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

Optimal. Leaf size=66 \[ -\frac{a^2}{5 c^2 x^4 \sqrt{c x^2}}-\frac{a b}{2 c^2 x^3 \sqrt{c x^2}}-\frac{b^2}{3 c^2 x^2 \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0363574, antiderivative size = 66, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ -\frac{a^2}{5 c^2 x^4 \sqrt{c x^2}}-\frac{a b}{2 c^2 x^3 \sqrt{c x^2}}-\frac{b^2}{3 c^2 x^2 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 17.0638, size = 61, normalized size = 0.92 \[ - \frac{a^{2} \sqrt{c x^{2}}}{5 c^{3} x^{6}} - \frac{a b \sqrt{c x^{2}}}{2 c^{3} x^{5}} - \frac{b^{2} \sqrt{c x^{2}}}{3 c^{3} x^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(5*c**3*x**6) - a*b*sqrt(c*x**2)/(2*c**3*x**5) - b**2*sqrt(c*
x**2)/(3*c**3*x**4)

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Mathematica [A]  time = 0.0221972, size = 38, normalized size = 0.58 \[ -\frac{\sqrt{c x^2} \left (6 a^2+15 a b x+10 b^2 x^2\right )}{30 c^3 x^6} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(6*a^2 + 15*a*b*x + 10*b^2*x^2))/(30*c^3*x^6)

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Maple [A]  time = 0.006, size = 29, normalized size = 0.4 \[ -{\frac{10\,{b}^{2}{x}^{2}+15\,abx+6\,{a}^{2}}{30} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*x^2+15*a*b*x+6*a^2)/(c*x^2)^(5/2)

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Maxima [A]  time = 1.32071, size = 50, normalized size = 0.76 \[ -\frac{b^{2}}{3 \, \left (c x^{2}\right )^{\frac{3}{2}} c} - \frac{a b}{2 \, c^{\frac{5}{2}} x^{4}} - \frac{a^{2}}{5 \, c^{\frac{5}{2}} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/3*b^2/((c*x^2)^(3/2)*c) - 1/2*a*b/(c^(5/2)*x^4) - 1/5*a^2/(c^(5/2)*x^5)

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Fricas [A]  time = 0.20799, size = 46, normalized size = 0.7 \[ -\frac{{\left (10 \, b^{2} x^{2} + 15 \, a b x + 6 \, a^{2}\right )} \sqrt{c x^{2}}}{30 \, c^{3} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/30*(10*b^2*x^2 + 15*a*b*x + 6*a^2)*sqrt(c*x^2)/(c^3*x^6)

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Sympy [A]  time = 3.97685, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{a b x}{2 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b^{2} x^{2}}{3 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2}}{\left (c x^{2}\right )^{\frac{5}{2}} x}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

integrate((b*x + a)^2/((c*x^2)^(5/2)*x), x)