3.4.3 \(\int (a+b x)^{5/2} \, dx\) [303]

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

[Out]

2/7*(b*x+a)^(7/2)/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)^{7/2}}{7 b} \end {gather*}

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*}

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Mathematica [A]
time = 0.01, size = 16, normalized size = 1.00 \begin {gather*} \frac {2 (a+b x)^{7/2}}{7 b} \end {gather*}

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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Mathics [A]
time = 1.58, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a+b x\right )^{\frac {7}{2}}}{7 b} \end {gather*}

Antiderivative was successfully verified.

[In]

mathics('Integrate[x^0*(a + b*x)^(5/2),x]')

[Out]

2 (a + b x) ^ (7 / 2) / (7 b)

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

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7 b}\)	\(13\)
derivativedivides	\(\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7 b}\)	\(13\)
default	\(\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7 b}\)	\(13\)
trager	\(\frac {2 \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {b x +a}}{7 b}\)	\(40\)
risch	\(\frac {2 \left (b^{3} x^{3}+3 a \,b^{2} x^{2}+3 a^{2} b x +a^{3}\right ) \sqrt {b x +a}}{7 b}\)	\(40\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]
time = 0.27, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 39 vs. \(2 (12) = 24\).
time = 0.30, size = 39, normalized size = 2.44 \begin {gather*} \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 {gather*}

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 = 0.03, size = 12, normalized size = 0.75 \begin {gather*} \frac {2 \left (a + b x\right )^{\frac {7}{2}}}{7 b} \end {gather*}

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] Leaf count of result is larger than twice the leaf count of optimal. 95 vs. \(2 (12) = 24\).
time = 0.00, size = 199, normalized size = 12.44 \begin {gather*} \frac {\frac {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 {6 a 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}}+6 a^{2} \left (\frac {1}{3} \sqrt {a+b x} \left (a+b x\right )-a \sqrt {a+b x}\right )+2 a^{3} \sqrt {a+b x}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/35*(5*(b*x + a)^(7/2) - 21*(b*x + a)^(5/2)*a + 35*(b*x + a)^(3/2)*a^2 + 35*((b*x + a)^(3/2) - 3*sqrt(b*x + a
)*a)*a^2 + 7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*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 )}^{7/2}}{7\,b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + b*x)^(5/2),x)

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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