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

Optimal. Leaf size=37 \[ \frac{1}{4} a c x^3 \sqrt{c x^2}+\frac{1}{5} b c x^4 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0254089, antiderivative size = 37, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133 \[ \frac{1}{4} a c x^3 \sqrt{c x^2}+\frac{1}{5} b c x^4 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[(c*x^2)^(3/2)*(a + b*x),x]

[Out]

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

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Rubi in Sympy [A]  time = 5.60317, size = 32, normalized size = 0.86 \[ \frac{a c x^{3} \sqrt{c x^{2}}}{4} + \frac{b c x^{4} \sqrt{c x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00726745, size = 22, normalized size = 0.59 \[ \frac{1}{20} x \left (c x^2\right )^{3/2} (5 a+4 b x) \]

Antiderivative was successfully verified.

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

[Out]

(x*(c*x^2)^(3/2)*(5*a + 4*b*x))/20

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Maple [A]  time = 0.005, size = 19, normalized size = 0.5 \[{\frac{x \left ( 4\,bx+5\,a \right ) }{20} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)*(b*x+a),x)

[Out]

1/20*x*(4*b*x+5*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, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.204298, size = 32, normalized size = 0.86 \[ \frac{1}{20} \,{\left (4 \, b c x^{4} + 5 \, a c x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(4*b*c*x^4 + 5*a*c*x^3)*sqrt(c*x^2)

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Sympy [A]  time = 1.59594, size = 34, normalized size = 0.92 \[ \frac{a c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} + \frac{b c^{\frac{3}{2}} x^{2} \left (x^{2}\right )^{\frac{3}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.205249, size = 30, normalized size = 0.81 \[ \frac{1}{20} \,{\left (4 \, b x^{5}{\rm sign}\left (x\right ) + 5 \, a x^{4}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/20*(4*b*x^5*sign(x) + 5*a*x^4*sign(x))*c^(3/2)