3.2.79 \(\int x^2 \sqrt {a+b x} \, dx\) [179]

Optimal. Leaf size=53 \[ \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]
time = 0.01, antiderivative size = 53, normalized size of antiderivative = 1.00, 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 = {45} \begin {gather*} \frac {2 a^2 (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{7/2}}{7 b^3}-\frac {4 a (a+b x)^{5/2}}{5 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2*Sqrt[a + b*x],x]

[Out]

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

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

________________________________________________________________________________________

Mathematica [A]
time = 0.02, size = 35, normalized size = 0.66 \begin {gather*} \frac {2 (a+b x)^{3/2} \left (8 a^2-12 a b x+15 b^2 x^2\right )}{105 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2*Sqrt[a + b*x],x]

[Out]

(2*(a + b*x)^(3/2)*(8*a^2 - 12*a*b*x + 15*b^2*x^2))/(105*b^3)

________________________________________________________________________________________

Mathics [B] Leaf count is larger than twice the leaf count of optimal. \(200\) vs. \(2(53)=106\).
time = 7.66, size = 184, normalized size = 3.47 \begin {gather*} \frac {2 \sqrt {a} \left (8 a^6 \left (-1+\sqrt {\frac {a+b x}{a}}\right )+4 a^5 b x \left (-6+5 \sqrt {\frac {a+b x}{a}}\right )+3 a^4 b^2 x^2 \left (-8+5 \sqrt {\frac {a+b x}{a}}\right )+10 a^2 b^3 x^3 \left (2 a+5 b x\right ) \sqrt {\frac {a+b x}{a}}-8 a^3 b^3 x^3+48 a b^5 x^5 \sqrt {\frac {a+b x}{a}}+15 b^6 x^6 \sqrt {\frac {a+b x}{a}}\right )}{105 b^3 \left (a^3+3 a^2 b x+3 a b^2 x^2+b^3 x^3\right )} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

mathics('Integrate[x^2*Sqrt[a+b*x],x]')

[Out]

2 Sqrt[a] (8 a ^ 6 (-1 + Sqrt[(a + b x) / a]) + 4 a ^ 5 b x (-6 + 5 Sqrt[(a + b x) / a]) + 3 a ^ 4 b ^ 2 x ^ 2
(-8 + 5 Sqrt[(a + b x) / a]) + 10 a ^ 2 b ^ 3 x ^ 3 (2 a + 5 b x) Sqrt[(a + b x) / a] - 8 a ^ 3 b ^ 3 x ^ 3 +
48 a b ^ 5 x ^ 5 Sqrt[(a + b x) / a] + 15 b ^ 6 x ^ 6 Sqrt[(a + b x) / a]) / (105 b ^ 3 (a ^ 3 + 3 a ^ 2 b x
+ 3 a b ^ 2 x ^ 2 + b ^ 3 x ^ 3))

________________________________________________________________________________________

Maple [A]
time = 0.04, size = 38, normalized size = 0.72

method	result	size
gosper	\(\frac {2 \left (b x +a \right )^{\frac {3}{2}} \left (15 x^{2} b^{2}-12 a b x +8 a^{2}\right )}{105 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\)	\(38\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {7}{2}}}{7}-\frac {4 a \left (b x +a \right )^{\frac {5}{2}}}{5}+\frac {2 a^{2} \left (b x +a \right )^{\frac {3}{2}}}{3}}{b^{3}}\)	\(38\)
trager	\(\frac {2 \left (15 b^{3} x^{3}+3 a \,b^{2} x^{2}-4 a^{2} b x +8 a^{3}\right ) \sqrt {b x +a}}{105 b^{3}}\)	\(43\)
risch	\(\frac {2 \left (15 b^{3} x^{3}+3 a \,b^{2} x^{2}-4 a^{2} b x +8 a^{3}\right ) \sqrt {b x +a}}{105 b^{3}}\)	\(43\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/b^3*(1/7*(b*x+a)^(7/2)-2/5*a*(b*x+a)^(5/2)+1/3*a^2*(b*x+a)^(3/2))

