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

Optimal. Leaf size=119 \[ \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b} \sqrt {b c-a d}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )} \]

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Rubi [A]  time = 0.11, antiderivative size = 119, normalized size of antiderivative = 1.00, number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {446, 99, 156, 63, 208} \begin {gather*} \frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{2 a^2 \sqrt {b} \sqrt {b c-a d}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]

[Out]

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

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 99

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

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

Rule 208

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

Rule 446

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol] :> Dist[1/n, Subst[Int
[x^(Simplify[(m + 1)/n] - 1)*(a + b*x)^p*(c + d*x)^q, x], x, x^n], x] /; FreeQ[{a, b, c, d, m, n, p, q}, x] &&
NeQ[b*c - a*d, 0] && IntegerQ[Simplify[(m + 1)/n]]

Rubi steps

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

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Mathematica [A]  time = 0.22, size = 112, normalized size = 0.94 \begin {gather*} \frac {\frac {a \sqrt {c+d x^2}}{a+b x^2}+\frac {(2 b c-a d) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {b} \sqrt {b c-a d}}-2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{2 a^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.53, size = 129, normalized size = 1.08 \begin {gather*} \frac {(2 b c-a d) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2} \sqrt {a d-b c}}{b c-a d}\right )}{2 a^2 \sqrt {b} \sqrt {a d-b c}}-\frac {\sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c+d x^2}}{\sqrt {c}}\right )}{a^2}+\frac {\sqrt {c+d x^2}}{2 a \left (a+b x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[Sqrt[c + d*x^2]/(x*(a + b*x^2)^2),x]

[Out]

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

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fricas [B]  time = 1.51, size = 1054, normalized size = 8.86 \begin {gather*} \left [-\frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) - 4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {8 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) - {\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {b^{2} c - a b d} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {b^{2} c - a b d} \sqrt {d x^{2} + c}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{8 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {c} \log \left (-\frac {d x^{2} - 2 \, \sqrt {d x^{2} + c} \sqrt {c} + 2 \, c}{x^{2}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}, \frac {{\left (2 \, a b c - a^{2} d + {\left (2 \, b^{2} c - a b d\right )} x^{2}\right )} \sqrt {-b^{2} c + a b d} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {-b^{2} c + a b d} \sqrt {d x^{2} + c}}{2 \, {\left (b^{2} c^{2} - a b c d + {\left (b^{2} c d - a b d^{2}\right )} x^{2}\right )}}\right ) + 4 \, {\left (a b^{2} c - a^{2} b d + {\left (b^{3} c - a b^{2} d\right )} x^{2}\right )} \sqrt {-c} \arctan \left (\frac {\sqrt {-c}}{\sqrt {d x^{2} + c}}\right ) + 2 \, {\left (a b^{2} c - a^{2} b d\right )} \sqrt {d x^{2} + c}}{4 \, {\left (a^{3} b^{2} c - a^{4} b d + {\left (a^{2} b^{3} c - a^{3} b^{2} d\right )} x^{2}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*((2*a*b*c - a^2*d + (2*b^2*c - a*b*d)*x^2)*sqrt(b^2*c - a*b*d)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d
+ a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^
2*x^4 + 2*a*b*x^2 + a^2)) - 4*(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 +
c)*sqrt(c) + 2*c)/x^2) - 4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c))/(a^3*b^2*c - a^4*b*d + (a^2*b^3*c - a^3*b^2*d
)*x^2), 1/8*(8*(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 + c)) - (2*a*b*
c - a^2*d + (2*b^2*c - a*b*d)*x^2)*sqrt(b^2*c - a*b*d)*log((b^2*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*
(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b*d*x^2 + 2*b*c - a*d)*sqrt(b^2*c - a*b*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*
x^2 + a^2)) + 4*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c))/(a^3*b^2*c - a^4*b*d + (a^2*b^3*c - a^3*b^2*d)*x^2), 1/4*
((2*a*b*c - a^2*d + (2*b^2*c - a*b*d)*x^2)*sqrt(-b^2*c + a*b*d)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*
c + a*b*d)*sqrt(d*x^2 + c)/(b^2*c^2 - a*b*c*d + (b^2*c*d - a*b*d^2)*x^2)) + 2*(a*b^2*c - a^2*b*d + (b^3*c - a*
b^2*d)*x^2)*sqrt(c)*log(-(d*x^2 - 2*sqrt(d*x^2 + c)*sqrt(c) + 2*c)/x^2) + 2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c
))/(a^3*b^2*c - a^4*b*d + (a^2*b^3*c - a^3*b^2*d)*x^2), 1/4*((2*a*b*c - a^2*d + (2*b^2*c - a*b*d)*x^2)*sqrt(-b
^2*c + a*b*d)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(-b^2*c + a*b*d)*sqrt(d*x^2 + c)/(b^2*c^2 - a*b*c*d + (b
^2*c*d - a*b*d^2)*x^2)) + 4*(a*b^2*c - a^2*b*d + (b^3*c - a*b^2*d)*x^2)*sqrt(-c)*arctan(sqrt(-c)/sqrt(d*x^2 +
c)) + 2*(a*b^2*c - a^2*b*d)*sqrt(d*x^2 + c))/(a^3*b^2*c - a^4*b*d + (a^2*b^3*c - a^3*b^2*d)*x^2)]

