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

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

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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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IntegrateAlgebraic [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]

IntegrateAlgebraic[(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2))/(7*b)

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fricas [B]  time = 0.98, 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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giac [B]  time = 0.98, size = 95, normalized size = 5.94 \begin {gather*} \frac {2 \, {\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 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{2} + 7 \, {\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\right )}}{35 \, b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[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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maple [A]  time = 0.00, size = 13, normalized size = 0.81 \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]

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

[Out]

2/7*(b*x+a)^(7/2)/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 {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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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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sympy [A]  time = 0.08, 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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