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

Optimal. Leaf size=92 \[ \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}} \]

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Rubi [A]  time = 0.10, antiderivative size = 92, 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 = {446, 105, 63, 217, 206, 93, 208} \begin {gather*} \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b} \sqrt {c+d x^2}}\right )}{\sqrt {d}}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((Sqrt[a]*ArcTanh[(Sqrt[c]*Sqrt[a + b*x^2])/(Sqrt[a]*Sqrt[c + d*x^2])])/Sqrt[c]) + (Sqrt[b]*ArcTanh[(Sqrt[d]*
Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/Sqrt[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 93

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> With[{q = Denomin
ator[m]}, Dist[q, Subst[Int[x^(q*(m + 1) - 1)/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^
(1/q)], x]] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && LtQ[-1, m, 0] && SimplerQ[
a + b*x, c + d*x]

Rule 105

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x_)), x_Symbol] :> Dist[b/f, Int[(a
+ b*x)^(m - 1)*(c + d*x)^n, x], x] - Dist[(b*e - a*f)/f, Int[((a + b*x)^(m - 1)*(c + d*x)^n)/(e + f*x), x], x]
/; FreeQ[{a, b, c, d, e, f, m, n}, x] && IGtQ[Simplify[m + n + 1], 0] && (GtQ[m, 0] || ( !RationalQ[m] && (Su
mSimplerQ[m, -1] ||  !SumSimplerQ[n, -1])))

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 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 217

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

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

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Mathematica [A]  time = 0.14, size = 129, normalized size = 1.40 \begin {gather*} \frac {\sqrt {b c-a d} \sqrt {\frac {b \left (c+d x^2\right )}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x^2}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {c+d x^2}}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x^2}}{\sqrt {a} \sqrt {c+d x^2}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 0.64, size = 92, normalized size = 1.00 \begin {gather*} \frac {\sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {d} \sqrt {a+b x^2}}\right )}{\sqrt {d}}-\frac {\sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {a} \sqrt {c+d x^2}}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.88, size = 777, normalized size = 8.45 \begin {gather*} \left [\frac {1}{4} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ) + \frac {1}{4} \, \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ), -\frac {1}{2} \, \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right ) + \frac {1}{4} \, \sqrt {\frac {a}{c}} \log \left (\frac {{\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {a}{c}}}{x^{4}}\right ), \frac {1}{2} \, \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) + \frac {1}{4} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x^{2} + 4 \, {\left (2 \, b d^{2} x^{2} + b c d + a d^{2}\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {\frac {b}{d}}\right ), \frac {1}{2} \, \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left ({\left (b c + a d\right )} x^{2} + 2 \, a c\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{4} + a^{2} c + {\left (a b c + a^{2} d\right )} x^{2}\right )}}\right ) - \frac {1}{2} \, \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt {b x^{2} + a} \sqrt {d x^{2} + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{4} + a b c + {\left (b^{2} c + a b d\right )} x^{2}\right )}}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[1/4*sqrt(b/d)*log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(b^2*c*d + a*b*d^2)*x^2 + 4*(2*b*d^2*x^2
+ b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b/d)) + 1/4*sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^
2)*x^4 + 8*a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 - 4*(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2
+ c)*sqrt(a/c))/x^4), -1/2*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(
-b/d)/(b^2*d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2)) + 1/4*sqrt(a/c)*log(((b^2*c^2 + 6*a*b*c*d + a^2*d^2)*x^4 + 8*
a^2*c^2 + 8*(a*b*c^2 + a^2*c*d)*x^2 - 4*(2*a*c^2 + (b*c^2 + a*c*d)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(a
/c))/x^4), 1/2*sqrt(-a/c)*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a*b
*d*x^4 + a^2*c + (a*b*c + a^2*d)*x^2)) + 1/4*sqrt(b/d)*log(8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2*d^2 + 8*(
b^2*c*d + a*b*d^2)*x^2 + 4*(2*b*d^2*x^2 + b*c*d + a*d^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(b/d)), 1/2*sqrt(
-a/c)*arctan(1/2*((b*c + a*d)*x^2 + 2*a*c)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-a/c)/(a*b*d*x^4 + a^2*c + (a*
b*c + a^2*d)*x^2)) - 1/2*sqrt(-b/d)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b
/d)/(b^2*d*x^4 + a*b*c + (b^2*c + a*b*d)*x^2))]

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giac [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: TypeError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: TypeError >> An error occurred running a Giac command:INPUT:sage2:=int(sage0,x):;OUTPUT:

