3.31.11 \(\int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx\)

Optimal. Leaf size=407 \[ \frac {2 \left (\sqrt {2} a^3+\sqrt {2} a c \sqrt {a^2+4 b}+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}+\sqrt {2} a^2 c+3 \sqrt {2} a b+2 \sqrt {2} b c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-\sqrt {a^2+4 b}-a-2 c}}+\frac {2 \left (-\sqrt {2} a^3+\sqrt {2} a c \sqrt {a^2+4 b}+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}-\sqrt {2} a^2 c-3 \sqrt {2} a b-2 \sqrt {2} b c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}-a-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {\sqrt {a^2+4 b}-a-2 c}}+\frac {4}{3} (3 a+c) \sqrt {\sqrt {a x+b}+c}+\frac {4}{3} \sqrt {a x+b} \sqrt {\sqrt {a x+b}+c} \]

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Rubi [A]  time = 4.65, antiderivative size = 255, normalized size of antiderivative = 0.63, number of steps used = 7, number of rules used = 4, integrand size = 40, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {897, 1287, 1166, 206} \begin {gather*} -\frac {2 \sqrt {2} \left (a^2-\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {-\sqrt {a^2+4 b}+a+2 c}}-\frac {2 \sqrt {2} \left (a^2+\frac {a^3+a^2 c+3 a b+2 b c}{\sqrt {a^2+4 b}}+a c+b\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {\sqrt {a x+b}+c}}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {\sqrt {a^2+4 b}+a+2 c}}+\frac {4}{3} \left (\sqrt {a x+b}+c\right )^{3/2}+4 a \sqrt {\sqrt {a x+b}+c} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

4*a*Sqrt[c + Sqrt[b + a*x]] + (4*(c + Sqrt[b + a*x])^(3/2))/3 - (2*Sqrt[2]*(a^2 + b + a*c - (a^3 + 3*a*b + a^2
*c + 2*b*c)/Sqrt[a^2 + 4*b])*ArcTanh[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[a - Sqrt[a^2 + 4*b] + 2*c]])/Sqrt[
a - Sqrt[a^2 + 4*b] + 2*c] - (2*Sqrt[2]*(a^2 + b + a*c + (a^3 + 3*a*b + a^2*c + 2*b*c)/Sqrt[a^2 + 4*b])*ArcTan
h[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[a + Sqrt[a^2 + 4*b] + 2*c]])/Sqrt[a + Sqrt[a^2 + 4*b] + 2*c]

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 897

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))^(n_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :
> With[{q = Denominator[m]}, Dist[q/e, Subst[Int[x^(q*(m + 1) - 1)*((e*f - d*g)/e + (g*x^q)/e)^n*((c*d^2 - b*d
*e + a*e^2)/e^2 - ((2*c*d - b*e)*x^q)/e^2 + (c*x^(2*q))/e^2)^p, x], x, (d + e*x)^(1/q)], x]] /; FreeQ[{a, b, c
, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && IntegersQ[n,
p] && FractionQ[m]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 1287

Int[(((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(q_.))/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> Int[Ex
pandIntegrand[((f*x)^m*(d + e*x^2)^q)/(a + b*x^2 + c*x^4), x], x] /; FreeQ[{a, b, c, d, e, f, m}, x] && NeQ[b^
2 - 4*a*c, 0] && IntegerQ[q] && IntegerQ[m]

Rubi steps

\begin {align*} \int \frac {\sqrt {b+a x} \sqrt {c+\sqrt {b+a x}}}{x-\sqrt {b+a x}} \, dx &=-\left (2 \operatorname {Subst}\left (\int \frac {x^2 \sqrt {c+x}}{b+a x-x^2} \, dx,x,\sqrt {b+a x}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \frac {x^2 \left (-c+x^2\right )^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right )\\ &=-\left (4 \operatorname {Subst}\left (\int \left (-a-x^2+\frac {a (b-c (a+c))+\left (a^2+b+a c\right ) x^2}{b-a c-c^2+(a+2 c) x^2-x^4}\right ) \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\right )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-4 \operatorname {Subst}\left (\int \frac {a (b-c (a+c))+\left (a^2+b+a c\right ) x^2}{b-a c-c^2+(a+2 c) x^2-x^4} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\left (2 \left (a^2+b+a c-\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{-\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )-\left (2 \left (a^2+b+a c+\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{\frac {1}{2} \sqrt {a^2+4 b}+\frac {1}{2} (a+2 c)-x^2} \, dx,x,\sqrt {c+\sqrt {b+a x}}\right )\\ &=4 a \sqrt {c+\sqrt {b+a x}}+\frac {4}{3} \left (c+\sqrt {b+a x}\right )^{3/2}-\frac {2 \sqrt {2} \left (a^2+b+a c-\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a-\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a-\sqrt {a^2+4 b}+2 c}}-\frac {2 \sqrt {2} \left (a^2+b+a c+\frac {a^3+3 a b+a^2 c+2 b c}{\sqrt {a^2+4 b}}\right ) \tanh ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {a+\sqrt {a^2+4 b}+2 c}}\right )}{\sqrt {a+\sqrt {a^2+4 b}+2 c}}\\ \end {align*}

