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

Optimal. Leaf size=35 \[ \frac{1}{4} a x^3 \sqrt{c x^2}+\frac{1}{5} b x^4 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0260764, antiderivative size = 35, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{1}{4} a x^3 \sqrt{c x^2}+\frac{1}{5} b x^4 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 6.72724, size = 29, normalized size = 0.83 \[ \frac{a x^{3} \sqrt{c x^{2}}}{4} + \frac{b x^{4} \sqrt{c x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00533284, size = 24, normalized size = 0.69 \[ \frac{1}{20} x^3 \sqrt{c x^2} (5 a+4 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*Sqrt[c*x^2]*(5*a + 4*b*x))/20

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Maple [A]  time = 0.006, size = 21, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 4\,bx+5\,a \right ) }{20}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*x^3*(4*b*x+5*a)*(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)*x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204281, size = 30, normalized size = 0.86 \[ \frac{1}{20} \,{\left (4 \, b x^{4} + 5 \, a x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(4*b*x^4 + 5*a*x^3)*sqrt(c*x^2)

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Sympy [A]  time = 0.922941, size = 36, normalized size = 1.03 \[ \frac{a \sqrt{c} x^{3} \sqrt{x^{2}}}{4} + \frac{b \sqrt{c} x^{4} \sqrt{x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209117, size = 30, normalized size = 0.86 \[ \frac{1}{20} \,{\left (4 \, b x^{5}{\rm sign}\left (x\right ) + 5 \, a x^{4}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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