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

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

[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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Rubi [A]  time = 0.08, antiderivative size = 111, normalized size of antiderivative = 1.00, 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 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])

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]]

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*}

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Mathematica [A]  time = 0.11, size = 102, normalized size = 0.92 \[ \frac {\frac {60 a^7}{a+b \sqrt {x}}+420 a^6 \log \left (a+b \sqrt {x}\right )-360 a^5 b \sqrt {x}+150 a^4 b^2 x-80 a^3 b^3 x^{3/2}+45 a^2 b^4 x^2-24 a b^5 x^{5/2}+10 b^6 x^3}{30 b^8} \]

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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fricas [A]  time = 0.96, size = 128, normalized size = 1.15 \[ \frac {10 \, b^{8} x^{4} + 35 \, a^{2} b^{6} x^{3} + 105 \, a^{4} b^{4} x^{2} - 150 \, a^{6} b^{2} x - 60 \, a^{8} + 420 \, {\left (a^{6} b^{2} x - a^{8}\right )} \log \left (b \sqrt {x} + a\right ) - 4 \, {\left (6 \, a b^{7} x^{3} + 14 \, a^{3} b^{5} x^{2} + 70 \, a^{5} b^{3} x - 105 \, a^{7} b\right )} \sqrt {x}}{30 \, {\left (b^{10} x - a^{2} b^{8}\right )}} \]

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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giac [A]  time = 0.21, size = 100, normalized size = 0.90 \[ \frac {14 \, a^{6} \log \left ({\left | b \sqrt {x} + a \right |}\right )}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} + \frac {10 \, b^{10} x^{3} - 24 \, a b^{9} x^{\frac {5}{2}} + 45 \, a^{2} b^{8} x^{2} - 80 \, a^{3} b^{7} x^{\frac {3}{2}} + 150 \, a^{4} b^{6} x - 360 \, a^{5} b^{5} \sqrt {x}}{30 \, b^{12}} \]

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

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maple [A]  time = 0.01, size = 94, normalized size = 0.85 \[ \frac {x^{3}}{3 b^{2}}-\frac {4 a \,x^{\frac {5}{2}}}{5 b^{3}}+\frac {3 a^{2} x^{2}}{2 b^{4}}-\frac {8 a^{3} x^{\frac {3}{2}}}{3 b^{5}}+\frac {2 a^{7}}{\left (b \sqrt {x}+a \right ) b^{8}}+\frac {14 a^{6} \ln \left (b \sqrt {x}+a \right )}{b^{8}}+\frac {5 a^{4} x}{b^{6}}-\frac {12 a^{5} \sqrt {x}}{b^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

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maxima [A]  time = 0.91, size = 129, normalized size = 1.16 \[ \frac {14 \, a^{6} \log \left (b \sqrt {x} + a\right )}{b^{8}} + \frac {{\left (b \sqrt {x} + a\right )}^{6}}{3 \, b^{8}} - \frac {14 \, {\left (b \sqrt {x} + a\right )}^{5} a}{5 \, b^{8}} + \frac {21 \, {\left (b \sqrt {x} + a\right )}^{4} a^{2}}{2 \, b^{8}} - \frac {70 \, {\left (b \sqrt {x} + a\right )}^{3} a^{3}}{3 \, b^{8}} + \frac {35 \, {\left (b \sqrt {x} + a\right )}^{2} a^{4}}{b^{8}} - \frac {42 \, {\left (b \sqrt {x} + a\right )} a^{5}}{b^{8}} + \frac {2 \, a^{7}}{{\left (b \sqrt {x} + a\right )} b^{8}} \]

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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mupad [B]  time = 1.11, size = 99, normalized size = 0.89 \[ \frac {x^3}{3\,b^2}+\frac {2\,a^7}{b\,\left (a\,b^7+b^8\,\sqrt {x}\right )}-\frac {4\,a\,x^{5/2}}{5\,b^3}+\frac {5\,a^4\,x}{b^6}+\frac {14\,a^6\,\ln \left (a+b\,\sqrt {x}\right )}{b^8}+\frac {3\,a^2\,x^2}{2\,b^4}-\frac {8\,a^3\,x^{3/2}}{3\,b^5}-\frac {12\,a^5\,\sqrt {x}}{b^7} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(a + b*x^(1/2))^2,x)

[Out]

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

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sympy [A]  time = 2.61, size = 272, normalized size = 2.45 \[ \begin {cases} \frac {420 a^{7} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{7}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {420 a^{6} b \sqrt {x} \log {\left (\frac {a}{b} + \sqrt {x} \right )}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {210 a^{5} b^{2} x}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {70 a^{4} b^{3} x^{\frac {3}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {35 a^{3} b^{4} x^{2}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {21 a^{2} b^{5} x^{\frac {5}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} - \frac {14 a b^{6} x^{3}}{30 a b^{8} + 30 b^{9} \sqrt {x}} + \frac {10 b^{7} x^{\frac {7}{2}}}{30 a b^{8} + 30 b^{9} \sqrt {x}} & \text {for}\: b \neq 0 \\\frac {x^{4}}{4 a^{2}} & \text {otherwise} \end {cases} \]

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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