3.779 \(\int \frac{\left (c x^2\right )^{5/2} (a+b x)}{x^4} \, dx\)

Optimal. Leaf size=39 \[ \frac{1}{2} a c^2 x \sqrt{c x^2}+\frac{1}{3} b c^2 x^2 \sqrt{c x^2} \]

[Out]

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

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Rubi [A]  time = 0.0228237, antiderivative size = 39, 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 \[ \frac{1}{2} a c^2 x \sqrt{c x^2}+\frac{1}{3} b c^2 x^2 \sqrt{c x^2} \]

Antiderivative was successfully verified.

[In]  Int[((c*x^2)^(5/2)*(a + b*x))/x^4,x]

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(5/2)*(b*x+a)/x**4,x)

[Out]

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

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Mathematica [A]  time = 0.00357037, size = 25, normalized size = 0.64 \[ \frac{1}{6} c^2 x \sqrt{c x^2} (3 a+2 b x) \]

Antiderivative was successfully verified.

[In]  Integrate[((c*x^2)^(5/2)*(a + b*x))/x^4,x]

[Out]

(c^2*x*Sqrt[c*x^2]*(3*a + 2*b*x))/6

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Maple [A]  time = 0.005, size = 21, normalized size = 0.5 \[{\frac{2\,bx+3\,a}{6\,{x}^{3}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(5/2)*(b*x+a)/x^4,x)

[Out]

1/6/x^3*(2*b*x+3*a)*(c*x^2)^(5/2)

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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)^(5/2)*(b*x + a)/x^4,x, algorithm="maxima")

[Out]

Exception raised: RuntimeError

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Fricas [A]  time = 0.20597, size = 35, normalized size = 0.9 \[ \frac{1}{6} \,{\left (2 \, b c^{2} x^{2} + 3 \, a c^{2} x\right )} \sqrt{c x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)/x^4,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 5.98195, size = 36, normalized size = 0.92 \[ \frac{a c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{2 x^{3}} + \frac{b c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}}{3 x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(5/2)*(b*x+a)/x**4,x)

[Out]

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

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GIAC/XCAS [A]  time = 0.20626, size = 38, normalized size = 0.97 \[ \frac{1}{6} \,{\left (2 \, b c^{2} x^{3}{\rm sign}\left (x\right ) + 3 \, a c^{2} x^{2}{\rm sign}\left (x\right )\right )} \sqrt{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x^2)^(5/2)*(b*x + a)/x^4,x, algorithm="giac")

[Out]

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