3.2214 \(\int \frac{1}{(a+b \sqrt{x})^3 x^4} \, dx\)

Optimal. Leaf size=139 \[ \frac{20 b^3}{3 a^6 x^{3/2}}-\frac{3 b^2}{a^5 x^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}+\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{15 b^4}{a^7 x}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{1}{3 a^3 x^3} \]

[Out]

b^6/(a^7*(a + b*Sqrt[x])^2) + (14*b^6)/(a^8*(a + b*Sqrt[x])) - 1/(3*a^3*x^3) + (6*b)/(5*a^4*x^(5/2)) - (3*b^2)
/(a^5*x^2) + (20*b^3)/(3*a^6*x^(3/2)) - (15*b^4)/(a^7*x) + (42*b^5)/(a^8*Sqrt[x]) - (56*b^6*Log[a + b*Sqrt[x]]
)/a^9 + (28*b^6*Log[x])/a^9

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Rubi [A]  time = 0.0993551, antiderivative size = 139, 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} \[ \frac{20 b^3}{3 a^6 x^{3/2}}-\frac{3 b^2}{a^5 x^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}+\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{15 b^4}{a^7 x}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{1}{3 a^3 x^3} \]

Antiderivative was successfully verified.

[In]

Int[1/((a + b*Sqrt[x])^3*x^4),x]

[Out]

b^6/(a^7*(a + b*Sqrt[x])^2) + (14*b^6)/(a^8*(a + b*Sqrt[x])) - 1/(3*a^3*x^3) + (6*b)/(5*a^4*x^(5/2)) - (3*b^2)
/(a^5*x^2) + (20*b^3)/(3*a^6*x^(3/2)) - (15*b^4)/(a^7*x) + (42*b^5)/(a^8*Sqrt[x]) - (56*b^6*Log[a + b*Sqrt[x]]
)/a^9 + (28*b^6*Log[x])/a^9

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 )^3 x^4} \, dx &=2 \operatorname{Subst}\left (\int \frac{1}{x^7 (a+b x)^3} \, dx,x,\sqrt{x}\right )\\ &=2 \operatorname{Subst}\left (\int \left (\frac{1}{a^3 x^7}-\frac{3 b}{a^4 x^6}+\frac{6 b^2}{a^5 x^5}-\frac{10 b^3}{a^6 x^4}+\frac{15 b^4}{a^7 x^3}-\frac{21 b^5}{a^8 x^2}+\frac{28 b^6}{a^9 x}-\frac{b^7}{a^7 (a+b x)^3}-\frac{7 b^7}{a^8 (a+b x)^2}-\frac{28 b^7}{a^9 (a+b x)}\right ) \, dx,x,\sqrt{x}\right )\\ &=\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}-\frac{1}{3 a^3 x^3}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{3 b^2}{a^5 x^2}+\frac{20 b^3}{3 a^6 x^{3/2}}-\frac{15 b^4}{a^7 x}+\frac{42 b^5}{a^8 \sqrt{x}}-\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}\\ \end{align*}

Mathematica [A]  time = 0.153078, size = 128, normalized size = 0.92 \[ \frac{\frac{a \left (28 a^4 b^3 x^{3/2}-70 a^3 b^4 x^2+280 a^2 b^5 x^{5/2}-14 a^5 b^2 x+8 a^6 b \sqrt{x}-5 a^7+1260 a b^6 x^3+840 b^7 x^{7/2}\right )}{x^3 \left (a+b \sqrt{x}\right )^2}-840 b^6 \log \left (a+b \sqrt{x}\right )+420 b^6 \log (x)}{15 a^9} \]

Antiderivative was successfully verified.

[In]

Integrate[1/((a + b*Sqrt[x])^3*x^4),x]

[Out]

((a*(-5*a^7 + 8*a^6*b*Sqrt[x] - 14*a^5*b^2*x + 28*a^4*b^3*x^(3/2) - 70*a^3*b^4*x^2 + 280*a^2*b^5*x^(5/2) + 126
0*a*b^6*x^3 + 840*b^7*x^(7/2)))/((a + b*Sqrt[x])^2*x^3) - 840*b^6*Log[a + b*Sqrt[x]] + 420*b^6*Log[x])/(15*a^9
)

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Maple [A]  time = 0.011, size = 122, normalized size = 0.9 \begin{align*} -{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{6\,b}{5\,{a}^{4}}{x}^{-{\frac{5}{2}}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}+{\frac{20\,{b}^{3}}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}-15\,{\frac{{b}^{4}}{{a}^{7}x}}+28\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{9}}}-56\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{9}}}+42\,{\frac{{b}^{5}}{{a}^{8}\sqrt{x}}}+{\frac{{b}^{6}}{{a}^{7}} \left ( a+b\sqrt{x} \right ) ^{-2}}+14\,{\frac{{b}^{6}}{{a}^{8} \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))^3,x)

[Out]

-1/3/a^3/x^3+6/5*b/a^4/x^(5/2)-3*b^2/a^5/x^2+20/3*b^3/a^6/x^(3/2)-15*b^4/a^7/x+28*b^6*ln(x)/a^9-56*b^6*ln(a+b*
x^(1/2))/a^9+42*b^5/a^8/x^(1/2)+b^6/a^7/(a+b*x^(1/2))^2+14*b^6/a^8/(a+b*x^(1/2))

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Maxima [A]  time = 0.983174, size = 178, normalized size = 1.28 \begin{align*} \frac{840 \, b^{7} x^{\frac{7}{2}} + 1260 \, a b^{6} x^{3} + 280 \, a^{2} b^{5} x^{\frac{5}{2}} - 70 \, a^{3} b^{4} x^{2} + 28 \, a^{4} b^{3} x^{\frac{3}{2}} - 14 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt{x} - 5 \, a^{7}}{15 \,{\left (a^{8} b^{2} x^{4} + 2 \, a^{9} b x^{\frac{7}{2}} + a^{10} x^{3}\right )}} - \frac{56 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*x^(1/2))^3,x, algorithm="maxima")

