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

Optimal. Leaf size=85 \[ \frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}} \]

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Rubi [A]  time = 0.05, antiderivative size = 85, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 6, integrand size = 22, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.273, Rules used = {105, 63, 217, 206, 93, 208} \begin {gather*} \frac {2 \sqrt {b} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {d}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rubi steps

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

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Mathematica [A]  time = 0.18, size = 120, normalized size = 1.41 \begin {gather*} \frac {2 \sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {d} \sqrt {c+d x}}-\frac {2 \sqrt {a} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [B]  time = 0.38, size = 200, normalized size = 2.35 \begin {gather*} -2 \sqrt {\frac {b}{d}} \log \left (\sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}-\sqrt {\frac {b}{d}} \sqrt {c+d x}\right )-\frac {2 \sqrt {a} \sqrt {d} \sqrt {\frac {b}{d}} \tanh ^{-1}\left (-\frac {\sqrt {b} (c+d x)}{\sqrt {a} \sqrt {c} \sqrt {d}}+\frac {\sqrt {d} \sqrt {\frac {b}{d}} \sqrt {c+d x} \sqrt {a+\frac {b (c+d x)}{d}-\frac {b c}{d}}}{\sqrt {a} \sqrt {b} \sqrt {c}}+\frac {\sqrt {b} \sqrt {c}}{\sqrt {a} \sqrt {d}}\right )}{\sqrt {b} \sqrt {c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 1.52, size = 711, normalized size = 8.36 \begin {gather*} \left [\frac {1}{2} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {a}{c}} \log \left (\frac {8 \, a^{2} c^{2} + {\left (b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2}\right )} x^{2} - 4 \, {\left (2 \, a c^{2} + {\left (b c^{2} + a c d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {a}{c}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {b}{d}} \log \left (8 \, b^{2} d^{2} x^{2} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 4 \, {\left (2 \, b d^{2} x + b c d + a d^{2}\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {b}{d}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ), \sqrt {-\frac {a}{c}} \arctan \left (\frac {{\left (2 \, a c + {\left (b c + a d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {a}{c}}}{2 \, {\left (a b d x^{2} + a^{2} c + {\left (a b c + a^{2} d\right )} x\right )}}\right ) - \sqrt {-\frac {b}{d}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {b}{d}}}{2 \, {\left (b^{2} d x^{2} + a b c + {\left (b^{2} c + a b d\right )} x\right )}}\right )\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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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+a)^(1/2)/x/(d*x+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 = 0.02, size = 133, normalized size = 1.56 \begin {gather*} \frac {\sqrt {b x +a}\, \sqrt {d x +c}\, \left (-\sqrt {b d}\, a \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+\sqrt {a c}\, b \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )\right )}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(b*x+a)^(1/2)*(d*x+c)^(1/2)*(ln(1/2*(2*b*d*x+a*d+b*c+2*((b*x+a)*(d*x+c))^(1/2)*(b*d)^(1/2))/(b*d)^(1/2))*b*(a*
c)^(1/2)-ln((a*d*x+b*c*x+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2)+2*a*c)/x)*a*(b*d)^(1/2))/((b*x+a)*(d*x+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+a)^(1/2)/x/(d*x+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 = 16.93, size = 4312, normalized size = 50.73

