3.805 \(\int x^2 \sqrt{c x^2} (a+b x)^2 \, dx\)

Optimal. Leaf size=57 \[ \frac{1}{4} a^2 x^3 \sqrt{c x^2}+\frac{2}{5} a b x^4 \sqrt{c x^2}+\frac{1}{6} b^2 x^5 \sqrt{c x^2} \]

[Out]

(a^2*x^3*Sqrt[c*x^2])/4 + (2*a*b*x^4*Sqrt[c*x^2])/5 + (b^2*x^5*Sqrt[c*x^2])/6

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Rubi [A]  time = 0.0398942, antiderivative size = 57, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{1}{4} a^2 x^3 \sqrt{c x^2}+\frac{2}{5} a b x^4 \sqrt{c x^2}+\frac{1}{6} b^2 x^5 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[x^2*Sqrt[c*x^2]*(a + b*x)^2,x]

[Out]

(a^2*x^3*Sqrt[c*x^2])/4 + (2*a*b*x^4*Sqrt[c*x^2])/5 + (b^2*x^5*Sqrt[c*x^2])/6

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Rubi in Sympy [A]  time = 12.6069, size = 51, normalized size = 0.89 \[ \frac{a^{2} x^{3} \sqrt{c x^{2}}}{4} + \frac{2 a b x^{4} \sqrt{c x^{2}}}{5} + \frac{b^{2} x^{5} \sqrt{c x^{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**3*sqrt(c*x**2)/4 + 2*a*b*x**4*sqrt(c*x**2)/5 + b**2*x**5*sqrt(c*x**2)/6

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Mathematica [A]  time = 0.00858386, size = 35, normalized size = 0.61 \[ \frac{1}{60} x^3 \sqrt{c x^2} \left (15 a^2+24 a b x+10 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[c*x^2]*(15*a^2 + 24*a*b*x + 10*b^2*x^2))/60

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Maple [A]  time = 0.006, size = 32, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 10\,{b}^{2}{x}^{2}+24\,abx+15\,{a}^{2} \right ) }{60}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2*(b*x+a)^2*(c*x^2)^(1/2),x)

[Out]

1/60*x^3*(10*b^2*x^2+24*a*b*x+15*a^2)*(c*x^2)^(1/2)

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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(c*x^2)*(b*x + a)^2*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.194943, size = 45, normalized size = 0.79 \[ \frac{1}{60} \,{\left (10 \, b^{2} x^{5} + 24 \, a b x^{4} + 15 \, a^{2} x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*b^2*x^5 + 24*a*b*x^4 + 15*a^2*x^3)*sqrt(c*x^2)

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Sympy [A]  time = 1.38453, size = 61, normalized size = 1.07 \[ \frac{a^{2} \sqrt{c} x^{3} \sqrt{x^{2}}}{4} + \frac{2 a b \sqrt{c} x^{4} \sqrt{x^{2}}}{5} + \frac{b^{2} \sqrt{c} x^{5} \sqrt{x^{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*x**3*sqrt(x**2)/4 + 2*a*b*sqrt(c)*x**4*sqrt(x**2)/5 + b**2*sqrt(c)*
x**5*sqrt(x**2)/6

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GIAC/XCAS [A]  time = 0.20649, size = 47, normalized size = 0.82 \[ \frac{1}{60} \,{\left (10 \, b^{2} x^{6}{\rm sign}\left (x\right ) + 24 \, a b x^{5}{\rm sign}\left (x\right ) + 15 \, a^{2} x^{4}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/60*(10*b^2*x^6*sign(x) + 24*a*b*x^5*sign(x) + 15*a^2*x^4*sign(x))*sqrt(c)