3.780 \(\int \frac{x^3 (a+b x)}{\sqrt{c x^2}} \, dx\)

Optimal. Leaf size=35 \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0236339, 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 \[ \frac{a x^4}{3 \sqrt{c x^2}}+\frac{b x^5}{4 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 9.6738, size = 32, normalized size = 0.91 \[ \frac{a x^{2} \sqrt{c x^{2}}}{3 c} + \frac{b x^{3} \sqrt{c x^{2}}}{4 c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**3*(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

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Mathematica [A]  time = 0.00748888, size = 24, normalized size = 0.69 \[ \frac{x^4 (4 a+3 b x)}{12 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

(x^4*(4*a + 3*b*x))/(12*Sqrt[c*x^2])

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Maple [A]  time = 0.005, size = 21, normalized size = 0.6 \[{\frac{{x}^{4} \left ( 3\,bx+4\,a \right ) }{12}{\frac{1}{\sqrt{c{x}^{2}}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^3*(b*x+a)/(c*x^2)^(1/2),x)

[Out]

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

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Maxima [A]  time = 1.35301, size = 45, normalized size = 1.29 \[ \frac{\sqrt{c x^{2}} b x^{3}}{4 \, c} + \frac{\sqrt{c x^{2}} a x^{2}}{3 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^3/sqrt(c*x^2),x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 0.203903, size = 34, normalized size = 0.97 \[ \frac{{\left (3 \, b x^{3} + 4 \, a x^{2}\right )} \sqrt{c x^{2}}}{12 \, c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^3/sqrt(c*x^2),x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 2.16656, size = 36, normalized size = 1.03 \[ \frac{a x^{4}}{3 \sqrt{c} \sqrt{x^{2}}} + \frac{b x^{5}}{4 \sqrt{c} \sqrt{x^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**3*(b*x+a)/(c*x**2)**(1/2),x)

[Out]

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

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GIAC/XCAS [A]  time = 0.211573, size = 35, normalized size = 1. \[ \frac{1}{12} \, \sqrt{c x^{2}}{\left (\frac{3 \, b x}{c} + \frac{4 \, a}{c}\right )} x^{2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x + a)*x^3/sqrt(c*x^2),x, algorithm="giac")

[Out]

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