\(\int \frac {x^3}{a+b \sqrt {x}} \, dx\) [2191]

Optimal result

Integrand size = 15, antiderivative size = 107 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {2 a^6 \sqrt {x}}{b^7}-\frac {a^5 x}{b^6}+\frac {2 a^4 x^{3/2}}{3 b^5}-\frac {a^3 x^2}{2 b^4}+\frac {2 a^2 x^{5/2}}{5 b^3}-\frac {a x^3}{3 b^2}+\frac {2 x^{7/2}}{7 b}-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 107, 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} \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8}+\frac {2 a^6 \sqrt {x}}{b^7}-\frac {a^5 x}{b^6}+\frac {2 a^4 x^{3/2}}{3 b^5}-\frac {a^3 x^2}{2 b^4}+\frac {2 a^2 x^{5/2}}{5 b^3}-\frac {a x^3}{3 b^2}+\frac {2 x^{7/2}}{7 b} \]

[In]

Int[x^3/(a + b*Sqrt[x]),x]

[Out]

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

Mathematica [A] (verified)

Time = 0.06 (sec) , antiderivative size = 100, normalized size of antiderivative = 0.93 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {\sqrt {x} \left (420 a^6-210 a^5 b \sqrt {x}+140 a^4 b^2 x-105 a^3 b^3 x^{3/2}+84 a^2 b^4 x^2-70 a b^5 x^{5/2}+60 b^6 x^3\right )}{210 b^7}-\frac {2 a^7 \log \left (a+b \sqrt {x}\right )}{b^8} \]

[In]

Integrate[x^3/(a + b*Sqrt[x]),x]

[Out]

(Sqrt[x]*(420*a^6 - 210*a^5*b*Sqrt[x] + 140*a^4*b^2*x - 105*a^3*b^3*x^(3/2) + 84*a^2*b^4*x^2 - 70*a*b^5*x^(5/2
) + 60*b^6*x^3))/(210*b^7) - (2*a^7*Log[a + b*Sqrt[x]])/b^8

Maple [A] (verified)

Time = 3.63 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82

method	result	size
derivativedivides	\(\frac {\frac {2 x^{\frac {7}{2}} b^{6}}{7}-\frac {a \,x^{3} b^{5}}{3}+\frac {2 a^{2} x^{\frac {5}{2}} b^{4}}{5}-\frac {a^{3} b^{3} x^{2}}{2}+\frac {2 a^{4} x^{\frac {3}{2}} b^{2}}{3}-a^{5} x b +2 a^{6} \sqrt {x}}{b^{7}}-\frac {2 a^{7} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\)	\(88\)
default	\(\frac {\frac {2 x^{\frac {7}{2}} b^{6}}{7}-\frac {a \,x^{3} b^{5}}{3}+\frac {2 a^{2} x^{\frac {5}{2}} b^{4}}{5}-\frac {a^{3} b^{3} x^{2}}{2}+\frac {2 a^{4} x^{\frac {3}{2}} b^{2}}{3}-a^{5} x b +2 a^{6} \sqrt {x}}{b^{7}}-\frac {2 a^{7} \ln \left (a +b \sqrt {x}\right )}{b^{8}}\)	\(88\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 88, normalized size of antiderivative = 0.82 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {70 \, a b^{6} x^{3} + 105 \, a^{3} b^{4} x^{2} + 210 \, a^{5} b^{2} x + 420 \, a^{7} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (15 \, b^{7} x^{3} + 21 \, a^{2} b^{5} x^{2} + 35 \, a^{4} b^{3} x + 105 \, a^{6} b\right )} \sqrt {x}}{210 \, b^{8}} \]

[In]

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

[Out]

-1/210*(70*a*b^6*x^3 + 105*a^3*b^4*x^2 + 210*a^5*b^2*x + 420*a^7*log(b*sqrt(x) + a) - 4*(15*b^7*x^3 + 21*a^2*b
^5*x^2 + 35*a^4*b^3*x + 105*a^6*b)*sqrt(x))/b^8

Sympy [A] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 109, normalized size of antiderivative = 1.02 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\begin {cases} - \frac {2 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{b^{8}} + \frac {2 a^{6} \sqrt {x}}{b^{7}} - \frac {a^{5} x}{b^{6}} + \frac {2 a^{4} x^{\frac {3}{2}}}{3 b^{5}} - \frac {a^{3} x^{2}}{2 b^{4}} + \frac {2 a^{2} x^{\frac {5}{2}}}{5 b^{3}} - \frac {a x^{3}}{3 b^{2}} + \frac {2 x^{\frac {7}{2}}}{7 b} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a} & \text {otherwise} \end {cases} \]

[In]

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

[Out]

Piecewise((-2*a**7*log(a/b + sqrt(x))/b**8 + 2*a**6*sqrt(x)/b**7 - a**5*x/b**6 + 2*a**4*x**(3/2)/(3*b**5) - a*
*3*x**2/(2*b**4) + 2*a**2*x**(5/2)/(5*b**3) - a*x**3/(3*b**2) + 2*x**(7/2)/(7*b), Ne(b, 0)), (x**4/(4*a), True
))

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.21 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{7} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {2 \, {\left (b \sqrt {x} + a\right )}^{7}}{7 \, b^{8}} - \frac {7 \, {\left (b \sqrt {x} + a\right )}^{6} a}{3 \, b^{8}} + \frac {42 \, {\left (b \sqrt {x} + a\right )}^{5} a^{2}}{5 \, b^{8}} - \frac {35 \, {\left (b \sqrt {x} + a\right )}^{4} a^{3}}{2 \, b^{8}} + \frac {70 \, {\left (b \sqrt {x} + a\right )}^{3} a^{4}}{3 \, b^{8}} - \frac {21 \, {\left (b \sqrt {x} + a\right )}^{2} a^{5}}{b^{8}} + \frac {14 \, {\left (b \sqrt {x} + a\right )} a^{6}}{b^{8}} \]

[In]

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

[Out]

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

Giac [A] (verification not implemented)

none

Time = 0.26 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.83 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=-\frac {2 \, a^{7} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {60 \, b^{6} x^{\frac {7}{2}} - 70 \, a b^{5} x^{3} + 84 \, a^{2} b^{4} x^{\frac {5}{2}} - 105 \, a^{3} b^{3} x^{2} + 140 \, a^{4} b^{2} x^{\frac {3}{2}} - 210 \, a^{5} b x + 420 \, a^{6} \sqrt {x}}{210 \, b^{7}} \]

[In]

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

[Out]

-2*a^7*log(abs(b*sqrt(x) + a))/b^8 + 1/210*(60*b^6*x^(7/2) - 70*a*b^5*x^3 + 84*a^2*b^4*x^(5/2) - 105*a^3*b^3*x
^2 + 140*a^4*b^2*x^(3/2) - 210*a^5*b*x + 420*a^6*sqrt(x))/b^7

Mupad [B] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.81 \[ \int \frac {x^3}{a+b \sqrt {x}} \, dx=\frac {2\,x^{7/2}}{7\,b}-\frac {a\,x^3}{3\,b^2}-\frac {a^5\,x}{b^6}-\frac {2\,a^7\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}-\frac {a^3\,x^2}{2\,b^4}+\frac {2\,a^2\,x^{5/2}}{5\,b^3}+\frac {2\,a^4\,x^{3/2}}{3\,b^5}+\frac {2\,a^6\,\sqrt {x}}{b^7} \]

[In]

int(x^3/(a + b*x^(1/2)),x)

[Out]

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