3.900 \(\int \frac{\sqrt{c x^2}}{x^4 (a+b x)^2} \, dx\)

Optimal. Leaf size=112 \[ \frac{3 b^2 \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b^2 \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{2 b \sqrt{c x^2}}{a^3 x^2}-\frac{\sqrt{c x^2}}{2 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0883547, antiderivative size = 112, 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 \[ \frac{3 b^2 \sqrt{c x^2} \log (x)}{a^4 x}-\frac{3 b^2 \sqrt{c x^2} \log (a+b x)}{a^4 x}+\frac{b^2 \sqrt{c x^2}}{a^3 x (a+b x)}+\frac{2 b \sqrt{c x^2}}{a^3 x^2}-\frac{\sqrt{c x^2}}{2 a^2 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 23.5787, size = 104, normalized size = 0.93 \[ - \frac{\sqrt{c x^{2}}}{2 a^{2} x^{3}} + \frac{b^{2} \sqrt{c x^{2}}}{a^{3} x \left (a + b x\right )} + \frac{2 b \sqrt{c x^{2}}}{a^{3} x^{2}} + \frac{3 b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{4} x} - \frac{3 b^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{4} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((c*x**2)**(1/2)/x**4/(b*x+a)**2,x)

[Out]

-sqrt(c*x**2)/(2*a**2*x**3) + b**2*sqrt(c*x**2)/(a**3*x*(a + b*x)) + 2*b*sqrt(c*
x**2)/(a**3*x**2) + 3*b**2*sqrt(c*x**2)*log(x)/(a**4*x) - 3*b**2*sqrt(c*x**2)*lo
g(a + b*x)/(a**4*x)

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Mathematica [A]  time = 0.043294, size = 82, normalized size = 0.73 \[ \frac{\sqrt{c x^2} \left (a \left (-a^2+3 a b x+6 b^2 x^2\right )+6 b^2 x^2 \log (x) (a+b x)-6 b^2 x^2 (a+b x) \log (a+b x)\right )}{2 a^4 x^3 (a+b x)} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.01, size = 95, normalized size = 0.9 \[{\frac{6\,{b}^{3}\ln \left ( x \right ){x}^{3}-6\,{b}^{3}\ln \left ( bx+a \right ){x}^{3}+6\,\ln \left ( x \right ){x}^{2}a{b}^{2}-6\,\ln \left ( bx+a \right ){x}^{2}a{b}^{2}+6\,a{b}^{2}{x}^{2}+3\,{a}^{2}bx-{a}^{3}}{2\,{x}^{3}{a}^{4} \left ( bx+a \right ) }\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((c*x^2)^(1/2)/x^4/(b*x+a)^2,x)

[Out]

1/2*(c*x^2)^(1/2)*(6*b^3*ln(x)*x^3-6*b^3*ln(b*x+a)*x^3+6*ln(x)*x^2*a*b^2-6*ln(b*
x+a)*x^2*a*b^2+6*a*b^2*x^2+3*a^2*b*x-a^3)/x^3/a^4/(b*x+a)

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Maxima [A]  time = 1.33582, size = 107, normalized size = 0.96 \[ \frac{6 \, b^{2} \sqrt{c} x^{2} + 3 \, a b \sqrt{c} x - a^{2} \sqrt{c}}{2 \,{\left (a^{3} b x^{3} + a^{4} x^{2}\right )}} - \frac{3 \, b^{2} \sqrt{c} \log \left (b x + a\right )}{a^{4}} + \frac{3 \, b^{2} \sqrt{c} \log \left (x\right )}{a^{4}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*b^2*sqrt(c)*x^2 + 3*a*b*sqrt(c)*x - a^2*sqrt(c))/(a^3*b*x^3 + a^4*x^2) -
3*b^2*sqrt(c)*log(b*x + a)/a^4 + 3*b^2*sqrt(c)*log(x)/a^4

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Fricas [A]  time = 0.225983, size = 104, normalized size = 0.93 \[ \frac{{\left (6 \, a b^{2} x^{2} + 3 \, a^{2} b x - a^{3} + 6 \,{\left (b^{3} x^{3} + a b^{2} x^{2}\right )} \log \left (\frac{x}{b x + a}\right )\right )} \sqrt{c x^{2}}}{2 \,{\left (a^{4} b x^{4} + a^{5} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

1/2*(6*a*b^2*x^2 + 3*a^2*b*x - a^3 + 6*(b^3*x^3 + a*b^2*x^2)*log(x/(b*x + a)))*s
qrt(c*x^2)/(a^4*b*x^4 + a^5*x^3)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{\sqrt{c x^{2}}}{x^{4} \left (a + b x\right )^{2}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((c*x**2)**(1/2)/x**4/(b*x+a)**2,x)

[Out]

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

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError