3.2.84 \(\int \frac {x^2}{\sqrt {a+b x}} \, dx\) [184]

Optimal. Leaf size=51 \[ \frac {2 a^2 \sqrt {a+b x}}{b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3} \]

[Out]

-4/3*a*(b*x+a)^(3/2)/b^3+2/5*(b*x+a)^(5/2)/b^3+2*a^2*(b*x+a)^(1/2)/b^3

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Rubi [A]
time = 0.01, antiderivative size = 51, 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 \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{5/2}}{5 b^3}-\frac {4 a (a+b x)^{3/2}}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/Sqrt[a + b*x],x]

[Out]

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

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Mathematica [A]
time = 0.02, size = 35, normalized size = 0.69 \begin {gather*} \frac {2 \sqrt {a+b x} \left (8 a^2-4 a b x+3 b^2 x^2\right )}{15 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/Sqrt[a + b*x],x]

[Out]

(2*Sqrt[a + b*x]*(8*a^2 - 4*a*b*x + 3*b^2*x^2))/(15*b^3)

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Maple [A]
time = 0.03, size = 37, normalized size = 0.73

method	result	size
gosper	\(\frac {2 \sqrt {b x +a}\, \left (3 x^{2} b^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\)	\(32\)
trager	\(\frac {2 \sqrt {b x +a}\, \left (3 x^{2} b^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\)	\(32\)
risch	\(\frac {2 \sqrt {b x +a}\, \left (3 x^{2} b^{2}-4 a b x +8 a^{2}\right )}{15 b^{3}}\)	\(32\)
derivativedivides	\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {b x +a}}{b^{3}}\)	\(37\)
default	\(\frac {\frac {2 \left (b x +a \right )^{\frac {5}{2}}}{5}-\frac {4 a \left (b x +a \right )^{\frac {3}{2}}}{3}+2 a^{2} \sqrt {b x +a}}{b^{3}}\)	\(37\)

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

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Maxima [A]
time = 2.04, size = 41, normalized size = 0.80 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {5}{2}}}{5 \, b^{3}} - \frac {4 \, {\left (b x + a\right )}^{\frac {3}{2}} a}{3 \, b^{3}} + \frac {2 \, \sqrt {b x + a} a^{2}}{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]

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

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Fricas [A]
time = 0.61, size = 31, normalized size = 0.61 \begin {gather*} \frac {2 \, {\left (3 \, b^{2} x^{2} - 4 \, a b x + 8 \, a^{2}\right )} \sqrt {b x + a}}{15 \, 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]

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

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Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 600 vs. \(2 (48) = 96\).
time = 0.81, size = 600, normalized size = 11.76 \begin {gather*} \frac {16 a^{\frac {21}{2}} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {21}{2}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {40 a^{\frac {19}{2}} b x \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {19}{2}} b x}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {30 a^{\frac {17}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {48 a^{\frac {17}{2}} b^{2} x^{2}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {15}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} - \frac {16 a^{\frac {15}{2}} b^{3} x^{3}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {10 a^{\frac {13}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 a^{5} b^{6} x^{3}} + \frac {6 a^{\frac {11}{2}} b^{5} x^{5} \sqrt {1 + \frac {b x}{a}}}{15 a^{8} b^{3} + 45 a^{7} b^{4} x + 45 a^{6} b^{5} x^{2} + 15 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**(21/2)*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 16*a**(
21/2)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 40*a**(19/2)*b*x*sqrt(1 + b*x/
a)/(15*a**8*b**3 + 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 48*a**(19/2)*b*x/(15*a**8*b**3 +
45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 30*a**(17/2)*b**2*x**2*sqrt(1 + b*x/a)/(15*a**8*b**3
+ 45*a**7*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 48*a**(17/2)*b**2*x**2/(15*a**8*b**3 + 45*a**7*b*
*4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 10*a**(15/2)*b**3*x**3*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7
*b**4*x + 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) - 16*a**(15/2)*b**3*x**3/(15*a**8*b**3 + 45*a**7*b**4*x + 45*
a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 10*a**(13/2)*b**4*x**4*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x +
45*a**6*b**5*x**2 + 15*a**5*b**6*x**3) + 6*a**(11/2)*b**5*x**5*sqrt(1 + b*x/a)/(15*a**8*b**3 + 45*a**7*b**4*x
+ 45*a**6*b**5*x**2 + 15*a**5*b**6*x**3)

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Giac [A]
time = 0.46, size = 37, normalized size = 0.73 \begin {gather*} \frac {2 \, {\left (3 \, {\left (b x + a\right )}^{\frac {5}{2}} - 10 \, {\left (b x + a\right )}^{\frac {3}{2}} a + 15 \, \sqrt {b x + a} a^{2}\right )}}{15 \, 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="giac")

[Out]

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

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Mupad [B]
time = 0.04, size = 37, normalized size = 0.73 \begin {gather*} \frac {6\,{\left (a+b\,x\right )}^{5/2}-20\,a\,{\left (a+b\,x\right )}^{3/2}+30\,a^2\,\sqrt {a+b\,x}}{15\,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]

(6*(a + b*x)^(5/2) - 20*a*(a + b*x)^(3/2) + 30*a^2*(a + b*x)^(1/2))/(15*b^3)

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