Optimal. Leaf size=65 \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]
[Out]
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Rubi [A] time = 0.0792115, antiderivative size = 65, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 20, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.15 \[ -\frac{a d^2 x (d x)^{m-2}}{c (2-m) \sqrt{c x^2}}-\frac{b d x (d x)^{m-1}}{c (1-m) \sqrt{c x^2}} \]
Antiderivative was successfully verified.
[In] Int[((d*x)^m*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 17.742, size = 56, normalized size = 0.86 \[ - \frac{a d^{2} \sqrt{c x^{2}} \left (d x\right )^{m - 2}}{c^{2} x \left (- m + 2\right )} - \frac{b d \sqrt{c x^{2}} \left (d x\right )^{m - 1}}{c^{2} x \left (- m + 1\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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Mathematica [A] time = 0.03946, size = 32, normalized size = 0.49 \[ \frac{x (d x)^m \left (\frac{a}{m-2}+\frac{b x}{m-1}\right )}{\left (c x^2\right )^{3/2}} \]
Antiderivative was successfully verified.
[In] Integrate[((d*x)^m*(a + b*x))/(c*x^2)^(3/2),x]
[Out]
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Maple [A] time = 0.004, size = 40, normalized size = 0.6 \[{\frac{ \left ( bmx+am-2\,bx-a \right ) x \left ( dx \right ) ^{m}}{ \left ( -1+m \right ) \left ( -2+m \right ) } \left ( c{x}^{2} \right ) ^{-{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(b*x+a)/(c*x^2)^(3/2),x)
[Out]
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Maxima [A] time = 1.42635, size = 53, normalized size = 0.82 \[ \frac{b d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 1\right )} x} + \frac{a d^{m} x^{m}}{c^{\frac{3}{2}}{\left (m - 2\right )} x^{2}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x)^m/(c*x^2)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.23291, size = 72, normalized size = 1.11 \[ \frac{\sqrt{c x^{2}}{\left (a m +{\left (b m - 2 \, b\right )} x - a\right )} \left (d x\right )^{m}}{{\left (c^{2} m^{2} - 3 \, c^{2} m + 2 \, c^{2}\right )} x^{3}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x)^m/(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((d*x)**m*(b*x+a)/(c*x**2)**(3/2),x)
[Out]
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GIAC/XCAS [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{{\left (b x + a\right )} \left (d x\right )^{m}}{\left (c x^{2}\right )^{\frac{3}{2}}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((b*x + a)*(d*x)^m/(c*x^2)^(3/2),x, algorithm="giac")
[Out]