3.2206 \(\int \frac{1}{(a+b \sqrt{x})^2 x^4} \, dx\)

Optimal. Leaf size=123 \[ \frac{8 b^3}{3 a^5 x^{3/2}}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b^6}{a^7 \left (a+b \sqrt{x}\right )}+\frac{12 b^5}{a^7 \sqrt{x}}-\frac{5 b^4}{a^6 x}-\frac{14 b^6 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{4 b}{5 a^3 x^{5/2}}-\frac{1}{3 a^2 x^3} \]

[Out]

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

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Rubi [A]  time = 0.0792728, antiderivative size = 123, 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, 44} \[ \frac{8 b^3}{3 a^5 x^{3/2}}-\frac{3 b^2}{2 a^4 x^2}+\frac{2 b^6}{a^7 \left (a+b \sqrt{x}\right )}+\frac{12 b^5}{a^7 \sqrt{x}}-\frac{5 b^4}{a^6 x}-\frac{14 b^6 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}+\frac{4 b}{5 a^3 x^{5/2}}-\frac{1}{3 a^2 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^2*x^4),x]

[Out]

(2*b^6)/(a^7*(a + b*Sqrt[x])) - 1/(3*a^2*x^3) + (4*b)/(5*a^3*x^(5/2)) - (3*b^2)/(2*a^4*x^2) + (8*b^3)/(3*a^5*x
^(3/2)) - (5*b^4)/(a^6*x) + (12*b^5)/(a^7*Sqrt[x]) - (14*b^6*Log[a + b*Sqrt[x]])/a^8 + (7*b^6*Log[x])/a^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 44

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}, x] && NeQ[b*c - a*d, 0] && ILtQ[m, 0] && IntegerQ[n] &&  !(IGtQ[n, 0] && L
tQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\left (a+b \sqrt{x}\right )^2 x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)^2} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^2 x^7}-\frac{2 b}{a^3 x^6}+\frac{3 b^2}{a^4 x^5}-\frac{4 b^3}{a^5 x^4}+\frac{5 b^4}{a^6 x^3}-\frac{6 b^5}{a^7 x^2}+\frac{7 b^6}{a^8 x}-\frac{b^7}{a^7 (a+b x)^2}-\frac{7 b^7}{a^8 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{2 b^6}{a^7 \left (a+b \sqrt{x}\right )}-\frac{1}{3 a^2 x^3}+\frac{4 b}{5 a^3 x^{5/2}}-\frac{3 b^2}{2 a^4 x^2}+\frac{8 b^3}{3 a^5 x^{3/2}}-\frac{5 b^4}{a^6 x}+\frac{12 b^5}{a^7 \sqrt{x}}-\frac{14 b^6 \log \left (a+b \sqrt{x}\right )}{a^8}+\frac{7 b^6 \log (x)}{a^8}\\ \end{align*}

Mathematica [A]  time = 0.121038, size = 115, normalized size = 0.93 \[ \frac{\frac{a \left (35 a^3 b^3 x^{3/2}-70 a^2 b^4 x^2-21 a^4 b^2 x+14 a^5 b \sqrt{x}-10 a^6+210 a b^5 x^{5/2}+420 b^6 x^3\right )}{x^3 \left (a+b \sqrt{x}\right )}-420 b^6 \log \left (a+b \sqrt{x}\right )+210 b^6 \log (x)}{30 a^8} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^2*x^4),x]

[Out]

((a*(-10*a^6 + 14*a^5*b*Sqrt[x] - 21*a^4*b^2*x + 35*a^3*b^3*x^(3/2) - 70*a^2*b^4*x^2 + 210*a*b^5*x^(5/2) + 420
*b^6*x^3))/((a + b*Sqrt[x])*x^3) - 420*b^6*Log[a + b*Sqrt[x]] + 210*b^6*Log[x])/(30*a^8)

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Maple [A]  time = 0.013, size = 106, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{x}^{3}{a}^{2}}}+{\frac{4\,b}{5\,{a}^{3}}{x}^{-{\frac{5}{2}}}}-{\frac{3\,{b}^{2}}{2\,{a}^{4}{x}^{2}}}+{\frac{8\,{b}^{3}}{3\,{a}^{5}}{x}^{-{\frac{3}{2}}}}-5\,{\frac{{b}^{4}}{{a}^{6}x}}+7\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{8}}}-14\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{8}}}+12\,{\frac{{b}^{5}}{{a}^{7}\sqrt{x}}}+2\,{\frac{{b}^{6}}{{a}^{7} \left ( a+b\sqrt{x} \right ) }} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(1/x^4/(a+b*x^(1/2))^2,x)

[Out]

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

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Maxima [A]  time = 0.957791, size = 149, normalized size = 1.21 \begin{align*} \frac{420 \, b^{6} x^{3} + 210 \, a b^{5} x^{\frac{5}{2}} - 70 \, a^{2} b^{4} x^{2} + 35 \, a^{3} b^{3} x^{\frac{3}{2}} - 21 \, a^{4} b^{2} x + 14 \, a^{5} b \sqrt{x} - 10 \, a^{6}}{30 \,{\left (a^{7} b x^{\frac{7}{2}} + a^{8} x^{3}\right )}} - \frac{14 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{8}} + \frac{7 \, b^{6} \log \left (x\right )}{a^{8}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*x^(1/2))^2,x, algorithm="maxima")

