3.2220 \(\int \frac{1}{\left (a+b \sqrt{x}\right )^5 x^3} \, dx\)

Optimal. Leaf size=156 \[ -\frac{140 b^4 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{70 b^4 \log (x)}{a^9}+\frac{70 b^4}{a^8 \left (a+b \sqrt{x}\right )}+\frac{70 b^3}{a^8 \sqrt{x}}+\frac{15 b^4}{a^7 \left (a+b \sqrt{x}\right )^2}-\frac{15 b^2}{a^7 x}+\frac{10 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 b}{3 a^6 x^{3/2}}+\frac{b^4}{2 a^5 \left (a+b \sqrt{x}\right )^4}-\frac{1}{2 a^5 x^2} \]

[Out]

b^4/(2*a^5*(a + b*Sqrt[x])^4) + (10*b^4)/(3*a^6*(a + b*Sqrt[x])^3) + (15*b^4)/(a
^7*(a + b*Sqrt[x])^2) + (70*b^4)/(a^8*(a + b*Sqrt[x])) - 1/(2*a^5*x^2) + (10*b)/
(3*a^6*x^(3/2)) - (15*b^2)/(a^7*x) + (70*b^3)/(a^8*Sqrt[x]) - (140*b^4*Log[a + b
*Sqrt[x]])/a^9 + (70*b^4*Log[x])/a^9

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Rubi [A]  time = 0.267995, antiderivative size = 156, 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 \[ -\frac{140 b^4 \log \left (a+b \sqrt{x}\right )}{a^9}+\frac{70 b^4 \log (x)}{a^9}+\frac{70 b^4}{a^8 \left (a+b \sqrt{x}\right )}+\frac{70 b^3}{a^8 \sqrt{x}}+\frac{15 b^4}{a^7 \left (a+b \sqrt{x}\right )^2}-\frac{15 b^2}{a^7 x}+\frac{10 b^4}{3 a^6 \left (a+b \sqrt{x}\right )^3}+\frac{10 b}{3 a^6 x^{3/2}}+\frac{b^4}{2 a^5 \left (a+b \sqrt{x}\right )^4}-\frac{1}{2 a^5 x^2} \]

Antiderivative was successfully verified.

[In]  Int[1/((a + b*Sqrt[x])^5*x^3),x]

[Out]

b^4/(2*a^5*(a + b*Sqrt[x])^4) + (10*b^4)/(3*a^6*(a + b*Sqrt[x])^3) + (15*b^4)/(a
^7*(a + b*Sqrt[x])^2) + (70*b^4)/(a^8*(a + b*Sqrt[x])) - 1/(2*a^5*x^2) + (10*b)/
(3*a^6*x^(3/2)) - (15*b^2)/(a^7*x) + (70*b^3)/(a^8*Sqrt[x]) - (140*b^4*Log[a + b
*Sqrt[x]])/a^9 + (70*b^4*Log[x])/a^9

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Rubi in Sympy [A]  time = 67.0575, size = 155, normalized size = 0.99 \[ \frac{b^{4}}{2 a^{5} \left (a + b \sqrt{x}\right )^{4}} - \frac{1}{2 a^{5} x^{2}} + \frac{10 b^{4}}{3 a^{6} \left (a + b \sqrt{x}\right )^{3}} + \frac{10 b}{3 a^{6} x^{\frac{3}{2}}} + \frac{15 b^{4}}{a^{7} \left (a + b \sqrt{x}\right )^{2}} - \frac{15 b^{2}}{a^{7} x} + \frac{70 b^{4}}{a^{8} \left (a + b \sqrt{x}\right )} + \frac{70 b^{3}}{a^{8} \sqrt{x}} + \frac{140 b^{4} \log{\left (\sqrt{x} \right )}}{a^{9}} - \frac{140 b^{4} \log{\left (a + b \sqrt{x} \right )}}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(1/x**3/(a+b*x**(1/2))**5,x)

[Out]

b**4/(2*a**5*(a + b*sqrt(x))**4) - 1/(2*a**5*x**2) + 10*b**4/(3*a**6*(a + b*sqrt
(x))**3) + 10*b/(3*a**6*x**(3/2)) + 15*b**4/(a**7*(a + b*sqrt(x))**2) - 15*b**2/
(a**7*x) + 70*b**4/(a**8*(a + b*sqrt(x))) + 70*b**3/(a**8*sqrt(x)) + 140*b**4*lo
g(sqrt(x))/a**9 - 140*b**4*log(a + b*sqrt(x))/a**9

