Optimal. Leaf size=92 \[ -\frac{a^3 c^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}+\frac{a^2 c^2 \sqrt{c x^2}}{b^3}-\frac{a c^2 x \sqrt{c x^2}}{2 b^2}+\frac{c^2 x^2 \sqrt{c x^2}}{3 b} \]
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Rubi [A] time = 0.0651437, antiderivative size = 92, 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 \[ -\frac{a^3 c^2 \sqrt{c x^2} \log (a+b x)}{b^4 x}+\frac{a^2 c^2 \sqrt{c x^2}}{b^3}-\frac{a c^2 x \sqrt{c x^2}}{2 b^2}+\frac{c^2 x^2 \sqrt{c x^2}}{3 b} \]
Antiderivative was successfully verified.
[In] Int[(c*x^2)^(5/2)/(x^2*(a + b*x)),x]
[Out]
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Rubi in Sympy [F] time = 0., size = 0, normalized size = 0. \[ - \frac{a^{3} c^{2} \sqrt{c x^{2}} \log{\left (a + b x \right )}}{b^{4} x} - \frac{a c^{2} \sqrt{c x^{2}} \int x\, dx}{b^{2} x} + \frac{c^{2} x^{2} \sqrt{c x^{2}}}{3 b} + \frac{c^{2} \sqrt{c x^{2}} \int a^{2}\, dx}{b^{3} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(5/2)/x**2/(b*x+a),x)
[Out]
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Mathematica [A] time = 0.00850899, size = 54, normalized size = 0.59 \[ \frac{c \left (c x^2\right )^{3/2} \left (b x \left (6 a^2-3 a b x+2 b^2 x^2\right )-6 a^3 \log (a+b x)\right )}{6 b^4 x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(c*x^2)^(5/2)/(x^2*(a + b*x)),x]
[Out]
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Maple [A] time = 0.009, size = 52, normalized size = 0.6 \[ -{\frac{-2\,{b}^{3}{x}^{3}+3\,a{b}^{2}{x}^{2}+6\,{a}^{3}\ln \left ( bx+a \right ) -6\,{a}^{2}bx}{6\,{x}^{5}{b}^{4}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(5/2)/x^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((c*x^2)^(5/2)/((b*x + a)*x^2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.213613, size = 85, normalized size = 0.92 \[ \frac{{\left (2 \, b^{3} c^{2} x^{3} - 3 \, a b^{2} c^{2} x^{2} + 6 \, a^{2} b c^{2} x - 6 \, a^{3} c^{2} \log \left (b x + a\right )\right )} \sqrt{c x^{2}}}{6 \, b^{4} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)/((b*x + a)*x^2),x, algorithm="fricas")
[Out]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}}{x^{2} \left (a + b x\right )}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(5/2)/x**2/(b*x+a),x)
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GIAC/XCAS [A] time = 0.232947, size = 113, normalized size = 1.23 \[ -\frac{1}{6} \,{\left (\frac{6 \, a^{3} c^{2}{\rm ln}\left ({\left | b x + a \right |}\right ){\rm sign}\left (x\right )}{b^{4}} - \frac{6 \, a^{3} c^{2}{\rm ln}\left ({\left | a \right |}\right ){\rm sign}\left (x\right )}{b^{4}} - \frac{2 \, b^{2} c^{2} x^{3}{\rm sign}\left (x\right ) - 3 \, a b c^{2} x^{2}{\rm sign}\left (x\right ) + 6 \, a^{2} c^{2} x{\rm sign}\left (x\right )}{b^{3}}\right )} \sqrt{c} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)/((b*x + a)*x^2),x, algorithm="giac")
[Out]