Optimal. Leaf size=65 \[ \frac{x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0495366, antiderivative size = 65, 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 \[ \frac{x (a+b x)^{n+2}}{b^2 c (n+2) \sqrt{c x^2}}-\frac{a x (a+b x)^{n+1}}{b^2 c (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^4*(a + b*x)^n)/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 19.4463, size = 58, normalized size = 0.89 \[ - \frac{a \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b^{2} c^{2} x \left (n + 1\right )} + \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 2}}{b^{2} c^{2} x \left (n + 2\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**4*(b*x+a)**n/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0373625, size = 45, normalized size = 0.69 \[ \frac{x^3 (a+b x)^{n+1} (b (n+1) x-a)}{b^2 (n+1) (n+2) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^4*(a + b*x)^n)/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.003, size = 46, normalized size = 0.7 \[ -{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{3} \left ( -bxn-bx+a \right ) }{{b}^{2} \left ({n}^{2}+3\,n+2 \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^4*(b*x+a)^n/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.36547, size = 61, normalized size = 0.94 \[ \frac{{\left (b^{2}{\left (n + 1\right )} x^{2} + a b n x - a^{2}\right )}{\left (b x + a\right )}^{n}}{{\left (n^{2} + 3 \, n + 2\right )} b^{2} c^{\frac{3}{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.241518, size = 97, normalized size = 1.49 \[ \frac{{\left (a b n x +{\left (b^{2} n + b^{2}\right )} x^{2} - a^{2}\right )} \sqrt{c x^{2}}{\left (b x + a\right )}^{n}}{{\left (b^{2} c^{2} n^{2} + 3 \, b^{2} c^{2} n + 2 \, b^{2} c^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [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(x**4*(b*x+a)**n/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{n} x^{4}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^4/(c*x^2)^(3/2),x, algorithm="giac")
[Out]