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

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

[Out]

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

Antiderivative was successfully verified.

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

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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Mathics [A]
time = 1.75, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c+d x\right )^{\frac {7}{2}}}{7 d} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[(a + b*x)^0*(c + d*x)^(5/2),x]')

[Out]

2 (c + d x) ^ (7 / 2) / (7 d)

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Maple [A]
time = 0.14, size = 13, normalized size = 0.81

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.26, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (d x + c\right )}^{\frac {7}{2}}}{7 \, d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 0.29, size = 39, normalized size = 2.44 \begin {gather*} \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} \end {gather*}

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 = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (c + d x\right )^{\frac {7}{2}}}{7 d} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (12) = 24\).
time = 0.00, size = 199, normalized size = 12.44 \begin {gather*} \frac {\frac {2 d^{3} \left (\frac {1}{7} \sqrt {c+d x} \left (c+d x\right )^{3}-\frac {3}{5} \sqrt {c+d x} \left (c+d x\right )^{2} c+\sqrt {c+d x} \left (c+d x\right ) c^{2}-\sqrt {c+d x} c^{3}\right )}{d^{3}}+\frac {6 c 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}}+6 c^{2} \left (\frac {1}{3} \sqrt {c+d x} \left (c+d x\right )-c \sqrt {c+d x}\right )+2 c^{3} \sqrt {c+d x}}{d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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

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Mupad [B]
time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (c+d\,x\right )}^{7/2}}{7\,d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c + d*x)^(5/2),x)

[Out]

(2*(c + d*x)^(7/2))/(7*d)

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