3.438 \(\int \sqrt{x} (a+b x)^2 \, dx\)
Optimal. Leaf size=36 \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{7} b^2 x^{7/2} \]
[Out]
(2*a^2*x^(3/2))/3 + (4*a*b*x^(5/2))/5 + (2*b^2*x^(7/2))/7
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Rubi [A] time = 0.0073302, antiderivative size = 36, normalized size of antiderivative = 1.,
number of steps used = 2, number of rules used = 1, integrand size = 13, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.077, Rules used =
{43} \[ \frac{2}{3} a^2 x^{3/2}+\frac{4}{5} a b x^{5/2}+\frac{2}{7} b^2 x^{7/2} \]
Antiderivative was successfully verified.
[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 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 \sqrt{x} (a+b x)^2 \, dx &=\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] time = 0.0072735, size = 28, normalized size = 0.78 \[ \frac{2}{105} x^{3/2} \left (35 a^2+42 a b x+15 b^2 x^2\right ) \]
Antiderivative was successfully verified.
[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
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Maple [A] time = 0.003, size = 25, normalized size = 0.7 \begin{align*}{\frac{30\,{b}^{2}{x}^{2}+84\,abx+70\,{a}^{2}}{105}{x}^{{\frac{3}{2}}}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[In]
int((b*x+a)^2*x^(1/2),x)
[Out]
2/105*x^(3/2)*(15*b^2*x^2+42*a*b*x+35*a^2)
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Maxima [A] time = 1.09635, size = 32, normalized size = 0.89 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Fricas [A] time = 1.46079, size = 70, normalized size = 1.94 \begin{align*} \frac{2}{105} \,{\left (15 \, b^{2} x^{3} + 42 \, a b x^{2} + 35 \, a^{2} x\right )} \sqrt{x} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)
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Sympy [C] time = 2.77118, size = 1853, normalized size = 51.47 \begin{align*} \text{result too large to display} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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))/Abs(a) > 1), (16*I*a**(23/2)*sqrt(1 - b*(a/b + x)/a)/(-10
5*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/(-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*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)
+ 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), True))
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Giac [A] time = 1.18563, size = 32, normalized size = 0.89 \begin{align*} \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}} \end{align*}
Verification of antiderivative is not currently implemented for this CAS.
[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)