________________________________________________________________________________________

Maxima [A]
time = 0.29, size = 41, normalized size = 0.77 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {7}{2}}}{7 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {5}{2}} a}{5 \, b^{3}} + \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}} a^{2}}{3 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

________________________________________________________________________________________

Fricas [A]
time = 0.30, size = 42, normalized size = 0.79 \begin {gather*} \frac {2 \, {\left (15 \, b^{3} x^{3} + 3 \, a b^{2} x^{2} - 4 \, a^{2} b x + 8 \, a^{3}\right )} \sqrt {b x + a}}{105 \, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(15*b^3*x^3 + 3*a*b^2*x^2 - 4*a^2*b*x + 8*a^3)*sqrt(b*x + a)/b^3

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 666 vs. \(2 (49) = 98\)
time = 0.85, size = 666, normalized size = 12.57 \begin {gather*} \frac {16 a^{\frac {23}{2}} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {23}{2}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {21}{2}} b x \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {21}{2}} b x}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {19}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b^{2} x^{2}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {17}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {17}{2}} b^{3} x^{3}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {100 a^{\frac {15}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {96 a^{\frac {13}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {11}{2}} b^{6} x^{6} \sqrt {1 + \frac {b x}{a}}}{105 a^{8} b^{3} + 315 a^{7} b^{4} x + 315 a^{6} b^{5} x^{2} + 105 a^{5} b^{6} x^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

16*a**(23/2)*sqrt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 16*
a**(23/2)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(21/2)*b*x*sqrt(
1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(21/2)*b*x/(105
*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 30*a**(19/2)*b**2*x**2*sqrt(1 + b*x/
a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 48*a**(19/2)*b**2*x**2/(105*a
**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 40*a**(17/2)*b**3*x**3*sqrt(1 + b*x/a)
/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) - 16*a**(17/2)*b**3*x**3/(105*a**
8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 100*a**(15/2)*b**4*x**4*sqrt(1 + b*x/a)/
(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 96*a**(13/2)*b**5*x**5*sqrt(1 +
b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3) + 30*a**(11/2)*b**6*x**6*sq
rt(1 + b*x/a)/(105*a**8*b**3 + 315*a**7*b**4*x + 315*a**6*b**5*x**2 + 105*a**5*b**6*x**3)

________________________________________________________________________________________

Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 93 vs. \(2 (41) = 82\).
time = 0.00, size = 145, normalized size = 2.74 \begin {gather*} \frac {\frac {2 b \left (\frac {1}{7} \sqrt {a+b x} \left (a+b x\right )^{3}-\frac {3}{5} \sqrt {a+b x} \left (a+b x\right )^{2} a+\sqrt {a+b x} \left (a+b x\right ) a^{2}-\sqrt {a+b x} a^{3}\right )}{b^{3}}+\frac {2 a \left (\frac {1}{5} \sqrt {a+b x} \left (a+b x\right )^{2}-\frac {2}{3} \sqrt {a+b x} \left (a+b x\right ) a+\sqrt {a+b x} a^{2}\right )}{b^{2}}}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/105*(7*(3*(b*x + a)^(5/2) - 10*(b*x + a)^(3/2)*a + 15*sqrt(b*x + a)*a^2)*a/b^2 + 3*(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)/b^2)/b

________________________________________________________________________________________

Mupad [B]
time = 0.15, size = 37, normalized size = 0.70 \begin {gather*} \frac {30\,{\left (a+b\,x\right )}^{7/2}-84\,a\,{\left (a+b\,x\right )}^{5/2}+70\,a^2\,{\left (a+b\,x\right )}^{3/2}}{105\,b^3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*(a + b*x)^(1/2),x)

[Out]

(30*(a + b*x)^(7/2) - 84*a*(a + b*x)^(5/2) + 70*a^2*(a + b*x)^(3/2))/(105*b^3)

________________________________________________________________________________________