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giac [A]  time = 0.36, size = 113, normalized size = 0.95 \begin {gather*} \frac {\sqrt {d x^{2} + c} d}{2 \, {\left ({\left (d x^{2} + c\right )} b - b c + a d\right )} a} - \frac {{\left (2 \, b c - a d\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{2 \, \sqrt {-b^{2} c + a b d} a^{2}} + \frac {c \arctan \left (\frac {\sqrt {d x^{2} + c}}{\sqrt {-c}}\right )}{a^{2} \sqrt {-c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [B]  time = 0.01, size = 2585, normalized size = 21.72

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 1.16, size = 996, normalized size = 8.37 \begin {gather*} \frac {d\,\sqrt {d\,x^2+c}}{2\,a\,\left (b\,\left (d\,x^2+c\right )+a\,d-b\,c\right )}-\frac {\sqrt {c}\,\mathrm {atanh}\left (\frac {\sqrt {d\,x^2+c}}{\sqrt {c}}\right )}{a^2}-\frac {\mathrm {atan}\left (\frac {\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}-\frac {\left (2\,a\,b^2\,c\,d^3-\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}+\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}+\frac {\left (2\,a\,b^2\,c\,d^3+\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}}{\frac {b^2\,c^2\,d^3-\frac {a\,b\,c\,d^4}{2}}{a^3}+\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}-\frac {\left (2\,a\,b^2\,c\,d^3-\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}-\frac {\left (\frac {\sqrt {d\,x^2+c}\,\left (a^2\,b\,d^4-4\,a\,b^2\,c\,d^3+8\,b^3\,c^2\,d^2\right )}{2\,a^2}+\frac {\left (2\,a\,b^2\,c\,d^3+\frac {\left (16\,a^5\,b^2\,d^3-32\,a^4\,b^3\,c\,d^2\right )\,\sqrt {d\,x^2+c}\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{8\,a^2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )}{4\,\left (a^2\,b^2\,c-a^3\,b\,d\right )}}\right )\,\sqrt {-b\,\left (a\,d-b\,c\right )}\,\left (a\,d-2\,b\,c\right )\,1{}\mathrm {i}}{2\,\left (a^2\,b^2\,c-a^3\,b\,d\right )} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*(c + d*x^2)^(1/2))/(2*a*(b*(c + d*x^2) + a*d - b*c)) - (c^(1/2)*atanh((c + d*x^2)^(1/2)/c^(1/2)))/a^2 - (at
an((((((c + d*x^2)^(1/2)*(a^2*b*d^4 + 8*b^3*c^2*d^2 - 4*a*b^2*c*d^3))/(2*a^2) - ((2*a*b^2*c*d^3 - ((16*a^5*b^2
*d^3 - 32*a^4*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(8*a^2*(a^2*b^2*c - a^3*b*d))
)*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(4*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c)*1i)/(4
*(a^2*b^2*c - a^3*b*d)) + ((((c + d*x^2)^(1/2)*(a^2*b*d^4 + 8*b^3*c^2*d^2 - 4*a*b^2*c*d^3))/(2*a^2) + ((2*a*b^
2*c*d^3 + ((16*a^5*b^2*d^3 - 32*a^4*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(8*a^2*
(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(4*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2
)*(a*d - 2*b*c)*1i)/(4*(a^2*b^2*c - a^3*b*d)))/((b^2*c^2*d^3 - (a*b*c*d^4)/2)/a^3 + ((((c + d*x^2)^(1/2)*(a^2*
b*d^4 + 8*b^3*c^2*d^2 - 4*a*b^2*c*d^3))/(2*a^2) - ((2*a*b^2*c*d^3 - ((16*a^5*b^2*d^3 - 32*a^4*b^3*c*d^2)*(c +
d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(8*a^2*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2)*(a*d
- 2*b*c))/(4*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(4*(a^2*b^2*c - a^3*b*d)) - ((((c +
d*x^2)^(1/2)*(a^2*b*d^4 + 8*b^3*c^2*d^2 - 4*a*b^2*c*d^3))/(2*a^2) + ((2*a*b^2*c*d^3 + ((16*a^5*b^2*d^3 - 32*a
^4*b^3*c*d^2)*(c + d*x^2)^(1/2)*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(8*a^2*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d
- b*c))^(1/2)*(a*d - 2*b*c))/(4*(a^2*b^2*c - a^3*b*d)))*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c))/(4*(a^2*b^2*c -
a^3*b*d))))*(-b*(a*d - b*c))^(1/2)*(a*d - 2*b*c)*1i)/(2*(a^2*b^2*c - a^3*b*d))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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