Optimal. Leaf size=61 \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
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Rubi [A] time = 0.0674009, antiderivative size = 61, 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 \[ \frac{a c \sqrt{c x^2} (d x)^{m+4}}{d^4 (m+4) x}+\frac{b c \sqrt{c x^2} (d x)^{m+5}}{d^5 (m+5) x} \]
Antiderivative was successfully verified.
[In] Int[(d*x)^m*(c*x^2)^(3/2)*(a + b*x),x]
[Out]
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Rubi in Sympy [A] time = 16.5407, size = 53, normalized size = 0.87 \[ \frac{a c \sqrt{c x^{2}} \left (d x\right )^{m + 4}}{d^{4} x \left (m + 4\right )} + \frac{b c \sqrt{c x^{2}} \left (d x\right )^{m + 5}}{d^{5} x \left (m + 5\right )} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((d*x)**m*(c*x**2)**(3/2)*(b*x+a),x)
[Out]
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Mathematica [A] time = 0.0301478, size = 39, normalized size = 0.64 \[ \frac{\left (c x^2\right )^{3/2} (d x)^m \left (\frac{a x^4}{m+4}+\frac{b x^5}{m+5}\right )}{x^3} \]
Antiderivative was successfully verified.
[In] Integrate[(d*x)^m*(c*x^2)^(3/2)*(a + b*x),x]
[Out]
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Maple [A] time = 0.003, size = 40, normalized size = 0.7 \[{\frac{ \left ( bmx+am+4\,bx+5\,a \right ) x \left ( dx \right ) ^{m}}{ \left ( 5+m \right ) \left ( 4+m \right ) } \left ( c{x}^{2} \right ) ^{{\frac{3}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((d*x)^m*(c*x^2)^(3/2)*(b*x+a),x)
[Out]
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Maxima [A] time = 1.37678, size = 53, normalized size = 0.87 \[ \frac{b c^{\frac{3}{2}} d^{m} x^{5} x^{m}}{m + 5} + \frac{a c^{\frac{3}{2}} d^{m} x^{4} x^{m}}{m + 4} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(b*x + a)*(d*x)^m,x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.222854, size = 68, normalized size = 1.11 \[ \frac{{\left ({\left (b c m + 4 \, b c\right )} x^{4} +{\left (a c m + 5 \, a c\right )} x^{3}\right )} \sqrt{c x^{2}} \left (d x\right )^{m}}{m^{2} + 9 \, m + 20} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c*x^2)^(3/2)*(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)**(3/2)*(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((c*x^2)^(3/2)*(b*x + a)*(d*x)^m,x, algorithm="giac")
[Out]