3.766 \(\int x \left (c x^2\right )^{3/2} (a+b x) \, dx\)

Optimal. Leaf size=37 \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0273144, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 16, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.125 \[ \frac{1}{5} a c x^4 \sqrt{c x^2}+\frac{1}{6} b c x^5 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int x \left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(x*(c*x**2)**(3/2)*(a + b*x), x)

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Mathematica [A]  time = 0.0075964, size = 24, normalized size = 0.65 \[ \frac{1}{30} x^2 \left (c x^2\right )^{3/2} (6 a+5 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^2*(c*x^2)^(3/2)*(6*a + 5*b*x))/30

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{2} \left ( 5\,bx+6\,a \right ) }{30} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x*(c*x^2)^(3/2)*(b*x+a),x)

[Out]

1/30*x^2*(5*b*x+6*a)*(c*x^2)^(3/2)

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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((c*x^2)^(3/2)*(b*x + a)*x,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.207279, size = 32, normalized size = 0.86 \[ \frac{1}{30} \,{\left (5 \, b c x^{5} + 6 \, a c x^{4}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(5*b*c*x^5 + 6*a*c*x^4)*sqrt(c*x^2)

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Sympy [A]  time = 2.13254, size = 36, normalized size = 0.97 \[ \frac{a c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} + \frac{b c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.20631, size = 30, normalized size = 0.81 \[ \frac{1}{30} \,{\left (5 \, b x^{6}{\rm sign}\left (x\right ) + 6 \, a x^{5}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/30*(5*b*x^6*sign(x) + 6*a*x^5*sign(x))*c^(3/2)