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

Optimal. Leaf size=66 \[ -\frac{a^2}{6 c^2 x^5 \sqrt{c x^2}}-\frac{2 a b}{5 c^2 x^4 \sqrt{c x^2}}-\frac{b^2}{4 c^2 x^3 \sqrt{c x^2}} \]

[Out]

-a^2/(6*c^2*x^5*Sqrt[c*x^2]) - (2*a*b)/(5*c^2*x^4*Sqrt[c*x^2]) - b^2/(4*c^2*x^3*
Sqrt[c*x^2])

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Rubi [A]  time = 0.0365168, 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 \[ -\frac{a^2}{6 c^2 x^5 \sqrt{c x^2}}-\frac{2 a b}{5 c^2 x^4 \sqrt{c x^2}}-\frac{b^2}{4 c^2 x^3 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(6*c^2*x^5*Sqrt[c*x^2]) - (2*a*b)/(5*c^2*x^4*Sqrt[c*x^2]) - b^2/(4*c^2*x^3*
Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 16.8138, size = 63, normalized size = 0.95 \[ - \frac{a^{2} \sqrt{c x^{2}}}{6 c^{3} x^{7}} - \frac{2 a b \sqrt{c x^{2}}}{5 c^{3} x^{6}} - \frac{b^{2} \sqrt{c x^{2}}}{4 c^{3} x^{5}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(6*c**3*x**7) - 2*a*b*sqrt(c*x**2)/(5*c**3*x**6) - b**2*sqrt(
c*x**2)/(4*c**3*x**5)

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Mathematica [A]  time = 0.0179024, size = 38, normalized size = 0.58 \[ -\frac{\sqrt{c x^2} \left (10 a^2+24 a b x+15 b^2 x^2\right )}{60 c^3 x^7} \]

Antiderivative was successfully verified.

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

[Out]

-(Sqrt[c*x^2]*(10*a^2 + 24*a*b*x + 15*b^2*x^2))/(60*c^3*x^7)

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Maple [A]  time = 0.009, size = 32, normalized size = 0.5 \[ -{\frac{15\,{b}^{2}{x}^{2}+24\,abx+10\,{a}^{2}}{60\,x} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(15*b^2*x^2+24*a*b*x+10*a^2)/x/(c*x^2)^(5/2)

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Maxima [A]  time = 1.33985, size = 45, normalized size = 0.68 \[ -\frac{b^{2}}{4 \, c^{\frac{5}{2}} x^{4}} - \frac{2 \, a b}{5 \, c^{\frac{5}{2}} x^{5}} - \frac{a^{2}}{6 \, c^{\frac{5}{2}} x^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/4*b^2/(c^(5/2)*x^4) - 2/5*a*b/(c^(5/2)*x^5) - 1/6*a^2/(c^(5/2)*x^6)

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Fricas [A]  time = 0.209239, size = 46, normalized size = 0.7 \[ -\frac{{\left (15 \, b^{2} x^{2} + 24 \, a b x + 10 \, a^{2}\right )} \sqrt{c x^{2}}}{60 \, c^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/60*(15*b^2*x^2 + 24*a*b*x + 10*a^2)*sqrt(c*x^2)/(c^3*x^7)

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Sympy [A]  time = 5.13168, size = 56, normalized size = 0.85 \[ - \frac{a^{2}}{6 c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}} - \frac{2 a b}{5 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b^{2} x}{4 c^{\frac{5}{2}} \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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GIAC/XCAS [A]  time = 0.532116, size = 4, normalized size = 0.06 \[ \mathit{sage}_{0} x \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

sage0*x