\(\int x^{3/2} (a+b x)^2 \, dx\) [437]

Optimal result

Integrand size = 13, antiderivative size = 36 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2} \]

[Out]

2/5*a^2*x^(5/2)+4/7*a*b*x^(7/2)+2/9*b^2*x^(9/2)

Rubi [A] (verified)

Time = 0.00 (sec) , antiderivative size = 36, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.077, Rules used = {45} \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2} \]

[In]

Int[x^(3/2)*(a + b*x)^2,x]

[Out]

(2*a^2*x^(5/2))/5 + (4*a*b*x^(7/2))/7 + (2*b^2*x^(9/2))/9

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 (a^2 x^{3/2}+2 a b x^{5/2}+b^2 x^{7/2}\right ) \, dx \\ & = \frac {2}{5} a^2 x^{5/2}+\frac {4}{7} a b x^{7/2}+\frac {2}{9} b^2 x^{9/2} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{315} x^{5/2} \left (63 a^2+90 a b x+35 b^2 x^2\right ) \]

[In]

Integrate[x^(3/2)*(a + b*x)^2,x]

[Out]

(2*x^(5/2)*(63*a^2 + 90*a*b*x + 35*b^2*x^2))/315

Maple [A] (verified)

Time = 0.07 (sec) , antiderivative size = 25, normalized size of antiderivative = 0.69

method	result	size
gosper	\(\frac {2 x^{\frac {5}{2}} \left (35 b^{2} x^{2}+90 a b x +63 a^{2}\right )}{315}\)	\(25\)
derivativedivides	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {9}{2}}}{9}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {5}{2}}}{5}+\frac {4 a b \,x^{\frac {7}{2}}}{7}+\frac {2 b^{2} x^{\frac {9}{2}}}{9}\)	\(25\)
trager	\(\frac {2 x^{\frac {5}{2}} \left (35 b^{2} x^{2}+90 a b x +63 a^{2}\right )}{315}\)	\(25\)
risch	\(\frac {2 x^{\frac {5}{2}} \left (35 b^{2} x^{2}+90 a b x +63 a^{2}\right )}{315}\)	\(25\)

[In]

int(x^(3/2)*(b*x+a)^2,x,method=_RETURNVERBOSE)

[Out]

2/315*x^(5/2)*(35*b^2*x^2+90*a*b*x+63*a^2)

Fricas [A] (verification not implemented)

none

Time = 0.23 (sec) , antiderivative size = 29, normalized size of antiderivative = 0.81 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{315} \, {\left (35 \, b^{2} x^{4} + 90 \, a b x^{3} + 63 \, a^{2} x^{2}\right )} \sqrt {x} \]

[In]

integrate(x^(3/2)*(b*x+a)^2,x, algorithm="fricas")

[Out]

2/315*(35*b^2*x^4 + 90*a*b*x^3 + 63*a^2*x^2)*sqrt(x)

Sympy [A] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 34, normalized size of antiderivative = 0.94 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2 a^{2} x^{\frac {5}{2}}}{5} + \frac {4 a b x^{\frac {7}{2}}}{7} + \frac {2 b^{2} x^{\frac {9}{2}}}{9} \]

[In]

integrate(x**(3/2)*(b*x+a)**2,x)

[Out]

2*a**2*x**(5/2)/5 + 4*a*b*x**(7/2)/7 + 2*b**2*x**(9/2)/9

Maxima [A] (verification not implemented)

none

Time = 0.20 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

[In]

integrate(x^(3/2)*(b*x+a)^2,x, algorithm="maxima")

[Out]

2/9*b^2*x^(9/2) + 4/7*a*b*x^(7/2) + 2/5*a^2*x^(5/2)

Giac [A] (verification not implemented)

none

Time = 0.30 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2}{9} \, b^{2} x^{\frac {9}{2}} + \frac {4}{7} \, a b x^{\frac {7}{2}} + \frac {2}{5} \, a^{2} x^{\frac {5}{2}} \]

[In]

integrate(x^(3/2)*(b*x+a)^2,x, algorithm="giac")

[Out]

2/9*b^2*x^(9/2) + 4/7*a*b*x^(7/2) + 2/5*a^2*x^(5/2)

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int x^{3/2} (a+b x)^2 \, dx=\frac {2\,x^{5/2}\,\left (63\,a^2+90\,a\,b\,x+35\,b^2\,x^2\right )}{315} \]

[In]

int(x^(3/2)*(a + b*x)^2,x)

[Out]

(2*x^(5/2)*(63*a^2 + 35*b^2*x^2 + 90*a*b*x))/315