Optimal. Leaf size=59 \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]
[Out]
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Rubi [A] time = 0.0606329, antiderivative size = 59, 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 \[ \frac{a \sqrt{c x^2} (d x)^{m+2}}{d^2 (m+2) x}+\frac{b \sqrt{c x^2} (d x)^{m+3}}{d^3 (m+3) x} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*Sqrt[c*x^2]*(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 15.6165, size = 49, normalized size = 0.83 \[ \frac{a \sqrt{c x^{2}} \left (d x\right )^{m + 2}}{d^{2} x \left (m + 2\right )} + \frac{b \sqrt{c x^{2}} \left (d x\right )^{m + 3}}{d^{3} x \left (m + 3\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2)**(1/2)*(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0256086, size = 39, normalized size = 0.66 \[ \frac{\sqrt{c x^2} (d x)^m \left (\frac{a x^2}{m+2}+\frac{b x^3}{m+3}\right )}{x} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*Sqrt[c*x^2]*(a + b*x),x]
[Out]
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Maple [A] time = 0.004, size = 40, normalized size = 0.7 \[{\frac{ \left ( bmx+am+2\,bx+3\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 3+m \right ) \left ( 2+m \right ) }\sqrt{c{x}^{2}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2)^(1/2)*(b*x+a),x)
[Out]
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Maxima [A] time = 1.37139, size = 53, normalized size = 0.9 \[ \frac{b \sqrt{c} d^{m} x^{3} x^{m}}{m + 3} + \frac{a \sqrt{c} d^{m} x^{2} x^{m}}{m + 2} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.232352, size = 59, normalized size = 1. \[ \frac{{\left ({\left (b m + 2 \, b\right )} x^{2} +{\left (a m + 3 \, a\right )} x\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 5 \, m + 6} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(sqrt(c*x^2)*(b*x + a)*(d*x)^m,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*(c*x**2)**(1/2)*(b*x+a),x)
[Out]
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GIAC/XCAS [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(sqrt(c*x^2)*(b*x + a)*(d*x)^m,x, algorithm="giac")
[Out]