\(\int (c+d x)^{5/2} \, dx\) [1404]

Optimal result

Integrand size = 9, antiderivative size = 16 \[ \int (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2}}{7 d} \]

[Out]

2/7*(d*x+c)^(7/2)/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} \[ \int (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2}}{7 d} \]

[In]

Int[(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2))/(7*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)^{7/2}}{7 d} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.00 (sec) , antiderivative size = 16, normalized size of antiderivative = 1.00 \[ \int (c+d x)^{5/2} \, dx=\frac {2 (c+d x)^{7/2}}{7 d} \]

[In]

Integrate[(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2))/(7*d)

Maple [A] (verified)

Time = 0.23 (sec) , antiderivative size = 13, normalized size of antiderivative = 0.81

method	result	size
gosper	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 d}\)	\(13\)
derivativedivides	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 d}\)	\(13\)
default	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 d}\)	\(13\)
pseudoelliptic	\(\frac {2 \left (d x +c \right )^{\frac {7}{2}}}{7 d}\)	\(13\)
trager	\(\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right ) \sqrt {d x +c}}{7 d}\)	\(40\)
risch	\(\frac {2 \left (d^{3} x^{3}+3 c \,d^{2} x^{2}+3 c^{2} d x +c^{3}\right ) \sqrt {d x +c}}{7 d}\)	\(40\)

[In]

int((d*x+c)^(5/2),x,method=_RETURNVERBOSE)

[Out]

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

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).

Time = 0.21 (sec) , antiderivative size = 39, normalized size of antiderivative = 2.44 \[ \int (c+d x)^{5/2} \, dx=\frac {2 \, {\left (d^{3} x^{3} + 3 \, c d^{2} x^{2} + 3 \, c^{2} d x + c^{3}\right )} \sqrt {d x + c}}{7 \, d} \]

[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

Sympy [A] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{5/2} \, dx=\frac {2 \left (c + d x\right )^{\frac {7}{2}}}{7 d} \]

[In]

integrate((d*x+c)**(5/2),x)

[Out]

2*(c + d*x)**(7/2)/(7*d)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{5/2} \, dx=\frac {2 \, {\left (d x + c\right )}^{\frac {7}{2}}}{7 \, d} \]

[In]

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

[Out]

2/7*(d*x + c)^(7/2)/d

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (12) = 24\).

Time = 0.32 (sec) , antiderivative size = 95, normalized size of antiderivative = 5.94 \[ \int (c+d x)^{5/2} \, dx=\frac {2 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} + 35 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} c^{2} + 7 \, {\left (3 \, {\left (d x + c\right )}^{\frac {5}{2}} - 10 \, {\left (d x + c\right )}^{\frac {3}{2}} c + 15 \, \sqrt {d x + c} c^{2}\right )} c\right )}}{35 \, d} \]

[In]

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

[Out]

2/35*(5*(d*x + c)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 + 35*((d*x + c)^(3/2) - 3*sqrt(d*x + c
)*c)*c^2 + 7*(3*(d*x + c)^(5/2) - 10*(d*x + c)^(3/2)*c + 15*sqrt(d*x + c)*c^2)*c)/d

Mupad [B] (verification not implemented)

Time = 0.02 (sec) , antiderivative size = 12, normalized size of antiderivative = 0.75 \[ \int (c+d x)^{5/2} \, dx=\frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,d} \]

[In]

int((c + d*x)^(5/2),x)

[Out]

(2*(c + d*x)^(7/2))/(7*d)