\(\int \sqrt {x} (a+b x)^2 \, dx\) [438]

Optimal result

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

[Out]

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

Rubi [A] (verified)

Time = 0.01 (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} \[ \int \sqrt {x} (a+b x)^2 \, dx=\frac {2}{3} a^2 x^{3/2}+\frac {4}{5} a b x^{5/2}+\frac {2}{7} b^2 x^{7/2} \]

[In]

Int[Sqrt[x]*(a + b*x)^2,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.01 (sec) , antiderivative size = 28, normalized size of antiderivative = 0.78 \[ \int \sqrt {x} (a+b x)^2 \, dx=\frac {2}{105} x^{3/2} \left (35 a^2+42 a b x+15 b^2 x^2\right ) \]

[In]

Integrate[Sqrt[x]*(a + b*x)^2,x]

[Out]

(2*x^(3/2)*(35*a^2 + 42*a*b*x + 15*b^2*x^2))/105

Maple [A] (verified)

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

method	result	size
gosper	\(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} x^{2}+42 a b x +35 a^{2}\right )}{105}\)	\(25\)
derivativedivides	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a b \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} x^{\frac {7}{2}}}{7}\)	\(25\)
default	\(\frac {2 a^{2} x^{\frac {3}{2}}}{3}+\frac {4 a b \,x^{\frac {5}{2}}}{5}+\frac {2 b^{2} x^{\frac {7}{2}}}{7}\)	\(25\)
trager	\(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} x^{2}+42 a b x +35 a^{2}\right )}{105}\)	\(25\)
risch	\(\frac {2 x^{\frac {3}{2}} \left (15 b^{2} x^{2}+42 a b x +35 a^{2}\right )}{105}\)	\(25\)

[In]

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

[Out]

2/105*x^(3/2)*(15*b^2*x^2+42*a*b*x+35*a^2)

Fricas [A] (verification not implemented)

none

Time = 0.22 (sec) , antiderivative size = 27, normalized size of antiderivative = 0.75 \[ \int \sqrt {x} (a+b x)^2 \, dx=\frac {2}{105} \, {\left (15 \, b^{2} x^{3} + 42 \, a b x^{2} + 35 \, a^{2} x\right )} \sqrt {x} \]

[In]

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

[Out]

2/105*(15*b^2*x^3 + 42*a*b*x^2 + 35*a^2*x)*sqrt(x)

Sympy [C] (verification not implemented)

Result contains complex when optimal does not.

Time = 81.19 (sec) , antiderivative size = 1851, normalized size of antiderivative = 51.42 \[ \int \sqrt {x} (a+b x)^2 \, dx=\text {Too large to display} \]

[In]

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

[Out]

Piecewise((16*a**(23/2)*sqrt(-1 + b*(a/b + x)/a)/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*
b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) - 16*I*a**(23/2)/(-105*a**8*b**(3/2) + 315*a**7*b**(5/
2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) - 40*a**(21/2)*b*sqrt(-1 + b*(
a/b + x)/a)*(a/b + x)/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105
*a**5*b**(9/2)*(a/b + x)**3) + 48*I*a**(21/2)*b*(a/b + x)/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) -
315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) + 30*a**(19/2)*b**2*sqrt(-1 + b*(a/b + x)/a)*
(a/b + x)**2/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**
(9/2)*(a/b + x)**3) - 48*I*a**(19/2)*b**2*(a/b + x)**2/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315
*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) - 40*a**(17/2)*b**3*sqrt(-1 + b*(a/b + x)/a)*(a/
b + x)**3/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/
2)*(a/b + x)**3) + 16*I*a**(17/2)*b**3*(a/b + x)**3/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a*
*6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) + 100*a**(15/2)*b**4*sqrt(-1 + b*(a/b + x)/a)*(a/b
+ x)**4/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)
*(a/b + x)**3) - 96*a**(13/2)*b**5*sqrt(-1 + b*(a/b + x)/a)*(a/b + x)**5/(-105*a**8*b**(3/2) + 315*a**7*b**(5/
2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) + 30*a**(11/2)*b**6*sqrt(-1 +
b*(a/b + x)/a)*(a/b + x)**6/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2
+ 105*a**5*b**(9/2)*(a/b + x)**3), Abs(b*(a/b + x)/a) > 1), (16*I*a**(23/2)*sqrt(1 - b*(a/b + x)/a)/(-105*a**
8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) -
16*I*a**(23/2)/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b
**(9/2)*(a/b + x)**3) - 40*I*a**(21/2)*b*sqrt(1 - b*(a/b + x)/a)*(a/b + x)/(-105*a**8*b**(3/2) + 315*a**7*b**(
5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) + 48*I*a**(21/2)*b*(a/b + x)
/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b +
x)**3) + 30*I*a**(19/2)*b**2*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**2/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/
b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) - 48*I*a**(19/2)*b**2*(a/b + x)**2/(
-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x
)**3) - 40*I*a**(17/2)*b**3*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**3/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b
+ x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) + 16*I*a**(17/2)*b**3*(a/b + x)**3/(-1
05*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)*
*3) + 100*I*a**(15/2)*b**4*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**4/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b +
x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3) - 96*I*a**(13/2)*b**5*sqrt(1 - b*(a/b +
x)/a)*(a/b + x)**5/(-105*a**8*b**(3/2) + 315*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a
**5*b**(9/2)*(a/b + x)**3) + 30*I*a**(11/2)*b**6*sqrt(1 - b*(a/b + x)/a)*(a/b + x)**6/(-105*a**8*b**(3/2) + 31
5*a**7*b**(5/2)*(a/b + x) - 315*a**6*b**(7/2)*(a/b + x)**2 + 105*a**5*b**(9/2)*(a/b + x)**3), True))

Maxima [A] (verification not implemented)

none

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

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \sqrt {x} (a+b x)^2 \, dx=\frac {2}{7} \, b^{2} x^{\frac {7}{2}} + \frac {4}{5} \, a b x^{\frac {5}{2}} + \frac {2}{3} \, a^{2} x^{\frac {3}{2}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 24, normalized size of antiderivative = 0.67 \[ \int \sqrt {x} (a+b x)^2 \, dx=\frac {2\,x^{3/2}\,\left (35\,a^2+42\,a\,b\,x+15\,b^2\,x^2\right )}{105} \]

[In]

int(x^(1/2)*(a + b*x)^2,x)

[Out]

(2*x^(3/2)*(35*a^2 + 15*b^2*x^2 + 42*a*b*x))/105