3.2986 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^4} \, dx\)

Optimal. Leaf size=172 \[ \frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]

[Out]

(4*a^5*Sqrt[a + b*Sqrt[c/x]])/(b^6*c^3) - (20*a^4*(a + b*Sqrt[c/x])^(3/2))/(3*b^
6*c^3) + (8*a^3*(a + b*Sqrt[c/x])^(5/2))/(b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x])^(
7/2))/(7*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(9/2))/(9*b^6*c^3) - (4*(a + b*Sqrt[
c/x])^(11/2))/(11*b^6*c^3)

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Rubi [A]  time = 0.223686, antiderivative size = 172, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 3, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.143 \[ \frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]

Antiderivative was successfully verified.

[In]  Int[1/(Sqrt[a + b*Sqrt[c/x]]*x^4),x]

[Out]

(4*a^5*Sqrt[a + b*Sqrt[c/x]])/(b^6*c^3) - (20*a^4*(a + b*Sqrt[c/x])^(3/2))/(3*b^
6*c^3) + (8*a^3*(a + b*Sqrt[c/x])^(5/2))/(b^6*c^3) - (40*a^2*(a + b*Sqrt[c/x])^(
7/2))/(7*b^6*c^3) + (20*a*(a + b*Sqrt[c/x])^(9/2))/(9*b^6*c^3) - (4*(a + b*Sqrt[
c/x])^(11/2))/(11*b^6*c^3)

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Rubi in Sympy [A]  time = 29.6582, size = 153, normalized size = 0.89 \[ \frac{4 a^{5} \sqrt{a + b \sqrt{\frac{c}{x}}}}{b^{6} c^{3}} - \frac{20 a^{4} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{3}{2}}}{3 b^{6} c^{3}} + \frac{8 a^{3} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{5}{2}}}{b^{6} c^{3}} - \frac{40 a^{2} \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{7}{2}}}{7 b^{6} c^{3}} + \frac{20 a \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{9}{2}}}{9 b^{6} c^{3}} - \frac{4 \left (a + b \sqrt{\frac{c}{x}}\right )^{\frac{11}{2}}}{11 b^{6} c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4*a**5*sqrt(a + b*sqrt(c/x))/(b**6*c**3) - 20*a**4*(a + b*sqrt(c/x))**(3/2)/(3*b
**6*c**3) + 8*a**3*(a + b*sqrt(c/x))**(5/2)/(b**6*c**3) - 40*a**2*(a + b*sqrt(c/
x))**(7/2)/(7*b**6*c**3) + 20*a*(a + b*sqrt(c/x))**(9/2)/(9*b**6*c**3) - 4*(a +
b*sqrt(c/x))**(11/2)/(11*b**6*c**3)

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Mathematica [A]  time = 0.120352, size = 111, normalized size = 0.65 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (256 a^5 x^2-128 a^4 b x^2 \sqrt{\frac{c}{x}}+96 a^3 b^2 c x-80 a^2 b^3 c x \sqrt{\frac{c}{x}}+70 a b^4 c^2-63 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{693 b^6 c^3 x^2} \]

Antiderivative was successfully verified.

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

[Out]

(4*Sqrt[a + b*Sqrt[c/x]]*(70*a*b^4*c^2 + 96*a^3*b^2*c*x - 80*a^2*b^3*c*Sqrt[c/x]
*x - 63*b^5*c*(c/x)^(3/2)*x + 256*a^5*x^2 - 128*a^4*b*Sqrt[c/x]*x^2))/(693*b^6*c
^3*x^2)

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Maple [C]  time = 0.072, size = 392, normalized size = 2.3 \[ -{\frac{1}{693\,{b}^{7}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{11/2}\sqrt{{\frac{c}{x}}}{x}^{4}b-693\,\ln \left ( 1/2\,{\frac{1}{\sqrt{a}} \left ( b\sqrt{{\frac{c}{x}}}\sqrt{x}+2\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }\sqrt{a}+2\,a\sqrt{x} \right ) } \right ){a}^{11/2}\sqrt{{\frac{c}{x}}}{x}^{4}b+252\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2} \left ({\frac{c}{x}} \right ) ^{5/2}{b}^{5}+852\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2} \left ({\frac{c}{x}} \right ) ^{3/2}{a}^{2}{b}^{3}+1748\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2}\sqrt{{\frac{c}{x}}}{a}^{4}b+1386\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{x}^{7/2}{a}^{6}+1386\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{x}^{7/2}{a}^{6}-2772\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{5/2}{a}^{5}-1236\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{x}^{3/2}{a}^{3}{b}^{2}c-532\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{x}a{b}^{4}{c}^{2} \right ){x}^{-{\frac{13}{2}}}{\frac{1}{\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }}} \left ({\frac{c}{x}} \right ) ^{-{\frac{7}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(1/x^4/(a+b*(c/x)^(1/2))^(1/2),x)

