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

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

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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 = 0.00, 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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IntegrateAlgebraic [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]

IntegrateAlgebraic[(c + d*x)^(3/2),x]

[Out]

(2*(c + d*x)^(5/2))/(5*d)

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fricas [B]  time = 1.20, 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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giac [B]  time = 1.24, size = 58, normalized size = 3.62 \begin {gather*} \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} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[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

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \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]

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

[Out]

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

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maxima [A]  time = 1.36, 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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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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sympy [A]  time = 0.06, 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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