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

Optimal. Leaf size=57 \[ \frac{1}{5} a^2 x^4 \sqrt{c x^2}+\frac{1}{3} a b x^5 \sqrt{c x^2}+\frac{1}{7} b^2 x^6 \sqrt{c x^2} \]

[Out]

(a^2*x^4*Sqrt[c*x^2])/5 + (a*b*x^5*Sqrt[c*x^2])/3 + (b^2*x^6*Sqrt[c*x^2])/7

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Rubi [A]  time = 0.0418007, 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 \[ \frac{1}{5} a^2 x^4 \sqrt{c x^2}+\frac{1}{3} a b x^5 \sqrt{c x^2}+\frac{1}{7} b^2 x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a^2*x^4*Sqrt[c*x^2])/5 + (a*b*x^5*Sqrt[c*x^2])/3 + (b^2*x^6*Sqrt[c*x^2])/7

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Rubi in Sympy [A]  time = 17.7589, size = 49, normalized size = 0.86 \[ \frac{a^{2} x^{4} \sqrt{c x^{2}}}{5} + \frac{a b x^{5} \sqrt{c x^{2}}}{3} + \frac{b^{2} x^{6} \sqrt{c x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*x**4*sqrt(c*x**2)/5 + a*b*x**5*sqrt(c*x**2)/3 + b**2*x**6*sqrt(c*x**2)/7

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Mathematica [A]  time = 0.00886833, size = 35, normalized size = 0.61 \[ \frac{1}{105} x^4 \sqrt{c x^2} \left (21 a^2+35 a b x+15 b^2 x^2\right ) \]

Antiderivative was successfully verified.

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

[Out]

(x^4*Sqrt[c*x^2]*(21*a^2 + 35*a*b*x + 15*b^2*x^2))/105

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Maple [A]  time = 0.007, size = 32, normalized size = 0.6 \[{\frac{{x}^{4} \left ( 15\,{b}^{2}{x}^{2}+35\,abx+21\,{a}^{2} \right ) }{105}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)^2*(c*x^2)^(1/2),x)

[Out]

1/105*x^4*(15*b^2*x^2+35*a*b*x+21*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.199967, size = 45, normalized size = 0.79 \[ \frac{1}{105} \,{\left (15 \, b^{2} x^{6} + 35 \, a b x^{5} + 21 \, a^{2} x^{4}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*b^2*x^6 + 35*a*b*x^5 + 21*a^2*x^4)*sqrt(c*x^2)

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Sympy [A]  time = 1.77094, size = 60, normalized size = 1.05 \[ \frac{a^{2} \sqrt{c} x^{4} \sqrt{x^{2}}}{5} + \frac{a b \sqrt{c} x^{5} \sqrt{x^{2}}}{3} + \frac{b^{2} \sqrt{c} x^{6} \sqrt{x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a**2*sqrt(c)*x**4*sqrt(x**2)/5 + a*b*sqrt(c)*x**5*sqrt(x**2)/3 + b**2*sqrt(c)*x*
*6*sqrt(x**2)/7

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GIAC/XCAS [A]  time = 0.207127, size = 47, normalized size = 0.82 \[ \frac{1}{105} \,{\left (15 \, b^{2} x^{7}{\rm sign}\left (x\right ) + 35 \, a b x^{6}{\rm sign}\left (x\right ) + 21 \, a^{2} x^{5}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/105*(15*b^2*x^7*sign(x) + 35*a*b*x^6*sign(x) + 21*a^2*x^5*sign(x))*sqrt(c)