3.4.45 \(\int \frac {x^2}{(a+b x)^{3/2}} \, dx\)

Optimal. Leaf size=49 \[ -\frac {2 a^2}{b^3 \sqrt {a+b x}}-\frac {4 a \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{3/2}}{3 b^3} \]

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Rubi [A]  time = 0.01, antiderivative size = 49, 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 = {43} \begin {gather*} -\frac {2 a^2}{b^3 \sqrt {a+b x}}-\frac {4 a \sqrt {a+b x}}{b^3}+\frac {2 (a+b x)^{3/2}}{3 b^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[x^2/(a + b*x)^(3/2),x]

[Out]

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

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

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Mathematica [A]  time = 0.02, size = 34, normalized size = 0.69 \begin {gather*} \frac {2 \left (-8 a^2-4 a b x+b^2 x^2\right )}{3 b^3 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[x^2/(a + b*x)^(3/2),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 0.02, size = 37, normalized size = 0.76 \begin {gather*} \frac {2 \left (-3 a^2-6 a (a+b x)+(a+b x)^2\right )}{3 b^3 \sqrt {a+b x}} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[x^2/(a + b*x)^(3/2),x]

[Out]

(2*(-3*a^2 - 6*a*(a + b*x) + (a + b*x)^2))/(3*b^3*Sqrt[a + b*x])

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fricas [A]  time = 0.84, size = 40, normalized size = 0.82 \begin {gather*} \frac {2 \, {\left (b^{2} x^{2} - 4 \, a b x - 8 \, a^{2}\right )} \sqrt {b x + a}}{3 \, {\left (b^{4} x + a b^{3}\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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giac [A]  time = 1.14, size = 46, normalized size = 0.94 \begin {gather*} -\frac {2 \, a^{2}}{\sqrt {b x + a} b^{3}} + \frac {2 \, {\left ({\left (b x + a\right )}^{\frac {3}{2}} b^{6} - 6 \, \sqrt {b x + a} a b^{6}\right )}}{3 \, b^{9}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*a^2/(sqrt(b*x + a)*b^3) + 2/3*((b*x + a)^(3/2)*b^6 - 6*sqrt(b*x + a)*a*b^6)/b^9

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maple [A]  time = 0.01, size = 32, normalized size = 0.65 \begin {gather*} -\frac {2 \left (-b^{2} x^{2}+4 a b x +8 a^{2}\right )}{3 \sqrt {b x +a}\, b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(b*x+a)^(3/2),x)

[Out]

-2/3/(b*x+a)^(1/2)*(-b^2*x^2+4*a*b*x+8*a^2)/b^3

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maxima [A]  time = 1.29, size = 41, normalized size = 0.84 \begin {gather*} \frac {2 \, {\left (b x + a\right )}^{\frac {3}{2}}}{3 \, b^{3}} - \frac {4 \, \sqrt {b x + a} a}{b^{3}} - \frac {2 \, a^{2}}{\sqrt {b x + a} b^{3}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 0.04, size = 35, normalized size = 0.71 \begin {gather*} -\frac {12\,a\,\left (a+b\,x\right )-2\,{\left (a+b\,x\right )}^2+6\,a^2}{3\,b^3\,\sqrt {a+b\,x}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2/(a + b*x)^(3/2),x)

[Out]

-(12*a*(a + b*x) - 2*(a + b*x)^2 + 6*a^2)/(3*b^3*(a + b*x)^(1/2))

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sympy [B]  time = 1.83, size = 534, normalized size = 10.90 \begin {gather*} - \frac {16 a^{\frac {19}{2}} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {16 a^{\frac {19}{2}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {40 a^{\frac {17}{2}} b x \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {48 a^{\frac {17}{2}} b x}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {30 a^{\frac {15}{2}} b^{2} x^{2} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {48 a^{\frac {15}{2}} b^{2} x^{2}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} - \frac {4 a^{\frac {13}{2}} b^{3} x^{3} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {16 a^{\frac {13}{2}} b^{3} x^{3}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 a^{5} b^{6} x^{3}} + \frac {2 a^{\frac {11}{2}} b^{4} x^{4} \sqrt {1 + \frac {b x}{a}}}{3 a^{8} b^{3} + 9 a^{7} b^{4} x + 9 a^{6} b^{5} x^{2} + 3 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)**(3/2),x)

[Out]

-16*a**(19/2)*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 16*a**(19/
2)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 40*a**(17/2)*b*x*sqrt(1 + b*x/a)/(3*a
**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(17/2)*b*x/(3*a**8*b**3 + 9*a**7*b**4*
x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) - 30*a**(15/2)*b**2*x**2*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x
+ 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3) + 48*a**(15/2)*b**2*x**2/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x*
*2 + 3*a**5*b**6*x**3) - 4*a**(13/2)*b**3*x**3*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2
+ 3*a**5*b**6*x**3) + 16*a**(13/2)*b**3*x**3/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x*
*3) + 2*a**(11/2)*b**4*x**4*sqrt(1 + b*x/a)/(3*a**8*b**3 + 9*a**7*b**4*x + 9*a**6*b**5*x**2 + 3*a**5*b**6*x**3
)

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