3.316 \(\int (a+b x)^{9/2} \, dx\)

Optimal. Leaf size=16 \[ \frac{2 (a+b x)^{11/2}}{11 b} \]

[Out]

(2*(a + b*x)^(11/2))/(11*b)

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Rubi [A]  time = 0.0015142, 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} \[ \frac{2 (a+b x)^{11/2}}{11 b} \]

Antiderivative was successfully verified.

[In]

Int[(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2))/(11*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)^{9/2} \, dx &=\frac{2 (a+b x)^{11/2}}{11 b}\\ \end{align*}

Mathematica [A]  time = 0.0177189, size = 16, normalized size = 1. \[ \frac{2 (a+b x)^{11/2}}{11 b} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2))/(11*b)

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Maple [A]  time = 0.002, size = 13, normalized size = 0.8 \begin{align*}{\frac{2}{11\,b} \left ( bx+a \right ) ^{{\frac{11}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((b*x+a)^(9/2),x)

[Out]

2/11*(b*x+a)^(11/2)/b

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Maxima [A]  time = 1.13427, size = 16, normalized size = 1. \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{11}{2}}}{11 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2),x, algorithm="maxima")

[Out]

2/11*(b*x + a)^(11/2)/b

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Fricas [B]  time = 1.52622, size = 132, normalized size = 8.25 \begin{align*} \frac{2 \,{\left (b^{5} x^{5} + 5 \, a b^{4} x^{4} + 10 \, a^{2} b^{3} x^{3} + 10 \, a^{3} b^{2} x^{2} + 5 \, a^{4} b x + a^{5}\right )} \sqrt{b x + a}}{11 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2),x, algorithm="fricas")

[Out]

2/11*(b^5*x^5 + 5*a*b^4*x^4 + 10*a^2*b^3*x^3 + 10*a^3*b^2*x^2 + 5*a^4*b*x + a^5)*sqrt(b*x + a)/b

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Sympy [A]  time = 0.080241, size = 12, normalized size = 0.75 \begin{align*} \frac{2 \left (a + b x\right )^{\frac{11}{2}}}{11 b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)**(9/2),x)

[Out]

2*(a + b*x)**(11/2)/(11*b)

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Giac [B]  time = 1.14426, size = 230, normalized size = 14.38 \begin{align*} \frac{2 \,{\left (315 \,{\left (b x + a\right )}^{\frac{11}{2}} - 1540 \,{\left (b x + a\right )}^{\frac{9}{2}} a + 2970 \,{\left (b x + a\right )}^{\frac{7}{2}} a^{2} - 2772 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{3} + 2310 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{4} + 924 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a^{3} + 198 \,{\left (15 \,{\left (b x + a\right )}^{\frac{7}{2}} - 42 \,{\left (b x + a\right )}^{\frac{5}{2}} a + 35 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{2}\right )} a^{2} + 44 \,{\left (35 \,{\left (b x + a\right )}^{\frac{9}{2}} - 135 \,{\left (b x + a\right )}^{\frac{7}{2}} a + 189 \,{\left (b x + a\right )}^{\frac{5}{2}} a^{2} - 105 \,{\left (b x + a\right )}^{\frac{3}{2}} a^{3}\right )} a\right )}}{3465 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((b*x+a)^(9/2),x, algorithm="giac")

[Out]

2/3465*(315*(b*x + a)^(11/2) - 1540*(b*x + a)^(9/2)*a + 2970*(b*x + a)^(7/2)*a^2 - 2772*(b*x + a)^(5/2)*a^3 +
2310*(b*x + a)^(3/2)*a^4 + 924*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*a^3 + 198*(15*(b*x + a)^(7/2) - 42*(b
*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2)*a^2 + 44*(35*(b*x + a)^(9/2) - 135*(b*x + a)^(7/2)*a + 189*(b*x + a)
^(5/2)*a^2 - 105*(b*x + a)^(3/2)*a^3)*a)/b