3.2993 \(\int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^4} \, dx\)

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

[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 = 0.103033, 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, Rules used = {369, 266, 43} \[ -\frac{40 a^2 \left (a+b \sqrt{\frac{c}{x}}\right )^{7/2}}{7 b^6 c^3}+\frac{8 a^3 \left (a+b \sqrt{\frac{c}{x}}\right )^{5/2}}{b^6 c^3}-\frac{20 a^4 \left (a+b \sqrt{\frac{c}{x}}\right )^{3/2}}{3 b^6 c^3}+\frac{4 a^5 \sqrt{a+b \sqrt{\frac{c}{x}}}}{b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}+\frac{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3} \]

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)

Rule 369

Int[((d_.)*(x_))^(m_.)*((a_) + (b_.)*((c_.)*(x_)^(q_))^(n_))^(p_.), x_Symbol] :> With[{k = Denominator[n]}, Su
bst[Int[(d*x)^m*(a + b*c^n*x^(n*q))^p, x], x^(1/k), (c*x^q)^(1/k)/(c^(1/k)*(x^(1/k))^(q - 1))]] /; FreeQ[{a, b
, c, d, m, p, q}, x] && FractionQ[n]

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 43

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, n}, x] && NeQ[b*c - a*d, 0] && IGtQ[m, 0] && ( !IntegerQ[n] || (EqQ[c, 0]
&& LeQ[7*m + 4*n + 4, 0]) || LtQ[9*m + 5*(n + 1), 0] || GtQ[m + n + 2, 0])

Rubi steps

\begin{align*} \int \frac{1}{\sqrt{a+b \sqrt{\frac{c}{x}}} x^4} \, dx &=\operatorname{Subst}\left (\int \frac{1}{\sqrt{a+\frac{b \sqrt{c}}{\sqrt{x}}} x^4} \, dx,\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \frac{x^5}{\sqrt{a+b \sqrt{c} x}} \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=-\operatorname{Subst}\left (2 \operatorname{Subst}\left (\int \left (-\frac{a^5}{b^5 c^{5/2} \sqrt{a+b \sqrt{c} x}}+\frac{5 a^4 \sqrt{a+b \sqrt{c} x}}{b^5 c^{5/2}}-\frac{10 a^3 \left (a+b \sqrt{c} x\right )^{3/2}}{b^5 c^{5/2}}+\frac{10 a^2 \left (a+b \sqrt{c} x\right )^{5/2}}{b^5 c^{5/2}}-\frac{5 a \left (a+b \sqrt{c} x\right )^{7/2}}{b^5 c^{5/2}}+\frac{\left (a+b \sqrt{c} x\right )^{9/2}}{b^5 c^{5/2}}\right ) \, dx,x,\frac{1}{\sqrt{x}}\right ),\sqrt{x},\frac{\sqrt{\frac{c}{x}} x}{\sqrt{c}}\right )\\ &=\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{20 a \left (a+b \sqrt{\frac{c}{x}}\right )^{9/2}}{9 b^6 c^3}-\frac{4 \left (a+b \sqrt{\frac{c}{x}}\right )^{11/2}}{11 b^6 c^3}\\ \end{align*}

Mathematica [A]  time = 0.0873034, size = 111, normalized size = 0.65 \[ \frac{4 \sqrt{a+b \sqrt{\frac{c}{x}}} \left (96 a^3 b^2 c x-80 a^2 b^3 c x \sqrt{\frac{c}{x}}-128 a^4 b x^2 \sqrt{\frac{c}{x}}+256 a^5 x^2+70 a b^4 c^2-63 b^5 c x \left (\frac{c}{x}\right )^{3/2}\right )}{693 b^6 c^3 x^2} \]

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 = 0.051, size = 400, normalized size = 2.3 \begin{align*} -{\frac{1}{693\,{b}^{7}}\sqrt{a+b\sqrt{{\frac{c}{x}}}} \left ( 1386\,\sqrt{ax+b\sqrt{{\frac{c}{x}}}x}{a}^{13/2}{x}^{7/2}+1386\,\sqrt{x \left ( a+b\sqrt{{\frac{c}{x}}} \right ) }{a}^{13/2}{x}^{7/2}+252\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2} \left ({\frac{c}{x}} \right ) ^{5/2}\sqrt{a}{x}^{5/2}{b}^{5}+852\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2} \left ({\frac{c}{x}} \right ) ^{3/2}{a}^{5/2}{x}^{5/2}{b}^{3}+1748\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}\sqrt{{\frac{c}{x}}}{a}^{9/2}{x}^{5/2}b-2772\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{11/2}{x}^{5/2}-1236\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{7/2}{x}^{3/2}{b}^{2}c-532\, \left ( ax+b\sqrt{{\frac{c}{x}}}x \right ) ^{3/2}{a}^{3/2}\sqrt{x}{b}^{4}{c}^{2}+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 ) \sqrt{{\frac{c}{x}}}{x}^{4}{a}^{6}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 ) \sqrt{{\frac{c}{x}}}{x}^{4}{a}^{6}b \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}}}{\frac{1}{\sqrt{a}}}} \end{align*}

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)*(1386*(a*x+b*(c/x)^(1/2)*x)^(1/2)*a^(13/2)*x^(7/2)+1386*(x*(a+b*(c/x)^(1/2)))^(
1/2)*a^(13/2)*x^(7/2)+252*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(c/x)^(5/2)*a^(1/2)*x^(5/2)*b^5+852*(a*x+b*(c/x)^(1/2)*x
)^(3/2)*(c/x)^(3/2)*a^(5/2)*x^(5/2)*b^3+1748*(a*x+b*(c/x)^(1/2)*x)^(3/2)*(c/x)^(1/2)*a^(9/2)*x^(5/2)*b-2772*(a
*x+b*(c/x)^(1/2)*x)^(3/2)*a^(11/2)*x^(5/2)-1236*(a*x+b*(c/x)^(1/2)*x)^(3/2)*a^(7/2)*x^(3/2)*b^2*c-532*(a*x+b*(
c/x)^(1/2)*x)^(3/2)*a^(3/2)*x^(1/2)*b^4*c^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))*(c/x)^(1/2)*x^4*a^6*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))*(c/x)^(1/2)*x^4*a^6*b)/x^(13/2)/(x*(a+b*(c/x)^(1/2)))^(1/2)/b^7/(c/x)^(7/2)/
a^(1/2)

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Maxima [A]  time = 1.01123, size = 171, normalized size = 0.99 \begin{align*} -\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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*(c/x)^(1/2))^(1/2),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 + 990*(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 = 1.57795, size = 198, normalized size = 1.15 \begin{align*} \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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(1/x^4/(a+b*(c/x)^(1/2))^(1/2),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. \begin{align*} \int \frac{1}{x^{4} \sqrt{a + b \sqrt{\frac{c}{x}}}}\, dx \end{align*}

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 [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{1}{\sqrt{b \sqrt{\frac{c}{x}} + a} x^{4}}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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