[Out]

1/30*(420*b^6*x^3 + 210*a*b^5*x^(5/2) - 70*a^2*b^4*x^2 + 35*a^3*b^3*x^(3/2) - 21*a^4*b^2*x + 14*a^5*b*sqrt(x)
- 10*a^6)/(a^7*b*x^(7/2) + a^8*x^3) - 14*b^6*log(b*sqrt(x) + a)/a^8 + 7*b^6*log(x)/a^8

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Fricas [A]  time = 1.30267, size = 343, normalized size = 2.79 \begin{align*} -\frac{210 \, a^{2} b^{6} x^{3} - 105 \, a^{4} b^{4} x^{2} - 35 \, a^{6} b^{2} x - 10 \, a^{8} + 420 \,{\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (b \sqrt{x} + a\right ) - 420 \,{\left (b^{8} x^{4} - a^{2} b^{6} x^{3}\right )} \log \left (\sqrt{x}\right ) - 4 \,{\left (105 \, a b^{7} x^{3} - 70 \, a^{3} b^{5} x^{2} - 14 \, a^{5} b^{3} x - 6 \, a^{7} b\right )} \sqrt{x}}{30 \,{\left (a^{8} b^{2} x^{4} - a^{10} x^{3}\right )}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*x^(1/2))^2,x, algorithm="fricas")

[Out]

-1/30*(210*a^2*b^6*x^3 - 105*a^4*b^4*x^2 - 35*a^6*b^2*x - 10*a^8 + 420*(b^8*x^4 - a^2*b^6*x^3)*log(b*sqrt(x) +
a) - 420*(b^8*x^4 - a^2*b^6*x^3)*log(sqrt(x)) - 4*(105*a*b^7*x^3 - 70*a^3*b^5*x^2 - 14*a^5*b^3*x - 6*a^7*b)*s
qrt(x))/(a^8*b^2*x^4 - a^10*x^3)

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Sympy [A]  time = 8.62884, size = 396, normalized size = 3.22 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{4}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{1}{4 b^{2} x^{4}} & \text{for}\: a = 0 \\- \frac{1}{3 a^{2} x^{3}} & \text{for}\: b = 0 \\- \frac{10 a^{7} \sqrt{x}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{14 a^{6} b x}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{21 a^{5} b^{2} x^{\frac{3}{2}}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{35 a^{4} b^{3} x^{2}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{70 a^{3} b^{4} x^{\frac{5}{2}}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 a^{2} b^{5} x^{3}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 a b^{6} x^{\frac{7}{2}} \log{\left (x \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 a b^{6} x^{\frac{7}{2}} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} + \frac{210 b^{7} x^{4} \log{\left (x \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 b^{7} x^{4} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} - \frac{420 b^{7} x^{4}}{30 a^{9} x^{\frac{7}{2}} + 30 a^{8} b x^{4}} & \text{otherwise} \end{cases} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+b*x**(1/2))**2,x)

[Out]

Piecewise((zoo/x**4, Eq(a, 0) & Eq(b, 0)), (-1/(4*b**2*x**4), Eq(a, 0)), (-1/(3*a**2*x**3), Eq(b, 0)), (-10*a*
*7*sqrt(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 14*a**6*b*x/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 21*a**5*b**
2*x**(3/2)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 35*a**4*b**3*x**2/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 70*a*
*3*b**4*x**(5/2)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) + 210*a**2*b**5*x**3/(30*a**9*x**(7/2) + 30*a**8*b*x**4)
+ 210*a*b**6*x**(7/2)*log(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 420*a*b**6*x**(7/2)*log(a/b + sqrt(x))/(30*
a**9*x**(7/2) + 30*a**8*b*x**4) + 210*b**7*x**4*log(x)/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 420*b**7*x**4*log
(a/b + sqrt(x))/(30*a**9*x**(7/2) + 30*a**8*b*x**4) - 420*b**7*x**4/(30*a**9*x**(7/2) + 30*a**8*b*x**4), True)
)

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Giac [A]  time = 1.12052, size = 151, normalized size = 1.23 \begin{align*} -\frac{14 \, b^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{8}} + \frac{7 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{8}} + \frac{420 \, a b^{6} x^{3} + 210 \, a^{2} b^{5} x^{\frac{5}{2}} - 70 \, a^{3} b^{4} x^{2} + 35 \, a^{4} b^{3} x^{\frac{3}{2}} - 21 \, a^{5} b^{2} x + 14 \, a^{6} b \sqrt{x} - 10 \, a^{7}}{30 \,{\left (b \sqrt{x} + a\right )} a^{8} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*x^(1/2))^2,x, algorithm="giac")

[Out]

-14*b^6*log(abs(b*sqrt(x) + a))/a^8 + 7*b^6*log(abs(x))/a^8 + 1/30*(420*a*b^6*x^3 + 210*a^2*b^5*x^(5/2) - 70*a
^3*b^4*x^2 + 35*a^4*b^3*x^(3/2) - 21*a^5*b^2*x + 14*a^6*b*sqrt(x) - 10*a^7)/((b*sqrt(x) + a)*a^8*x^3)