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

Optimal. Leaf size=29 \[ a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0133267, antiderivative size = 29, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 18, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.056 \[ a c \sqrt{c x^2}+\frac{1}{2} b c x \sqrt{c x^2} \]

Antiderivative was successfully verified.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.00454536, size = 21, normalized size = 0.72 \[ \frac{1}{2} c \sqrt{c x^2} (2 a+b x) \]

Antiderivative was successfully verified.

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

[Out]

(c*Sqrt[c*x^2]*(2*a + b*x))/2

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Maple [A]  time = 0.003, size = 20, normalized size = 0.7 \[{\frac{bx+2\,a}{2\,{x}^{2}} \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: RuntimeError} \]

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: RuntimeError

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Fricas [A]  time = 0.202718, size = 24, normalized size = 0.83 \[ \frac{1}{2} \,{\left (b c x + 2 \, a c\right )} \sqrt{c x^{2}} \]

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]

1/2*(b*c*x + 2*a*c)*sqrt(c*x^2)

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Sympy [A]  time = 2.41714, size = 32, normalized size = 1.1 \[ \frac{a c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{x^{2}} + \frac{b c^{\frac{3}{2}} \left (x^{2}\right )^{\frac{3}{2}}}{2 x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.208048, size = 23, normalized size = 0.79 \[ \frac{1}{2} \,{\left (b x^{2} + 2 \, a x\right )} c^{\frac{3}{2}}{\rm sign}\left (x\right ) \]

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]

1/2*(b*x^2 + 2*a*x)*c^(3/2)*sign(x)