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

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

[Out]

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

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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} \[ \frac {2 \sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b} \sqrt {c+d x}}\right )}{\sqrt {b}}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

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Mathematica [A]  time = 0.31, size = 120, normalized size = 1.41 \[ \frac {2 \sqrt {d} \sqrt {c+d x} \sinh ^{-1}\left (\frac {\sqrt {d} \sqrt {a+b x}}{\sqrt {b c-a d}}\right )}{\sqrt {b c-a d} \sqrt {\frac {b (c+d x)}{b c-a d}}}-\frac {2 \sqrt {c} \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {a+b x}}{\sqrt {a} \sqrt {c+d x}}\right )}{\sqrt {a}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.97, size = 711, normalized size = 8.36 \[ \left [\frac {1}{2} \, \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), -\sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {c}{a}} \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^{2} c + {\left (a b c + a^{2} d\right )} x\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {c}{a}} + 8 \, {\left (a b c^{2} + a^{2} c d\right )} x}{x^{2}}\right ), \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) + \frac {1}{2} \, \sqrt {\frac {d}{b}} \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^{2} d x + b^{2} c + a b d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {\frac {d}{b}} + 8 \, {\left (b^{2} c d + a b d^{2}\right )} x\right ), \sqrt {-\frac {c}{a}} \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 {c}{a}}}{2 \, {\left (b c d x^{2} + a c^{2} + {\left (b c^{2} + a c d\right )} x\right )}}\right ) - \sqrt {-\frac {d}{b}} \arctan \left (\frac {{\left (2 \, b d x + b c + a d\right )} \sqrt {b x + a} \sqrt {d x + c} \sqrt {-\frac {d}{b}}}{2 \, {\left (b d^{2} x^{2} + a c d + {\left (b c d + a d^{2}\right )} x\right )}}\right )\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/x/(b*x+a)^(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 \[ \frac {\sqrt {d x +c}\, \sqrt {b x +a}\, \left (-\sqrt {b d}\, c \ln \left (\frac {a d x +b c x +2 a c +2 \sqrt {a c}\, \sqrt {\left (b x +a \right ) \left (d x +c \right )}}{x}\right )+\sqrt {a c}\, d \ln \left (\frac {2 b d x +a d +b c +2 \sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}}{2 \sqrt {b d}}\right )\right )}{\sqrt {\left (b x +a \right ) \left (d x +c \right )}\, \sqrt {b d}\, \sqrt {a c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d*x+c)^(1/2)*(b*x+a)^(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))*d*(a*
c)^(1/2)-ln((a*d*x+b*c*x+2*a*c+2*(a*c)^(1/2)*((b*x+a)*(d*x+c))^(1/2))/x)*c*(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 \[ \text {Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((d*x+c)^(1/2)/x/(b*x+a)^(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 = 18.14, size = 4311, normalized size = 50.72 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(c^(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)*log(((a + b*x)^(1/2) - a^(1/2))/
((c + d*x)^(1/2) - c^(1/2))))/a^(1/2) + (4*atanh((64*a^2*b^2*(b*d)^(1/2))/((200*a^(1/2)*b^4*c^(3/2))/d - 288*a
^(3/2)*b^3*c^(1/2) + (544*b^5*c^(5/2))/(a^(1/2)*d^2) - (528*b^6*c^(7/2))/(a^(3/2)*d^3) + (64*b^7*c^(9/2))/(a^(
5/2)*d^4) + (8*b^8*c^(11/2))/(a^(7/2)*d^5) + (64*a^2*b^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(
1/2)) - (864*b^4*c^2*((a + b*x)^(1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*c*((a + b*x)^(1
/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*b^5*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2
) - c^(1/2))) + (160*b^6*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^3*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^5*