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Mathematica [A]  time = 0.154801, size = 128, normalized size = 0.82 \[ \frac{\frac{a \left (-3 a^7+8 a^6 b \sqrt{x}-28 a^5 b^2 x+168 a^4 b^3 x^{3/2}+1750 a^3 b^4 x^2+3640 a^2 b^5 x^{5/2}+2940 a b^6 x^3+840 b^7 x^{7/2}\right )}{x^2 \left (a+b \sqrt{x}\right )^4}-840 b^4 \log \left (a+b \sqrt{x}\right )+420 b^4 \log (x)}{6 a^9} \]

Antiderivative was successfully verified.

[In]  Integrate[1/((a + b*Sqrt[x])^5*x^3),x]

[Out]

((a*(-3*a^7 + 8*a^6*b*Sqrt[x] - 28*a^5*b^2*x + 168*a^4*b^3*x^(3/2) + 1750*a^3*b^
4*x^2 + 3640*a^2*b^5*x^(5/2) + 2940*a*b^6*x^3 + 840*b^7*x^(7/2)))/((a + b*Sqrt[x
])^4*x^2) - 840*b^4*Log[a + b*Sqrt[x]] + 420*b^4*Log[x])/(6*a^9)

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Maple [A]  time = 0.02, size = 135, normalized size = 0.9 \[ -{\frac{1}{2\,{a}^{5}{x}^{2}}}+{\frac{10\,b}{3\,{a}^{6}}{x}^{-{\frac{3}{2}}}}-15\,{\frac{{b}^{2}}{{a}^{7}x}}+70\,{\frac{{b}^{4}\ln \left ( x \right ) }{{a}^{9}}}-140\,{\frac{{b}^{4}\ln \left ( a+b\sqrt{x} \right ) }{{a}^{9}}}+70\,{\frac{{b}^{3}}{{a}^{8}\sqrt{x}}}+{\frac{{b}^{4}}{2\,{a}^{5}} \left ( a+b\sqrt{x} \right ) ^{-4}}+{\frac{10\,{b}^{4}}{3\,{a}^{6}} \left ( a+b\sqrt{x} \right ) ^{-3}}+15\,{\frac{{b}^{4}}{{a}^{7} \left ( a+b\sqrt{x} \right ) ^{2}}}+70\,{\frac{{b}^{4}}{{a}^{8} \left ( a+b\sqrt{x} \right ) }} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^3/(a+b*x^(1/2))^5,x)

[Out]

-1/2/a^5/x^2+10/3*b/a^6/x^(3/2)-15*b^2/a^7/x+70*b^4*ln(x)/a^9-140*b^4*ln(a+b*x^(
1/2))/a^9+70*b^3/a^8/x^(1/2)+1/2*b^4/a^5/(a+b*x^(1/2))^4+10/3*b^4/a^6/(a+b*x^(1/
2))^3+15*b^4/a^7/(a+b*x^(1/2))^2+70*b^4/a^8/(a+b*x^(1/2))

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Maxima [A]  time = 1.46034, size = 208, normalized size = 1.33 \[ \frac{840 \, b^{7} x^{\frac{7}{2}} + 2940 \, a b^{6} x^{3} + 3640 \, a^{2} b^{5} x^{\frac{5}{2}} + 1750 \, a^{3} b^{4} x^{2} + 168 \, a^{4} b^{3} x^{\frac{3}{2}} - 28 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt{x} - 3 \, a^{7}}{6 \,{\left (a^{8} b^{4} x^{4} + 4 \, a^{9} b^{3} x^{\frac{7}{2}} + 6 \, a^{10} b^{2} x^{3} + 4 \, a^{11} b x^{\frac{5}{2}} + a^{12} x^{2}\right )}} - \frac{140 \, b^{4} \log \left (b \sqrt{x} + a\right )}{a^{9}} + \frac{70 \, b^{4} \log \left (x\right )}{a^{9}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^3),x, algorithm="maxima")

[Out]

