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

Optimal. Leaf size=371 \[ \frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \tanh ^{-1}\left (\frac {b d+2 c \sqrt {\frac {d}{x}}}{2 \sqrt {c} \sqrt {d} \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}\right )}{1024 c^{13/2}}+\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2+18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {11 b \left (\frac {d}{x}\right )^{3/2} \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2} \]

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Rubi [A]  time = 0.66, antiderivative size = 371, normalized size of antiderivative = 1.00, number of steps used = 9, number of rules used = 8, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.308, Rules used = {1970, 1357, 742, 832, 779, 612, 621, 206} \begin {gather*} -\frac {\left (1024 a^2 c^2+18 b c \sqrt {\frac {d}{x}} \left (148 a c-77 b^2 d\right )-3276 a b^2 c d+1155 b^4 d^2\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\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 )}{1024 c^{13/2}}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {11 b \left (\frac {d}{x}\right )^{3/2} \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

(b*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*(b*d + 2*c*Sqrt[d/x])*Sqrt[a + b*Sqrt[d/x] + c/x])/(512*c^6) - ((
1024*a^2*c^2 - 3276*a*b^2*c*d + 1155*b^4*d^2 + 18*b*c*(148*a*c - 77*b^2*d)*Sqrt[d/x])*(a + b*Sqrt[d/x] + c/x)^
(3/2))/(6720*c^5) + (11*b*(a + b*Sqrt[d/x] + c/x)^(3/2)*(d/x)^(3/2))/(42*c^2*d) - (2*(a + b*Sqrt[d/x] + c/x)^(
3/2))/(7*c*x^2) + ((32*a*c - 33*b^2*d)*(a + b*Sqrt[d/x] + c/x)^(3/2))/(140*c^3*x) + (b*Sqrt[d]*(4*a*c - b^2*d)
*(80*a^2*c^2 - 120*a*b^2*c*d + 33*b^4*d^2)*ArcTanh[(b*d + 2*c*Sqrt[d/x])/(2*Sqrt[c]*Sqrt[d]*Sqrt[a + b*Sqrt[d/
x] + c/x])])/(1024*c^(13/2))

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 832

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

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^4} \, dx &=-\frac {\operatorname {Subst}\left (\int x^2 \sqrt {a+b \sqrt {x}+\frac {c x}{d}} \, dx,x,\frac {d}{x}\right )}{d^3}\\ &=-\frac {2 \operatorname {Subst}\left (\int x^5 \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{d^3}\\ &=-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}-\frac {2 \operatorname {Subst}\left (\int x^3 \left (-4 a-\frac {11 b x}{2}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{7 c d^2}\\ &=\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}-\frac {\operatorname {Subst}\left (\int x^2 \left (\frac {33 a b}{2}-\frac {3 \left (32 a c-33 b^2 d\right ) x}{4 d}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{21 c^2 d}\\ &=\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}-\frac {\operatorname {Subst}\left (\int x \left (-\frac {3}{2} a \left (33 b^2-\frac {32 a c}{d}\right )+\frac {9 b \left (148 a c-77 b^2 d\right ) x}{8 d}\right ) \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{105 c^3}\\ &=-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname {Subst}\left (\int \sqrt {a+b x+\frac {c x^2}{d}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{128 c^5}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\sqrt {a+b x+\frac {c x^2}{d}}} \, dx,x,\sqrt {\frac {d}{x}}\right )}{1024 c^6}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {\left (b \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\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 )}{512 c^6}\\ &=\frac {b \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\right ) \left (b d+2 c \sqrt {\frac {d}{x}}\right ) \sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{512 c^6}-\frac {\left (1024 a^2 c^2-3276 a b^2 c d+1155 b^4 d^2+18 b c \left (148 a c-77 b^2 d\right ) \sqrt {\frac {d}{x}}\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{6720 c^5}+\frac {11 b \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2} \left (\frac {d}{x}\right )^{3/2}}{42 c^2 d}-\frac {2 \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{7 c x^2}+\frac {\left (32 a c-33 b^2 d\right ) \left (a+b \sqrt {\frac {d}{x}}+\frac {c}{x}\right )^{3/2}}{140 c^3 x}+\frac {b \sqrt {d} \left (4 a c-b^2 d\right ) \left (80 a^2 c^2-120 a b^2 c d+33 b^4 d^2\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 )}{1024 c^{13/2}}\\ \end {align*}

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Mathematica [F]  time = 0.14, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a+b \sqrt {\frac {d}{x}}+\frac {c}{x}}}{x^4} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4,x]

[Out]

Integrate[Sqrt[a + b*Sqrt[d/x] + c/x]/x^4, x]

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IntegrateAlgebraic [F]  time = 180.02, 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^4,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^4,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^4,x, algorithm="giac")

[Out]

