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

Optimal. Leaf size=35 \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0217553, antiderivative size = 35, 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 \[ \frac{1}{2} a c x \sqrt{c x^2}+\frac{1}{3} b c x^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00410058, size = 23, normalized size = 0.66 \[ \frac{1}{6} c x \sqrt{c x^2} (3 a+2 b x) \]

Antiderivative was successfully verified.

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

[Out]

(c*x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

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Maple [A]  time = 0.003, size = 21, normalized size = 0.6 \[{\frac{2\,bx+3\,a}{6\,x} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/6/x*(2*b*x+3*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 = 0.204503, size = 30, normalized size = 0.86 \[ \frac{1}{6} \,{\left (2 \, b c x^{2} + 3 \, a c x\right )} \sqrt{c x^{2}} \]

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]

1/6*(2*b*c*x^2 + 3*a*c*x)*sqrt(c*x^2)

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Sympy [A]  time = 1.63504, size = 31, normalized size = 0.89 \[ \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} + \frac{b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{3} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.209699, size = 30, normalized size = 0.86 \[ \frac{1}{6} \,{\left (2 \, b x^{3}{\rm sign}\left (x\right ) + 3 \, a x^{2}{\rm sign}\left (x\right )\right )} c^{\frac{3}{2}} \]

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]

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