\(\int x^3 \sqrt {a+b x} \, dx\) [284]

Optimal result

Integrand size = 13, antiderivative size = 72 \[ \int x^3 \sqrt {a+b x} \, dx=-\frac {2 a^3 (a+b x)^{3/2}}{3 b^4}+\frac {6 a^2 (a+b x)^{5/2}}{5 b^4}-\frac {6 a (a+b x)^{7/2}}{7 b^4}+\frac {2 (a+b x)^{9/2}}{9 b^4} \]

[Out]

-2/3*a^3*(b*x+a)^(3/2)/b^4+6/5*a^2*(b*x+a)^(5/2)/b^4-6/7*a*(b*x+a)^(7/2)/b^4+2/9*(b*x+a)^(9/2)/b^4

Rubi [A] (verified)

Time = 0.02 (sec) , antiderivative size = 72, 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} \[ \int x^3 \sqrt {a+b x} \, dx=-\frac {2 a^3 (a+b x)^{3/2}}{3 b^4}+\frac {6 a^2 (a+b x)^{5/2}}{5 b^4}+\frac {2 (a+b x)^{9/2}}{9 b^4}-\frac {6 a (a+b x)^{7/2}}{7 b^4} \]

[In]

Int[x^3*Sqrt[a + b*x],x]

[Out]

(-2*a^3*(a + b*x)^(3/2))/(3*b^4) + (6*a^2*(a + b*x)^(5/2))/(5*b^4) - (6*a*(a + b*x)^(7/2))/(7*b^4) + (2*(a + b
*x)^(9/2))/(9*b^4)

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^3 \sqrt {a+b x}}{b^3}+\frac {3 a^2 (a+b x)^{3/2}}{b^3}-\frac {3 a (a+b x)^{5/2}}{b^3}+\frac {(a+b x)^{7/2}}{b^3}\right ) \, dx \\ & = -\frac {2 a^3 (a+b x)^{3/2}}{3 b^4}+\frac {6 a^2 (a+b x)^{5/2}}{5 b^4}-\frac {6 a (a+b x)^{7/2}}{7 b^4}+\frac {2 (a+b x)^{9/2}}{9 b^4} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.02 (sec) , antiderivative size = 46, normalized size of antiderivative = 0.64 \[ \int x^3 \sqrt {a+b x} \, dx=\frac {2 (a+b x)^{3/2} \left (-16 a^3+24 a^2 b x-30 a b^2 x^2+35 b^3 x^3\right )}{315 b^4} \]

[In]

Integrate[x^3*Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2)*(-16*a^3 + 24*a^2*b*x - 30*a*b^2*x^2 + 35*b^3*x^3))/(315*b^4)

Maple [A] (verified)

Time = 0.11 (sec) , antiderivative size = 43, normalized size of antiderivative = 0.60

method	result	size
gosper	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-35 b^{3} x^{3}+30 a \,b^{2} x^{2}-24 a^{2} b x +16 a^{3}\right )}{315 b^{4}}\)	\(43\)
pseudoelliptic	\(-\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (-35 b^{3} x^{3}+30 a \,b^{2} x^{2}-24 a^{2} b x +16 a^{3}\right )}{315 b^{4}}\)	\(43\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {6 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{4}}\)	\(50\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {9}{2}}}{9}-\frac {6 a \left (b x +a \right )^{\frac {7}{2}}}{7}+\frac {6 a^{2} \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {2 a^{3} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{4}}\)	\(50\)
trager	\(-\frac {2 \left (-35 b^{4} x^{4}-5 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}-8 a^{3} b x +16 a^{4}\right ) \sqrt {b x +a}}{315 b^{4}}\)	\(54\)
risch	\(-\frac {2 \left (-35 b^{4} x^{4}-5 a \,b^{3} x^{3}+6 a^{2} b^{2} x^{2}-8 a^{3} b x +16 a^{4}\right ) \sqrt {b x +a}}{315 b^{4}}\)	\(54\)

[In]

int(x^3*(b*x+a)^(1/2),x,method=_RETURNVERBOSE)

[Out]

-2/315*(b*x+a)^(3/2)*(-35*b^3*x^3+30*a*b^2*x^2-24*a^2*b*x+16*a^3)/b^4

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 53, normalized size of antiderivative = 0.74 \[ \int x^3 \sqrt {a+b x} \, dx=\frac {2 \, {\left (35 \, b^{4} x^{4} + 5 \, a b^{3} x^{3} - 6 \, a^{2} b^{2} x^{2} + 8 \, a^{3} b x - 16 \, a^{4}\right )} \sqrt {b x + a}}{315 \, b^{4}} \]

[In]

integrate(x^3*(b*x+a)^(1/2),x, algorithm="fricas")

[Out]

2/315*(35*b^4*x^4 + 5*a*b^3*x^3 - 6*a^2*b^2*x^2 + 8*a^3*b*x - 16*a^4)*sqrt(b*x + a)/b^4

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1742 vs. \(2 (68) = 136\).

