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

Optimal. Leaf size=84 \[ \frac{b^2 \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b \sqrt{c x^2}}{a^2 x^2}-\frac{\sqrt{c x^2}}{2 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0566069, antiderivative size = 84, 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 \[ \frac{b^2 \sqrt{c x^2} \log (x)}{a^3 x}-\frac{b^2 \sqrt{c x^2} \log (a+b x)}{a^3 x}+\frac{b \sqrt{c x^2}}{a^2 x^2}-\frac{\sqrt{c x^2}}{2 a x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Rubi in Sympy [A]  time = 20.075, size = 75, normalized size = 0.89 \[ - \frac{\sqrt{c x^{2}}}{2 a x^{3}} + \frac{b \sqrt{c x^{2}}}{a^{2} x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} x} - \frac{b^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} x} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Mathematica [A]  time = 0.0230228, size = 53, normalized size = 0.63 \[ \frac{\sqrt{c x^2} \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 x^3} \]

Antiderivative was successfully verified.

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

[Out]

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

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Maple [A]  time = 0.008, size = 51, normalized size = 0.6 \[{\frac{2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2}}{2\,{a}^{3}{x}^{3}}\sqrt{c{x}^{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Maxima [A]  time = 1.34775, size = 70, normalized size = 0.83 \[ -\frac{b^{2} \sqrt{c} \log \left (b x + a\right )}{a^{3}} + \frac{b^{2} \sqrt{c} \log \left (x\right )}{a^{3}} + \frac{2 \, b \sqrt{c} x - a \sqrt{c}}{2 \, a^{2} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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Fricas [A]  time = 0.221737, size = 59, normalized size = 0.7 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(sqrt(c*x**2)/(x**4*(a + b*x)), 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)*x^4),x, algorithm="giac")

[Out]

Exception raised: TypeError