3.2216 \(\int \frac{x^3}{(a+b \sqrt{x})^5} \, dx\)

Optimal. Leaf size=131 \[ \frac{a^7}{2 b^8 \left (a+b \sqrt{x}\right )^4}-\frac{14 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^3}+\frac{21 a^5}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{70 a^4}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^2 \sqrt{x}}{b^7}-\frac{70 a^3 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{5 a x}{b^6}+\frac{2 x^{3/2}}{3 b^5} \]

[Out]

a^7/(2*b^8*(a + b*Sqrt[x])^4) - (14*a^6)/(3*b^8*(a + b*Sqrt[x])^3) + (21*a^5)/(b^8*(a + b*Sqrt[x])^2) - (70*a^
4)/(b^8*(a + b*Sqrt[x])) + (30*a^2*Sqrt[x])/b^7 - (5*a*x)/b^6 + (2*x^(3/2))/(3*b^5) - (70*a^3*Log[a + b*Sqrt[x
]])/b^8

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Rubi [A]  time = 0.104766, antiderivative size = 131, 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} \[ \frac{a^7}{2 b^8 \left (a+b \sqrt{x}\right )^4}-\frac{14 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^3}+\frac{21 a^5}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{70 a^4}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^2 \sqrt{x}}{b^7}-\frac{70 a^3 \log \left (a+b \sqrt{x}\right )}{b^8}-\frac{5 a x}{b^6}+\frac{2 x^{3/2}}{3 b^5} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(a + b*Sqrt[x])^5,x]

[Out]

a^7/(2*b^8*(a + b*Sqrt[x])^4) - (14*a^6)/(3*b^8*(a + b*Sqrt[x])^3) + (21*a^5)/(b^8*(a + b*Sqrt[x])^2) - (70*a^
4)/(b^8*(a + b*Sqrt[x])) + (30*a^2*Sqrt[x])/b^7 - (5*a*x)/b^6 + (2*x^(3/2))/(3*b^5) - (70*a^3*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 )^5} \, dx &=2 \operatorname{Subst}\left (\int \frac{x^7}{(a+b x)^5} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{15 a^2}{b^7}-\frac{5 a x}{b^6}+\frac{x^2}{b^5}-\frac{a^7}{b^7 (a+b x)^5}+\frac{7 a^6}{b^7 (a+b x)^4}-\frac{21 a^5}{b^7 (a+b x)^3}+\frac{35 a^4}{b^7 (a+b x)^2}-\frac{35 a^3}{b^7 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{a^7}{2 b^8 \left (a+b \sqrt{x}\right )^4}-\frac{14 a^6}{3 b^8 \left (a+b \sqrt{x}\right )^3}+\frac{21 a^5}{b^8 \left (a+b \sqrt{x}\right )^2}-\frac{70 a^4}{b^8 \left (a+b \sqrt{x}\right )}+\frac{30 a^2 \sqrt{x}}{b^7}-\frac{5 a x}{b^6}+\frac{2 x^{3/2}}{3 b^5}-\frac{70 a^3 \log \left (a+b \sqrt{x}\right )}{b^8}\\ \end{align*}

Mathematica [A]  time = 0.0922146, size = 126, normalized size = 0.96 \[ \frac{544 a^4 b^3 x^{3/2}+556 a^3 b^4 x^2+84 a^2 b^5 x^{5/2}-444 a^5 b^2 x-856 a^6 b \sqrt{x}-420 a^3 \left (a+b \sqrt{x}\right )^4 \log \left (a+b \sqrt{x}\right )-319 a^7-14 a b^6 x^3+4 b^7 x^{7/2}}{6 b^8 \left (a+b \sqrt{x}\right )^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(a + b*Sqrt[x])^5,x]

[Out]

