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

Optimal. Leaf size=34 \[ \frac{2 (a+b x)^{9/2}}{9 b^2}-\frac{2 a (a+b x)^{7/2}}{7 b^2} \]

[Out]

(-2*a*(a + b*x)^(7/2))/(7*b^2) + (2*(a + b*x)^(9/2))/(9*b^2)

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Rubi [A]  time = 0.0086827, antiderivative size = 34, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 11, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.091, Rules used = {43} \[ \frac{2 (a+b x)^{9/2}}{9 b^2}-\frac{2 a (a+b x)^{7/2}}{7 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x*(a + b*x)^(5/2),x]

[Out]

(-2*a*(a + b*x)^(7/2))/(7*b^2) + (2*(a + b*x)^(9/2))/(9*b^2)

Rule 43

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*x)^m*(c + d
*x)^n, x], x] /; FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int x (a+b x)^{5/2} \, dx &=\int \left (-\frac{a (a+b x)^{5/2}}{b}+\frac{(a+b x)^{7/2}}{b}\right ) \, dx\\ &=-\frac{2 a (a+b x)^{7/2}}{7 b^2}+\frac{2 (a+b x)^{9/2}}{9 b^2}\\ \end{align*}

Mathematica [A]  time = 0.0257153, size = 24, normalized size = 0.71 \[ \frac{2 (a+b x)^{7/2} (7 b x-2 a)}{63 b^2} \]

Antiderivative was successfully verified.

[In]

Integrate[x*(a + b*x)^(5/2),x]

[Out]

(2*(a + b*x)^(7/2)*(-2*a + 7*b*x))/(63*b^2)

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Maple [A]  time = 0.001, size = 21, normalized size = 0.6 \begin{align*} -{\frac{-14\,bx+4\,a}{63\,{b}^{2}} \left ( bx+a \right ) ^{{\frac{7}{2}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/63*(b*x+a)^(7/2)*(-7*b*x+2*a)/b^2

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Maxima [A]  time = 1.12851, size = 35, normalized size = 1.03 \begin{align*} \frac{2 \,{\left (b x + a\right )}^{\frac{9}{2}}}{9 \, b^{2}} - \frac{2 \,{\left (b x + a\right )}^{\frac{7}{2}} a}{7 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/9*(b*x + a)^(9/2)/b^2 - 2/7*(b*x + a)^(7/2)*a/b^2

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Fricas [A]  time = 1.47407, size = 116, normalized size = 3.41 \begin{align*} \frac{2 \,{\left (7 \, b^{4} x^{4} + 19 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + a^{3} b x - 2 \, a^{4}\right )} \sqrt{b x + a}}{63 \, b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/63*(7*b^4*x^4 + 19*a*b^3*x^3 + 15*a^2*b^2*x^2 + a^3*b*x - 2*a^4)*sqrt(b*x + a)/b^2

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Sympy [A]  time = 3.0582, size = 102, normalized size = 3. \begin{align*} \begin{cases} - \frac{4 a^{4} \sqrt{a + b x}}{63 b^{2}} + \frac{2 a^{3} x \sqrt{a + b x}}{63 b} + \frac{10 a^{2} x^{2} \sqrt{a + b x}}{21} + \frac{38 a b x^{3} \sqrt{a + b x}}{63} + \frac{2 b^{2} x^{4} \sqrt{a + b x}}{9} & \text{for}\: b \neq 0 \\\frac{a^{\frac{5}{2}} x^{2}}{2} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Piecewise((-4*a**4*sqrt(a + b*x)/(63*b**2) + 2*a**3*x*sqrt(a + b*x)/(63*b) + 10*a**2*x**2*sqrt(a + b*x)/21 + 3
8*a*b*x**3*sqrt(a + b*x)/63 + 2*b**2*x**4*sqrt(a + b*x)/9, Ne(b, 0)), (a**(5/2)*x**2/2, True))

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Giac [B]  time = 1.21709, size = 162, normalized size = 4.76 \begin{align*} \frac{2 \,{\left (\frac{21 \,{\left (3 \,{\left (b x + a\right )}^{\frac{5}{2}} - 5 \,{\left (b x + a\right )}^{\frac{3}{2}} a\right )} a^{2}}{b} + \frac{6 \,{\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}{b} + \frac{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}}{b}\right )}}{315 \, b} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/315*(21*(3*(b*x + a)^(5/2) - 5*(b*x + a)^(3/2)*a)*a^2/b + 6*(15*(b*x + a)^(7/2) - 42*(b*x + a)^(5/2)*a + 35*
(b*x + a)^(3/2)*a^2)*a/b + (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)/b)/b