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

Optimal. Leaf size=66 \[ -\frac{a^2}{8 c^2 x^7 \sqrt{c x^2}}-\frac{2 a b}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b^2}{6 c^2 x^5 \sqrt{c x^2}} \]

[Out]

-a^2/(8*c^2*x^7*Sqrt[c*x^2]) - (2*a*b)/(7*c^2*x^6*Sqrt[c*x^2]) - b^2/(6*c^2*x^5*
Sqrt[c*x^2])

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Rubi [A]  time = 0.0375986, 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 \[ -\frac{a^2}{8 c^2 x^7 \sqrt{c x^2}}-\frac{2 a b}{7 c^2 x^6 \sqrt{c x^2}}-\frac{b^2}{6 c^2 x^5 \sqrt{c x^2}} \]

Antiderivative was successfully verified.

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

[Out]

-a^2/(8*c^2*x^7*Sqrt[c*x^2]) - (2*a*b)/(7*c^2*x^6*Sqrt[c*x^2]) - b^2/(6*c^2*x^5*
Sqrt[c*x^2])

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Rubi in Sympy [A]  time = 16.9921, size = 63, normalized size = 0.95 \[ - \frac{a^{2} \sqrt{c x^{2}}}{8 c^{3} x^{9}} - \frac{2 a b \sqrt{c x^{2}}}{7 c^{3} x^{8}} - \frac{b^{2} \sqrt{c x^{2}}}{6 c^{3} x^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2*sqrt(c*x**2)/(8*c**3*x**9) - 2*a*b*sqrt(c*x**2)/(7*c**3*x**8) - b**2*sqrt(
c*x**2)/(6*c**3*x**7)

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Mathematica [A]  time = 0.0162651, size = 35, normalized size = 0.53 \[ \frac{-21 a^2-48 a b x-28 b^2 x^2}{168 x^3 \left (c x^2\right )^{5/2}} \]

Antiderivative was successfully verified.

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

[Out]

(-21*a^2 - 48*a*b*x - 28*b^2*x^2)/(168*x^3*(c*x^2)^(5/2))

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Maple [A]  time = 0.008, size = 32, normalized size = 0.5 \[ -{\frac{28\,{b}^{2}{x}^{2}+48\,abx+21\,{a}^{2}}{168\,{x}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(28*b^2*x^2+48*a*b*x+21*a^2)/x^3/(c*x^2)^(5/2)

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Maxima [A]  time = 1.33033, size = 45, normalized size = 0.68 \[ -\frac{b^{2}}{6 \, c^{\frac{5}{2}} x^{6}} - \frac{2 \, a b}{7 \, c^{\frac{5}{2}} x^{7}} - \frac{a^{2}}{8 \, c^{\frac{5}{2}} x^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/6*b^2/(c^(5/2)*x^6) - 2/7*a*b/(c^(5/2)*x^7) - 1/8*a^2/(c^(5/2)*x^8)

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Fricas [A]  time = 0.208186, size = 46, normalized size = 0.7 \[ -\frac{{\left (28 \, b^{2} x^{2} + 48 \, a b x + 21 \, a^{2}\right )} \sqrt{c x^{2}}}{168 \, c^{3} x^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-1/168*(28*b^2*x^2 + 48*a*b*x + 21*a^2)*sqrt(c*x^2)/(c^3*x^9)

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Sympy [A]  time = 7.0458, size = 61, normalized size = 0.92 \[ - \frac{a^{2}}{8 c^{\frac{5}{2}} x^{3} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{2 a b}{7 c^{\frac{5}{2}} x^{2} \left (x^{2}\right )^{\frac{5}{2}}} - \frac{b^{2}}{6 c^{\frac{5}{2}} x \left (x^{2}\right )^{\frac{5}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-a**2/(8*c**(5/2)*x**3*(x**2)**(5/2)) - 2*a*b/(7*c**(5/2)*x**2*(x**2)**(5/2)) -
b**2/(6*c**(5/2)*x*(x**2)**(5/2))

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GIAC/XCAS [A]  time = 0.537897, 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^4),x, algorithm="giac")

[Out]

sage0*x