3.1392 \(\int (c+d x)^{3/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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Rubi [A]  time = 0.00709082, antiderivative size = 16, normalized size of antiderivative = 1., number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.111 \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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Rubi in Sympy [A]  time = 1.33383, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(3/2),x)

[Out]

2*(c + d*x)**(5/2)/(5*d)

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Mathematica [A]  time = 0.00743417, size = 16, normalized size = 1. \[ \frac{2 (c+d x)^{5/2}}{5 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{2}{5\,d} \left ( dx+c \right ) ^{{\frac{5}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(3/2),x)

[Out]

2/5*(d*x+c)^(5/2)/d

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Maxima [A]  time = 1.36479, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{5}{2}}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2),x, algorithm="maxima")

[Out]

2/5*(d*x + c)^(5/2)/d

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Fricas [A]  time = 0.198105, size = 38, normalized size = 2.38 \[ \frac{2 \,{\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt{d x + c}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2),x, algorithm="fricas")

[Out]

2/5*(d^2*x^2 + 2*c*d*x + c^2)*sqrt(d*x + c)/d

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Sympy [A]  time = 0.03347, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{5}{2}}}{5 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(3/2),x)

[Out]

2*(c + d*x)**(5/2)/(5*d)

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GIAC/XCAS [A]  time = 0.218687, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{5}{2}}}{5 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(3/2),x, algorithm="giac")

[Out]

2/5*(d*x + c)^(5/2)/d