3.2199 \(\int \frac{x^3}{(a+b \sqrt{x})^2} \, dx\)

Optimal. Leaf size=111 \[ -\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}+\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]

[Out]

(2*a^7)/(b^8*(a + b*Sqrt[x])) - (12*a^5*Sqrt[x])/b^7 + (5*a^4*x)/b^6 - (8*a^3*x^(3/2))/(3*b^5) + (3*a^2*x^2)/(
2*b^4) - (4*a*x^(5/2))/(5*b^3) + x^3/(3*b^2) + (14*a^6*Log[a + b*Sqrt[x]])/b^8

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Rubi [A]  time = 0.0827748, antiderivative size = 111, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.133, Rules used = {266, 43} \[ -\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}+\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x])^2,x]

[Out]

(2*a^7)/(b^8*(a + b*Sqrt[x])) - (12*a^5*Sqrt[x])/b^7 + (5*a^4*x)/b^6 - (8*a^3*x^(3/2))/(3*b^5) + (3*a^2*x^2)/(
2*b^4) - (4*a*x^(5/2))/(5*b^3) + x^3/(3*b^2) + (14*a^6*Log[a + b*Sqrt[x]])/b^8

Rule 266

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplify[(m + 1)/n] - 1)*(a
+ b*x)^p, x], x, x^n], x] /; FreeQ[{a, b, m, n, p}, x] && IntegerQ[Simplify[(m + 1)/n]]

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^3}{\left (a+b \sqrt{x}\right )^2} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (-\frac{6 a^5}{b^7}+\frac{5 a^4 x}{b^6}-\frac{4 a^3 x^2}{b^5}+\frac{3 a^2 x^3}{b^4}-\frac{2 a x^4}{b^3}+\frac{x^5}{b^2}-\frac{a^7}{b^7 (a+b x)^2}+\frac{7 a^6}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 a^7}{b^8 \left (a+b \sqrt{x}\right )}-\frac{12 a^5 \sqrt{x}}{b^7}+\frac{5 a^4 x}{b^6}-\frac{8 a^3 x^{3/2}}{3 b^5}+\frac{3 a^2 x^2}{2 b^4}-\frac{4 a x^{5/2}}{5 b^3}+\frac{x^3}{3 b^2}+\frac{14 a^6 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.079838, size = 102, normalized size = 0.92 \[ \frac{-80 a^3 b^3 x^{3/2}+45 a^2 b^4 x^2+150 a^4 b^2 x+\frac{60 a^7}{a+b \sqrt{x}}-360 a^5 b \sqrt{x}+420 a^6 \log \left (a+b \sqrt{x}\right )-24 a b^5 x^{5/2}+10 b^6 x^3}{30 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x])^2,x]

[Out]

((60*a^7)/(a + b*Sqrt[x]) - 360*a^5*b*Sqrt[x] + 150*a^4*b^2*x - 80*a^3*b^3*x^(3/2) + 45*a^2*b^4*x^2 - 24*a*b^5
*x^(5/2) + 10*b^6*x^3 + 420*a^6*Log[a + b*Sqrt[x]])/(30*b^8)

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Maple [A]  time = 0.007, size = 94, normalized size = 0.9 \begin{align*} 5\,{\frac{{a}^{4}x}{{b}^{6}}}-{\frac{8\,{a}^{3}}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}+{\frac{3\,{a}^{2}{x}^{2}}{2\,{b}^{4}}}-{\frac{4\,a}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}+{\frac{{x}^{3}}{3\,{b}^{2}}}+14\,{\frac{{a}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}-12\,{\frac{{a}^{5}\sqrt{x}}{{b}^{7}}}+2\,{\frac{{a}^{7}}{{b}^{8} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a+b*x^(1/2))^2,x)

[Out]

5*a^4*x/b^6-8/3*a^3*x^(3/2)/b^5+3/2*a^2*x^2/b^4-4/5*a*x^(5/2)/b^3+1/3*x^3/b^2+14*a^6*ln(a+b*x^(1/2))/b^8-12*a^
5*x^(1/2)/b^7+2*a^7/b^8/(a+b*x^(1/2))

