 a^{3} b^{5} x^{14}+\frac {35}{6} a^{4} b^{4} x^{12}+\frac {28}{5} a^{5} b^{3} x^{10}+\frac {7}{2} a^{6} b^{2} x^{8}+\frac {4}{3} a^{7} b \,x^{6}+\frac {1}{4} a^{8} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(b*x^2+a)^8,x)

[Out]

1/20*b^8*x^20+4/9*a*b^7*x^18+7/4*a^2*b^6*x^16+4*a^3*b^5*x^14+35/6*a^4*b^4*x^12+28/5*a^5*b^3*x^10+7/2*a^6*b^2*x
^8+4/3*a^7*b*x^6+1/4*a^8*x^4

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maxima [B]  time = 1.43, size = 90, normalized size = 2.65 \begin {gather*} \frac {1}{20} \, b^{8} x^{20} + \frac {4}{9} \, a b^{7} x^{18} + \frac {7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac {35}{6} \, a^{4} b^{4} x^{12} + \frac {28}{5} \, a^{5} b^{3} x^{10} + \frac {7}{2} \, a^{6} b^{2} x^{8} + \frac {4}{3} \, a^{7} b x^{6} + \frac {1}{4} \, a^{8} x^{4} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3*(b*x^2+a)^8,x, algorithm="maxima")

[Out]

1/20*b^8*x^20 + 4/9*a*b^7*x^18 + 7/4*a^2*b^6*x^16 + 4*a^3*b^5*x^14 + 35/6*a^4*b^4*x^12 + 28/5*a^5*b^3*x^10 + 7
/2*a^6*b^2*x^8 + 4/3*a^7*b*x^6 + 1/4*a^8*x^4

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mupad [B]  time = 0.09, size = 90, normalized size = 2.65 \begin {gather*} \frac {a^8\,x^4}{4}+\frac {4\,a^7\,b\,x^6}{3}+\frac {7\,a^6\,b^2\,x^8}{2}+\frac {28\,a^5\,b^3\,x^{10}}{5}+\frac {35\,a^4\,b^4\,x^{12}}{6}+4\,a^3\,b^5\,x^{14}+\frac {7\,a^2\,b^6\,x^{16}}{4}+\frac {4\,a\,b^7\,x^{18}}{9}+\frac {b^8\,x^{20}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3*(a + b*x^2)^8,x)

[Out]

(a^8*x^4)/4 + (b^8*x^20)/20 + (4*a^7*b*x^6)/3 + (4*a*b^7*x^18)/9 + (7*a^6*b^2*x^8)/2 + (28*a^5*b^3*x^10)/5 + (
35*a^4*b^4*x^12)/6 + 4*a^3*b^5*x^14 + (7*a^2*b^6*x^16)/4

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sympy [B]  time = 0.08, size = 105, normalized size = 3.09 \begin {gather*} \frac {a^{8} x^{4}}{4} + \frac {4 a^{7} b x^{6}}{3} + \frac {7 a^{6} b^{2} x^{8}}{2} + \frac {28 a^{5} b^{3} x^{10}}{5} + \frac {35 a^{4} b^{4} x^{12}}{6} + 4 a^{3} b^{5} x^{14} + \frac {7 a^{2} b^{6} x^{16}}{4} + \frac {4 a b^{7} x^{18}}{9} + \frac {b^{8} x^{20}}{20} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3*(b*x**2+a)**8,x)

[Out]

a**8*x**4/4 + 4*a**7*b*x**6/3 + 7*a**6*b**2*x**8/2 + 28*a**5*b**3*x**10/5 + 35*a**4*b**4*x**12/6 + 4*a**3*b**5
*x**14 + 7*a**2*b**6*x**16/4 + 4*a*b**7*x**18/9 + b**8*x**20/20

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