3.25.47 \(\int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx\)

Optimal. Leaf size=233 \[ -\frac {b \sqrt {d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{9/2}}-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-35 b^2 d+42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x} \]

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Rubi [A]  time = 0.31, antiderivative size = 233, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 7, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.269, Rules used = {1970, 1357, 742, 779, 612, 621, 206} \begin {gather*} \frac {\left (32 a c-35 b^2 d+42 b c \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}-\frac {b \sqrt {d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{9/2}}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

-(b*(12*a*c - 7*b^2*d)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(64*c^4) + ((32*a*c - 35*b^2*d + 42*
b*c*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^(3/2))/(120*c^3) - (2*(a + b*Sqrt[d/x] + c/x)^(3/2))/(5*c*x) - (b*Sqrt[
d]*(12*a*c - 7*b^2*d)*(4*a*c - b^2*d)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/x] +
c/x])])/(128*c^(9/2))

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rule 612

Int[((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((b + 2*c*x)*(a + b*x + c*x^2)^p)/(2*c*(2*p +
1)), x] - Dist[(p*(b^2 - 4*a*c))/(2*c*(2*p + 1)), Int[(a + b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c}, x]
&& NeQ[b^2 - 4*a*c, 0] && GtQ[p, 0] && IntegerQ[4*p]

Rule 621

Int[1/Sqrt[(a_) + (b_.)*(x_) + (c_.)*(x_)^2], x_Symbol] :> Dist[2, Subst[Int[1/(4*c - x^2), x], x, (b + 2*c*x)
/Sqrt[a + b*x + c*x^2]], x] /; FreeQ[{a, b, c}, x] && NeQ[b^2 - 4*a*c, 0]

Rule 742

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(e*(d + e*x)^(m - 1)
*(a + b*x + c*x^2)^(p + 1))/(c*(m + 2*p + 1)), x] + Dist[1/(c*(m + 2*p + 1)), Int[(d + e*x)^(m - 2)*Simp[c*d^2
*(m + 2*p + 1) - e*(a*e*(m - 1) + b*d*(p + 1)) + e*(2*c*d - b*e)*(m + p)*x, x]*(a + b*x + c*x^2)^p, x], x] /;
FreeQ[{a, b, c, d, e, m, p}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]
&& If[RationalQ[m], GtQ[m, 1], SumSimplerQ[m, -2]] && NeQ[m + 2*p + 1, 0] && IntQuadraticQ[a, b, c, d, e, m,
p, x]

Rule 779

Int[((d_.) + (e_.)*(x_))*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((b
*e*g*(p + 2) - c*(e*f + d*g)*(2*p + 3) - 2*c*e*g*(p + 1)*x)*(a + b*x + c*x^2)^(p + 1))/(2*c^2*(p + 1)*(2*p + 3
)), x] + Dist[(b^2*e*g*(p + 2) - 2*a*c*e*g + c*(2*c*d*f - b*(e*f + d*g))*(2*p + 3))/(2*c^2*(2*p + 3)), Int[(a
+ b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] && NeQ[b^2 - 4*a*c, 0] &&  !LeQ[p, -1]

Rule 1357

Int[(x_)^(m_.)*((a_) + (c_.)*(x_)^(n2_.) + (b_.)*(x_)^(n_))^(p_.), x_Symbol] :> Dist[1/n, Subst[Int[x^(Simplif
y[(m + 1)/n] - 1)*(a + b*x + c*x^2)^p, x], x, x^n], x] /; FreeQ[{a, b, c, m, n, p}, x] && EqQ[n2, 2*n] && NeQ[
b^2 - 4*a*c, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rule 1970

Int[(x_)^(m_.)*((a_) + (b_.)*((d_.)/(x_))^(n_) + (c_.)*(x_)^(n2_.))^(p_.), x_Symbol] :> -Dist[d^(m + 1), Subst
[Int[(a + b*x^n + (c*x^(2*n))/d^(2*n))^p/x^(m + 2), x], x, d/x], x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[n2,
-2*n] && IntegerQ[2*n] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx &=-\frac {\operatorname {Subst}\left (\int x \sqrt {a+b \sqrt {x}+\frac {c x}{d}} \, dx,x,\frac {d}{x}\right )}{d^2}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^3 \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^2}\\ &=-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {2 \operatorname {Subst}\left (\int x \left (-2 a-\frac {7 b x}{2}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{5 c d}\\ &=\frac {\left (32 a c-7 b \left (5 b d-6 c \sqrt {\frac {d}{x}}\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {\left (b \left (12 a c-7 b^2 d\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{16 c^3}\\ &=-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-7 b \left (5 b d-6 c \sqrt {\frac {d}{x}}\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {\left (b \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{128 c^4}\\ &=-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-7 b \left (5 b d-6 c \sqrt {\frac {d}{x}}\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {\left (b \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {4 c}{d}-x^2} \, dx,x,\frac {b+\frac {2 c \sqrt {\frac {d}{x}}}{d}}{\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{64 c^4}\\ &=-\frac {b \left (12 a c-7 b^2 d\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{64 c^4}+\frac {\left (32 a c-7 b \left (5 b d-6 c \sqrt {\frac {d}{x}}\right )\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{120 c^3}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{5 c x}-\frac {b \sqrt {d} \left (12 a c-7 b^2 d\right ) \left (4 a c-b^2 d\right ) \tanh ^{-1}\left (\frac {\sqrt {d} \left (b+\frac {2 c \sqrt {\frac {d}{x}}}{d}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{128 c^{9/2}}\\ \end {align*}