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(4*atanh((64*b^10*c^2*(b*d)^(1/2))/(200*a^(3/2)*b^9*c^(1/2)*d^2 - 288*a^(1/2)*b^10*c^(3/2)*d + (544*a^(5/2)*b^
8*d^3)/c^(1/2) - (528*a^(7/2)*b^7*d^4)/c^(3/2) + (64*a^(9/2)*b^6*d^5)/c^(5/2) + (8*a^(11/2)*b^5*d^6)/c^(7/2) +
(64*b^10*c^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (864*a^2*b^8*d^3*((a + b*x)^(1/2) -
a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a^3*b^7*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c
^(1/2))) + (160*a^4*b^6*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^5*b^5*d^6*(
(a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^9*c*d^2*((a + b*x)^(1/2) - a^(1/2)))/
((c + d*x)^(1/2) - c^(1/2))) - (864*b^8*(b*d)^(1/2))/((200*b^9*c^(1/2))/a^(1/2) + (544*a^(1/2)*b^8*d)/c^(1/2)
- (528*a^(3/2)*b^7*d^2)/c^(3/2) - (288*b^10*c^(3/2))/(a^(3/2)*d) + (64*a^(5/2)*b^6*d^3)/c^(5/2) + (8*a^(7/2)*b
^5*d^4)/c^(7/2) - (864*b^8*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*b^9*c*((a + b*x)^
(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^
(1/2) - c^(1/2))) + (160*a^2*b^6*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10
*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^4*((a + b*x)^(1/2) - a^(
1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2)))) + (320*a*b^7*(b*d)^(1/2))/(544*a^(1/2)*b^8*c^(1/2) - (528*a^(3/2)*b^
7*d)/c^(1/2) + (200*b^9*c^(3/2))/(a^(1/2)*d) + (64*a^(5/2)*b^6*d^2)/c^(3/2) - (288*b^10*c^(5/2))/(a^(3/2)*d^2)
+ (8*a^(7/2)*b^5*d^3)/c^(5/2) - (864*b^8*c*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a*
b^7*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (160*a^2*b^6*d^2*((a + b*x)^(1/2) - a^(1/2)))
/(c*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2)))
- (48*a^3*b^5*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^3*((a + b*x)^(1
/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1/2) - c^(1/2)))) + (368*b^9*c*(b*d)^(1/2))/(200*a^(1/2)*b^9*c^(1/2)*d -
(288*b^10*c^(3/2))/a^(1/2) + (544*a^(3/2)*b^8*d^2)/c^(1/2) - (528*a^(5/2)*b^7*d^3)/c^(3/2) + (64*a^(7/2)*b^6*d
^4)/c^(5/2) + (8*a^(9/2)*b^5*d^5)/c^(7/2) - (864*a*b^8*d^2*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(
1/2)) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*d*((a + b*x)^(1
/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a^2*b^7*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2
) - c^(1/2))) + (160*a^3*b^6*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^4*b^5*
d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2)))) + (160*a^2*b^6*(b*d)^(1/2))/((64*a^(5/2)*b
^6*d)/c^(1/2) - 528*a^(3/2)*b^7*c^(1/2) + (544*a^(1/2)*b^8*c^(3/2))/d + (200*b^9*c^(5/2))/(a^(1/2)*d^2) + (8*a
^(7/2)*b^5*d^2)/c^(3/2) - (288*b^10*c^(7/2))/(a^(3/2)*d^3) + (160*a^2*b^6*d*((a + b*x)^(1/2) - a^(1/2)))/((c +
d*x)^(1/2) - c^(1/2)) - (864*b^8*c^2*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^
7*c*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*a^3*b^5*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c
*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2)))
+ (64*b^10*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2)))) - (48*a^3*b^5*(b*d)^(1/2))/
(64*a^(5/2)*b^6*c^(1/2) + (8*a^(7/2)*b^5*d)/c^(1/2) - (528*a^(3/2)*b^7*c^(3/2))/d + (544*a^(1/2)*b^8*c^(5/2))/
d^2 + (200*b^9*c^(7/2))/(a^(1/2)*d^3) - (288*b^10*c^(9/2))/(a^(3/2)*d^4) + (160*a^2*b^6*c*((a + b*x)^(1/2) - a
^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*a^3*b^5*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))
- (864*b^8*c^3*((a + b*x)^(1/2) - a^(1/2)))/(d^2*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*c^2*((a + b*x)^(1/2
) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a*d^3*((c + d*x)^(1
/2) - c^(1/2))) + (64*b^10*c^5*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x)^(1/2) - c^(1/2)))) + (8*a^(7/2
)*b^4*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(7/2)*((c + d*x)^(1/2) - c^(1/2))*((8*a^(7/2)*b^5)/c^(7/2) +
(544*a^(1/2)*b^8)/(c^(1/2)*d^3) + (200*b^9*c^(1/2))/(a^(1/2)*d^4) - (528*a^(3/2)*b^7)/(c^(3/2)*d^2) + (64*a^(
5/2)*b^6)/(c^(5/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^5) - (864*b^8*((a + b*x)^(1/2) - a^(1/2)))/(d^3*((c + d*
