3.2207 \(\int \frac{x^3}{(a+b \sqrt{x})^3} \, dx\)

Optimal. Leaf size=114 \[ \frac{4 a^2 x^{3/2}}{b^5}+\frac{a^7}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{14 a^6}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^4 \sqrt{x}}{b^7}-\frac{10 a^3 x}{b^6}-\frac{42 a^5 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{3 a x^2}{2 b^4}+\frac{2 x^{5/2}}{5 b^3} \]

[Out]

a^7/(b^8*(a + b*Sqrt[x])^2) - (14*a^6)/(b^8*(a + b*Sqrt[x])) + (30*a^4*Sqrt[x])/b^7 - (10*a^3*x)/b^6 + (4*a^2*
x^(3/2))/b^5 - (3*a*x^2)/(2*b^4) + (2*x^(5/2))/(5*b^3) - (42*a^5*Log[a + b*Sqrt[x]])/b^8

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Rubi [A]  time = 0.087194, antiderivative size = 114, 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} \[ \frac{4 a^2 x^{3/2}}{b^5}+\frac{a^7}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{14 a^6}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^4 \sqrt{x}}{b^7}-\frac{10 a^3 x}{b^6}-\frac{42 a^5 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{3 a x^2}{2 b^4}+\frac{2 x^{5/2}}{5 b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x])^3,x]

[Out]

a^7/(b^8*(a + b*Sqrt[x])^2) - (14*a^6)/(b^8*(a + b*Sqrt[x])) + (30*a^4*Sqrt[x])/b^7 - (10*a^3*x)/b^6 + (4*a^2*
x^(3/2))/b^5 - (3*a*x^2)/(2*b^4) + (2*x^(5/2))/(5*b^3) - (42*a^5*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 )^3} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{15 a^4}{b^7}-\frac{10 a^3 x}{b^6}+\frac{6 a^2 x^2}{b^5}-\frac{3 a x^3}{b^4}+\frac{x^4}{b^3}-\frac{a^7}{b^7 (a+b x)^3}+\frac{7 a^6}{b^7 (a+b x)^2}-\frac{21 a^5}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^7}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{14 a^6}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^4 \sqrt{x}}{b^7}-\frac{10 a^3 x}{b^6}+\frac{4 a^2 x^{3/2}}{b^5}-\frac{3 a x^2}{2 b^4}+\frac{2 x^{5/2}}{5 b^3}-\frac{42 a^5 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.0910766, size = 107, normalized size = 0.94 \[ \frac{40 a^2 b^3 x^{3/2}-100 a^3 b^2 x+\frac{10 a^7}{\left (a+b \sqrt{x}\right )^2}-\frac{140 a^6}{a+b \sqrt{x}}+300 a^4 b \sqrt{x}-420 a^5 \log \left (a+b \sqrt{x}\right )-15 a b^4 x^2+4 b^5 x^{5/2}}{10 b^8} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x])^3,x]

[Out]

((10*a^7)/(a + b*Sqrt[x])^2 - (140*a^6)/(a + b*Sqrt[x]) + 300*a^4*b*Sqrt[x] - 100*a^3*b^2*x + 40*a^2*b^3*x^(3/
2) - 15*a*b^4*x^2 + 4*b^5*x^(5/2) - 420*a^5*Log[a + b*Sqrt[x]])/(10*b^8)

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Maple [A]  time = 0.008, size = 99, normalized size = 0.9 \begin{align*} -10\,{\frac{{a}^{3}x}{{b}^{6}}}+4\,{\frac{{a}^{2}{x}^{3/2}}{{b}^{5}}}-{\frac{3\,a{x}^{2}}{2\,{b}^{4}}}+{\frac{2}{5\,{b}^{3}}{x}^{{\frac{5}{2}}}}-42\,{\frac{{a}^{5}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+30\,{\frac{{a}^{4}\sqrt{x}}{{b}^{7}}}+{\frac{{a}^{7}}{{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-2}}-14\,{\frac{{a}^{6}}{{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))^3,x)

[Out]

-10*a^3*x/b^6+4*a^2*x^(3/2)/b^5-3/2*a*x^2/b^4+2/5*x^(5/2)/b^3-42*a^5*ln(a+b*x^(1/2))/b^8+30*a^4*x^(1/2)/b^7+a^
7/b^8/(a+b*x^(1/2))^2-14*a^6/b^8/(a+b*x^(1/2))