1/6*(840*b^7*x^(7/2) + 2940*a*b^6*x^3 + 3640*a^2*b^5*x^(5/2) + 1750*a^3*b^4*x^2
+ 168*a^4*b^3*x^(3/2) - 28*a^5*b^2*x + 8*a^6*b*sqrt(x) - 3*a^7)/(a^8*b^4*x^4 + 4
*a^9*b^3*x^(7/2) + 6*a^10*b^2*x^3 + 4*a^11*b*x^(5/2) + a^12*x^2) - 140*b^4*log(b
*sqrt(x) + a)/a^9 + 70*b^4*log(x)/a^9

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Fricas [A]  time = 0.255213, size = 346, normalized size = 2.22 \[ \frac{2940 \, a^{2} b^{6} x^{3} + 1750 \, a^{4} b^{4} x^{2} - 28 \, a^{6} b^{2} x - 3 \, a^{8} - 840 \,{\left (b^{8} x^{4} + 6 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2} + 4 \,{\left (a b^{7} x^{3} + a^{3} b^{5} x^{2}\right )} \sqrt{x}\right )} \log \left (b \sqrt{x} + a\right ) + 840 \,{\left (b^{8} x^{4} + 6 \, a^{2} b^{6} x^{3} + a^{4} b^{4} x^{2} + 4 \,{\left (a b^{7} x^{3} + a^{3} b^{5} x^{2}\right )} \sqrt{x}\right )} \log \left (\sqrt{x}\right ) + 8 \,{\left (105 \, a b^{7} x^{3} + 455 \, a^{3} b^{5} x^{2} + 21 \, a^{5} b^{3} x + a^{7} b\right )} \sqrt{x}}{6 \,{\left (a^{9} b^{4} x^{4} + 6 \, a^{11} b^{2} x^{3} + a^{13} x^{2} + 4 \,{\left (a^{10} b^{3} x^{3} + a^{12} b x^{2}\right )} \sqrt{x}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^3),x, algorithm="fricas")

[Out]

1/6*(2940*a^2*b^6*x^3 + 1750*a^4*b^4*x^2 - 28*a^6*b^2*x - 3*a^8 - 840*(b^8*x^4 +
6*a^2*b^6*x^3 + a^4*b^4*x^2 + 4*(a*b^7*x^3 + a^3*b^5*x^2)*sqrt(x))*log(b*sqrt(x
) + a) + 840*(b^8*x^4 + 6*a^2*b^6*x^3 + a^4*b^4*x^2 + 4*(a*b^7*x^3 + a^3*b^5*x^2
)*sqrt(x))*log(sqrt(x)) + 8*(105*a*b^7*x^3 + 455*a^3*b^5*x^2 + 21*a^5*b^3*x + a^
7*b)*sqrt(x))/(a^9*b^4*x^4 + 6*a^11*b^2*x^3 + a^13*x^2 + 4*(a^10*b^3*x^3 + a^12*
b*x^2)*sqrt(x))

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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**3/(a+b*x**(1/2))**5,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.252481, size = 161, normalized size = 1.03 \[ -\frac{140 \, b^{4}{\rm ln}\left ({\left | b \sqrt{x} + a \right |}\right )}{a^{9}} + \frac{70 \, b^{4}{\rm ln}\left ({\left | x \right |}\right )}{a^{9}} + \frac{840 \, b^{7} x^{\frac{7}{2}} + 2940 \, a b^{6} x^{3} + 3640 \, a^{2} b^{5} x^{\frac{5}{2}} + 1750 \, a^{3} b^{4} x^{2} + 168 \, a^{4} b^{3} x^{\frac{3}{2}} - 28 \, a^{5} b^{2} x + 8 \, a^{6} b \sqrt{x} - 3 \, a^{7}}{6 \,{\left (b x + a \sqrt{x}\right )}^{4} a^{8}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(1/((b*sqrt(x) + a)^5*x^3),x, algorithm="giac")

[Out]

-140*b^4*ln(abs(b*sqrt(x) + a))/a^9 + 70*b^4*ln(abs(x))/a^9 + 1/6*(840*b^7*x^(7/
2) + 2940*a*b^6*x^3 + 3640*a^2*b^5*x^(5/2) + 1750*a^3*b^4*x^2 + 168*a^4*b^3*x^(3
/2) - 28*a^5*b^2*x + 8*a^6*b*sqrt(x) - 3*a^7)/((b*x + a*sqrt(x))^4*a^8)