Timed out

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maple [B]  time = 0.15, size = 979, normalized size = 2.64 \begin {gather*} \frac {\sqrt {\frac {a x +\sqrt {\frac {d}{x}}\, b x +c}{x}}\, \left (-3465 \left (\frac {d}{x}\right )^{\frac {7}{2}} b^{7} \sqrt {c}\, x^{7} \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 )+26460 \left (\frac {d}{x}\right )^{\frac {5}{2}} a \,b^{5} c^{\frac {3}{2}} x^{6} \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 )+6930 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a \,b^{6} d^{3} x^{4}+6930 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {7}{2}} b^{7} x^{7}-58800 \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{2} b^{3} c^{\frac {5}{2}} 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 )-25200 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{2} b^{4} c \,d^{2} x^{4}-39060 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {5}{2}} a \,b^{5} c \,x^{6}+33600 \sqrt {\frac {d}{x}}\, a^{3} b \,c^{\frac {7}{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 )+16800 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, a^{3} b^{2} c^{2} d \,x^{4}+67200 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \left (\frac {d}{x}\right )^{\frac {3}{2}} a^{2} b^{3} c^{2} x^{5}-6930 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{6} d^{3} x^{3}-33600 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, \sqrt {\frac {d}{x}}\, a^{3} b \,c^{3} x^{4}+25200 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,b^{4} c \,d^{2} x^{3}+13860 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {5}{2}} b^{5} c \,x^{5}-16800 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{2} b^{2} c^{2} d \,x^{3}-50400 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \left (\frac {d}{x}\right )^{\frac {3}{2}} a \,b^{3} c^{2} x^{4}-18480 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{4} c^{2} d^{2} x^{2}+33600 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a^{2} b \,c^{3} x^{3}+52416 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,b^{2} c^{3} d \,x^{2}+22176 \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^{3} x^{3}-16384 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a^{2} c^{4} x^{2}-42624 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, a b \,c^{4} x^{2}-25344 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} b^{2} c^{4} d x +24576 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} a \,c^{5} x +28160 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} \sqrt {\frac {d}{x}}\, b \,c^{5} x -30720 \left (a x +\sqrt {\frac {d}{x}}\, b x +c \right )^{\frac {3}{2}} c^{6}\right )}{107520 \sqrt {a x +\sqrt {\frac {d}{x}}\, b x +c}\, c^{7} x^{3}} \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^4,x)

[Out]

1/107520*((a*x+(d/x)^(1/2)*b*x+c)/x)^(1/2)/x^3*(-16384*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a^2*x^2*c^4+24576*(a*x+(d
/x)^(1/2)*b*x+c)^(3/2)*a*x*c^5+6930*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*(d/x)^(7/2)*x^7*b^7-6930*(a*x+(d/x)^(1/2)*b*
x+c)^(3/2)*d^3*x^3*b^6-58800*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)^(3/2)*x^5*b^3+33600*a^3*c^(7/2)*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^4*b+67200*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^2*(d/x)^(3/2)*x^5*b^3*c^2-50400*(a*x+(d/x)^(1/2
)*b*x+c)^(3/2)*a*(d/x)^(3/2)*x^4*b^3*c^2-33600*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^3*(d/x)^(1/2)*x^4*b*c^3+33600*(
a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a^2*(d/x)^(1/2)*x^3*b*c^3-42624*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a*(d/x)^(1/2)*x^2*b
*c^4+26460*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)^(5/2)*x^6
*b^5+52416*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a*d*x^2*b^2*c^3-25200*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^2*d^2*x^4*b^4*c
+25200*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a*d^2*x^3*b^4*c+16800*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a^3*d*x^4*b^2*c^2-168
00*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*a^2*d*x^3*b^2*c^2-39060*(a*x+(d/x)^(1/2)*b*x+c)^(1/2)*a*(d/x)^(5/2)*x^6*b^5*c
+28160*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*(d/x)^(1/2)*x*b*c^5-30720*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*c^6-3465*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)^(7/2)*x^7*b^7+13860*(a*x+(d/x)
^(1/2)*b*x+c)^(3/2)*(d/x)^(5/2)*x^5*b^5*c+22176*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*(d/x)^(3/2)*x^3*b^3*c^3+6930*(a*
x+(d/x)^(1/2)*b*x+c)^(1/2)*a*d^3*x^4*b^6-18480*(a*x+(d/x)^(1/2)*b*x+c)^(3/2)*d^2*x^2*b^4*c^2-25344*(a*x+(d/x)^
(1/2)*b*x+c)^(3/2)*d*x*b^2*c^4)/(a*x+(d/x)^(1/2)*b*x+c)^(1/2)/c^7

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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^{4}}\,{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^4,x, algorithm="maxima")

[Out]

integrate(sqrt(b*sqrt(d/x) + a + c/x)/x^4, 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^4} \,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^4,x)

[Out]

int((a + c/x + b*(d/x)^(1/2))^(1/2)/x^4, 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^{4}}\, 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**4,x)

[Out]

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

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