3.57 \(\int x^3 \left (a+b x^2\right )^5 \, dx\)

Optimal. Leaf size=34 \[ \frac{\left (a+b x^2\right )^7}{14 b^2}-\frac{a \left (a+b x^2\right )^6}{12 b^2} \]

[Out]

-(a*(a + b*x^2)^6)/(12*b^2) + (a + b*x^2)^7/(14*b^2)

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Rubi [A]  time = 0.0983513, 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 \[ \frac{\left (a+b x^2\right )^7}{14 b^2}-\frac{a \left (a+b x^2\right )^6}{12 b^2} \]

Antiderivative was successfully verified.

[In]  Int[x^3*(a + b*x^2)^5,x]

[Out]

-(a*(a + b*x^2)^6)/(12*b^2) + (a + b*x^2)^7/(14*b^2)

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Rubi in Sympy [A]  time = 10.5751, size = 27, normalized size = 0.79 \[ - \frac{a \left (a + b x^{2}\right )^{6}}{12 b^{2}} + \frac{\left (a + b x^{2}\right )^{7}}{14 b^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x**2+a)**5,x)

[Out]

-a*(a + b*x**2)**6/(12*b**2) + (a + b*x**2)**7/(14*b**2)

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Mathematica [A]  time = 0.00341422, size = 66, normalized size = 1.94 \[ \frac{a^5 x^4}{4}+\frac{5}{6} a^4 b x^6+\frac{5}{4} a^3 b^2 x^8+a^2 b^3 x^{10}+\frac{5}{12} a b^4 x^{12}+\frac{b^5 x^{14}}{14} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*(a + b*x^2)^5,x]

[Out]

(a^5*x^4)/4 + (5*a^4*b*x^6)/6 + (5*a^3*b^2*x^8)/4 + a^2*b^3*x^10 + (5*a*b^4*x^12
)/12 + (b^5*x^14)/14

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Maple [A]  time = 0.002, size = 57, normalized size = 1.7 \[{\frac{{b}^{5}{x}^{14}}{14}}+{\frac{5\,a{b}^{4}{x}^{12}}{12}}+{a}^{2}{b}^{3}{x}^{10}+{\frac{5\,{a}^{3}{b}^{2}{x}^{8}}{4}}+{\frac{5\,{a}^{4}b{x}^{6}}{6}}+{\frac{{a}^{5}{x}^{4}}{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x^2+a)^5,x)

[Out]

1/14*b^5*x^14+5/12*a*b^4*x^12+a^2*b^3*x^10+5/4*a^3*b^2*x^8+5/6*a^4*b*x^6+1/4*a^5
*x^4

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Maxima [A]  time = 1.3438, size = 76, normalized size = 2.24 \[ \frac{1}{14} \, b^{5} x^{14} + \frac{5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^3,x, algorithm="maxima")

[Out]

1/14*b^5*x^14 + 5/12*a*b^4*x^12 + a^2*b^3*x^10 + 5/4*a^3*b^2*x^8 + 5/6*a^4*b*x^6
+ 1/4*a^5*x^4

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Fricas [A]  time = 0.194538, size = 1, normalized size = 0.03 \[ \frac{1}{14} x^{14} b^{5} + \frac{5}{12} x^{12} b^{4} a + x^{10} b^{3} a^{2} + \frac{5}{4} x^{8} b^{2} a^{3} + \frac{5}{6} x^{6} b a^{4} + \frac{1}{4} x^{4} a^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^3,x, algorithm="fricas")

[Out]

1/14*x^14*b^5 + 5/12*x^12*b^4*a + x^10*b^3*a^2 + 5/4*x^8*b^2*a^3 + 5/6*x^6*b*a^4
+ 1/4*x^4*a^5

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Sympy [A]  time = 0.119794, size = 63, normalized size = 1.85 \[ \frac{a^{5} x^{4}}{4} + \frac{5 a^{4} b x^{6}}{6} + \frac{5 a^{3} b^{2} x^{8}}{4} + a^{2} b^{3} x^{10} + \frac{5 a b^{4} x^{12}}{12} + \frac{b^{5} x^{14}}{14} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x**2+a)**5,x)

[Out]

a**5*x**4/4 + 5*a**4*b*x**6/6 + 5*a**3*b**2*x**8/4 + a**2*b**3*x**10 + 5*a*b**4*
x**12/12 + b**5*x**14/14

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GIAC/XCAS [A]  time = 0.207204, size = 76, normalized size = 2.24 \[ \frac{1}{14} \, b^{5} x^{14} + \frac{5}{12} \, a b^{4} x^{12} + a^{2} b^{3} x^{10} + \frac{5}{4} \, a^{3} b^{2} x^{8} + \frac{5}{6} \, a^{4} b x^{6} + \frac{1}{4} \, a^{5} x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^5*x^3,x, algorithm="giac")

[Out]

1/14*b^5*x^14 + 5/12*a*b^4*x^12 + a^2*b^3*x^10 + 5/4*a^3*b^2*x^8 + 5/6*a^4*b*x^6
+ 1/4*a^5*x^4