3.630 \(\int \frac{\sqrt{a+b \sqrt{c+d x}}}{x^2} \, dx\)

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

[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])

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Rubi [A]  time = 0.168879, antiderivative size = 137, normalized size of antiderivative = 1., 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 = {371, 1398, 821, 12, 708, 1093, 207} \[ -\frac{\sqrt{a+b \sqrt{c+d x}}}{x}+\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a-b \sqrt{c}}}-\frac{b d \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )}{2 \sqrt{c} \sqrt{a+b \sqrt{c}}} \]

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 371

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 1398

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]

Rule 821

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 12

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

Rule 708

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 1093

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 207

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

Rubi steps

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

Mathematica [A]  time = 0.189986, size = 181, normalized size = 1.32 \[ \frac{\left (a-b \sqrt{c}\right ) \left (2 \sqrt{c} \left (a+b \sqrt{c}\right ) \sqrt{a+b \sqrt{c+d x}}+b d x \sqrt{a+b \sqrt{c}} \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a+b \sqrt{c}}}\right )\right )-b d x \sqrt{a-b \sqrt{c}} \left (a+b \sqrt{c}\right ) \tanh ^{-1}\left (\frac{\sqrt{a+b \sqrt{c+d x}}}{\sqrt{a-b \sqrt{c}}}\right )}{2 \sqrt{c} x \left (b^2 c-a^2\right )} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-(b*Sqrt[a - b*Sqrt[c]]*(a + b*Sqrt[c])*d*x*ArcTanh[Sqrt[a + b*Sqrt[c + d*x]]/Sqrt[a - b*Sqrt[c]]]) + (a - b*
Sqrt[c])*(2*(a + b*Sqrt[c])*Sqrt[c]*Sqrt[a + b*Sqrt[c + d*x]] + b*Sqrt[a + b*Sqrt[c]]*d*x*ArcTanh[Sqrt[a + b*S
qrt[c + d*x]]/Sqrt[a + b*Sqrt[c]]]))/(2*Sqrt[c]*(-a^2 + b^2*c)*x)

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Maple [A]  time = 0.017, size = 151, normalized size = 1.1 \begin{align*} -{\frac{{b}^{2}d}{{b}^{2} \left ( dx+c \right ) -{b}^{2}c}\sqrt{a+b\sqrt{dx+c}}}+{\frac{{b}^{2}d}{2}\arctan \left ({\sqrt{a+b\sqrt{dx+c}}{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}}} \right ){\frac{1}{\sqrt{{b}^{2}c}}}{\frac{1}{\sqrt{-\sqrt{{b}^{2}c}-a}}}}-{\frac{{b}^{2}d}{2}\arctan \left ({\sqrt{a+b\sqrt{dx+c}}{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}}} \right ){\frac{1}{\sqrt{{b}^{2}c}}}{\frac{1}{\sqrt{\sqrt{{b}^{2}c}-a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-b^2*d*(a+b*(d*x+c)^(1/2))^(1/2)/(b^2*(d*x+c)-b^2*c)+1/2*b^2*d/(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))-1/2*b^2*d/(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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{\sqrt{d x + c} b + a}}{x^{2}}\,{d x} \end{align*}

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]  time = 2.57014, size = 1871, normalized size = 13.66 \begin{align*} \text{result too large to display} \end{align*}

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., size = 0, normalized size = 0. \begin{align*} \int \frac{\sqrt{a + b \sqrt{c + d x}}}{x^{2}}\, dx \end{align*}

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 [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: NotImplementedError} \end{align*}

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]

Exception raised: NotImplementedError