\(\int x (a+b x)^{5/2} \, dx\) [302]

Optimal result

Integrand size = 11, antiderivative size = 34 \[ \int x (a+b x)^{5/2} \, dx=-\frac {2 a (a+b x)^{7/2}}{7 b^2}+\frac {2 (a+b x)^{9/2}}{9 b^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.01 (sec) , antiderivative size = 34, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.091, Rules used = {45} \[ \int x (a+b x)^{5/2} \, dx=\frac {2 (a+b x)^{9/2}}{9 b^2}-\frac {2 a (a+b x)^{7/2}}{7 b^2} \]

[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 45

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*} \text {integral}& = \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] (verified)

Time = 0.02 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.71 \[ \int x (a+b x)^{5/2} \, dx=\frac {2 (a+b x)^{7/2} (-2 a+7 b x)}{63 b^2} \]

[In]

Integrate[x*(a + b*x)^(5/2),x]

[Out]

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

Maple [A] (verified)

Time = 0.09 (sec) , antiderivative size = 21, normalized size of antiderivative = 0.62

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-7 b x +2 a \right )}{63 b^{2}}\)	\(21\)
pseudoelliptic	\(-\frac {2 \left (b x +a \right )^{\frac {7}{2}} \left (-7 b x +2 a \right )}{63 b^{2}}\)	\(21\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{2}}\)	\(26\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {2 a \left (b x +a \right )^{\frac {7}{2}}}{7}}{b^{2}}\)	\(26\)
trager	\(-\frac {2 \left (-7 b^{4} x^{4}-19 a \,b^{3} x^{3}-15 a^{2} b^{2} x^{2}-a^{3} b x +2 a^{4}\right ) \sqrt {b x +a}}{63 b^{2}}\)	\(54\)
risch	\(-\frac {2 \left (-7 b^{4} x^{4}-19 a \,b^{3} x^{3}-15 a^{2} b^{2} x^{2}-a^{3} b x +2 a^{4}\right ) \sqrt {b x +a}}{63 b^{2}}\)	\(54\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 52, normalized size of antiderivative = 1.53 \[ \int x (a+b x)^{5/2} \, dx=\frac {2 \, {\left (7 \, b^{4} x^{4} + 19 \, a b^{3} x^{3} + 15 \, a^{2} b^{2} x^{2} + a^{3} b x - 2 \, a^{4}\right )} \sqrt {b x + a}}{63 \, b^{2}} \]

[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

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 102 vs. \(2 (31) = 62\).

Time = 0.52 (sec) , antiderivative size = 102, normalized size of antiderivative = 3.00 \[ \int x (a+b x)^{5/2} \, dx=\begin {cases} - \frac {4 a^{4} \sqrt {a + b x}}{63 b^{2}} + \frac {2 a^{3} x \sqrt {a + b x}}{63 b} + \frac {10 a^{2} x^{2} \sqrt {a + b x}}{21} + \frac {38 a b x^{3} \sqrt {a + b x}}{63} + \frac {2 b^{2} x^{4} \sqrt {a + b x}}{9} & \text {for}\: b \neq 0 \\\frac {a^{\frac {5}{2}} x^{2}}{2} & \text {otherwise} \end {cases} \]

[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))

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 26, normalized size of antiderivative = 0.76 \[ \int x (a+b x)^{5/2} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}}}{9 \, b^{2}} - \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}} a}{7 \, b^{2}} \]

[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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 182 vs. \(2 (26) = 52\).

Time = 0.32 (sec) , antiderivative size = 182, normalized size of antiderivative = 5.35 \[ \int x (a+b x)^{5/2} \, dx=\frac {2 \, {\left (\frac {105 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} - 3 \, \sqrt {b x + a} a\right )} a^{3}}{b} + \frac {63 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )} a^{2}}{b} + \frac {27 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a}{b} + \frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}}{b}\right )}}{315 \, b} \]

[In]

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

[Out]

2/315*(105*((b*x + a)^(3/2) - 3*sqrt(b*x + a)*a)*a^3/b + 63*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqr
t(b*x + a)*a^2)*a^2/b + 27*(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/b + (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)/b)/b

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.74 \[ \int x (a+b x)^{5/2} \, dx=-\frac {18\,a\,{\left (a+b\,x\right )}^{7/2}-14\,{\left (a+b\,x\right )}^{9/2}}{63\,b^2} \]

[In]

int(x*(a + b*x)^(5/2),x)

[Out]

-(18*a*(a + b*x)^(7/2) - 14*(a + b*x)^(9/2))/(63*b^2)