[Out]

-1/693*(a+b*(c/x)^(1/2))^(1/2)*(693*ln(1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(a*x+b*(c/x)
^(1/2)*x)^(1/2)*a^(1/2)+2*a*x^(1/2))/a^(1/2))*a^(11/2)*(c/x)^(1/2)*x^4*b-693*ln(
1/2*(b*(c/x)^(1/2)*x^(1/2)+2*(x*(a+b*(c/x)^(1/2)))^(1/2)*a^(1/2)+2*a*x^(1/2))/a^
(1/2))*a^(11/2)*(c/x)^(1/2)*x^4*b+252*(a*x+b*(c/x)^(1/2)*x)^(3/2)*x^(5/2)*(c/x)^
(5/2)*b^5+852*(a*x+b*(c/x)^(1/2)*x)^(3/2)*x^(5/2)*(c/x)^(3/2)*a^2*b^3+1748*(a*x+
b*(c/x)^(1/2)*x)^(3/2)*x^(5/2)*(c/x)^(1/2)*a^4*b+1386*(a*x+b*(c/x)^(1/2)*x)^(1/2
)*x^(7/2)*a^6+1386*(x*(a+b*(c/x)^(1/2)))^(1/2)*x^(7/2)*a^6-2772*(a*x+b*(c/x)^(1/
2)*x)^(3/2)*x^(5/2)*a^5-1236*(a*x+b*(c/x)^(1/2)*x)^(3/2)*x^(3/2)*a^3*b^2*c-532*(
a*x+b*(c/x)^(1/2)*x)^(3/2)*x^(1/2)*a*b^4*c^2)/x^(13/2)/(x*(a+b*(c/x)^(1/2)))^(1/
2)/b^7/(c/x)^(7/2)

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Maxima [A]  time = 1.33538, size = 171, normalized size = 0.99 \[ -\frac{4 \,{\left (\frac{63 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{11}{2}}}{b^{6}} - \frac{385 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{9}{2}} a}{b^{6}} + \frac{990 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{7}{2}} a^{2}}{b^{6}} - \frac{1386 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{5}{2}} a^{3}}{b^{6}} + \frac{1155 \,{\left (b \sqrt{\frac{c}{x}} + a\right )}^{\frac{3}{2}} a^{4}}{b^{6}} - \frac{693 \, \sqrt{b \sqrt{\frac{c}{x}} + a} a^{5}}{b^{6}}\right )}}{693 \, c^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

-4/693*(63*(b*sqrt(c/x) + a)^(11/2)/b^6 - 385*(b*sqrt(c/x) + a)^(9/2)*a/b^6 + 99
0*(b*sqrt(c/x) + a)^(7/2)*a^2/b^6 - 1386*(b*sqrt(c/x) + a)^(5/2)*a^3/b^6 + 1155*
(b*sqrt(c/x) + a)^(3/2)*a^4/b^6 - 693*sqrt(b*sqrt(c/x) + a)*a^5/b^6)/c^3

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Fricas [A]  time = 0.247496, size = 120, normalized size = 0.7 \[ \frac{4 \,{\left (70 \, a b^{4} c^{2} + 96 \, a^{3} b^{2} c x + 256 \, a^{5} x^{2} -{\left (63 \, b^{5} c^{2} + 80 \, a^{2} b^{3} c x + 128 \, a^{4} b x^{2}\right )} \sqrt{\frac{c}{x}}\right )} \sqrt{b \sqrt{\frac{c}{x}} + a}}{693 \, b^{6} c^{3} x^{2}} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

4/693*(70*a*b^4*c^2 + 96*a^3*b^2*c*x + 256*a^5*x^2 - (63*b^5*c^2 + 80*a^2*b^3*c*
x + 128*a^4*b*x^2)*sqrt(c/x))*sqrt(b*sqrt(c/x) + a)/(b^6*c^3*x^2)

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Sympy [F]  time = 0., size = 0, normalized size = 0. \[ \int \frac{1}{x^{4} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Integral(1/(x**4*sqrt(a + b*sqrt(c/x))), x)

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GIAC/XCAS [F(-2)]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: TypeError} \]

Verification of antiderivative is not currently implemented for this CAS.

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

[Out]

Exception raised: TypeError