\(\int \frac {x^3}{(a+b \sqrt {x})^3} \, dx\) [2207]

Optimal result

Integrand size = 15, antiderivative size = 114 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {a^7}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {14 a^6}{b^8 \left (a+b \sqrt {x}\right )}+\frac {30 a^4 \sqrt {x}}{b^7}-\frac {10 a^3 x}{b^6}+\frac {4 a^2 x^{3/2}}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {2 x^{5/2}}{5 b^3}-\frac {42 a^5 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[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))

Rubi [A] (verified)

Time = 0.07 (sec) , antiderivative size = 114, normalized size of antiderivative = 1.00, number of steps used = 3, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.133, Rules used = {272, 45} \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {a^7}{b^8 \left (a+b \sqrt {x}\right )^2}-\frac {14 a^6}{b^8 \left (a+b \sqrt {x}\right )}-\frac {42 a^5 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {30 a^4 \sqrt {x}}{b^7}-\frac {10 a^3 x}{b^6}+\frac {4 a^2 x^{3/2}}{b^5}-\frac {3 a x^2}{2 b^4}+\frac {2 x^{5/2}}{5 b^3} \]

[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 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])

Rule 272

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]]

Rubi steps \begin{align*} \text {integral}& = 2 \text {Subst}\left (\int \frac {x^7}{(a+b x)^3} \, dx,x,\sqrt {x}\right ) \\ & = 2 \text {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] (verified)

Time = 0.09 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.04 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {-130 a^7+160 a^6 b \sqrt {x}+500 a^5 b^2 x+140 a^4 b^3 x^{3/2}-35 a^3 b^4 x^2+14 a^2 b^5 x^{5/2}-7 a b^6 x^3+4 b^7 x^{7/2}}{10 b^8 \left (a+b \sqrt {x}\right )^2}-\frac {42 a^5 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[In]

Integrate[x^3/(a + b*Sqrt[x])^3,x]

[Out]

(-130*a^7 + 160*a^6*b*Sqrt[x] + 500*a^5*b^2*x + 140*a^4*b^3*x^(3/2) - 35*a^3*b^4*x^2 + 14*a^2*b^5*x^(5/2) - 7*
a*b^6*x^3 + 4*b^7*x^(7/2))/(10*b^8*(a + b*Sqrt[x])^2) - (42*a^5*Log[a + b*Sqrt[x]])/b^8

Maple [A] (verified)

Time = 3.71 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.88

method	result	size
derivativedivides	\(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a \,b^{3} x^{2}}{2}+4 a^{2} x^{\frac {3}{2}} b^{2}-10 a^{3} b x +30 a^{4} \sqrt {x}}{b^{7}}-\frac {14 a^{6}}{b^{8} \left (a +b \sqrt {x}\right )}-\frac {42 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{8}}+\frac {a^{7}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}\)	\(100\)
default	\(\frac {\frac {2 x^{\frac {5}{2}} b^{4}}{5}-\frac {3 a \,b^{3} x^{2}}{2}+4 a^{2} x^{\frac {3}{2}} b^{2}-10 a^{3} b x +30 a^{4} \sqrt {x}}{b^{7}}-\frac {14 a^{6}}{b^{8} \left (a +b \sqrt {x}\right )}-\frac {42 a^{5} \ln \left (a +b \sqrt {x}\right )}{b^{8}}+\frac {a^{7}}{b^{8} \left (a +b \sqrt {x}\right )^{2}}\)	\(100\)

[In]

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

[Out]

2/b^7*(1/5*x^(5/2)*b^4-3/4*a*b^3*x^2+2*a^2*x^(3/2)*b^2-5*a^3*b*x+15*a^4*x^(1/2))-14*a^6/b^8/(a+b*x^(1/2))-42*a
^5*ln(a+b*x^(1/2))/b^8+a^7/b^8/(a+b*x^(1/2))^2

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.38 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {15 \, a b^{8} x^{4} + 70 \, a^{3} b^{6} x^{3} - 185 \, a^{5} b^{4} x^{2} - 50 \, a^{7} b^{2} x + 130 \, a^{9} + 420 \, {\left (a^{5} b^{4} x^{2} - 2 \, a^{7} b^{2} x + a^{9}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (b^{9} x^{4} + 8 \, a^{2} b^{7} x^{3} + 56 \, a^{4} b^{5} x^{2} - 175 \, a^{6} b^{3} x + 105 \, a^{8} b\right )} \sqrt {x}}{10 \, {\left (b^{12} x^{2} - 2 \, a^{2} b^{10} x + a^{4} b^{8}\right )}} \]

[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)

Sympy [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 413 vs. \(2 (112) = 224\).

Time = 0.50 (sec) , antiderivative size = 413, normalized size of antiderivative = 3.62 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\begin {cases} - \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {630 a^{7}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {840 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {840 a^{6} b \sqrt {x}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {420 a^{5} b^{2} x \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {140 a^{4} b^{3} x^{\frac {3}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {35 a^{3} b^{4} x^{2}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {14 a^{2} b^{5} x^{\frac {5}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} - \frac {7 a b^{6} x^{3}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} + \frac {4 b^{7} x^{\frac {7}{2}}}{10 a^{2} b^{8} + 20 a b^{9} \sqrt {x} + 10 b^{10} x} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{3}} & \text {otherwise} \end {cases} \]

[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) - 630*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) - 840*a**6*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) + 140*a**4*b**3*x**(3/2)/(10*a**2*b**8 + 20*a*b**9*
sqrt(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))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 128, normalized size of antiderivative = 1.12 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {42 \, a^{5} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{5}}{5 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{4} a}{2 \, b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )}^{3} a^{2}}{b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{2} a^{3}}{b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )} a^{4}}{b^{8}} - \frac {14 \, a^{6}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 101, normalized size of antiderivative = 0.89 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=-\frac {42 \, a^{5} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} - \frac {14 \, a^{6} b \sqrt {x} + 13 \, a^{7}}{{\left (b \sqrt {x} + a\right )}^{2} b^{8}} + \frac {4 \, b^{12} x^{\frac {5}{2}} - 15 \, a b^{11} x^{2} + 40 \, a^{2} b^{10} x^{\frac {3}{2}} - 100 \, a^{3} b^{9} x + 300 \, a^{4} b^{8} \sqrt {x}}{10 \, b^{15}} \]

[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

Mupad [B] (verification not implemented)

Time = 5.62 (sec) , antiderivative size = 108, normalized size of antiderivative = 0.95 \[ \int \frac {x^3}{\left (a+b \sqrt {x}\right )^3} \, dx=\frac {2\,x^{5/2}}{5\,b^3}-\frac {\frac {13\,a^7}{b}+14\,a^6\,\sqrt {x}}{b^9\,x+a^2\,b^7+2\,a\,b^8\,\sqrt {x}}-\frac {3\,a\,x^2}{2\,b^4}-\frac {10\,a^3\,x}{b^6}-\frac {42\,a^5\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {4\,a^2\,x^{3/2}}{b^5}+\frac {30\,a^4\,\sqrt {x}}{b^7} \]

[In]

int(x^3/(a + b*x^(1/2))^3,x)

[Out]

(2*x^(5/2))/(5*b^3) - ((13*a^7)/b + 14*a^6*x^(1/2))/(b^9*x + a^2*b^7 + 2*a*b^8*x^(1/2)) - (3*a*x^2)/(2*b^4) -
(10*a^3*x)/b^6 - (42*a^5*log(a + b*x^(1/2)))/b^8 + (4*a^2*x^(3/2))/b^5 + (30*a^4*x^(1/2))/b^7