Optimal. Leaf size=31 \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]
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Rubi [A] time = 0.0198229, antiderivative size = 31, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.1 \[ \frac{x (a+b x)^{n+1}}{b c (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(a + b*x)^n)/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 13.1006, size = 26, normalized size = 0.84 \[ \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b c^{2} x \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(b*x+a)**n/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.0208168, size = 30, normalized size = 0.97 \[ \frac{x^3 (a+b x)^{n+1}}{b (n+1) \left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(a + b*x)^n)/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 29, normalized size = 0.9 \[{\frac{ \left ( bx+a \right ) ^{1+n}{x}^{3}}{b \left ( 1+n \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(b*x+a)^n/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.35654, size = 42, normalized size = 1.35 \[ \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b c^{2}{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^3/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.230468, size = 50, normalized size = 1.61 \[ \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b c^{2} n + b c^{2}\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^3/(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**3*(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. \[ \int \frac{{\left (b x + a\right )}^{n} x^{3}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^n*x^3/(c*x^2)^(3/2),x, algorithm="giac")
[Out]