Optimal. Leaf size=50 \[ -\frac{c^2 \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]
[Out]
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Rubi [A] time = 0.036072, antiderivative size = 50, 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 \[ -\frac{c^2 \sqrt{c x^2} (a+b x)^{n+1} \, _2F_1\left (1,n+1;n+2;\frac{b x}{a}+1\right )}{a (n+1) x} \]
Antiderivative was successfully verified.
[In] Int[((c*x^2)^(5/2)*(a + b*x)^n)/x^6,x]
[Out]
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Rubi in Sympy [A] time = 17.127, size = 39, normalized size = 0.78 \[ - \frac{c^{2} \sqrt{c x^{2}} \left (a + b x\right )^{n + 1}{{}_{2}F_{1}\left (\begin{matrix} 1, n + 1 \\ n + 2 \end{matrix}\middle |{1 + \frac{b x}{a}} \right )}}{a x \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c*x**2)**(5/2)*(b*x+a)**n/x**6,x)
[Out]
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Mathematica [A] time = 0.0208223, size = 58, normalized size = 1.16 \[ \frac{\left (c x^2\right )^{5/2} \left (\frac{a}{b x}+1\right )^{-n} (a+b x)^n \, _2F_1\left (-n,-n;1-n;-\frac{a}{b x}\right )}{n x^5} \]
Antiderivative was successfully verified.
[In] Integrate[((c*x^2)^(5/2)*(a + b*x)^n)/x^6,x]
[Out]
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Maple [F] time = 0.031, size = 0, normalized size = 0. \[ \int{\frac{ \left ( bx+a \right ) ^{n}}{{x}^{6}} \left ( c{x}^{2} \right ) ^{{\frac{5}{2}}}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c*x^2)^(5/2)*(b*x+a)^n/x^6,x)
[Out]
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Maxima [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)^n/x^6,x, algorithm="maxima")
[Out]
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Fricas [F] time = 0., size = 0, normalized size = 0. \[{\rm integral}\left (\frac{\sqrt{c x^{2}}{\left (b x + a\right )}^{n} c^{2}}{x^{2}}, x\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)^n/x^6,x, algorithm="fricas")
[Out]
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Sympy [F(-1)] time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x**2)**(5/2)*(b*x+a)**n/x**6,x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{\left (c x^{2}\right )^{\frac{5}{2}}{\left (b x + a\right )}^{n}}{x^{6}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(5/2)*(b*x + a)^n/x^6,x, algorithm="giac")
[Out]