(-319*a^7 - 856*a^6*b*Sqrt[x] - 444*a^5*b^2*x + 544*a^4*b^3*x^(3/2) + 556*a^3*b^4*x^2 + 84*a^2*b^5*x^(5/2) - 1
4*a*b^6*x^3 + 4*b^7*x^(7/2) - 420*a^3*(a + b*Sqrt[x])^4*Log[a + b*Sqrt[x]])/(6*b^8*(a + b*Sqrt[x])^4)

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Maple [A]  time = 0.009, size = 112, normalized size = 0.9 \begin{align*} -5\,{\frac{ax}{{b}^{6}}}+{\frac{2}{3\,{b}^{5}}{x}^{{\frac{3}{2}}}}-70\,{\frac{{a}^{3}\ln \left ( a+b\sqrt{x} \right ) }{{b}^{8}}}+30\,{\frac{{a}^{2}\sqrt{x}}{{b}^{7}}}+{\frac{{a}^{7}}{2\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-4}}-{\frac{14\,{a}^{6}}{3\,{b}^{8}} \left ( a+b\sqrt{x} \right ) ^{-3}}+21\,{\frac{{a}^{5}}{{b}^{8} \left ( a+b\sqrt{x} \right ) ^{2}}}-70\,{\frac{{a}^{4}}{{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))^5,x)

[Out]

-5*a*x/b^6+2/3*x^(3/2)/b^5-70*a^3*ln(a+b*x^(1/2))/b^8+30*a^2*x^(1/2)/b^7+1/2*a^7/b^8/(a+b*x^(1/2))^4-14/3*a^6/
b^8/(a+b*x^(1/2))^3+21*a^5/b^8/(a+b*x^(1/2))^2-70*a^4/b^8/(a+b*x^(1/2))

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Maxima [A]  time = 0.972562, size = 174, normalized size = 1.33 \begin{align*} -\frac{70 \, a^{3} \log \left (b \sqrt{x} + a\right )}{b^{8}} + \frac{2 \,{\left (b \sqrt{x} + a\right )}^{3}}{3 \, b^{8}} - \frac{7 \,{\left (b \sqrt{x} + a\right )}^{2} a}{b^{8}} + \frac{42 \,{\left (b \sqrt{x} + a\right )} a^{2}}{b^{8}} - \frac{70 \, a^{4}}{{\left (b \sqrt{x} + a\right )} b^{8}} + \frac{21 \, a^{5}}{{\left (b \sqrt{x} + a\right )}^{2} b^{8}} - \frac{14 \, a^{6}}{3 \,{\left (b \sqrt{x} + a\right )}^{3} b^{8}} + \frac{a^{7}}{2 \,{\left (b \sqrt{x} + a\right )}^{4} b^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="maxima")

[Out]

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

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Fricas [B]  time = 1.31233, size = 502, normalized size = 3.83 \begin{align*} -\frac{30 \, a b^{10} x^{5} - 120 \, a^{3} b^{8} x^{4} - 366 \, a^{5} b^{6} x^{3} + 1179 \, a^{7} b^{4} x^{2} - 1066 \, a^{9} b^{2} x + 319 \, a^{11} + 420 \,{\left (a^{3} b^{8} x^{4} - 4 \, a^{5} b^{6} x^{3} + 6 \, a^{7} b^{4} x^{2} - 4 \, a^{9} b^{2} x + a^{11}\right )} \log \left (b \sqrt{x} + a\right ) - 4 \,{\left (b^{11} x^{5} + 41 \, a^{2} b^{9} x^{4} - 279 \, a^{4} b^{7} x^{3} + 511 \, a^{6} b^{5} x^{2} - 385 \, a^{8} b^{3} x + 105 \, a^{10} b\right )} \sqrt{x}}{6 \,{\left (b^{16} x^{4} - 4 \, a^{2} b^{14} x^{3} + 6 \, a^{4} b^{12} x^{2} - 4 \, a^{6} b^{10} x + a^{8} b^{8}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="fricas")

[Out]

