3.4.16 \(\int (a+b x)^{9/2} \, dx\) [316]

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

[Out]

2/11*(b*x+a)^(11/2)/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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Mathics [A]
time = 1.60, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a+b x\right )^{\frac {11}{2}}}{11 b} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^0*(a + b*x)^(9/2),x]')

[Out]

2 (a + b x) ^ (11 / 2) / (11 b)

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Maple [A]
time = 0.10, size = 13, normalized size = 0.81

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11 b}\)	\(13\)
derivativedivides	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11 b}\)	\(13\)
default	\(\frac {2 \left (b x +a \right )^{\frac {11}{2}}}{11 b}\)	\(13\)
trager	\(\frac {2 \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right ) \sqrt {b x +a}}{11 b}\)	\(62\)
risch	\(\frac {2 \left (b^{5} x^{5}+5 a \,b^{4} x^{4}+10 a^{2} b^{3} x^{3}+10 a^{3} b^{2} x^{2}+5 a^{4} b x +a^{5}\right ) \sqrt {b x +a}}{11 b}\)	\(62\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.25, 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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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 61 vs. \(2 (12) = 24\).
time = 0.30, 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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Sympy [A]
time = 0.04, 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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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (12) = 24\).
time = 0.00, size = 445, normalized size = 27.81 \begin {gather*} \frac {\frac {2 b^{5} \left (\frac {1}{11} \sqrt {a+b x} \left (a+b x\right )^{5}-\frac {5}{9} \sqrt {a+b x} \left (a+b x\right )^{4} a+\frac {10}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a^{2}-2 \sqrt {a+b x} \left (a+b x\right )^{2} a^{3}+\frac {5}{3} \sqrt {a+b x} \left (a+b x\right ) a^{4}-\sqrt {a+b x} a^{5}\right )}{b^{5}}+\frac {10 a b^{4} \left (\frac {1}{9} \sqrt {a+b x} \left (a+b x\right )^{4}-\frac {4}{7} \sqrt {a+b x} \left (a+b x\right )^{3} a+\frac {6}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a^{2}-\frac {4}{3} \sqrt {a+b x} \left (a+b x\right ) a^{3}+\sqrt {a+b x} a^{4}\right )}{b^{4}}+\frac {20 a^{2} b^{3} \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}+\frac {20 a^{3} b^{2} \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}+10 a^{4} \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )+2 a^{5} \sqrt {a+b x}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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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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