3.777 \(\int \frac{\left (c x^2\right )^{5/2} (a+b x)}{x^2} \, dx\)

Optimal. Leaf size=41 \[ \frac{1}{4} a c^2 x^3 \sqrt{c x^2}+\frac{1}{5} b c^2 x^4 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0274776, antiderivative size = 41, 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 \[ \frac{1}{4} a c^2 x^3 \sqrt{c x^2}+\frac{1}{5} b c^2 x^4 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[((c*x^2)^(5/2)*(a + b*x))/x^2,x]

[Out]

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

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Rubi in Sympy [A]  time = 9.07657, size = 36, normalized size = 0.88 \[ \frac{a c^{2} x^{3} \sqrt{c x^{2}}}{4} + \frac{b c^{2} x^{4} \sqrt{c x^{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(5/2)*(b*x+a)/x**2,x)

[Out]

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

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Mathematica [A]  time = 0.00702267, size = 23, normalized size = 0.56 \[ \frac{1}{20} c x \left (c x^2\right )^{3/2} (5 a+4 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x^2)^(5/2)*(a + b*x))/x^2,x]

[Out]

(c*x*(c*x^2)^(3/2)*(5*a + 4*b*x))/20

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Maple [A]  time = 0.005, size = 21, normalized size = 0.5 \[{\frac{4\,bx+5\,a}{20\,x} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a)/x^2,x)

[Out]

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

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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)^(5/2)*(b*x + a)/x^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.206782, size = 38, normalized size = 0.93 \[ \frac{1}{20} \,{\left (4 \, b c^{2} x^{4} + 5 \, a c^{2} x^{3}\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)/x^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.11859, size = 31, normalized size = 0.76 \[ \frac{a c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{4 x} + \frac{b c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{5} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(5/2)*(b*x+a)/x**2,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.205322, size = 38, normalized size = 0.93 \[ \frac{1}{20} \,{\left (4 \, b c^{2} x^{5}{\rm sign}\left (x\right ) + 5 \, a c^{2} x^{4}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)/x^2,x, algorithm="giac")

[Out]

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