-1/6*(30*a*b^10*x^5 - 120*a^3*b^8*x^4 - 366*a^5*b^6*x^3 + 1179*a^7*b^4*x^2 - 1066*a^9*b^2*x + 319*a^11 + 420*(
a^3*b^8*x^4 - 4*a^5*b^6*x^3 + 6*a^7*b^4*x^2 - 4*a^9*b^2*x + a^11)*log(b*sqrt(x) + a) - 4*(b^11*x^5 + 41*a^2*b^
9*x^4 - 279*a^4*b^7*x^3 + 511*a^6*b^5*x^2 - 385*a^8*b^3*x + 105*a^10*b)*sqrt(x))/(b^16*x^4 - 4*a^2*b^14*x^3 +
6*a^4*b^12*x^2 - 4*a^6*b^10*x + a^8*b^8)

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Sympy [A]  time = 3.94302, size = 818, normalized size = 6.24 \begin{align*} \begin{cases} - \frac{420 a^{7} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{105 a^{7}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{1680 a^{6} b \sqrt{x} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{2520 a^{5} b^{2} x \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} + \frac{840 a^{5} b^{2} x}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{1680 a^{4} b^{3} x^{\frac{3}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} + \frac{1400 a^{4} b^{3} x^{\frac{3}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{420 a^{3} b^{4} x^{2} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} + \frac{770 a^{3} b^{4} x^{2}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} + \frac{84 a^{2} b^{5} x^{\frac{5}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} - \frac{14 a b^{6} x^{3}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} + \frac{4 b^{7} x^{\frac{7}{2}}}{6 a^{4} b^{8} + 24 a^{3} b^{9} \sqrt{x} + 36 a^{2} b^{10} x + 24 a b^{11} x^{\frac{3}{2}} + 6 b^{12} x^{2}} & \text{for}\: b \neq 0 \\\frac{x^{4}}{4 a^{5}} & \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))**5,x)

[Out]

Piecewise((-420*a**7*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**
(3/2) + 6*b**12*x**2) - 105*a**7/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) +
6*b**12*x**2) - 1680*a**6*b*sqrt(x)*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x
+ 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 2520*a**5*b**2*x*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x
) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) + 840*a**5*b**2*x/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x
) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 1680*a**4*b**3*x**(3/2)*log(a/b + sqrt(x))/(6*a**4
*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) + 1400*a**4*b**3*x**(3/2)
/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x**2) - 420*a**3*b**4*x
**2*log(a/b + sqrt(x))/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**12*x
**2) + 770*a**3*b**4*x**2/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6*b**1
2*x**2) + 84*a**2*b**5*x**(5/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) +
6*b**12*x**2) - 14*a*b**6*x**3/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6
*b**12*x**2) + 4*b**7*x**(7/2)/(6*a**4*b**8 + 24*a**3*b**9*sqrt(x) + 36*a**2*b**10*x + 24*a*b**11*x**(3/2) + 6
*b**12*x**2), Ne(b, 0)), (x**4/(4*a**5), True))

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Giac [A]  time = 1.12651, size = 134, normalized size = 1.02 \begin{align*} -\frac{70 \, a^{3} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{b^{8}} - \frac{420 \, a^{4} b^{3} x^{\frac{3}{2}} + 1134 \, a^{5} b^{2} x + 1036 \, a^{6} b \sqrt{x} + 319 \, a^{7}}{6 \,{\left (b \sqrt{x} + a\right )}^{4} b^{8}} + \frac{2 \, b^{10} x^{\frac{3}{2}} - 15 \, a b^{9} x + 90 \, a^{2} b^{8} \sqrt{x}}{3 \, b^{15}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(a+b*x^(1/2))^5,x, algorithm="giac")

[Out]

-70*a^3*log(abs(b*sqrt(x) + a))/b^8 - 1/6*(420*a^4*b^3*x^(3/2) + 1134*a^5*b^2*x + 1036*a^6*b*sqrt(x) + 319*a^7
)/((b*sqrt(x) + a)^4*b^8) + 1/3*(2*b^10*x^(3/2) - 15*a*b^9*x + 90*a^2*b^8*sqrt(x))/b^15