Optimal. Leaf size=30 \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.0172442, antiderivative size = 30, 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 \[ \frac{\sqrt{c x^2} (a+b x)^{n+1}}{b (n+1) x} \]
Antiderivative was successfully verified.
[In] Int[(Sqrt[c*x^2]*(a + b*x)^n)/x,x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 13.2308, size = 22, normalized size = 0.73 \[ \frac{\sqrt{c x^{2}} \left (a + b x\right )^{n + 1}}{b x \left (n + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((b*x+a)**n*(c*x**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0179754, size = 29, normalized size = 0.97 \[ \frac{c x (a+b x)^{n+1}}{b (n+1) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Integrate[(Sqrt[c*x^2]*(a + b*x)^n)/x,x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.003, size = 29, normalized size = 1. \[{\frac{ \left ( bx+a \right ) ^{1+n}}{b \left ( 1+n \right ) x}\sqrt{c{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((b*x+a)^n*(c*x^2)^(1/2)/x,x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.35338, size = 38, normalized size = 1.27 \[ \frac{{\left (b \sqrt{c} x + a \sqrt{c}\right )}{\left (b x + a\right )}^{n}}{b{\left (n + 1\right )}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x,x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.225717, size = 41, normalized size = 1.37 \[ \frac{\sqrt{c x^{2}}{\left (b x + a\right )}{\left (b x + a\right )}^{n}}{{\left (b n + b\right )} x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x,x, algorithm="fricas")
[Out]
_______________________________________________________________________________________
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((b*x+a)**n*(c*x**2)**(1/2)/x,x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [A] time = 0.203946, size = 57, normalized size = 1.9 \[ -\sqrt{c}{\left (\frac{a^{n + 1}{\rm sign}\left (x\right )}{b n + b} - \frac{{\left (b x + a\right )}^{n + 1}{\rm sign}\left (x\right )}{b{\left (n + 1\right )}}\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)^n/x,x, algorithm="giac")
[Out]