3.2198 \(\int \frac{1}{(a+b \sqrt{x}) x^4} \, dx\)

Optimal. Leaf size=103 \[ \frac{2 b^3}{3 a^4 x^{3/2}}-\frac{b^2}{2 a^3 x^2}+\frac{2 b^5}{a^6 \sqrt{x}}-\frac{b^4}{a^5 x}-\frac{2 b^6 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{2 b}{5 a^2 x^{5/2}}-\frac{1}{3 a x^3} \]

[Out]

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

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Rubi [A]  time = 0.0515842, antiderivative size = 103, 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} \[ \frac{2 b^3}{3 a^4 x^{3/2}}-\frac{b^2}{2 a^3 x^2}+\frac{2 b^5}{a^6 \sqrt{x}}-\frac{b^4}{a^5 x}-\frac{2 b^6 \log \left (a+b \sqrt{x}\right )}{a^7}+\frac{b^6 \log (x)}{a^7}+\frac{2 b}{5 a^2 x^{5/2}}-\frac{1}{3 a x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])*x^4),x]

[Out]

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

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

Mathematica [A]  time = 0.0573427, size = 93, normalized size = 0.9 \[ \frac{\frac{a \left (20 a^2 b^3 x^{3/2}-15 a^3 b^2 x+12 a^4 b \sqrt{x}-10 a^5-30 a b^4 x^2+60 b^5 x^{5/2}\right )}{x^3}-60 b^6 \log \left (a+b \sqrt{x}\right )+30 b^6 \log (x)}{30 a^7} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])*x^4),x]

[Out]

((a*(-10*a^5 + 12*a^4*b*Sqrt[x] - 15*a^3*b^2*x + 20*a^2*b^3*x^(3/2) - 30*a*b^4*x^2 + 60*b^5*x^(5/2)))/x^3 - 60
*b^6*Log[a + b*Sqrt[x]] + 30*b^6*Log[x])/(30*a^7)

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Maple [A]  time = 0.01, size = 88, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,a{x}^{3}}}+{\frac{2\,b}{5\,{a}^{2}}{x}^{-{\frac{5}{2}}}}-{\frac{{b}^{2}}{2\,{x}^{2}{a}^{3}}}+{\frac{2\,{b}^{3}}{3\,{a}^{4}}{x}^{-{\frac{3}{2}}}}-{\frac{{b}^{4}}{x{a}^{5}}}+{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{7}}}-2\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{7}}}+2\,{\frac{{b}^{5}}{{a}^{6}\sqrt{x}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 0.969261, size = 116, normalized size = 1.13 \begin{align*} -\frac{2 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{7}} + \frac{b^{6} \log \left (x\right )}{a^{7}} + \frac{60 \, b^{5} x^{\frac{5}{2}} - 30 \, a b^{4} x^{2} + 20 \, a^{2} b^{3} x^{\frac{3}{2}} - 15 \, a^{3} b^{2} x + 12 \, a^{4} b \sqrt{x} - 10 \, a^{5}}{30 \, a^{6} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^6*log(b*sqrt(x) + a)/a^7 + b^6*log(x)/a^7 + 1/30*(60*b^5*x^(5/2) - 30*a*b^4*x^2 + 20*a^2*b^3*x^(3/2) - 15
*a^3*b^2*x + 12*a^4*b*sqrt(x) - 10*a^5)/(a^6*x^3)

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Fricas [A]  time = 1.28383, size = 225, normalized size = 2.18 \begin{align*} -\frac{60 \, b^{6} x^{3} \log \left (b \sqrt{x} + a\right ) - 60 \, b^{6} x^{3} \log \left (\sqrt{x}\right ) + 30 \, a^{2} b^{4} x^{2} + 15 \, a^{4} b^{2} x + 10 \, a^{6} - 4 \,{\left (15 \, a b^{5} x^{2} + 5 \, a^{3} b^{3} x + 3 \, a^{5} b\right )} \sqrt{x}}{30 \, a^{7} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/30*(60*b^6*x^3*log(b*sqrt(x) + a) - 60*b^6*x^3*log(sqrt(x)) + 30*a^2*b^4*x^2 + 15*a^4*b^2*x + 10*a^6 - 4*(1
5*a*b^5*x^2 + 5*a^3*b^3*x + 3*a^5*b)*sqrt(x))/(a^7*x^3)

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Sympy [A]  time = 9.54432, size = 126, normalized size = 1.22 \begin{align*} \begin{cases} \frac{\tilde{\infty }}{x^{\frac{7}{2}}} & \text{for}\: a = 0 \wedge b = 0 \\- \frac{2}{7 b x^{\frac{7}{2}}} & \text{for}\: a = 0 \\- \frac{1}{3 a x^{3}} & \text{for}\: b = 0 \\- \frac{1}{3 a x^{3}} + \frac{2 b}{5 a^{2} x^{\frac{5}{2}}} - \frac{b^{2}}{2 a^{3} x^{2}} + \frac{2 b^{3}}{3 a^{4} x^{\frac{3}{2}}} - \frac{b^{4}}{a^{5} x} + \frac{2 b^{5}}{a^{6} \sqrt{x}} + \frac{b^{6} \log{\left (x \right )}}{a^{7}} - \frac{2 b^{6} \log{\left (\frac{a}{b} + \sqrt{x} \right )}}{a^{7}} & \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)),x)

[Out]

Piecewise((zoo/x**(7/2), Eq(a, 0) & Eq(b, 0)), (-2/(7*b*x**(7/2)), Eq(a, 0)), (-1/(3*a*x**3), Eq(b, 0)), (-1/(
3*a*x**3) + 2*b/(5*a**2*x**(5/2)) - b**2/(2*a**3*x**2) + 2*b**3/(3*a**4*x**(3/2)) - b**4/(a**5*x) + 2*b**5/(a*
*6*sqrt(x)) + b**6*log(x)/a**7 - 2*b**6*log(a/b + sqrt(x))/a**7, True))

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Giac [A]  time = 1.11855, size = 123, normalized size = 1.19 \begin{align*} -\frac{2 \, b^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{7}} + \frac{b^{6} \log \left ({\left | x \right |}\right )}{a^{7}} + \frac{60 \, a b^{5} x^{\frac{5}{2}} - 30 \, a^{2} b^{4} x^{2} + 20 \, a^{3} b^{3} x^{\frac{3}{2}} - 15 \, a^{4} b^{2} x + 12 \, a^{5} b \sqrt{x} - 10 \, a^{6}}{30 \, a^{7} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2*b^6*log(abs(b*sqrt(x) + a))/a^7 + b^6*log(abs(x))/a^7 + 1/30*(60*a*b^5*x^(5/2) - 30*a^2*b^4*x^2 + 20*a^3*b^
3*x^(3/2) - 15*a^4*b^2*x + 12*a^5*b*sqrt(x) - 10*a^6)/(a^7*x^3)