[Out]

1/15*(840*b^7*x^(7/2) + 1260*a*b^6*x^3 + 280*a^2*b^5*x^(5/2) - 70*a^3*b^4*x^2 + 28*a^4*b^3*x^(3/2) - 14*a^5*b^
2*x + 8*a^6*b*sqrt(x) - 5*a^7)/(a^8*b^2*x^4 + 2*a^9*b*x^(7/2) + a^10*x^3) - 56*b^6*log(b*sqrt(x) + a)/a^9 + 28
*b^6*log(x)/a^9

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Fricas [A]  time = 1.35731, size = 462, normalized size = 3.32 \begin{align*} -\frac{420 \, a^{2} b^{8} x^{4} - 630 \, a^{4} b^{6} x^{3} + 140 \, a^{6} b^{4} x^{2} + 35 \, a^{8} b^{2} x + 5 \, a^{10} + 840 \,{\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (b \sqrt{x} + a\right ) - 840 \,{\left (b^{10} x^{5} - 2 \, a^{2} b^{8} x^{4} + a^{4} b^{6} x^{3}\right )} \log \left (\sqrt{x}\right ) - 2 \,{\left (420 \, a b^{9} x^{4} - 700 \, a^{3} b^{7} x^{3} + 224 \, a^{5} b^{5} x^{2} + 32 \, a^{7} b^{3} x + 9 \, a^{9} b\right )} \sqrt{x}}{15 \,{\left (a^{9} b^{4} x^{5} - 2 \, a^{11} b^{2} x^{4} + a^{13} 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))^3,x, algorithm="fricas")

[Out]

-1/15*(420*a^2*b^8*x^4 - 630*a^4*b^6*x^3 + 140*a^6*b^4*x^2 + 35*a^8*b^2*x + 5*a^10 + 840*(b^10*x^5 - 2*a^2*b^8
*x^4 + a^4*b^6*x^3)*log(b*sqrt(x) + a) - 840*(b^10*x^5 - 2*a^2*b^8*x^4 + a^4*b^6*x^3)*log(sqrt(x)) - 2*(420*a*
b^9*x^4 - 700*a^3*b^7*x^3 + 224*a^5*b^5*x^2 + 32*a^7*b^3*x + 9*a^9*b)*sqrt(x))/(a^9*b^4*x^5 - 2*a^11*b^2*x^4 +
a^13*x^3)

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Sympy [A]  time = 98.2033, size = 707, normalized size = 5.09 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x**4/(a+b*x**(1/2))**3,x)

[Out]

Piecewise((zoo/x**(9/2), Eq(a, 0) & Eq(b, 0)), (-2/(9*b**3*x**(9/2)), Eq(a, 0)), (-1/(3*a**3*x**3), Eq(b, 0)),
(-5*a**8*sqrt(x)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 8*a**7*b*x/(15*a**11*x**(7/2
) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 14*a**6*b**2*x**(3/2)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15
*a**9*b**2*x**(9/2)) + 28*a**5*b**3*x**2/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 70*a*
*4*b**4*x**(5/2)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 280*a**3*b**5*x**3/(15*a**11*
x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 420*a**2*b**6*x**(7/2)*log(x)/(15*a**11*x**(7/2) + 30*a*
*10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 840*a**2*b**6*x**(7/2)*log(a/b + sqrt(x))/(15*a**11*x**(7/2) + 30*a**10*
b*x**4 + 15*a**9*b**2*x**(9/2)) + 1260*a**2*b**6*x**(7/2)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*
x**(9/2)) + 840*a*b**7*x**4*log(x)/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 1680*a*b**7
*x**4*log(a/b + sqrt(x))/(15*a**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 840*a*b**7*x**4/(15*a
**11*x**(7/2) + 30*a**10*b*x**4 + 15*a**9*b**2*x**(9/2)) + 420*b**8*x**(9/2)*log(x)/(15*a**11*x**(7/2) + 30*a*
*10*b*x**4 + 15*a**9*b**2*x**(9/2)) - 840*b**8*x**(9/2)*log(a/b + sqrt(x))/(15*a**11*x**(7/2) + 30*a**10*b*x**
4 + 15*a**9*b**2*x**(9/2)), True))

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Giac [A]  time = 1.11283, size = 166, normalized size = 1.19 \begin{align*} -\frac{56 \, b^{6} \log \left ({\left | b \sqrt{x} + a \right |}\right )}{a^{9}} + \frac{28 \, b^{6} \log \left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{\frac{7}{2}} + 1260 \, a^{2} b^{6} x^{3} + 280 \, a^{3} b^{5} x^{\frac{5}{2}} - 70 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} x^{\frac{3}{2}} - 14 \, a^{6} b^{2} x + 8 \, a^{7} b \sqrt{x} - 5 \, a^{8}}{15 \,{\left (b \sqrt{x} + a\right )}^{2} a^{9} x^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*x^(1/2))^3,x, algorithm="giac")

[Out]

-56*b^6*log(abs(b*sqrt(x) + a))/a^9 + 28*b^6*log(abs(x))/a^9 + 1/15*(840*a*b^7*x^(7/2) + 1260*a^2*b^6*x^3 + 28
0*a^3*b^5*x^(5/2) - 70*a^4*b^4*x^2 + 28*a^5*b^3*x^(3/2) - 14*a^6*b^2*x + 8*a^7*b*sqrt(x) - 5*a^8)/((b*sqrt(x)
+ a)^2*a^9*x^3)