x)^(1/2) - c^(1/2))) - (48*a^3*b^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7
*((a + b*x)^(1/2) - a^(1/2)))/(c*d^2*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a
*d^4*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*d*((c + d*x)^(1/2) - c^(1/2
))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^5*((c + d*x)^(1/2) - c^(1/2))))) + (544*a^(1/2)*b^7*(b*
d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(1/2)*((c + d*x)^(1/2) - c^(1/2))*((544*a^(1/2)*b^8)/c^(1/2) - (864*b
^8*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (528*a^(3/2)*b^7*d)/c^(3/2) + (200*b^9*c^(1/2))/
(a^(1/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^2) + (64*a^(5/2)*b^6*d^2)/c^(5/2) + (8*a^(7/2)*b^5*d^3)/c^(7/2) +
(368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d^2*((a + b*x)^(1/2)
- a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(
1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^
7*d*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))))) - (528*a^(3/2)*b^6*(b*d)^(1/2)*((a + b*x)^(
1/2) - a^(1/2)))/(c^(3/2)*((c + d*x)^(1/2) - c^(1/2))*((64*a^(5/2)*b^6*d)/c^(5/2) - (528*a^(3/2)*b^7)/c^(3/2)
+ (544*a^(1/2)*b^8)/(c^(1/2)*d) + (200*b^9*c^(1/2))/(a^(1/2)*d^2) - (288*b^10*c^(3/2))/(a^(3/2)*d^3) + (8*a^(7
/2)*b^5*d^2)/c^(7/2) - (864*b^8*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*((a
+ b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((
c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2))) - (48
*a^3*b^5*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) -
a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2))))) + (200*b^8*c^(1/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/
(a^(1/2)*((c + d*x)^(1/2) - c^(1/2))*((200*b^9*c^(1/2))/a^(1/2) + (544*a^(1/2)*b^8*d)/c^(1/2) - (528*a^(3/2)*b
^7*d^2)/c^(3/2) - (288*b^10*c^(3/2))/(a^(3/2)*d) + (64*a^(5/2)*b^6*d^3)/c^(5/2) + (8*a^(7/2)*b^5*d^4)/c^(7/2)
- (864*b^8*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2))
)/(a*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^7*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))
) + (160*a^2*b^6*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^
(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c
+ d*x)^(1/2) - c^(1/2))))) + (64*a^(5/2)*b^5*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(5/2)*((c + d*x)^(1/2
) - c^(1/2))*((64*a^(5/2)*b^6)/c^(5/2) + (8*a^(7/2)*b^5*d)/c^(7/2) + (544*a^(1/2)*b^8)/(c^(1/2)*d^2) + (200*b^
9*c^(1/2))/(a^(1/2)*d^3) - (528*a^(3/2)*b^7)/(c^(3/2)*d) - (288*b^10*c^(3/2))/(a^(3/2)*d^4) - (864*b^8*((a + b
*x)^(1/2) - a^(1/2)))/(d^2*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c +
d*x)^(1/2) - c^(1/2))) + (320*a*b^7*((a + b*x)^(1/2) - a^(1/2)))/(c*d*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*
b^5*d*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d*x)^(1/2) - c^(1/2))) + (368*b^9*c*((a + b*x)^(1/2) - a^(1/2)))
/(a*d^3*((c + d*x)^(1/2) - c^(1/2))) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^4*((c + d*x)^(1/2) - c
^(1/2))))) - (288*b^9*c^(3/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(a^(3/2)*((c + d*x)^(1/2) - c^(1/2))*((
200*b^9*c^(1/2)*d)/a^(1/2) - (288*b^10*c^(3/2))/a^(3/2) + (544*a^(1/2)*b^8*d^2)/c^(1/2) - (528*a^(3/2)*b^7*d^3
)/c^(3/2) + (64*a^(5/2)*b^6*d^4)/c^(5/2) + (8*a^(7/2)*b^5*d^5)/c^(7/2) - (864*b^8*d^2*((a + b*x)^(1/2) - a^(1/
2)))/((c + d*x)^(1/2) - c^(1/2)) + (64*b^10*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*((c + d*x)^(1/2) - c^(1/2)))
+ (320*a*b^7*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (160*a^2*b^6*d^4*((a + b*x)^(
1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*a^3*b^5*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^3*((c + d
*x)^(1/2) - c^(1/2))) + (368*b^9*c*d*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))))))*(b*d)^(1/
2))/d - (a^(1/2)*log(((a + b*x)^(1/2) - a^(1/2))/((c + d*x)^(1/2) - c^(1/2))) - a^(1/2)*log(((c^(1/2)*(a + b*x
)^(1/2) - a^(1/2)*(c + d*x)^(1/2))*(b*c^(1/2) - (a^(1/2)*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(
1/2))))/((c + d*x)^(1/2) - c^(1/2))))/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}}{x \sqrt {c + d x}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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