3.138 \(\int x^3 \sqrt{a^2+2 a b x+b^2 x^2} \, dx\)

Optimal. Leaf size=71 \[ \frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

[Out]

(a*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (b*x^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(5*(a + b*x))

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Rubi [A]  time = 0.0799356, antiderivative size = 71, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.083 \[ \frac{b x^5 \sqrt{a^2+2 a b x+b^2 x^2}}{5 (a+b x)}+\frac{a x^4 \sqrt{a^2+2 a b x+b^2 x^2}}{4 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Int[x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(a*x^4*Sqrt[a^2 + 2*a*b*x + b^2*x^2])/(4*(a + b*x)) + (b*x^5*Sqrt[a^2 + 2*a*b*x
+ b^2*x^2])/(5*(a + b*x))

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Rubi in Sympy [A]  time = 8.79901, size = 58, normalized size = 0.82 \[ \frac{a x^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{20 \left (a + b x\right )} + \frac{x^{4} \sqrt{a^{2} + 2 a b x + b^{2} x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*((b*x+a)**2)**(1/2),x)

[Out]

a*x**4*sqrt(a**2 + 2*a*b*x + b**2*x**2)/(20*(a + b*x)) + x**4*sqrt(a**2 + 2*a*b*
x + b**2*x**2)/5

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Mathematica [A]  time = 0.015013, size = 33, normalized size = 0.46 \[ \frac{x^4 \sqrt{(a+b x)^2} (5 a+4 b x)}{20 (a+b x)} \]

Antiderivative was successfully verified.

[In]  Integrate[x^3*Sqrt[a^2 + 2*a*b*x + b^2*x^2],x]

[Out]

(x^4*Sqrt[(a + b*x)^2]*(5*a + 4*b*x))/(20*(a + b*x))

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Maple [A]  time = 0.005, size = 30, normalized size = 0.4 \[{\frac{{x}^{4} \left ( 4\,bx+5\,a \right ) }{20\,bx+20\,a}\sqrt{ \left ( bx+a \right ) ^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*((b*x+a)^2)^(1/2),x)

[Out]

1/20*x^4*(4*b*x+5*a)*((b*x+a)^2)^(1/2)/(b*x+a)

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.218791, size = 18, normalized size = 0.25 \[ \frac{1}{5} \, b x^{5} + \frac{1}{4} \, a x^{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5 + 1/4*a*x^4

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Sympy [A]  time = 0.142807, size = 12, normalized size = 0.17 \[ \frac{a x^{4}}{4} + \frac{b x^{5}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*((b*x+a)**2)**(1/2),x)

[Out]

a*x**4/4 + b*x**5/5

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GIAC/XCAS [A]  time = 0.208437, size = 53, normalized size = 0.75 \[ \frac{1}{5} \, b x^{5}{\rm sign}\left (b x + a\right ) + \frac{1}{4} \, a x^{4}{\rm sign}\left (b x + a\right ) - \frac{a^{5}{\rm sign}\left (b x + a\right )}{20 \, b^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/5*b*x^5*sign(b*x + a) + 1/4*a*x^4*sign(b*x + a) - 1/20*a^5*sign(b*x + a)/b^4