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Maxima [A]  time = 0.964366, size = 174, normalized size = 1.57 \begin{align*} \frac{14 \, a^{6} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{{\left (b \sqrt{x} + a\right )}^{6}}{3 \, b^{8}} - \frac{14 \,{\left (b \sqrt{x} + a\right )}^{5} a}{5 \, b^{8}} + \frac{21 \,{\left (b \sqrt{x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac{70 \,{\left (b \sqrt{x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac{35 \,{\left (b \sqrt{x} + a\right )}^{2} a^{4}}{b^{8}} - \frac{42 \,{\left (b \sqrt{x} + a\right )} a^{5}}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="maxima")

[Out]

14*a^6*log(b*sqrt(x) + a)/b^8 + 1/3*(b*sqrt(x) + a)^6/b^8 - 14/5*(b*sqrt(x) + a)^5*a/b^8 + 21/2*(b*sqrt(x) + a
)^4*a^2/b^8 - 70/3*(b*sqrt(x) + a)^3*a^3/b^8 + 35*(b*sqrt(x) + a)^2*a^4/b^8 - 42*(b*sqrt(x) + a)*a^5/b^8 + 2*a
^7/((b*sqrt(x) + a)*b^8)

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Fricas [A]  time = 1.31609, size = 285, normalized size = 2.57 \begin{align*} \frac{10 \, b^{8} x^{4} + 35 \, a^{2} b^{6} x^{3} + 105 \, a^{4} b^{4} x^{2} - 150 \, a^{6} b^{2} x - 60 \, a^{8} + 420 \,{\left (a^{6} b^{2} x - a^{8}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (6 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 70 \, a^{5} b^{3} x - 105 \, a^{7} b\right )} \sqrt{x}}{30 \,{\left (b^{10} x - a^{2} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="fricas")

[Out]

1/30*(10*b^8*x^4 + 35*a^2*b^6*x^3 + 105*a^4*b^4*x^2 - 150*a^6*b^2*x - 60*a^8 + 420*(a^6*b^2*x - a^8)*log(b*sqr
t(x) + a) - 4*(6*a*b^7*x^3 + 14*a^3*b^5*x^2 + 70*a^5*b^3*x - 105*a^7*b)*sqrt(x))/(b^10*x - a^2*b^8)

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Sympy [A]  time = 2.40098, size = 272, normalized size = 2.45 \begin{align*} \begin{cases} \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{420 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{70 a^{4} b^{3} x^{\frac{3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{21 a^{2} b^{5} x^{\frac{5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} - \frac{14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt{x}} + \frac{10 b^{7} x^{\frac{7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt{x}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{2}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(a+b*x**(1/2))**2,x)

[Out]

Piecewise((420*a**7*log(a/b + sqrt(x))/(30*a*b**8 + 30*b**9*sqrt(x)) + 420*a**7/(30*a*b**8 + 30*b**9*sqrt(x))
+ 420*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(30*a*b**8 + 30*b**9*sqrt(x)) - 210*a**5*b**2*x/(30*a*b**8 + 30*b**9*s
qrt(x)) + 70*a**4*b**3*x**(3/2)/(30*a*b**8 + 30*b**9*sqrt(x)) - 35*a**3*b**4*x**2/(30*a*b**8 + 30*b**9*sqrt(x)
) + 21*a**2*b**5*x**(5/2)/(30*a*b**8 + 30*b**9*sqrt(x)) - 14*a*b**6*x**3/(30*a*b**8 + 30*b**9*sqrt(x)) + 10*b*
*7*x**(7/2)/(30*a*b**8 + 30*b**9*sqrt(x)), Ne(b, 0)), (x**4/(4*a**2), True))

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Giac [A]  time = 1.09456, size = 135, normalized size = 1.22 \begin{align*} \frac{14 \, a^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} + \frac{2 \, a^{7}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac{5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac{3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt{x}}{30 \, b^{12}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^2,x, algorithm="giac")

[Out]

14*a^6*log(abs(b*sqrt(x) + a))/b^8 + 2*a^7/((b*sqrt(x) + a)*b^8) + 1/30*(10*b^10*x^3 - 24*a*b^9*x^(5/2) + 45*a
^2*b^8*x^2 - 80*a^3*b^7*x^(3/2) + 150*a^4*b^6*x - 360*a^5*b^5*sqrt(x))/b^12