Optimal. Leaf size=23 \[ \frac{2 (d (a+b x)+c)^{5/2}}{5 b d} \]
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Rubi [A] time = 0.0243161, antiderivative size = 23, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 2, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154 \[ \frac{2 (d (a+b x)+c)^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
[In] Int[(c + d*(a + b*x))^(3/2),x]
[Out]
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Rubi in Sympy [A] time = 2.16235, size = 17, normalized size = 0.74 \[ \frac{2 \left (c + d \left (a + b x\right )\right )^{\frac{5}{2}}}{5 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate((c+d*(b*x+a))**(3/2),x)
[Out]
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Mathematica [A] time = 0.0260178, size = 23, normalized size = 1. \[ \frac{2 (d (a+b x)+c)^{5/2}}{5 b d} \]
Antiderivative was successfully verified.
[In] Integrate[(c + d*(a + b*x))^(3/2),x]
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Maple [A] time = 0.004, size = 20, normalized size = 0.9 \[{\frac{2}{5\,db} \left ( bdx+ad+c \right ) ^{{\frac{5}{2}}}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int((c+d*(b*x+a))^(3/2),x)
[Out]
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Maxima [A] time = 1.3382, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left ({\left (b x + a\right )} d + c\right )}^{\frac{5}{2}}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)*d + c)^(3/2),x, algorithm="maxima")
[Out]
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Fricas [A] time = 0.198016, size = 80, normalized size = 3.48 \[ \frac{2 \,{\left (b^{2} d^{2} x^{2} + a^{2} d^{2} + 2 \, a c d + c^{2} + 2 \,{\left (a b d^{2} + b c d\right )} x\right )} \sqrt{b d x + a d + c}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)*d + c)^(3/2),x, algorithm="fricas")
[Out]
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Sympy [A] time = 14.1343, size = 156, normalized size = 6.78 \[ \begin{cases} c^{\frac{3}{2}} x & \text{for}\: b = 0 \wedge d = 0 \\x \left (a d + c\right )^{\frac{3}{2}} & \text{for}\: b = 0 \\c^{\frac{3}{2}} x & \text{for}\: d = 0 \\\frac{2 a^{2} d \sqrt{a d + b d x + c}}{5 b} + \frac{4 a d x \sqrt{a d + b d x + c}}{5} + \frac{4 a c \sqrt{a d + b d x + c}}{5 b} + \frac{2 b d x^{2} \sqrt{a d + b d x + c}}{5} + \frac{4 c x \sqrt{a d + b d x + c}}{5} + \frac{2 c^{2} \sqrt{a d + b d x + c}}{5 b d} & \text{otherwise} \end{cases} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((c+d*(b*x+a))**(3/2),x)
[Out]
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GIAC/XCAS [A] time = 0.210198, size = 26, normalized size = 1.13 \[ \frac{2 \,{\left (b d x + a d + c\right )}^{\frac{5}{2}}}{5 \, b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(((b*x + a)*d + c)^(3/2),x, algorithm="giac")
[Out]