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Mathematica [A]  time = 4.45, size = 328, normalized size = 0.81 \begin {gather*} \frac {\sqrt {\sqrt {a x+b}+c} \left (-2 a \left (-3 \sqrt {a^2+4 b}+\sqrt {a x+b}+c\right )+2 a \left (3 \sqrt {a^2+4 b}+\sqrt {a x+b}+c\right )-\frac {3 \left (a \sqrt {a^2+4 b}-a^2-2 b\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {a x+b}+c}{-\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {\frac {\sqrt {a x+b}+c}{-2 \sqrt {a^2+4 b}+2 a+4 c}}}-\frac {3 \sqrt {2} \left (a \sqrt {a^2+4 b}+a^2+2 b\right ) \tanh ^{-1}\left (\sqrt {2} \sqrt {\frac {\sqrt {a x+b}+c}{\sqrt {a^2+4 b}+a+2 c}}\right )}{\sqrt {\frac {\sqrt {a x+b}+c}{\sqrt {a^2+4 b}+a+2 c}}}+4 c \sqrt {a^2+4 b}+4 \sqrt {a^2+4 b} \sqrt {a x+b}\right )}{3 \sqrt {a^2+4 b}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[c + Sqrt[b + a*x]]*(4*Sqrt[a^2 + 4*b]*c + 4*Sqrt[a^2 + 4*b]*Sqrt[b + a*x] - 2*a*(-3*Sqrt[a^2 + 4*b] + c
+ Sqrt[b + a*x]) + 2*a*(3*Sqrt[a^2 + 4*b] + c + Sqrt[b + a*x]) - (3*(-a^2 - 2*b + a*Sqrt[a^2 + 4*b])*ArcTanh[S
qrt[2]*Sqrt[(c + Sqrt[b + a*x])/(a - Sqrt[a^2 + 4*b] + 2*c)]])/Sqrt[(c + Sqrt[b + a*x])/(2*a - 2*Sqrt[a^2 + 4*
b] + 4*c)] - (3*Sqrt[2]*(a^2 + 2*b + a*Sqrt[a^2 + 4*b])*ArcTanh[Sqrt[2]*Sqrt[(c + Sqrt[b + a*x])/(a + Sqrt[a^2
+ 4*b] + 2*c)]])/Sqrt[(c + Sqrt[b + a*x])/(a + Sqrt[a^2 + 4*b] + 2*c)]))/(3*Sqrt[a^2 + 4*b])

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IntegrateAlgebraic [A]  time = 0.89, size = 388, normalized size = 0.95 \begin {gather*} \frac {4}{3} \sqrt {c+\sqrt {b+a x}} \left (3 a+c+\sqrt {b+a x}\right )+\frac {2 \left (\sqrt {2} a^3+3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}+\sqrt {2} a^2 c+2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a-\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a-\sqrt {a^2+4 b}-2 c}}+\frac {2 \left (-\sqrt {2} a^3-3 \sqrt {2} a b+\sqrt {2} a^2 \sqrt {a^2+4 b}+\sqrt {2} b \sqrt {a^2+4 b}-\sqrt {2} a^2 c-2 \sqrt {2} b c+\sqrt {2} a \sqrt {a^2+4 b} c\right ) \tan ^{-1}\left (\frac {\sqrt {2} \sqrt {c+\sqrt {b+a x}}}{\sqrt {-a+\sqrt {a^2+4 b}-2 c}}\right )}{\sqrt {a^2+4 b} \sqrt {-a+\sqrt {a^2+4 b}-2 c}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(4*Sqrt[c + Sqrt[b + a*x]]*(3*a + c + Sqrt[b + a*x]))/3 + (2*(Sqrt[2]*a^3 + 3*Sqrt[2]*a*b + Sqrt[2]*a^2*Sqrt[a
^2 + 4*b] + Sqrt[2]*b*Sqrt[a^2 + 4*b] + Sqrt[2]*a^2*c + 2*Sqrt[2]*b*c + Sqrt[2]*a*Sqrt[a^2 + 4*b]*c)*ArcTan[(S
qrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a - Sqrt[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a - Sqrt[a^2 + 4*b]
- 2*c]) + (2*(-(Sqrt[2]*a^3) - 3*Sqrt[2]*a*b + Sqrt[2]*a^2*Sqrt[a^2 + 4*b] + Sqrt[2]*b*Sqrt[a^2 + 4*b] - Sqrt[
2]*a^2*c - 2*Sqrt[2]*b*c + Sqrt[2]*a*Sqrt[a^2 + 4*b]*c)*ArcTan[(Sqrt[2]*Sqrt[c + Sqrt[b + a*x]])/Sqrt[-a + Sqr
t[a^2 + 4*b] - 2*c]])/(Sqrt[a^2 + 4*b]*Sqrt[-a + Sqrt[a^2 + 4*b] - 2*c])

