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

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

[Out]

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

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Rubi [A]
time = 0.00, antiderivative size = 16, normalized size of antiderivative = 1.00, number of steps used = 1, number of rules used = 1, integrand size = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.111, Rules used = {32} \begin {gather*} \frac {2 (c+d x)^{5/2}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[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*} \int (c+d x)^{3/2} \, dx &=\frac {2 (c+d x)^{5/2}}{5 d}\\ \end {align*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (c+d x)^{5/2}}{5 d} \end {gather*}

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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Mathics [A]
time = 1.63, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c+d x\right )^{\frac {5}{2}}}{5 d} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^0*(c + d*x)^(3/2),x]')

[Out]

2 (c + d x) ^ (5 / 2) / (5 d)

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Maple [A]
time = 0.14, size = 13, normalized size = 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	\(\frac {2 \left (d x +c \right )^{\frac {5}{2}}}{5 d}\)	\(13\)
trager	\(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\)	\(29\)
risch	\(\frac {2 \left (d^{2} x^{2}+2 c d x +c^{2}\right ) \sqrt {d x +c}}{5 d}\)	\(29\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.28, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {5}{2}}}{5 \, d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 28 vs. \(2 (12) = 24\).
time = 0.30, size = 28, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (d^{2} x^{2} + 2 \, c d x + c^{2}\right )} \sqrt {d x + c}}{5 \, d} \end {gather*}

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 = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c + d x\right )^{\frac {5}{2}}}{5 d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 58 vs. \(2 (12) = 24\).
time = 0.00, size = 113, normalized size = 7.06 \begin {gather*} \frac {\frac {2 d^{2} \left (\frac {1}{5} \sqrt {c+d x} \left (c+d x\right )^{2}-\frac {2}{3} \sqrt {c+d x} \left (c+d x\right ) c+\sqrt {c+d x} c^{2}\right )}{d^{2}}+4 c \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+2 c^{2} \sqrt {c+d x}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{5/2}}{5\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(3/2),x)

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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