3.2118 \(\int \left (a+b \sqrt{x}\right )^2 x^4 \, dx\)

Optimal. Leaf size=32 \[ \frac{a^2 x^5}{5}+\frac{4}{11} a b x^{11/2}+\frac{b^2 x^6}{6} \]

[Out]

(a^2*x^5)/5 + (4*a*b*x^(11/2))/11 + (b^2*x^6)/6

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Rubi [A]  time = 0.0716848, antiderivative size = 32, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{a^2 x^5}{5}+\frac{4}{11} a b x^{11/2}+\frac{b^2 x^6}{6} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*Sqrt[x])^2*x^4,x]

[Out]

(a^2*x^5)/5 + (4*a*b*x^(11/2))/11 + (b^2*x^6)/6

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Rubi in Sympy [A]  time = 9.49705, size = 27, normalized size = 0.84 \[ \frac{a^{2} x^{5}}{5} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**4*(a+b*x**(1/2))**2,x)

[Out]

a**2*x**5/5 + 4*a*b*x**(11/2)/11 + b**2*x**6/6

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Mathematica [A]  time = 0.0130854, size = 28, normalized size = 0.88 \[ \frac{1}{330} x^5 \left (66 a^2+120 a b \sqrt{x}+55 b^2 x\right ) \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*Sqrt[x])^2*x^4,x]

[Out]

(x^5*(66*a^2 + 120*a*b*Sqrt[x] + 55*b^2*x))/330

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Maple [A]  time = 0.002, size = 25, normalized size = 0.8 \[{\frac{{x}^{5}{a}^{2}}{5}}+{\frac{4\,ab}{11}{x}^{{\frac{11}{2}}}}+{\frac{{b}^{2}{x}^{6}}{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^4*(a+b*x^(1/2))^2,x)

[Out]

1/5*x^5*a^2+4/11*a*b*x^(11/2)+1/6*b^2*x^6

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Maxima [A]  time = 1.45135, size = 224, normalized size = 7. \[ \frac{{\left (b \sqrt{x} + a\right )}^{12}}{6 \, b^{10}} - \frac{18 \,{\left (b \sqrt{x} + a\right )}^{11} a}{11 \, b^{10}} + \frac{36 \,{\left (b \sqrt{x} + a\right )}^{10} a^{2}}{5 \, b^{10}} - \frac{56 \,{\left (b \sqrt{x} + a\right )}^{9} a^{3}}{3 \, b^{10}} + \frac{63 \,{\left (b \sqrt{x} + a\right )}^{8} a^{4}}{2 \, b^{10}} - \frac{36 \,{\left (b \sqrt{x} + a\right )}^{7} a^{5}}{b^{10}} + \frac{28 \,{\left (b \sqrt{x} + a\right )}^{6} a^{6}}{b^{10}} - \frac{72 \,{\left (b \sqrt{x} + a\right )}^{5} a^{7}}{5 \, b^{10}} + \frac{9 \,{\left (b \sqrt{x} + a\right )}^{4} a^{8}}{2 \, b^{10}} - \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3} a^{9}}{3 \, b^{10}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2*x^4,x, algorithm="maxima")

[Out]

1/6*(b*sqrt(x) + a)^12/b^10 - 18/11*(b*sqrt(x) + a)^11*a/b^10 + 36/5*(b*sqrt(x)
+ a)^10*a^2/b^10 - 56/3*(b*sqrt(x) + a)^9*a^3/b^10 + 63/2*(b*sqrt(x) + a)^8*a^4/
b^10 - 36*(b*sqrt(x) + a)^7*a^5/b^10 + 28*(b*sqrt(x) + a)^6*a^6/b^10 - 72/5*(b*s
qrt(x) + a)^5*a^7/b^10 + 9/2*(b*sqrt(x) + a)^4*a^8/b^10 - 2/3*(b*sqrt(x) + a)^3*
a^9/b^10

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Fricas [A]  time = 0.232691, size = 32, normalized size = 1. \[ \frac{1}{6} \, b^{2} x^{6} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2*x^4,x, algorithm="fricas")

[Out]

1/6*b^2*x^6 + 4/11*a*b*x^(11/2) + 1/5*a^2*x^5

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Sympy [A]  time = 4.02337, size = 27, normalized size = 0.84 \[ \frac{a^{2} x^{5}}{5} + \frac{4 a b x^{\frac{11}{2}}}{11} + \frac{b^{2} x^{6}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**4*(a+b*x**(1/2))**2,x)

[Out]

a**2*x**5/5 + 4*a*b*x**(11/2)/11 + b**2*x**6/6

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GIAC/XCAS [A]  time = 0.216225, size = 32, normalized size = 1. \[ \frac{1}{6} \, b^{2} x^{6} + \frac{4}{11} \, a b x^{\frac{11}{2}} + \frac{1}{5} \, a^{2} x^{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*sqrt(x) + a)^2*x^4,x, algorithm="giac")

[Out]

1/6*b^2*x^6 + 4/11*a*b*x^(11/2) + 1/5*a^2*x^5