Optimal. Leaf size=89 \[ \frac{b^2 x \log (x)}{a^3 c \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 c \sqrt{c x^2}}+\frac{b}{a^2 c \sqrt{c x^2}}-\frac{1}{2 a c x \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0609574, antiderivative size = 89, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.118 \[ \frac{b^2 x \log (x)}{a^3 c \sqrt{c x^2}}-\frac{b^2 x \log (a+b x)}{a^3 c \sqrt{c x^2}}+\frac{b}{a^2 c \sqrt{c x^2}}-\frac{1}{2 a c x \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[1/((c*x^2)^(3/2)*(a + b*x)),x]
[Out]
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Rubi in Sympy [A] time = 15.1481, size = 88, normalized size = 0.99 \[ - \frac{\sqrt{c x^{2}}}{2 a c^{2} x^{3}} + \frac{b \sqrt{c x^{2}}}{a^{2} c^{2} x^{2}} + \frac{b^{2} \sqrt{c x^{2}} \log{\left (x \right )}}{a^{3} c^{2} x} - \frac{b^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{a^{3} c^{2} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(1/(c*x**2)**(3/2)/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0103348, size = 51, normalized size = 0.57 \[ \frac{x \left (-2 b^2 x^2 \log (a+b x)-a (a-2 b x)+2 b^2 x^2 \log (x)\right )}{2 a^3 \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[1/((c*x^2)^(3/2)*(a + b*x)),x]
[Out]
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Maple [A] time = 0.007, size = 49, normalized size = 0.6 \[{\frac{x \left ( 2\,{b}^{2}\ln \left ( x \right ){x}^{2}-2\,{b}^{2}\ln \left ( bx+a \right ){x}^{2}+2\,abx-{a}^{2} \right ) }{2\,{a}^{3}} \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(1/(c*x^2)^(3/2)/(b*x+a),x)
[Out]
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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(1/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.220869, size = 63, normalized size = 0.71 \[ \frac{{\left (2 \, b^{2} x^{2} \log \left (\frac{x}{b x + a}\right ) + 2 \, a b x - a^{2}\right )} \sqrt{c x^{2}}}{2 \, a^{3} c^{2} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{\left (c x^{2}\right )^{\frac{3}{2}} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/(c*x**2)**(3/2)/(b*x+a),x)
[Out]
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GIAC/XCAS [A] time = 0.505189, size = 4, normalized size = 0.04 \[ \mathit{sage}_{0} x \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(1/((c*x^2)^(3/2)*(b*x + a)),x, algorithm="giac")
[Out]