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fricas [B]  time = 0.91, size = 2014, normalized size = 4.95

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

sqrt(2)*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b
^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b
)))/(a^2 + 4*b))*log(8*sqrt(2)*(a^7 + 7*a^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6 + 6*a^4*b + 8*a^2*b^2)*c - (a^4 +
6*a^2*b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7
+ 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c +
(a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5
*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^4*b^2 + 3*a^2*b^3 + b^4 + (a^3*b^2 + 2*a*b
^3)*c)*sqrt(c + sqrt(a*x + b))) - sqrt(2)*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4
*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b +
7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(-8*sqrt(2)*(a^7 + 7*a^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6
+ 6*a^4*b + 8*a^2*b^2)*c - (a^4 + 6*a^2*b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6
+ 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))*sqrt((a^5 + 5*a^3*b + 5*
a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c + (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*
a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^4*b^2 +
3*a^2*b^3 + b^4 + (a^3*b^2 + 2*a*b^3)*c)*sqrt(c + sqrt(a*x + b))) + sqrt(2)*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (
a^4 + 4*a^2*b + 2*b^2)*c - (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4
*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b))*log(8*sqrt(2)*(a^7 + 7*a
^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6 + 6*a^4*b + 8*a^2*b^2)*c + (a^4 + 6*a^2*b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 11
*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2
+ 4*b)))*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c - (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4
*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4
*b)))/(a^2 + 4*b)) + 32*(a^4*b^2 + 3*a^2*b^3 + b^4 + (a^3*b^2 + 2*a*b^3)*c)*sqrt(c + sqrt(a*x + b))) - sqrt(2)
*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c - (a^2 + 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*
a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^
2 + 4*b))*log(-8*sqrt(2)*(a^7 + 7*a^5*b + 13*a^3*b^2 + 4*a*b^3 + (a^6 + 6*a^4*b + 8*a^2*b^2)*c + (a^4 + 6*a^2*
b + 8*b^2)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a
^5*b + 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))*sqrt((a^5 + 5*a^3*b + 5*a*b^2 + (a^4 + 4*a^2*b + 2*b^2)*c - (a^2
+ 4*b)*sqrt((a^8 + 6*a^6*b + 11*a^4*b^2 + 6*a^2*b^3 + b^4 + (a^6 + 4*a^4*b + 4*a^2*b^2)*c^2 + 2*(a^7 + 5*a^5*b
+ 7*a^3*b^2 + 2*a*b^3)*c)/(a^2 + 4*b)))/(a^2 + 4*b)) + 32*(a^4*b^2 + 3*a^2*b^3 + b^4 + (a^3*b^2 + 2*a*b^3)*c)
*sqrt(c + sqrt(a*x + b))) + 4/3*(3*a + c + sqrt(a*x + b))*sqrt(c + sqrt(a*x + b))

