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

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

[Out]

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

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Rubi [A]  time = 0.024273, 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 \[ \frac{1}{3} a c x^2 \sqrt{c x^2}+\frac{1}{4} b c x^3 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 8.47228, size = 32, normalized size = 0.86 \[ \frac{a c x^{2} \sqrt{c x^{2}}}{3} + \frac{b c x^{3} \sqrt{c x^{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.002988, size = 25, normalized size = 0.68 \[ \frac{1}{12} c x^2 \sqrt{c x^2} (4 a+3 b x) \]

Antiderivative was successfully verified.

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

[Out]

(c*x^2*Sqrt[c*x^2]*(4*a + 3*b*x))/12

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Maple [A]  time = 0.004, size = 18, normalized size = 0.5 \[{\frac{3\,bx+4\,a}{12} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(3/2)*(b*x+a)/x,x)

[Out]

1/12*(3*b*x+4*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 = 0.205819, size = 32, normalized size = 0.86 \[ \frac{1}{12} \,{\left (3 \, b c x^{3} + 4 \, a c x^{2}\right )} \sqrt{c x^{2}} \]

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]

1/12*(3*b*c*x^3 + 4*a*c*x^2)*sqrt(c*x^2)

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Sympy [A]  time = 1.55931, size = 31, normalized size = 0.84 \[ \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} + \frac{b c^{\frac{3}{2}} x \left (x^{2}\right )^{\frac{3}{2}}}{4} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.20843, size = 30, normalized size = 0.81 \[ \frac{1}{12} \,{\left (3 \, b x^{4}{\rm sign}\left (x\right ) + 4 \, a x^{3}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

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]

1/12*(3*b*x^4*sign(x) + 4*a*x^3*sign(x))*c^(3/2)