3.303 \(\int (a+b x)^{5/2} \, dx\)

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

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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Rubi [A]  time = 0.0013822, 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} \[ \frac{2 (a+b x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2))/(7*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)^{5/2} \, dx &=\frac{2 (a+b x)^{7/2}}{7 b}\\ \end{align*}

Mathematica [A]  time = 0.0072039, size = 16, normalized size = 1. \[ \frac{2 (a+b x)^{7/2}}{7 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{7\,b} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/7*(b*x+a)^(7/2)/b

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Maxima [A]  time = 1.09342, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}}}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2),x, algorithm="maxima")

[Out]

2/7*(b*x + a)^(7/2)/b

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Fricas [B]  time = 1.5099, size = 85, normalized size = 5.31 \begin{align*} \frac{2 \,{\left (b^{3} x^{3} + 3 \, a b^{2} x^{2} + 3 \, a^{2} b x + a^{3}\right )} \sqrt{b x + a}}{7 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2),x, algorithm="fricas")

[Out]

2/7*(b^3*x^3 + 3*a*b^2*x^2 + 3*a^2*b*x + a^3)*sqrt(b*x + a)/b

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Sympy [A]  time = 0.107884, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (a + b x\right )^{\frac{7}{2}}}{7 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(5/2),x)

[Out]

2*(a + b*x)**(7/2)/(7*b)

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Giac [B]  time = 1.25108, size = 81, normalized size = 5.06 \begin{align*} \frac{2 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 70 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2} + 14 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a\right )}}{105 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(5/2),x, algorithm="giac")

[Out]

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