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

Optimal. Leaf size=37 \[ \frac{1}{6} a c x^5 \sqrt{c x^2}+\frac{1}{7} b c x^6 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0316252, antiderivative size = 37, 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 \[ \frac{1}{6} a c x^5 \sqrt{c x^2}+\frac{1}{7} b c x^6 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 7.28452, size = 32, normalized size = 0.86 \[ \frac{a c x^{5} \sqrt{c x^{2}}}{6} + \frac{b c x^{6} \sqrt{c x^{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00787798, size = 24, normalized size = 0.65 \[ \frac{1}{42} x^3 \left (c x^2\right )^{3/2} (7 a+6 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^3*(c*x^2)^(3/2)*(7*a + 6*b*x))/42

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{3} \left ( 6\,bx+7\,a \right ) }{42} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*x^3*(6*b*x+7*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^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.202245, size = 32, normalized size = 0.86 \[ \frac{1}{42} \,{\left (6 \, b c x^{6} + 7 \, a c x^{5}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*(6*b*c*x^6 + 7*a*c*x^5)*sqrt(c*x^2)

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Sympy [A]  time = 3.05869, size = 36, normalized size = 0.97 \[ \frac{a c^{\frac{3}{2}} x^{3} \left (x^{2}\right )^{\frac{3}{2}}}{6} + \frac{b c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209427, size = 30, normalized size = 0.81 \[ \frac{1}{42} \,{\left (6 \, b x^{7}{\rm sign}\left (x\right ) + 7 \, a x^{6}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/42*(6*b*x^7*sign(x) + 7*a*x^6*sign(x))*c^(3/2)