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

Optimal. Leaf size=37 \[ \frac{1}{7} a c x^6 \sqrt{c x^2}+\frac{1}{8} b c x^7 \sqrt{c x^2} \]

[Out]

(a*c*x^6*Sqrt[c*x^2])/7 + (b*c*x^7*Sqrt[c*x^2])/8

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Rubi [A]  time = 0.0339374, 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 \[ \frac{1}{7} a c x^6 \sqrt{c x^2}+\frac{1}{8} b c x^7 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

(a*c*x^6*Sqrt[c*x^2])/7 + (b*c*x^7*Sqrt[c*x^2])/8

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Rubi in Sympy [A]  time = 9.19798, size = 32, normalized size = 0.86 \[ \frac{a c x^{6} \sqrt{c x^{2}}}{7} + \frac{b c x^{7} \sqrt{c x^{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c*x**6*sqrt(c*x**2)/7 + b*c*x**7*sqrt(c*x**2)/8

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Mathematica [A]  time = 0.00918447, size = 24, normalized size = 0.65 \[ \frac{1}{56} x^4 \left (c x^2\right )^{3/2} (8 a+7 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(c*x^2)^(3/2)*(8*a + 7*b*x))/56

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Maple [A]  time = 0.004, size = 21, normalized size = 0.6 \[{\frac{{x}^{4} \left ( 7\,bx+8\,a \right ) }{56} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*x^4*(7*b*x+8*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^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.210216, size = 32, normalized size = 0.86 \[ \frac{1}{56} \,{\left (7 \, b c x^{7} + 8 \, a c x^{6}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*(7*b*c*x^7 + 8*a*c*x^6)*sqrt(c*x^2)

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Sympy [A]  time = 4.01013, size = 36, normalized size = 0.97 \[ \frac{a c^{\frac{3}{2}} x^{4} \left (x^{2}\right )^{\frac{3}{2}}}{7} + \frac{b c^{\frac{3}{2}} x^{5} \left (x^{2}\right )^{\frac{3}{2}}}{8} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

a*c**(3/2)*x**4*(x**2)**(3/2)/7 + b*c**(3/2)*x**5*(x**2)**(3/2)/8

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GIAC/XCAS [A]  time = 0.20359, size = 30, normalized size = 0.81 \[ \frac{1}{56} \,{\left (7 \, b x^{8}{\rm sign}\left (x\right ) + 8 \, a x^{7}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/56*(7*b*x^8*sign(x) + 8*a*x^7*sign(x))*c^(3/2)