3.1404 \(\int (c+d x)^{5/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (c+d x)^{7/2}}{7 d} \]

[Out]

(2*(c + d*x)^(7/2))/(7*d)

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Rubi [A]  time = 0.00720378, 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 \[ \frac{2 (c+d x)^{7/2}}{7 d} \]

Antiderivative was successfully verified.

[In]  Int[(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2))/(7*d)

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Rubi in Sympy [A]  time = 1.39628, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((d*x+c)**(5/2),x)

[Out]

2*(c + d*x)**(7/2)/(7*d)

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Mathematica [A]  time = 0.00754488, size = 16, normalized size = 1. \[ \frac{2 (c+d x)^{7/2}}{7 d} \]

Antiderivative was successfully verified.

[In]  Integrate[(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2))/(7*d)

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Maple [A]  time = 0.003, size = 13, normalized size = 0.8 \[{\frac{2}{7\,d} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((d*x+c)^(5/2),x)

[Out]

2/7*(d*x+c)^(7/2)/d

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Maxima [A]  time = 1.34753, size = 16, normalized size = 1. \[ \frac{2 \,{\left (d x + c\right )}^{\frac{7}{2}}}{7 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2),x, algorithm="maxima")

[Out]

2/7*(d*x + c)^(7/2)/d

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Fricas [A]  time = 0.199671, size = 53, normalized size = 3.31 \[ \frac{2 \,{\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt{d x + c}}{7 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2),x, algorithm="fricas")

[Out]

2/7*(d^3*x^3 + 3*c*d^2*x^2 + 3*c^2*d*x + c^3)*sqrt(d*x + c)/d

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Sympy [A]  time = 0.041956, size = 12, normalized size = 0.75 \[ \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x+c)**(5/2),x)

[Out]

2*(c + d*x)**(7/2)/(7*d)

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GIAC/XCAS [A]  time = 0.217361, size = 116, normalized size = 7.25 \[ \frac{2 \,{\left (35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} + 14 \,{\left (3 \,{\left (d x + c\right )}^{\frac{5}{2}} - 5 \,{\left (d x + c\right )}^{\frac{3}{2}} c\right )} c + \frac{15 \,{\left (d x + c\right )}^{\frac{7}{2}} d^{12} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c d^{12} + 35 \,{\left (d x + c\right )}^{\frac{3}{2}} c^{2} d^{12}}{d^{12}}\right )}}{105 \, d} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((d*x + c)^(5/2),x, algorithm="giac")

[Out]

2/105*(35*(d*x + c)^(3/2)*c^2 + 14*(3*(d*x + c)^(5/2) - 5*(d*x + c)^(3/2)*c)*c +
(15*(d*x + c)^(7/2)*d^12 - 42*(d*x + c)^(5/2)*c*d^12 + 35*(d*x + c)^(3/2)*c^2*d
^12)/d^12)/d