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Maxima [A]  time = 0.992341, size = 173, normalized size = 1.52 \begin{align*} -\frac{42 \, a^{5} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{5}}{5 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{4} a}{2 \, b^{8}} + \frac{14 \,{\left (b \sqrt{x} + a\right )}^{3} a^{2}}{b^{8}} - \frac{35 \,{\left (b \sqrt{x} + a\right )}^{2} a^{3}}{b^{8}} + \frac{70 \,{\left (b \sqrt{x} + a\right )} a^{4}}{b^{8}} - \frac{14 \, a^{6}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 1.36042, size = 351, normalized size = 3.08 \begin{align*} -\frac{15 \, a b^{8} x^{4} + 70 \, a^{3} b^{6} x^{3} - 185 \, a^{5} b^{4} x^{2} - 50 \, a^{7} b^{2} x + 130 \, a^{9} + 420 \,{\left (a^{5} b^{4} x^{2} - 2 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (b^{9} x^{4} + 8 \, a^{2} b^{7} x^{3} + 56 \, a^{4} b^{5} x^{2} - 175 \, a^{6} b^{3} x + 105 \, a^{8} b\right )} \sqrt{x}}{10 \,{\left (b^{12} x^{2} - 2 \, a^{2} b^{10} x + a^{4} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/10*(15*a*b^8*x^4 + 70*a^3*b^6*x^3 - 185*a^5*b^4*x^2 - 50*a^7*b^2*x + 130*a^9 + 420*(a^5*b^4*x^2 - 2*a^7*b^2
*x + a^9)*log(b*sqrt(x) + a) - 4*(b^9*x^4 + 8*a^2*b^7*x^3 + 56*a^4*b^5*x^2 - 175*a^6*b^3*x + 105*a^8*b)*sqrt(x
))/(b^12*x^2 - 2*a^2*b^10*x + a^4*b^8)

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Sympy [A]  time = 3.16482, size = 411, normalized size = 3.61 \begin{align*} \begin{cases} - \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{210 a^{7}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{840 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{420 a^{5} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{420 a^{5} b^{2} x}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{140 a^{4} b^{3} x^{\frac{3}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{35 a^{3} b^{4} x^{2}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{14 a^{2} b^{5} x^{\frac{5}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} - \frac{7 a b^{6} x^{3}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} + \frac{4 b^{7} x^{\frac{7}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt{x} + 10 b^{10} x} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{3}} & \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))**3,x)

[Out]

Piecewise((-420*a**7*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 210*a**7/(10*a**2*b*
*8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 840*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqrt(x)
+ 10*b**10*x) - 420*a**5*b**2*x*log(a/b + sqrt(x))/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) + 420*a**5
*b**2*x/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) + 140*a**4*b**3*x**(3/2)/(10*a**2*b**8 + 20*a*b**9*sqr
t(x) + 10*b**10*x) - 35*a**3*b**4*x**2/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) + 14*a**2*b**5*x**(5/2)
/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x) - 7*a*b**6*x**3/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*
x) + 4*b**7*x**(7/2)/(10*a**2*b**8 + 20*a*b**9*sqrt(x) + 10*b**10*x), Ne(b, 0)), (x**4/(4*a**3), True))

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Giac [A]  time = 1.12699, size = 136, normalized size = 1.19 \begin{align*} -\frac{42 \, a^{5} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} - \frac{14 \, a^{6} b \sqrt{x} + 13 \, a^{7}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} + \frac{4 \, b^{12} x^{\frac{5}{2}} - 15 \, a b^{11} x^{2} + 40 \, a^{2} b^{10} x^{\frac{3}{2}} - 100 \, a^{3} b^{9} x + 300 \, a^{4} b^{8} \sqrt{x}}{10 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-42*a^5*log(abs(b*sqrt(x) + a))/b^8 - (14*a^6*b*sqrt(x) + 13*a^7)/((b*sqrt(x) + a)^2*b^8) + 1/10*(4*b^12*x^(5/
2) - 15*a*b^11*x^2 + 40*a^2*b^10*x^(3/2) - 100*a^3*b^9*x + 300*a^4*b^8*sqrt(x))/b^15