3.90 \(\int x^3 \left (a+b x^2\right )^8 \, dx\)

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

[Out]

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

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Rubi [A]  time = 0.118407, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{\left (a+b x^2\right )^{10}}{20 b^2}-\frac{a \left (a+b x^2\right )^9}{18 b^2} \]

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)

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Rubi in Sympy [A]  time = 13.6076, size = 27, normalized size = 0.79 \[ - \frac{a \left (a + b x^{2}\right )^{9}}{18 b^{2}} + \frac{\left (a + b x^{2}\right )^{10}}{20 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**8,x)

[Out]

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

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Mathematica [B]  time = 0.00426665, size = 106, normalized size = 3.12 \[ \frac{a^8 x^4}{4}+\frac{4}{3} a^7 b x^6+\frac{7}{2} a^6 b^2 x^8+\frac{28}{5} a^5 b^3 x^{10}+\frac{35}{6} a^4 b^4 x^{12}+4 a^3 b^5 x^{14}+\frac{7}{4} a^2 b^6 x^{16}+\frac{4}{9} a b^7 x^{18}+\frac{b^8 x^{20}}{20} \]

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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Maple [B]  time = 0.003, size = 91, normalized size = 2.7 \[{\frac{{b}^{8}{x}^{20}}{20}}+{\frac{4\,a{b}^{7}{x}^{18}}{9}}+{\frac{7\,{a}^{2}{b}^{6}{x}^{16}}{4}}+4\,{a}^{3}{b}^{5}{x}^{14}+{\frac{35\,{a}^{4}{b}^{4}{x}^{12}}{6}}+{\frac{28\,{a}^{5}{b}^{3}{x}^{10}}{5}}+{\frac{7\,{a}^{6}{b}^{2}{x}^{8}}{2}}+{\frac{4\,{a}^{7}b{x}^{6}}{3}}+{\frac{{a}^{8}{x}^{4}}{4}} \]

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+2
8/5*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 [A]  time = 1.34712, size = 122, normalized size = 3.59 \[ \frac{1}{20} \, b^{8} x^{20} + \frac{4}{9} \, a b^{7} x^{18} + \frac{7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{4}{3} \, a^{7} b x^{6} + \frac{1}{4} \, a^{8} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^3,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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Fricas [A]  time = 0.175892, size = 1, normalized size = 0.03 \[ \frac{1}{20} x^{20} b^{8} + \frac{4}{9} x^{18} b^{7} a + \frac{7}{4} x^{16} b^{6} a^{2} + 4 x^{14} b^{5} a^{3} + \frac{35}{6} x^{12} b^{4} a^{4} + \frac{28}{5} x^{10} b^{3} a^{5} + \frac{7}{2} x^{8} b^{2} a^{6} + \frac{4}{3} x^{6} b a^{7} + \frac{1}{4} x^{4} a^{8} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^3,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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Sympy [A]  time = 0.154378, size = 105, normalized size = 3.09 \[ \frac{a^{8} x^{4}}{4} + \frac{4 a^{7} b x^{6}}{3} + \frac{7 a^{6} b^{2} x^{8}}{2} + \frac{28 a^{5} b^{3} x^{10}}{5} + \frac{35 a^{4} b^{4} x^{12}}{6} + 4 a^{3} b^{5} x^{14} + \frac{7 a^{2} b^{6} x^{16}}{4} + \frac{4 a b^{7} x^{18}}{9} + \frac{b^{8} x^{20}}{20} \]

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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GIAC/XCAS [A]  time = 0.207723, size = 122, normalized size = 3.59 \[ \frac{1}{20} \, b^{8} x^{20} + \frac{4}{9} \, a b^{7} x^{18} + \frac{7}{4} \, a^{2} b^{6} x^{16} + 4 \, a^{3} b^{5} x^{14} + \frac{35}{6} \, a^{4} b^{4} x^{12} + \frac{28}{5} \, a^{5} b^{3} x^{10} + \frac{7}{2} \, a^{6} b^{2} x^{8} + \frac{4}{3} \, a^{7} b x^{6} + \frac{1}{4} \, a^{8} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^8*x^3,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