3.889 \(\int \frac{x^2}{\left (c x^2\right )^{3/2} (a+b x)} \, dx\)

Optimal. Leaf size=44 \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]

[Out]

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

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Rubi [A]  time = 0.0250121, antiderivative size = 44, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.2 \[ \frac{x \log (x)}{a c \sqrt{c x^2}}-\frac{x \log (a+b x)}{a c \sqrt{c x^2}} \]

Antiderivative was successfully verified.

[In]  Int[x^2/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

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Rubi in Sympy [A]  time = 11.281, size = 39, normalized size = 0.89 \[ \frac{\sqrt{c x^{2}} \log{\left (x \right )}}{a c^{2} x} - \frac{\sqrt{c x^{2}} \log{\left (a + b x \right )}}{a c^{2} x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**2/(c*x**2)**(3/2)/(b*x+a),x)

[Out]

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

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Mathematica [A]  time = 0.0105271, size = 27, normalized size = 0.61 \[ \frac{x^3 (\log (x)-\log (a+b x))}{a \left (c x^2\right )^{3/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[x^2/((c*x^2)^(3/2)*(a + b*x)),x]

[Out]

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

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Maple [A]  time = 0.006, size = 26, normalized size = 0.6 \[{\frac{{x}^{3} \left ( \ln \left ( x \right ) -\ln \left ( bx+a \right ) \right ) }{a} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^2/(c*x^2)^(3/2)/(b*x+a),x)

[Out]

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

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Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 0.216421, size = 1, normalized size = 0.02 \[ \left [\frac{\sqrt{c x^{2}} \log \left (\frac{x}{b x + a}\right )}{a c^{2} x}, \frac{2 \, \sqrt{-c} \arctan \left (\frac{\sqrt{c x^{2}}{\left (2 \, b x + a\right )} \sqrt{-c}}{a c x}\right )}{a c^{2}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="fricas")

[Out]

[sqrt(c*x^2)*log(x/(b*x + a))/(a*c^2*x), 2*sqrt(-c)*arctan(sqrt(c*x^2)*(2*b*x +
a)*sqrt(-c)/(a*c*x))/(a*c^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{2}}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**2/(c*x**2)**(3/2)/(b*x+a),x)

[Out]

Integral(x**2/((c*x**2)**(3/2)*(a + b*x)), x)

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GIAC/XCAS [A]  time = 0.218624, size = 85, normalized size = 1.93 \[ \frac{\frac{{\rm ln}\left ({\left | -{\left (\sqrt{c} x - \sqrt{c x^{2}}\right )} b - 2 \, a \sqrt{c} \right |}\right )}{a \sqrt{c}} - \frac{{\rm ln}\left ({\left | -\sqrt{c} x + \sqrt{c x^{2}} \right |}\right )}{a \sqrt{c}}}{c} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x^2/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="giac")

[Out]

(ln(abs(-(sqrt(c)*x - sqrt(c*x^2))*b - 2*a*sqrt(c)))/(a*sqrt(c)) - ln(abs(-sqrt(
c)*x + sqrt(c*x^2)))/(a*sqrt(c)))/c