((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^4*((c + d*x)^(1/2) - c^(1/2)))) - (864*b^4*(b*d)^(1/2))/((544*b^5*c^(1/2))
/a^(1/2) + (200*a^(1/2)*b^4*d)/c^(1/2) - (288*a^(3/2)*b^3*d^2)/c^(3/2) - (528*b^6*c^(3/2))/(a^(3/2)*d) + (64*b
^7*c^(5/2))/(a^(5/2)*d^2) + (8*b^8*c^(7/2))/(a^(7/2)*d^3) - (864*b^4*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)
^(1/2) - c^(1/2)) + (320*b^5*c*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d^2*(
(a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^2*b^2*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^
2*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2)))
- (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^2*((c + d*x)^(1/2) - c^(1/2)))) + (368*a*b^3*(b*d)^(1/2))/(
200*a^(1/2)*b^4*c^(1/2) - (288*a^(3/2)*b^3*d)/c^(1/2) + (544*b^5*c^(3/2))/(a^(1/2)*d) - (528*b^6*c^(5/2))/(a^(
3/2)*d^2) + (64*b^7*c^(7/2))/(a^(5/2)*d^3) + (8*b^8*c^(9/2))/(a^(7/2)*d^4) - (864*b^4*c*((a + b*x)^(1/2) - a^(
1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*a*b^3*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (
64*a^2*b^2*d^2*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (320*b^5*c^2*((a + b*x)^(1/2) -
a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1
/2) - c^(1/2))) - (48*b^7*c^4*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^3*((c + d*x)^(1/2) - c^(1/2)))) + (320*b^5*c
*(b*d)^(1/2))/(544*a^(1/2)*b^5*c^(1/2)*d - (528*b^6*c^(3/2))/a^(1/2) + (200*a^(3/2)*b^4*d^2)/c^(1/2) - (288*a^
(5/2)*b^3*d^3)/c^(3/2) + (64*b^7*c^(5/2))/(a^(3/2)*d) + (8*b^8*c^(7/2))/(a^(5/2)*d^2) - (864*a*b^4*d^2*((a + b
*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (160*b^6*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1
/2) - c^(1/2))) + (320*b^5*c*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*a^2*b^3*d^3*((a
+ b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^3*b^2*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^2*
((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2)))) -
(48*b^7*c^3*(b*d)^(1/2))/((8*b^8*c^(7/2))/a^(1/2) + 64*a^(1/2)*b^7*c^(5/2)*d - 528*a^(3/2)*b^6*c^(3/2)*d^2 +
544*a^(5/2)*b^5*c^(1/2)*d^3 + (200*a^(7/2)*b^4*d^4)/c^(1/2) - (288*a^(9/2)*b^3*d^5)/c^(3/2) - (48*b^7*c^3*d*((
a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (864*a^3*b^4*d^4*((a + b*x)^(1/2) - a^(1/2)))/((c + d
*x)^(1/2) - c^(1/2)) + (160*a*b^6*c^2*d^2*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (320*a^2*
b^5*c*d^3*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (368*a^4*b^3*d^5*((a + b*x)^(1/2) - a^(1/
2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^5*b^2*d^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^
(1/2)))) + (160*b^6*c^2*(b*d)^(1/2))/((64*b^7*c^(5/2))/a^(1/2) - 528*a^(1/2)*b^6*c^(3/2)*d + 544*a^(3/2)*b^5*c
^(1/2)*d^2 + (200*a^(5/2)*b^4*d^3)/c^(1/2) - (288*a^(7/2)*b^3*d^4)/c^(3/2) + (8*b^8*c^(7/2))/(a^(3/2)*d) + (16
0*b^6*c^2*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2))
)/(a*((c + d*x)^(1/2) - c^(1/2))) - (864*a^2*b^4*d^3*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2))
+ (368*a^3*b^3*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^4*b^2*d^5*((a + b*x)^(
1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (320*a*b^5*c*d^2*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^
(1/2) - c^(1/2))) - (288*a^(3/2)*b^2*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(c^(3/2)*((c + d*x)^(1/2) - c^(1
/2))*((200*a^(1/2)*b^4)/(c^(1/2)*d) - (288*a^(3/2)*b^3)/c^(3/2) + (544*b^5*c^(1/2))/(a^(1/2)*d^2) - (528*b^6*c
^(3/2))/(a^(3/2)*d^3) + (64*b^7*c^(5/2))/(a^(5/2)*d^4) + (8*b^8*c^(7/2))/(a^(7/2)*d^5) - (864*b^4*((a + b*x)^(
1/2) - a^(1/2)))/(d*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2)
- c^(1/2))) + (64*a^2*b^2*d*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) + (320*b^5*c*((a +
b*x)^(1/2) - a^(1/2)))/(a*d^2*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d
^3*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^4*((c + d*x)^(1/2) - c^(1/2)
