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

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

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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)^{11/2}}{11 b} \end {gather*}

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*}

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Mathematica [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{11/2}}{11 b} \end {gather*}

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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IntegrateAlgebraic [A]  time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{11/2}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(a + b*x)^(9/2),x]

[Out]

(2*(a + b*x)^(11/2))/(11*b)

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fricas [B]  time = 1.05, size = 61, normalized size = 3.81 \begin {gather*} \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 {gather*}

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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giac [B]  time = 0.99, size = 229, normalized size = 14.31 \begin {gather*} \frac {2 \, {\left (63 \, {\left (b x + a\right )}^{\frac {11}{2}} - 385 \, {\left (b x + a\right )}^{\frac {9}{2}} a + 990 \, {\left (b x + a\right )}^{\frac {7}{2}} a^{2} - 1386 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{3} + 1155 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{4} + 1155 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{4} + 462 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{3} + 198 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a^{2} + 11 \, {\left (35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}\right )} a\right )}}{693 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/693*(63*(b*x + a)^(11/2) - 385*(b*x + a)^(9/2)*a + 990*(b*x + a)^(7/2)*a^2 - 1386*(b*x + a)^(5/2)*a^3 + 1155
*(b*x + a)^(3/2)*a^4 + 1155*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*a^4 + 462*(3*(b*x + a)^(5/2) - 10*(b*x + a)^
(3/2)*a + 15*sqrt(b*x + a)*a^2)*a^3 + 198*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 -
35*sqrt(b*x + a)*a^3)*a^2 + 11*(35*(b*x + a)^(9/2) - 180*(b*x + a)^(7/2)*a + 378*(b*x + a)^(5/2)*a^2 - 420*(b
*x + a)^(3/2)*a^3 + 315*sqrt(b*x + a)*a^4)*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 {11}{2}}}{11 b} \end {gather*}

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 = 1.34, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {11}{2}}}{11 \, b} \end {gather*}

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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mupad [B]  time = 0.02, size = 12, normalized size = 0.75 \begin {gather*} \frac {2\,{\left (a+b\,x\right )}^{11/2}}{11\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(9/2),x)

[Out]

(2*(a + b*x)^(11/2))/(11*b)

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sympy [A]  time = 0.08, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a + b x\right )^{\frac {11}{2}}}{11 b} \end {gather*}

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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