3.2212 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^3 x^4} \, dx\)

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

[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 = 0.234081, 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 \[ -\frac{56 b^6 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{28 b^6 \log (x)}{a^9}+\frac{14 b^6}{a^8 \left (a+b \sqrt{x}\right )}+\frac{42 b^5}{a^8 \sqrt{x}}+\frac{b^6}{a^7 \left (a+b \sqrt{x}\right )^2}-\frac{15 b^4}{a^7 x}+\frac{20 b^3}{3 a^6 x^{3/2}}-\frac{3 b^2}{a^5 x^2}+\frac{6 b}{5 a^4 x^{5/2}}-\frac{1}{3 a^3 x^3} \]

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

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Rubi in Sympy [A]  time = 39.7535, size = 141, normalized size = 1.01 \[ - \frac{1}{3 a^{3} x^{3}} + \frac{6 b}{5 a^{4} x^{\frac{5}{2}}} - \frac{3 b^{2}}{a^{5} x^{2}} + \frac{20 b^{3}}{3 a^{6} x^{\frac{3}{2}}} + \frac{b^{6}}{a^{7} \left (a + b \sqrt{x}\right )^{2}} - \frac{15 b^{4}}{a^{7} x} + \frac{14 b^{6}}{a^{8} \left (a + b \sqrt{x}\right )} + \frac{42 b^{5}}{a^{8} \sqrt{x}} + \frac{56 b^{6} \log{\left (\sqrt{x} \right )}}{a^{9}} - \frac{56 b^{6} \log{\left (a + b \sqrt{x} \right )}}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**4/(a+b*x**(1/2))**3,x)

[Out]

-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)) + b**6/(a**7*(a + b*sqrt(x))**2) - 15*b**4/(a**7*x) + 14*b**6/(a**8*(a
+ b*sqrt(x))) + 42*b**5/(a**8*sqrt(x)) + 56*b**6*log(sqrt(x))/a**9 - 56*b**6*lo
g(a + b*sqrt(x))/a**9

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Mathematica [A]  time = 0.276557, size = 128, normalized size = 0.92 \[ \frac{\frac{a \left (-5 a^7+8 a^6 b \sqrt{x}-14 a^5 b^2 x+28 a^4 b^3 x^{3/2}-70 a^3 b^4 x^2+280 a^2 b^5 x^{5/2}+1260 a b^6 x^3+840 b^7 x^{7/2}\right )}{x^3 \left (a+b \sqrt{x}\right )^2}-840 b^6 \log \left (a+b \sqrt{x}\right )+420 b^6 \log (x)}{15 a^9} \]

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) + 1260*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 = 0.019, size = 122, normalized size = 0.9 \[ -{\frac{1}{3\,{a}^{3}{x}^{3}}}+{\frac{6\,b}{5\,{a}^{4}}{x}^{-{\frac{5}{2}}}}-3\,{\frac{{b}^{2}}{{a}^{5}{x}^{2}}}+{\frac{20\,{b}^{3}}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}-15\,{\frac{{b}^{4}}{{a}^{7}x}}+28\,{\frac{{b}^{6}\ln \left ( x \right ) }{{a}^{9}}}-56\,{\frac{{b}^{6}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{9}}}+42\,{\frac{{b}^{5}}{{a}^{8}\sqrt{x}}}+{\frac{{b}^{6}}{{a}^{7}} \left ( a+b\sqrt{x} \right ) ^{-2}}+14\,{\frac{{b}^{6}}{{a}^{8} \left ( a+b\sqrt{x} \right ) }} \]

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+2
8*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 = 1.44213, size = 178, normalized size = 1.28 \[ \frac{840 \, b^{7} x^{\frac{7}{2}} + 1260 \, a b^{6} x^{3} + 280 \, a^{2} b^{5} x^{\frac{5}{2}} - 70 \, a^{3} b^{4} x^{2} + 28 \, a^{4} b^{3} x^{\frac{3}{2}} - 14 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt{x} - 5 \, a^{7}}{15 \,{\left (a^{8} b^{2} x^{4} + 2 \, a^{9} b x^{\frac{7}{2}} + a^{10} x^{3}\right )}} - \frac{56 \, b^{6} \log \left (b \sqrt{x} + a\right )}{a^{9}} + \frac{28 \, b^{6} \log \left (x\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^4),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 = 0.244592, size = 242, normalized size = 1.74 \[ \frac{1260 \, a^{2} b^{6} x^{3} - 70 \, a^{4} b^{4} x^{2} - 14 \, a^{6} b^{2} x - 5 \, a^{8} - 840 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{\frac{7}{2}} + a^{2} b^{6} x^{3}\right )} \log \left (b \sqrt{x} + a\right ) + 840 \,{\left (b^{8} x^{4} + 2 \, a b^{7} x^{\frac{7}{2}} + a^{2} b^{6} x^{3}\right )} \log \left (\sqrt{x}\right ) + 4 \,{\left (210 \, a b^{7} x^{3} + 70 \, a^{3} b^{5} x^{2} + 7 \, a^{5} b^{3} x + 2 \, a^{7} b\right )} \sqrt{x}}{15 \,{\left (a^{9} b^{2} x^{4} + 2 \, a^{10} b x^{\frac{7}{2}} + a^{11} x^{3}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^4),x, algorithm="fricas")

[Out]

1/15*(1260*a^2*b^6*x^3 - 70*a^4*b^4*x^2 - 14*a^6*b^2*x - 5*a^8 - 840*(b^8*x^4 +
2*a*b^7*x^(7/2) + a^2*b^6*x^3)*log(b*sqrt(x) + a) + 840*(b^8*x^4 + 2*a*b^7*x^(7/
2) + a^2*b^6*x^3)*log(sqrt(x)) + 4*(210*a*b^7*x^3 + 70*a^3*b^5*x^2 + 7*a^5*b^3*x
+ 2*a^7*b)*sqrt(x))/(a^9*b^2*x^4 + 2*a^10*b*x^(7/2) + a^11*x^3)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.278928, size = 166, normalized size = 1.19 \[ -\frac{56 \, b^{6}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{9}} + \frac{28 \, b^{6}{\rm ln}\left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, a b^{7} x^{\frac{7}{2}} + 1260 \, a^{2} b^{6} x^{3} + 280 \, a^{3} b^{5} x^{\frac{5}{2}} - 70 \, a^{4} b^{4} x^{2} + 28 \, a^{5} b^{3} x^{\frac{3}{2}} - 14 \, a^{6} b^{2} x + 8 \, a^{7} b \sqrt{x} - 5 \, a^{8}}{15 \,{\left (b \sqrt{x} + a\right )}^{2} a^{9} x^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^3*x^4),x, algorithm="giac")

[Out]

-56*b^6*ln(abs(b*sqrt(x) + a))/a^9 + 28*b^6*ln(abs(x))/a^9 + 1/15*(840*a*b^7*x^(
7/2) + 1260*a^2*b^6*x^3 + 280*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)