)))) + (8*b^7*c^(7/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(a^(7/2)*((c + d*x)^(1/2) - c^(1/2))*((8*b^8*c^
(7/2))/a^(7/2) + (64*b^7*c^(5/2)*d)/a^(5/2) + (544*b^5*c^(1/2)*d^3)/a^(1/2) + (200*a^(1/2)*b^4*d^4)/c^(1/2) -
(528*b^6*c^(3/2)*d^2)/a^(3/2) - (288*a^(3/2)*b^3*d^5)/c^(3/2) - (864*b^4*d^4*((a + b*x)^(1/2) - a^(1/2)))/((c
+ d*x)^(1/2) - c^(1/2)) + (368*a*b^3*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (320*b
^5*c*d^3*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^3*d*((a + b*x)^(1/2) - a^(1/
2)))/(a^3*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*d^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*((c + d*x)^(1/2) -
c^(1/2))) + (64*a^2*b^2*d^6*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))))) + (544*b^4*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))*((544*b^5*c^(1/2))/a^(1/2) +
(200*a^(1/2)*b^4*d)/c^(1/2) - (288*a^(3/2)*b^3*d^2)/c^(3/2) - (528*b^6*c^(3/2))/(a^(3/2)*d) + (64*b^7*c^(5/2))
/(a^(5/2)*d^2) + (8*b^8*c^(7/2))/(a^(7/2)*d^3) - (864*b^4*d*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^
(1/2)) + (320*b^5*c*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d^2*((a + b*x)^(
1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (64*a^2*b^2*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x
)^(1/2) - c^(1/2))) + (160*b^6*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*
c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^2*((c + d*x)^(1/2) - c^(1/2))))) + (200*a^(1/2)*b^3*(b*d)^(1/2)*((a +
b*x)^(1/2) - a^(1/2)))/(c^(1/2)*((c + d*x)^(1/2) - c^(1/2))*((200*a^(1/2)*b^4)/c^(1/2) - (864*b^4*((a + b*x)^(
1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (288*a^(3/2)*b^3*d)/c^(3/2) + (544*b^5*c^(1/2))/(a^(1/2)*d) - (
528*b^6*c^(3/2))/(a^(3/2)*d^2) + (64*b^7*c^(5/2))/(a^(5/2)*d^3) + (8*b^8*c^(7/2))/(a^(7/2)*d^4) + (320*b^5*c*(
(a + b*x)^(1/2) - a^(1/2)))/(a*d*((c + d*x)^(1/2) - c^(1/2))) + (64*a^2*b^2*d^2*((a + b*x)^(1/2) - a^(1/2)))/(
c^2*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*((a + b*x)^(1/2) - a^(1/2)))/(a^2*d^2*((c + d*x)^(1/2) - c^(1/
2))) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*d^3*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d*((a + b*x
)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))))) - (528*b^5*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))*((544*b^5*c^(1/2)*d)/a^(1/2) - (528*b^6*c^(3/2))/a^(3/2) + (200*a^(1/
2)*b^4*d^2)/c^(1/2) - (288*a^(3/2)*b^3*d^3)/c^(3/2) + (64*b^7*c^(5/2))/(a^(5/2)*d) + (8*b^8*c^(7/2))/(a^(7/2)*
d^2) - (864*b^4*d^2*((a + b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) + (160*b^6*c^2*((a + b*x)^(1/2) -
a^(1/2)))/(a^2*((c + d*x)^(1/2) - c^(1/2))) + (368*a*b^3*d^3*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2)
- c^(1/2))) + (64*a^2*b^2*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/2))) - (48*b^7*c^3*((
a + b*x)^(1/2) - a^(1/2)))/(a^3*d*((c + d*x)^(1/2) - c^(1/2))) + (320*b^5*c*d*((a + b*x)^(1/2) - a^(1/2)))/(a*
((c + d*x)^(1/2) - c^(1/2))))) + (64*b^6*c^(5/2)*(b*d)^(1/2)*((a + b*x)^(1/2) - a^(1/2)))/(a^(5/2)*((c + d*x)^
(1/2) - c^(1/2))*((64*b^7*c^(5/2))/a^(5/2) - (528*b^6*c^(3/2)*d)/a^(3/2) + (544*b^5*c^(1/2)*d^2)/a^(1/2) + (20
0*a^(1/2)*b^4*d^3)/c^(1/2) - (288*a^(3/2)*b^3*d^4)/c^(3/2) + (8*b^8*c^(7/2))/(a^(7/2)*d) - (864*b^4*d^3*((a +
b*x)^(1/2) - a^(1/2)))/((c + d*x)^(1/2) - c^(1/2)) - (48*b^7*c^3*((a + b*x)^(1/2) - a^(1/2)))/(a^3*((c + d*x)^
(1/2) - c^(1/2))) + (368*a*b^3*d^4*((a + b*x)^(1/2) - a^(1/2)))/(c*((c + d*x)^(1/2) - c^(1/2))) + (320*b^5*c*d
^2*((a + b*x)^(1/2) - a^(1/2)))/(a*((c + d*x)^(1/2) - c^(1/2))) + (160*b^6*c^2*d*((a + b*x)^(1/2) - a^(1/2)))/
(a^2*((c + d*x)^(1/2) - c^(1/2))) + (64*a^2*b^2*d^5*((a + b*x)^(1/2) - a^(1/2)))/(c^2*((c + d*x)^(1/2) - c^(1/
2))))))*(b*d)^(1/2))/b

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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