Optimal. Leaf size=93 \[ -\frac{a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}} \]
[Out]
_______________________________________________________________________________________
Rubi [A] time = 0.105703, antiderivative size = 93, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.136 \[ -\frac{a^2 d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{2 a b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}}+\frac{b^2 x (d x)^m}{c m \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[((d*x)^m*(a + b*x)^2)/(c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Rubi in Sympy [A] time = 29.2067, size = 83, normalized size = 0.89 \[ - \frac{a^{2} d^{2} \sqrt{c x^{2}} \left (d x\right )^{m - 2}}{c^{2} x \left (- m + 2\right )} - \frac{2 a b d \sqrt{c x^{2}} \left (d x\right )^{m - 1}}{c^{2} x \left (- m + 1\right )} + \frac{b^{2} \sqrt{c x^{2}} \left (d x\right )^{m}}{c^{2} m x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(b*x+a)**2/(c*x**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
Mathematica [A] time = 0.0552515, size = 50, normalized size = 0.54 \[ \frac{x^3 (d x)^m \left (\frac{a^2}{(m-2) x^2}+\frac{2 a b}{(m-1) x}+\frac{b^2}{m}\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((d*x)^m*(a + b*x)^2)/(c*x^2)^(3/2),x]
[Out]
_______________________________________________________________________________________
Maple [A] time = 0.006, size = 83, normalized size = 0.9 \[{\frac{ \left ({b}^{2}{m}^{2}{x}^{2}+2\,abx{m}^{2}-3\,{b}^{2}m{x}^{2}+{a}^{2}{m}^{2}-4\,abxm+2\,{b}^{2}{x}^{2}-{a}^{2}m \right ) x \left ( dx \right ) ^{m}}{m \left ( -1+m \right ) \left ( -2+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(b*x+a)^2/(c*x^2)^(3/2),x)
[Out]
_______________________________________________________________________________________
Maxima [A] time = 1.38114, size = 80, normalized size = 0.86 \[ \frac{b^{2} d^{m} x^{m}}{c^{\frac{3}{2}} m} + \frac{2 \, a b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a^{2} d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*(d*x)^m/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
_______________________________________________________________________________________
Fricas [A] time = 0.232149, size = 124, normalized size = 1.33 \[ \frac{{\left (a^{2} m^{2} - a^{2} m +{\left (b^{2} m^{2} - 3 \, b^{2} m + 2 \, b^{2}\right )} x^{2} + 2 \,{\left (a b m^{2} - 2 \, a b m\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{{\left (c^{2} m^{3} - 3 \, c^{2} m^{2} + 2 \, c^{2} m\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*(d*x)^m/(c*x^2)^(3/2),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((d*x)**m*(b*x+a)**2/(c*x**2)**(3/2),x)
[Out]
_______________________________________________________________________________________
GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )}^{2} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)^2*(d*x)^m/(c*x^2)^(3/2),x, algorithm="giac")
[Out]