3.7.30 \(\int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx\) [630]

Optimal. Leaf size=137 \[ -\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}} \sqrt {c}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}} \sqrt {c}} \]

[Out]

1/2*b*d*arctanh((a+b*(d*x+c)^(1/2))^(1/2)/(a-b*c^(1/2))^(1/2))/c^(1/2)/(a-b*c^(1/2))^(1/2)-1/2*b*d*arctanh((a+
b*(d*x+c)^(1/2))^(1/2)/(a+b*c^(1/2))^(1/2))/c^(1/2)/(a+b*c^(1/2))^(1/2)-(a+b*(d*x+c)^(1/2))^(1/2)/x

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Rubi [A]
time = 0.12, antiderivative size = 137, normalized size of antiderivative = 1.00, number of steps used = 8, number of rules used = 7, integrand size = 21, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {378, 1412, 835, 12, 722, 1107, 213} \begin {gather*} -\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a-b \sqrt {c}}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {c} \sqrt {a+b \sqrt {c}}} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[a + b*Sqrt[c + d*x]]/x^2,x]

[Out]

-(Sqrt[a + b*Sqrt[c + d*x]]/x) + (b*d*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]])/(2*Sqrt[a - b*Sq
rt[c]]*Sqrt[c]) - (b*d*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]])/(2*Sqrt[a + b*Sqrt[c]]*Sqrt[c])

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

Rule 213

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

Rule 378

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 722

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

Rule 835

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^m*(
a + c*x^2)^(p + 1)*((a*g - c*f*x)/(2*a*c*(p + 1))), x] - Dist[1/(2*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*
x^2)^(p + 1)*Simp[a*e*g*m - c*d*f*(2*p + 3) - c*e*f*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x
] && NeQ[c*d^2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])

Rule 1107

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

Rule 1412

Int[((a_) + (c_.)*(x_)^(n2_.))^(p_.)*((d_) + (e_.)*(x_)^(n_))^(q_.), x_Symbol] :> With[{g = Denominator[n]}, D
ist[g, Subst[Int[x^(g - 1)*(d + e*x^(g*n))^q*(a + c*x^(2*g*n))^p, x], x, x^(1/g)], x]] /; FreeQ[{a, c, d, e, p
, q}, x] && EqQ[n2, 2*n] && FractionQ[n]

Rubi steps

\begin {align*} \int \frac {\sqrt {a+b \sqrt {c+d x}}}{x^2} \, dx &=d \text {Subst}\left (\int \frac {\sqrt {a+b \sqrt {x}}}{(-c+x)^2} \, dx,x,c+d x\right )\\ &=(2 d) \text {Subst}\left (\int \frac {x \sqrt {a+b x}}{\left (-c+x^2\right )^2} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}-\frac {d \text {Subst}\left (\int -\frac {b c}{2 \sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )}{c}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {1}{2} (b d) \text {Subst}\left (\int \frac {1}{\sqrt {a+b x} \left (-c+x^2\right )} \, dx,x,\sqrt {c+d x}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\left (b^2 d\right ) \text {Subst}\left (\int \frac {1}{a^2-b^2 c-2 a x^2+x^4} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {(b d) \text {Subst}\left (\int \frac {1}{-a-b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \sqrt {c}}-\frac {(b d) \text {Subst}\left (\int \frac {1}{-a+b \sqrt {c}+x^2} \, dx,x,\sqrt {a+b \sqrt {c+d x}}\right )}{2 \sqrt {c}}\\ &=-\frac {\sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a-b \sqrt {c}}}\right )}{2 \sqrt {a-b \sqrt {c}} \sqrt {c}}-\frac {b d \tanh ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {a+b \sqrt {c}}}\right )}{2 \sqrt {a+b \sqrt {c}} \sqrt {c}}\\ \end {align*}

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Mathematica [A]
time = 0.39, size = 144, normalized size = 1.05 \begin {gather*} \frac {1}{2} \left (-\frac {2 \sqrt {a+b \sqrt {c+d x}}}{x}+\frac {b d \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a-b \sqrt {c}}}\right )}{\sqrt {-a-b \sqrt {c}} \sqrt {c}}-\frac {b d \tan ^{-1}\left (\frac {\sqrt {a+b \sqrt {c+d x}}}{\sqrt {-a+b \sqrt {c}}}\right )}{\sqrt {-a+b \sqrt {c}} \sqrt {c}}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[Sqrt[a + b*Sqrt[c + d*x]]/x^2,x]

[Out]

((-2*Sqrt[a + b*Sqrt[c + d*x]])/x + (b*d*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a - b*Sqrt[c]]])/(Sqrt[-a - b*
Sqrt[c]]*Sqrt[c]) - (b*d*ArcTan[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[-a + b*Sqrt[c]]])/(Sqrt[-a + b*Sqrt[c]]*Sqrt[c]
))/2