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giac [A]  time = 1.05, size = 259, normalized size = 0.64 \begin {gather*} \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c + 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c + \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{4} + 6 \, a^{2} b + 8 \, b^{2} + {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} - \frac {4 \, \sqrt {a^{2} + 4 \, b} \sqrt {-2 \, a - 4 \, c - 2 \, \sqrt {a^{2} + 4 \, b}} b^{2} \arctan \left (\frac {\sqrt {c + \sqrt {a x + b}}}{\sqrt {-\frac {1}{2} \, a - c - \frac {1}{2} \, \sqrt {{\left (a + 2 \, c\right )}^{2} - 4 \, a c - 4 \, c^{2} + 4 \, b}}}\right )}{a^{4} + 6 \, a^{2} b + 8 \, b^{2} - {\left (a^{3} + 4 \, a b\right )} \sqrt {a^{2} + 4 \, b}} + 4 \, a \sqrt {c + \sqrt {a x + b}} + \frac {4}{3} \, {\left (c + \sqrt {a x + b}\right )}^{\frac {3}{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

4*sqrt(a^2 + 4*b)*sqrt(-2*a - 4*c + 2*sqrt(a^2 + 4*b))*b^2*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c + 1/
2*sqrt((a + 2*c)^2 - 4*a*c - 4*c^2 + 4*b)))/(a^4 + 6*a^2*b + 8*b^2 + (a^3 + 4*a*b)*sqrt(a^2 + 4*b)) - 4*sqrt(a
^2 + 4*b)*sqrt(-2*a - 4*c - 2*sqrt(a^2 + 4*b))*b^2*arctan(sqrt(c + sqrt(a*x + b))/sqrt(-1/2*a - c - 1/2*sqrt((
a + 2*c)^2 - 4*a*c - 4*c^2 + 4*b)))/(a^4 + 6*a^2*b + 8*b^2 - (a^3 + 4*a*b)*sqrt(a^2 + 4*b)) + 4*a*sqrt(c + sqr
t(a*x + b)) + 4/3*(c + sqrt(a*x + b))^(3/2)

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maple [A]  time = 0.29, size = 268, normalized size = 0.66

method	result	size
derivativedivides	\(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c +b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c +b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\)	\(268\)
default	\(\frac {4 \left (c +\sqrt {a x +b}\right )^{\frac {3}{2}}}{3}+4 a \sqrt {c +\sqrt {a x +b}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}+a^{3}+a^{2} c +b \sqrt {a^{2}+4 b}+3 a b +2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {-2 \sqrt {a^{2}+4 b}-2 a -4 c}}+\frac {4 \left (a^{2} \sqrt {a^{2}+4 b}+a c \sqrt {a^{2}+4 b}-a^{3}-a^{2} c +b \sqrt {a^{2}+4 b}-3 a b -2 b c \right ) \arctan \left (\frac {2 \sqrt {c +\sqrt {a x +b}}}{\sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\right )}{\sqrt {a^{2}+4 b}\, \sqrt {2 \sqrt {a^{2}+4 b}-2 a -4 c}}\)	\(268\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((a*x+b)^(1/2)*(c+(a*x+b)^(1/2))^(1/2)/(x-(a*x+b)^(1/2)),x,method=_RETURNVERBOSE)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