Time = 1.70 (sec) , antiderivative size = 1742, normalized size of antiderivative = 24.19 \[ \int x^3 \sqrt {a+b x} \, dx=\text {Too large to display} \]

[In]

integrate(x**3*(b*x+a)**(1/2),x)

[Out]

-32*a**(49/2)*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**
3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 32*a**(49/2)/(315*a**20*b**4 + 1890*
a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315
*a**14*b**10*x**6) - 176*a**(47/2)*b*x*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x
**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 192*a**(47/
2)*b*x/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**
4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) - 396*a**(45/2)*b**2*x**2*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1
890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 +
315*a**14*b**10*x**6) + 480*a**(45/2)*b**2*x**2/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 +
6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) - 462*a**(43/2)*b**
3*x**3*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 472
5*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 640*a**(43/2)*b**3*x**3/(315*a**20*b**4 + 1
890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 +
315*a**14*b**10*x**6) - 210*a**(41/2)*b**4*x**4*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a*
*18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 4
80*a**(41/2)*b**4*x**4/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 472
5*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 378*a**(39/2)*b**5*x**5*sqrt(1 + b*x/a)/(31
5*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a
**15*b**9*x**5 + 315*a**14*b**10*x**6) + 192*a**(39/2)*b**5*x**5/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a*
*18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 1
134*a**(37/2)*b**6*x**6*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**1
7*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 32*a**(37/2)*b**6*x**6/(31
5*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a
**15*b**9*x**5 + 315*a**14*b**10*x**6) + 1494*a**(35/2)*b**7*x**7*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19
*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**1
4*b**10*x**6) + 1098*a**(33/2)*b**8*x**8*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6
*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 430*a**(3
1/2)*b**9*x**9*sqrt(1 + b*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x*
*3 + 4725*a**16*b**8*x**4 + 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6) + 70*a**(29/2)*b**10*x**10*sqrt(1 + b
*x/a)/(315*a**20*b**4 + 1890*a**19*b**5*x + 4725*a**18*b**6*x**2 + 6300*a**17*b**7*x**3 + 4725*a**16*b**8*x**4
+ 1890*a**15*b**9*x**5 + 315*a**14*b**10*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int x^3 \sqrt {a+b x} \, dx=\frac {2 \, {\left (b x + a\right )}^{\frac {9}{2}}}{9 \, b^{4}} - \frac {6 \, {\left (b x + a\right )}^{\frac {7}{2}} a}{7 \, b^{4}} + \frac {6 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2}}{5 \, b^{4}} - \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3}}{3 \, b^{4}} \]

[In]

integrate(x^3*(b*x+a)^(1/2),x, algorithm="maxima")

[Out]

2/9*(b*x + a)^(9/2)/b^4 - 6/7*(b*x + a)^(7/2)*a/b^4 + 6/5*(b*x + a)^(5/2)*a^2/b^4 - 2/3*(b*x + a)^(3/2)*a^3/b^
4

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 116 vs. \(2 (56) = 112\).

Time = 0.30 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.61 \[ \int x^3 \sqrt {a+b x} \, dx=\frac {2 \, {\left (\frac {9 \, {\left (5 \, {\left (b x + a\right )}^{\frac {7}{2}} - 21 \, {\left (b x + a\right )}^{\frac {5}{2}} a + 35 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2} - 35 \, \sqrt {b x + a} a^{3}\right )} a}{b^{3}} + \frac {35 \, {\left (b x + a\right )}^{\frac {9}{2}} - 180 \, {\left (b x + a\right )}^{\frac {7}{2}} a + 378 \, {\left (b x + a\right )}^{\frac {5}{2}} a^{2} - 420 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{3} + 315 \, \sqrt {b x + a} a^{4}}{b^{3}}\right )}}{315 \, b} \]

[In]

integrate(x^3*(b*x+a)^(1/2),x, algorithm="giac")

[Out]

2/315*(9*(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^3 + (3
5*(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^3)/b

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 56, normalized size of antiderivative = 0.78 \[ \int x^3 \sqrt {a+b x} \, dx=\frac {2\,{\left (a+b\,x\right )}^{9/2}}{9\,b^4}-\frac {2\,a^3\,{\left (a+b\,x\right )}^{3/2}}{3\,b^4}+\frac {6\,a^2\,{\left (a+b\,x\right )}^{5/2}}{5\,b^4}-\frac {6\,a\,{\left (a+b\,x\right )}^{7/2}}{7\,b^4} \]

[In]

int(x^3*(a + b*x)^(1/2),x)

[Out]

(2*(a + b*x)^(9/2))/(9*b^4) - (2*a^3*(a + b*x)^(3/2))/(3*b^4) + (6*a^2*(a + b*x)^(5/2))/(5*b^4) - (6*a*(a + b*
x)^(7/2))/(7*b^4)