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maple [B]  time = 0.03, size = 177, normalized size = 1.92 \begin {gather*} \frac {\sqrt {b \,x^{2}+a}\, \sqrt {d \,x^{2}+c}\, \left (-\sqrt {b d}\, a \ln \left (\frac {a d \,x^{2}+b c \,x^{2}+2 a c +2 \sqrt {a c}\, \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}}{x^{2}}\right )+\sqrt {a c}\, b \ln \left (\frac {2 b d \,x^{2}+a d +b c +2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}}{2 \sqrt {b d}}\right )\right )}{2 \sqrt {x^{4} b d +a d \,x^{2}+b c \,x^{2}+a c}\, \sqrt {b d}\, \sqrt {a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c zero or nonzero?

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mupad [B]  time = 19.83, size = 4638, normalized size = 50.41

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(2*atanh((20*a*b^7*(b*d)^(1/2))/(34*a^(1/2)*b^8*c^(1/2) - (33*a^(3/2)*b^7*d)/c^(1/2) - (54*b^8*c*((a + b*x^2)^
(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) + (25*b^9*c^(3/2))/(2*a^(1/2)*d) + (4*a^(5/2)*b^6*d^2)/c^(3/2)
- (18*b^10*c^(5/2))/(a^(3/2)*d^2) + (a^(7/2)*b^5*d^3)/(2*c^(5/2)) + (20*a*b^7*d*((a + b*x^2)^(1/2) - a^(1/2))
)/((c + d*x^2)^(1/2) - c^(1/2)) + (10*a^2*b^6*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/
2))) + (23*b^9*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d^3*((a + b
*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^3*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*
d^2*((c + d*x^2)^(1/2) - c^(1/2)))) - (54*b^8*(b*d)^(1/2))/((25*b^9*c^(1/2))/(2*a^(1/2)) + (34*a^(1/2)*b^8*d)/
c^(1/2) - (54*b^8*d*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (33*a^(3/2)*b^7*d^2)/c^(3/2
) - (18*b^10*c^(3/2))/(a^(3/2)*d) + (4*a^(5/2)*b^6*d^3)/c^(5/2) + (a^(7/2)*b^5*d^4)/(2*c^(7/2)) + (23*b^9*c*((
a + b*x^2)^(1/2) - a^(1/2)))/(a*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^2*b^6*d^3*((a + b*x^2)^(1/2) - a^(1/2))
)/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x^2)^(1/2) -
c^(1/2))) - (3*a^3*b^5*d^4*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2))) + (20*a*b^7*d^2
*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2)))) + (23*b^9*c*(b*d)^(1/2))/((25*a^(1/2)*b^9*c
^(1/2)*d)/2 - (18*b^10*c^(3/2))/a^(1/2) + (34*a^(3/2)*b^8*d^2)/c^(1/2) - (33*a^(5/2)*b^7*d^3)/c^(3/2) + (4*a^(
7/2)*b^6*d^4)/c^(5/2) + (a^(9/2)*b^5*d^5)/(2*c^(7/2)) - (54*a*b^8*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x
^2)^(1/2) - c^(1/2)) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a*((c + d*x^2)^(1/2) - c^(1/2))) + (23*b^9*
c*d*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) + (20*a^2*b^7*d^3*((a + b*x^2)^(1/2) - a^(1/2
)))/(c*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^3*b^6*d^4*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2)
- c^(1/2))) - (3*a^4*b^5*d^5*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2)))) + (10*a^2*b^
6*(b*d)^(1/2))/((4*a^(5/2)*b^6*d)/c^(1/2) - 33*a^(3/2)*b^7*c^(1/2) + (34*a^(1/2)*b^8*c^(3/2))/d + (25*b^9*c^(5
/2))/(2*a^(1/2)*d^2) + (a^(7/2)*b^5*d^2)/(2*c^(3/2)) - (18*b^10*c^(7/2))/(a^(3/2)*d^3) + (10*a^2*b^6*d*((a + b
*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (54*b^8*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(d*((c + d*
x^2)^(1/2) - c^(1/2))) + (20*a*b^7*c*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (3*a^3*b^5