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Maple [A]
time = 0.14, size = 166, normalized size = 1.21

method	result	size
derivativedivides	\(4 d \,b^{2} \left (-\frac {\sqrt {a +b \sqrt {d x +c}}}{4 \left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )}+\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}-\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}\right )\)	\(166\)
default	\(4 d \,b^{2} \left (-\frac {\sqrt {a +b \sqrt {d x +c}}}{4 \left (\left (a +b \sqrt {d x +c}\right )^{2}-2 a \left (a +b \sqrt {d x +c}\right )-b^{2} c +a^{2}\right )}+\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {-\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {-\sqrt {b^{2} c}-a}}-\frac {\arctan \left (\frac {\sqrt {a +b \sqrt {d x +c}}}{\sqrt {\sqrt {b^{2} c}-a}}\right )}{8 \sqrt {b^{2} c}\, \sqrt {\sqrt {b^{2} c}-a}}\right )\)	\(166\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a+b*(d*x+c)^(1/2))^(1/2)/x^2,x,method=_RETURNVERBOSE)

[Out]

4*d*b^2*(-1/4*(a+b*(d*x+c)^(1/2))^(1/2)/((a+b*(d*x+c)^(1/2))^2-2*a*(a+b*(d*x+c)^(1/2))-b^2*c+a^2)+1/8/(b^2*c)^
(1/2)/(-(b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/(-(b^2*c)^(1/2)-a)^(1/2))-1/8/(b^2*c)^(1/2)/((
b^2*c)^(1/2)-a)^(1/2)*arctan((a+b*(d*x+c)^(1/2))^(1/2)/((b^2*c)^(1/2)-a)^(1/2)))

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(sqrt(d*x + c)*b + a)/x^2, x)

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1003 vs. \(2 (101) = 202\).
time = 0.40, size = 1003, normalized size = 7.32 \begin {gather*} -\frac {x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} + {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) - x \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}} \log \left (\sqrt {\sqrt {d x + c} b + a} b^{4} d^{3} - {\left (b^{4} c d^{2} + \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (a b^{2} c^{2} - a^{3} c\right )}\right )} \sqrt {-\frac {a b^{2} d^{2} - \sqrt {\frac {b^{6} d^{4}}{b^{4} c^{3} - 2 \, a^{2} b^{2} c^{2} + a^{4} c}} {\left (b^{2} c^{2} - a^{2} c\right )}}{b^{2} c^{2} - a^{2} c}}\right ) + 4 \, \sqrt {\sqrt {d x + c} b + a}}{4 \, x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/4*(x*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2 - a^2*c
))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 + (b^4*c*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(a*b^2
*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2
- a^2*c))) - x*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b^2*c^2
- a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 - (b^4*c*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))
*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b
^2*c^2 - a^2*c))) + x*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2*c))/(b
^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 + (b^4*c*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 +
a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2
*c))/(b^2*c^2 - a^2*c))) - x*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^2 - a^2
*c))/(b^2*c^2 - a^2*c))*log(sqrt(sqrt(d*x + c)*b + a)*b^4*d^3 - (b^4*c*d^2 + sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2
*c^2 + a^4*c))*(a*b^2*c^2 - a^3*c))*sqrt(-(a*b^2*d^2 - sqrt(b^6*d^4/(b^4*c^3 - 2*a^2*b^2*c^2 + a^4*c))*(b^2*c^
2 - a^2*c))/(b^2*c^2 - a^2*c))) + 4*sqrt(sqrt(d*x + c)*b + a))/x

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {\sqrt {a + b \sqrt {c + d x}}}{x^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Integral(sqrt(a + b*sqrt(c + d*x))/x**2, x)

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Giac [B] Leaf count of result is larger than twice the leaf count of optimal. 232 vs. \(2 (101) = 202\).
time = 3.65, size = 232, normalized size = 1.69 \begin {gather*} \frac {\frac {2 \, \sqrt {\sqrt {d x + c} b + a} b^{3} d^{2}}{b^{2} c - {\left (\sqrt {d x + c} b + a\right )}^{2} + 2 \, {\left (\sqrt {d x + c} b + a\right )} a - a^{2}} - \frac {{\left (b^{3} c d^{2} {\left | b \right |} + a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a + \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} + a c\right )} \sqrt {b \sqrt {c} - a} {\left | b \right |}} + \frac {{\left (b^{3} c d^{2} {\left | b \right |} - a b^{3} \sqrt {c} d^{2}\right )} \arctan \left (\frac {\sqrt {\sqrt {d x + c} b + a}}{\sqrt {-a - \sqrt {b^{2} c}}}\right )}{{\left (b c^{\frac {3}{2}} - a c\right )} \sqrt {-b \sqrt {c} - a} {\left | b \right |}}}{2 \, b d} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*(2*sqrt(sqrt(d*x + c)*b + a)*b^3*d^2/(b^2*c - (sqrt(d*x + c)*b + a)^2 + 2*(sqrt(d*x + c)*b + a)*a - a^2) -
(b^3*c*d^2*abs(b) + a*b^3*sqrt(c)*d^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sqrt(-a + sqrt(b^2*c)))/((b*c^(3/2) +
a*c)*sqrt(b*sqrt(c) - a)*abs(b)) + (b^3*c*d^2*abs(b) - a*b^3*sqrt(c)*d^2)*arctan(sqrt(sqrt(d*x + c)*b + a)/sq
rt(-a - sqrt(b^2*c)))/((b*c^(3/2) - a*c)*sqrt(-b*sqrt(c) - a)*abs(b)))/(b*d)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.01 \begin {gather*} \int \frac {\sqrt {a+b\,\sqrt {c+d\,x}}}{x^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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