3.3.95 \(\int (a+b x)^{3/2} \, dx\)

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

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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 (a+b x)^{5/2}}{5 b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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 (a+b x)^{3/2} \, dx &=\frac {2 (a+b x)^{5/2}}{5 b}\\ \end {align*}

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

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(3/2),x]

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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fricas [B]  time = 0.61, size = 28, normalized size = 1.75 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} + 2 \, a b x + a^{2}\right )} \sqrt {b x + a}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2),x, algorithm="fricas")

[Out]

2/5*(b^2*x^2 + 2*a*b*x + a^2)*sqrt(b*x + a)/b

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giac [B]  time = 1.13, size = 58, normalized size = 3.62 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 30 \, \sqrt {b x + a} a^{2} + 10 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a\right )}}{15 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2),x, algorithm="giac")

[Out]

2/15*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 30*sqrt(b*x + a)*a^2 + 10*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*
a)*a)/b

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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \begin {gather*} \frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(3/2),x)

[Out]

2/5*(b*x+a)^(5/2)/b

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maxima [A]  time = 1.29, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(3/2),x, algorithm="maxima")

[Out]

2/5*(b*x + a)^(5/2)/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(3/2),x)

[Out]

(2*(a + b*x)^(5/2))/(5*b)

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sympy [A]  time = 0.07, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a + b x\right )^{\frac {5}{2}}}{5 b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(3/2),x)

[Out]

2*(a + b*x)**(5/2)/(5*b)

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