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 = 0.0014248, 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, Rules used = {32} \[ \frac{2 (c+d x)^{7/2}}{7 d} \]

Antiderivative was successfully verified.

[In]

Int[(c + d*x)^(5/2),x]

[Out]

(2*(c + d*x)^(7/2))/(7*d)

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*}

Mathematica [A]  time = 0.006018, 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 = 0.002, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{7\,d} \left ( dx+c \right ) ^{{\frac{7}{2}}}} \end{align*}

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 = 0.952011, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (d x + c\right )}^{\frac{7}{2}}}{7 \, d} \end{align*}

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]  time = 1.73007, size = 85, normalized size = 5.31 \begin{align*} \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{align*}

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 = 0.059051, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (c + d x\right )^{\frac{7}{2}}}{7 d} \end{align*}

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]  time = 1.07311, size = 81, normalized size = 5.06 \begin{align*} \frac{2 \,{\left (15 \,{\left (d x + c\right )}^{\frac{7}{2}} - 42 \,{\left (d x + c\right )}^{\frac{5}{2}} c + 70 \,{\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\right )}}{105 \, d} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*(d*x + c)^(7/2) - 42*(d*x + c)^(5/2)*c + 70*(d*x + c)^(3/2)*c^2 + 14*(3*(d*x + c)^(5/2) - 5*(d*x + c
)^(3/2)*c)*c)/d