*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2))) + (23*b^9*c^3*((a + b*x^2)^(1/2) - a^(1/
2)))/(a*d^2*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^4*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*d^3*((c + d*x^2)^
(1/2) - c^(1/2)))) - (3*a^3*b^5*(b*d)^(1/2))/(4*a^(5/2)*b^6*c^(1/2) + (a^(7/2)*b^5*d)/(2*c^(1/2)) - (33*a^(3/2
)*b^7*c^(3/2))/d + (34*a^(1/2)*b^8*c^(5/2))/d^2 + (25*b^9*c^(7/2))/(2*a^(1/2)*d^3) - (18*b^10*c^(9/2))/(a^(3/2
)*d^4) + (10*a^2*b^6*c*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (3*a^3*b^5*d*((a + b*x^2
)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (54*b^8*c^3*((a + b*x^2)^(1/2) - a^(1/2)))/(d^2*((c + d*x^
2)^(1/2) - c^(1/2))) + (23*b^9*c^4*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d^3*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b
^10*c^5*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x^2)^(1/2) - c^(1/2))) + (20*a*b^7*c^2*((a + b*x^2)^(1
/2) - a^(1/2)))/(d*((c + d*x^2)^(1/2) - c^(1/2)))) + (4*b^10*c^2*(b*d)^(1/2))/((25*a^(3/2)*b^9*c^(1/2)*d^2)/2
- 18*a^(1/2)*b^10*c^(3/2)*d + (34*a^(5/2)*b^8*d^3)/c^(1/2) - (33*a^(7/2)*b^7*d^4)/c^(3/2) + (4*a^(9/2)*b^6*d^5
)/c^(5/2) + (a^(11/2)*b^5*d^6)/(2*c^(7/2)) + (4*b^10*c^2*d*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) -
c^(1/2)) - (54*a^2*b^8*d^3*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) + (20*a^3*b^7*d^4*((a
+ b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^4*b^6*d^5*((a + b*x^2)^(1/2) - a^(1/2)))
/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^5*b^5*d^6*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) -
c^(1/2))) + (23*a*b^9*c*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2))) + (34*a^(1/2)*b^7*(
b*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(c^(1/2)*((c + d*x^2)^(1/2) - c^(1/2))*((34*a^(1/2)*b^8)/c^(1/2) - (
54*b^8*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (33*a^(3/2)*b^7*d)/c^(3/2) + (25*b^9*c^(
1/2))/(2*a^(1/2)*d) - (18*b^10*c^(3/2))/(a^(3/2)*d^2) + (4*a^(5/2)*b^6*d^2)/c^(5/2) + (a^(7/2)*b^5*d^3)/(2*c^(
7/2)) + (10*a^2*b^6*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d^3*((
a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/
(a^2*d^2*((c + d*x^2)^(1/2) - c^(1/2))) + (20*a*b^7*d*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c
^(1/2))) + (23*b^9*c*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d*((c + d*x^2)^(1/2) - c^(1/2))))) - (33*a^(3/2)*b^6*(b
*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(c^(3/2)*((c + d*x^2)^(1/2) - c^(1/2))*((4*a^(5/2)*b^6*d)/c^(5/2) - (
33*a^(3/2)*b^7)/c^(3/2) + (34*a^(1/2)*b^8)/(c^(1/2)*d) + (25*b^9*c^(1/2))/(2*a^(1/2)*d^2) - (18*b^10*c^(3/2))/
(a^(3/2)*d^3) + (a^(7/2)*b^5*d^2)/(2*c^(7/2)) - (54*b^8*((a + b*x^2)^(1/2) - a^(1/2)))/(d*((c + d*x^2)^(1/2) -
c^(1/2))) + (20*a*b^7*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d^2*((a +
b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a^
2*d^3*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^2*b^6*d*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) -
c^(1/2))) + (23*b^9*c*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x^2)^(1/2) - c^(1/2))))) + (25*b^8*c^(1/2)
*(b*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(2*a^(1/2)*((c + d*x^2)^(1/2) - c^(1/2))*((25*b^9*c^(1/2))/(2*a^(1
/2)) + (34*a^(1/2)*b^8*d)/c^(1/2) - (54*b^8*d*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^(1/2) - c^(1/2)) - (