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Mathematica [F]  time = 0.13, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3, x]

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IntegrateAlgebraic [F]  time = 180.43, size = 0, normalized size = 0.00 \begin {gather*} \text {\$Aborted} \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[Sqrt[a + b*Sqrt[d/x] + c/x]/x^3,x]

[Out]

$Aborted

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fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((a+c/x+b*(d/x)^(1/2))^(1/2)/x^3,x, algorithm="fricas")

[Out]

Timed out

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giac [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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maple [B]  time = 0.14, size = 615, normalized size = 2.64 \begin {gather*} -\frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (105 \left (\frac {d}{x}\right )^{\frac {5}{2}} b^{5} \sqrt {c}\, x^{5} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )-600 \left (\frac {d}{x}\right )^{\frac {3}{2}} a \,b^{3} c^{\frac {3}{2}} x^{4} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )-210 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,b^{4} d^{2} x^{3}-210 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} b^{5} x^{5}+720 \sqrt {\frac {d}{x}}\, a^{2} b \,c^{\frac {5}{2}} x^{3} \ln \left (\frac {\sqrt {\frac {d}{x}}\, b x +2 c +2 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {c}}{\sqrt {x}}\right )+360 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{2} b^{2} c d \,x^{3}+780 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a \,b^{3} c \,x^{4}-720 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{2} b \,c^{2} x^{3}+210 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{4} d^{2} x^{2}-360 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,b^{2} c d \,x^{2}-420 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} b^{3} c \,x^{3}+720 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a b \,c^{2} x^{2}+560 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{2} c^{2} d x -512 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,c^{3} x -672 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, b \,c^{3} x +768 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} c^{4}\right )}{1920 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{5} x^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+(d/x)^(1/2)*b+c/x)^(1/2)/x^3,x)

[Out]

-1/1920*((a*x+(d/x)^(1/2)*b*x+c)/x)^(1/2)/x^2*(105*c^(1/2)*ln(((d/x)^(1/2)*b*x+2*c+2*(a*x+(d/x)^(1/2)*b*x+c)^(
1/2)*c^(1/2))/x^(1/2))*(d/x)^(5/2)*x^5*b^5-210*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(5/2)*x^5*b^5-600*a*c^(3/2)
*ln(((d/x)^(1/2)*b*x+2*c+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*c^(1/2))/x^(1/2))*(d/x)^(3/2)*x^4*b^3-210*a*(a*x+(d/x
)^(1/2)*b*x+c)^(1/2)*d^2*x^3*b^4+720*a^2*c^(5/2)*ln(((d/x)^(1/2)*b*x+2*c+2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*c^(1/
2))/x^(1/2))*(d/x)^(1/2)*x^3*b+780*a*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(3/2)*x^4*b^3*c+360*a^2*(a*x+(d/x)^(1
/2)*b*x+c)^(1/2)*d*x^3*b^2*c+210*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*d^2*x^2*b^4-420*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*(
d/x)^(3/2)*x^3*b^3*c-720*a^2*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(1/2)*x^3*b*c^2-360*a*(a*x+(d/x)^(1/2)*b*x+c)
^(3/2)*d*x^2*b^2*c+720*a*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*(d/x)^(1/2)*x^2*b*c^2+560*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)
*d*x*b^2*c^2-512*a*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*x*c^3-672*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^3+7
68*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*c^4)/(a*x+(d/x)^(1/2)*b*x+c)^(1/2)/c^5

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {b \sqrt {\frac {d}{x}} + a + \frac {c}{x}}}{x^{3}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^3, x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {a+\frac {c}{x}+b\,\sqrt {\frac {d}{x}}}}{x^3} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a + c/x + b*(d/x)^(1/2))^(1/2)/x^3,x)

[Out]

int((a + c/x + b*(d/x)^(1/2))^(1/2)/x^3, x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {\frac {d}{x}} + \frac {c}{x}}}{x^{3}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(d/x) + c/x)/x**3, x)

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