\(\int (c+d x)^{3/2} \, dx\) [1392]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2}}{5 d} \]

[Out]

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

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \[ \int (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2}}{5 d} \]

[In]

Int[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

Rule 32

Int[((a_.) + (b_.)*(x_))^(m_), x_Symbol] :> Simp[(a + b*x)^(m + 1)/(b*(m + 1)), x] /; FreeQ[{a, b, m}, x] && N
eQ[m, -1]

Rubi steps \begin{align*} \text {integral}& = \frac {2 (c+d x)^{5/2}}{5 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int (c+d x)^{3/2} \, dx=\frac {2 (c+d x)^{5/2}}{5 d} \]

[In]

Integrate[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

Maple [A] (verified)

Time = 0.22 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\)	\(13\)
derivativedivides	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\)	\(13\)
default	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\)	\(13\)
pseudoelliptic	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\)	\(13\)
trager	\(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\)	\(29\)
risch	\(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\)	\(29\)

[In]

int((d*x+c)^(3/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).

Time = 0.22 (sec) , antiderivative size = 28, normalized size of antiderivative = 1.75 \[ \int (c+d x)^{3/2} \, dx=\frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt {d x + c}}{5 \, d} \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{3/2} \, dx=\frac {2 \left (c + d x\right )^{\frac {5}{2}}}{5 d} \]

[In]

integrate((d*x+c)**(3/2),x)

[Out]

2*(c + d*x)**(5/2)/(5*d)

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{3/2} \, dx=\frac {2 \, {\left (d x + c\right )}^{\frac {5}{2}}}{5 \, d} \]

[In]

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

[Out]

2/5*(d*x + c)^(5/2)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).

Time = 0.29 (sec) , antiderivative size = 58, normalized size of antiderivative = 3.62 \[ \int (c+d x)^{3/2} \, dx=\frac {2 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 30 \, \sqrt {d x + c} c^{2} + 10 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} c\right )}}{15 \, d} \]

[In]

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

[Out]

2/15*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 30*sqrt(d*x + c)*c^2 + 10*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*
c)*c)/d

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{3/2} \, dx=\frac {2\,{\left (c+d\,x\right )}^{5/2}}{5\,d} \]

[In]

int((c + d*x)^(3/2),x)

[Out]

(2*(c + d*x)^(5/2))/(5*d)