33*a^(3/2)*b^7*d^2)/c^(3/2) - (18*b^10*c^(3/2))/(a^(3/2)*d) + (4*a^(5/2)*b^6*d^3)/c^(5/2) + (a^(7/2)*b^5*d^4)/
(2*c^(7/2)) + (23*b^9*c*((a + b*x^2)^(1/2) - a^(1/2)))/(a*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^2*b^6*d^3*((a
+ b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(
a^2*d*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d^4*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) -
c^(1/2))) + (20*a*b^7*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2))))) + (4*a^(5/2)*b^5
*(b*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(c^(5/2)*((c + d*x^2)^(1/2) - c^(1/2))*((4*a^(5/2)*b^6)/c^(5/2) +
(a^(7/2)*b^5*d)/(2*c^(7/2)) + (34*a^(1/2)*b^8)/(c^(1/2)*d^2) + (25*b^9*c^(1/2))/(2*a^(1/2)*d^3) - (33*a^(3/2)*
b^7)/(c^(3/2)*d) - (18*b^10*c^(3/2))/(a^(3/2)*d^4) - (54*b^8*((a + b*x^2)^(1/2) - a^(1/2)))/(d^2*((c + d*x^2)^
(1/2) - c^(1/2))) + (10*a^2*b^6*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c
^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x^2)^(1/2) - c^(1/2))) + (20*a*b^7*((a + b*x^2)^(1/2) - a^(
1/2)))/(c*d*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/
2) - c^(1/2))) + (23*b^9*c*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d^3*((c + d*x^2)^(1/2) - c^(1/2))))) - (18*b^9*c^
(3/2)*(b*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(a^(3/2)*((c + d*x^2)^(1/2) - c^(1/2))*((25*b^9*c^(1/2)*d)/(2
*a^(1/2)) - (18*b^10*c^(3/2))/a^(3/2) + (34*a^(1/2)*b^8*d^2)/c^(1/2) - (33*a^(3/2)*b^7*d^3)/c^(3/2) + (4*a^(5/
2)*b^6*d^4)/c^(5/2) + (a^(7/2)*b^5*d^5)/(2*c^(7/2)) - (54*b^8*d^2*((a + b*x^2)^(1/2) - a^(1/2)))/((c + d*x^2)^
(1/2) - c^(1/2)) + (4*b^10*c^2*((a + b*x^2)^(1/2) - a^(1/2)))/(a^2*((c + d*x^2)^(1/2) - c^(1/2))) + (10*a^2*b^
6*d^4*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*((c + d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*d^5*((a + b*x^2)^(1/2) -
a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2))) + (23*b^9*c*d*((a + b*x^2)^(1/2) - a^(1/2)))/(a*((c + d*x^2)^(1
/2) - c^(1/2))) + (20*a*b^7*d^3*((a + b*x^2)^(1/2) - a^(1/2)))/(c*((c + d*x^2)^(1/2) - c^(1/2))))) + (a^(7/2)*
b^4*(b*d)^(1/2)*((a + b*x^2)^(1/2) - a^(1/2)))/(2*c^(7/2)*((c + d*x^2)^(1/2) - c^(1/2))*((a^(7/2)*b^5)/(2*c^(7
/2)) + (34*a^(1/2)*b^8)/(c^(1/2)*d^3) + (25*b^9*c^(1/2))/(2*a^(1/2)*d^4) - (33*a^(3/2)*b^7)/(c^(3/2)*d^2) + (4
*a^(5/2)*b^6)/(c^(5/2)*d) - (18*b^10*c^(3/2))/(a^(3/2)*d^5) - (54*b^8*((a + b*x^2)^(1/2) - a^(1/2)))/(d^3*((c
+ d*x^2)^(1/2) - c^(1/2))) - (3*a^3*b^5*((a + b*x^2)^(1/2) - a^(1/2)))/(c^3*((c + d*x^2)^(1/2) - c^(1/2))) + (
10*a^2*b^6*((a + b*x^2)^(1/2) - a^(1/2)))/(c^2*d*((c + d*x^2)^(1/2) - c^(1/2))) + (4*b^10*c^2*((a + b*x^2)^(1/
2) - a^(1/2)))/(a^2*d^5*((c + d*x^2)^(1/2) - c^(1/2))) + (20*a*b^7*((a + b*x^2)^(1/2) - a^(1/2)))/(c*d^2*((c +
d*x^2)^(1/2) - c^(1/2))) + (23*b^9*c*((a + b*x^2)^(1/2) - a^(1/2)))/(a*d^4*((c + d*x^2)^(1/2) - c^(1/2))))))*
(b*d)^(1/2))/d - (a^(1/2)*log(((a + b*x^2)^(1/2) - a^(1/2))/((c + d*x^2)^(1/2) - c^(1/2))) - a^(1/2)*log(((c^(
1/2)*(a + b*x^2)^(1/2) - a^(1/2)*(c + d*x^2)^(1/2))*(b*c^(1/2) - (a^(1/2)*d*((a + b*x^2)^(1/2) - a^(1/2)))/((c
+ d*x^2)^(1/2) - c^(1/2))))/((c + d*x^2)^(1/2) - c^(1/2))))/(2*c^(1/2))

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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