integrate(sqrt(a*x + b)*sqrt(c + sqrt(a*x + b))/(x - sqrt(a*x + b)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {\sqrt {c+\sqrt {b+a\,x}}\,\sqrt {b+a\,x}}{x-\sqrt {b+a\,x}} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [B]  time = 123.16, size = 1095, normalized size = 2.69 \begin {gather*} - 4 a^{2} c \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a^{2} \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a b \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} - 4 a c^{2} \operatorname {RootSum} {\left (t^{4} \left (16 a^{5} c - 16 a^{4} b + 16 a^{4} c^{2} + 128 a^{3} b c - 128 a^{2} b^{2} + 128 a^{2} b c^{2} + 256 a b^{2} c - 256 b^{3} + 256 b^{2} c^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + 1, \left (t \mapsto t \log {\left (8 t^{3} a^{4} c - 8 t^{3} a^{3} b + 24 t^{3} a^{3} c^{2} + 16 t^{3} a^{2} b c + 16 t^{3} a^{2} c^{3} - 32 t^{3} a b^{2} + 96 t^{3} a b c^{2} - 64 t^{3} b^{2} c + 64 t^{3} b c^{3} - 2 t a^{2} - 4 t a c - 4 t b - 4 t c^{2} + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a c \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + 4 a \sqrt {c + \sqrt {a x + b}} + 4 b \operatorname {RootSum} {\left (t^{4} \left (16 a^{4} + 128 a^{2} b + 256 b^{2}\right ) + t^{2} \left (- 4 a^{3} - 8 a^{2} c - 16 a b - 32 b c\right ) + a c - b + c^{2}, \left (t \mapsto t \log {\left (- 16 t^{3} a^{2} - 64 t^{3} b + 2 t a + 4 t c + \sqrt {c + \sqrt {a x + b}} \right )} \right )\right )} + \frac {4 \left (c + \sqrt {a x + b}\right )^{\frac {3}{2}}}{3} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*a**2*c*RootSum(_t**4*(16*a**5*c - 16*a**4*b + 16*a**4*c**2 + 128*a**3*b*c - 128*a**2*b**2 + 128*a**2*b*c**2
+ 256*a*b**2*c - 256*b**3 + 256*b**2*c**2) + _t**2*(-4*a**3 - 8*a**2*c - 16*a*b - 32*b*c) + 1, Lambda(_t, _t*
log(8*_t**3*a**4*c - 8*_t**3*a**3*b + 24*_t**3*a**3*c**2 + 16*_t**3*a**2*b*c + 16*_t**3*a**2*c**3 - 32*_t**3*a
*b**2 + 96*_t**3*a*b*c**2 - 64*_t**3*b**2*c + 64*_t**3*b*c**3 - 2*_t*a**2 - 4*_t*a*c - 4*_t*b - 4*_t*c**2 + sq
rt(c + sqrt(a*x + b))))) + 4*a**2*RootSum(_t**4*(16*a**4 + 128*a**2*b + 256*b**2) + _t**2*(-4*a**3 - 8*a**2*c
- 16*a*b - 32*b*c) + a*c - b + c**2, Lambda(_t, _t*log(-16*_t**3*a**2 - 64*_t**3*b + 2*_t*a + 4*_t*c + sqrt(c
+ sqrt(a*x + b))))) + 4*a*b*RootSum(_t**4*(16*a**5*c - 16*a**4*b + 16*a**4*c**2 + 128*a**3*b*c - 128*a**2*b**2
+ 128*a**2*b*c**2 + 256*a*b**2*c - 256*b**3 + 256*b**2*c**2) + _t**2*(-4*a**3 - 8*a**2*c - 16*a*b - 32*b*c) +
1, Lambda(_t, _t*log(8*_t**3*a**4*c - 8*_t**3*a**3*b + 24*_t**3*a**3*c**2 + 16*_t**3*a**2*b*c + 16*_t**3*a**2
*c**3 - 32*_t**3*a*b**2 + 96*_t**3*a*b*c**2 - 64*_t**3*b**2*c + 64*_t**3*b*c**3 - 2*_t*a**2 - 4*_t*a*c - 4*_t*
b - 4*_t*c**2 + sqrt(c + sqrt(a*x + b))))) - 4*a*c**2*RootSum(_t**4*(16*a**5*c - 16*a**4*b + 16*a**4*c**2 + 12
8*a**3*b*c - 128*a**2*b**2 + 128*a**2*b*c**2 + 256*a*b**2*c - 256*b**3 + 256*b**2*c**2) + _t**2*(-4*a**3 - 8*a
**2*c - 16*a*b - 32*b*c) + 1, Lambda(_t, _t*log(8*_t**3*a**4*c - 8*_t**3*a**3*b + 24*_t**3*a**3*c**2 + 16*_t**
3*a**2*b*c + 16*_t**3*a**2*c**3 - 32*_t**3*a*b**2 + 96*_t**3*a*b*c**2 - 64*_t**3*b**2*c + 64*_t**3*b*c**3 - 2*
_t*a**2 - 4*_t*a*c - 4*_t*b - 4*_t*c**2 + sqrt(c + sqrt(a*x + b))))) + 4*a*c*RootSum(_t**4*(16*a**4 + 128*a**2
*b + 256*b**2) + _t**2*(-4*a**3 - 8*a**2*c - 16*a*b - 32*b*c) + a*c - b + c**2, Lambda(_t, _t*log(-16*_t**3*a*
*2 - 64*_t**3*b + 2*_t*a + 4*_t*c + sqrt(c + sqrt(a*x + b))))) + 4*a*sqrt(c + sqrt(a*x + b)) + 4*b*RootSum(_t*
*4*(16*a**4 + 128*a**2*b + 256*b**2) + _t**2*(-4*a**3 - 8*a**2*c - 16*a*b - 32*b*c) + a*c - b + c**2, Lambda(_
t, _t*log(-16*_t**3*a**2 - 64*_t**3*b + 2*_t*a + 4*_t*c + sqrt(c + sqrt(a*x + b))))) + 4*(c + sqrt(a*x + b))**
(3/2)/3

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