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

Optimal. Leaf size=747 \[ -\frac {4 b^2 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {4 b^2 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {4 a d^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{c^{3/2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {4 a d^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{c^{3/2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}-\frac {4 a^2 d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{c^{3/2} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {4 a^2 d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{c^{3/2} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2 \sqrt {a^2 x^2+b^2} (a c x-6 a d)+2 \left (a^2 c x^2-6 a^2 d x-b^2 c\right )}{3 a c \sqrt {\sqrt {a^2 x^2+b^2}+a x}} \]

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Rubi [A]  time = 0.89, antiderivative size = 279, normalized size of antiderivative = 0.37, number of steps used = 12, number of rules used = 8, integrand size = 38, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.210, Rules used = {6742, 2117, 14, 2119, 1628, 826, 1166, 205} \begin {gather*} \frac {4 d \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{c^{3/2}}+\frac {4 d \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{c^{3/2}}-\frac {4 d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{c}-\frac {b^2}{a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {\left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{3 a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 14

Int[(u_)*((c_.)*(x_))^(m_.), x_Symbol] :> Int[ExpandIntegrand[(c*x)^m*u, x], x] /; FreeQ[{c, m}, x] && SumQ[u]
&&  !LinearQ[u, x] &&  !MatchQ[u, (a_) + (b_.)*(v_) /; FreeQ[{a, b}, x] && InverseFunctionQ[v]]

Rule 205

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

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

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 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rule 2117

Int[((g_.) + (h_.)*((d_.) + (e_.)*(x_) + (f_.)*Sqrt[(a_) + (c_.)*(x_)^2])^(n_))^(p_.), x_Symbol] :> Dist[1/(2*
e), Subst[Int[((g + h*x^n)^p*(d^2 + a*f^2 - 2*d*x + x^2))/(d - x)^2, x], x, d + e*x + f*Sqrt[a + c*x^2]], x] /
; FreeQ[{a, c, d, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[p]

Rule 2119

Int[((g_.) + (h_.)*(x_))^(m_.)*((e_.)*(x_) + (f_.)*Sqrt[(a_.) + (c_.)*(x_)^2])^(n_.), x_Symbol] :> Dist[1/(2^(
m + 1)*e^(m + 1)), Subst[Int[x^(n - m - 2)*(a*f^2 + x^2)*(-(a*f^2*h) + 2*e*g*x + h*x^2)^m, x], x, e*x + f*Sqrt
[a + c*x^2]], x] /; FreeQ[{a, c, e, f, g, h, n}, x] && EqQ[e^2 - c*f^2, 0] && IntegerQ[m]

Rule 6742

Int[u_, x_Symbol] :> With[{v = ExpandIntegrand[u, x]}, Int[v, x] /; SumQ[v]]

Rubi steps

\begin {align*} \int \frac {(-d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx &=\int \left (\sqrt {a x+\sqrt {b^2+a^2 x^2}}-\frac {2 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x}\right ) \, dx\\ &=-\left ((2 d) \int \frac {\sqrt {a x+\sqrt {b^2+a^2 x^2}}}{d+c x} \, dx\right )+\int \sqrt {a x+\sqrt {b^2+a^2 x^2}} \, dx\\ &=\frac {\operatorname {Subst}\left (\int \frac {b^2+x^2}{x^{3/2}} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-(2 d) \operatorname {Subst}\left (\int \frac {b^2+x^2}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=\frac {\operatorname {Subst}\left (\int \left (\frac {b^2}{x^{3/2}}+\sqrt {x}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{2 a}-(2 d) \operatorname {Subst}\left (\int \left (\frac {1}{c \sqrt {x}}+\frac {2 \left (b^2 c-a d x\right )}{c \sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )}\right ) \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {(4 d) \operatorname {Subst}\left (\int \frac {b^2 c-a d x}{\sqrt {x} \left (-b^2 c+2 a d x+c x^2\right )} \, dx,x,a x+\sqrt {b^2+a^2 x^2}\right )}{c}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}-\frac {(8 d) \operatorname {Subst}\left (\int \frac {b^2 c-a d x^2}{-b^2 c+2 a d x^2+c x^4} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {\left (4 d \left (a d-\sqrt {b^2 c^2+a^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a d-\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}+\frac {\left (4 d \left (a d+\sqrt {b^2 c^2+a^2 d^2}\right )\right ) \operatorname {Subst}\left (\int \frac {1}{a d+\sqrt {b^2 c^2+a^2 d^2}+c x^2} \, dx,x,\sqrt {a x+\sqrt {b^2+a^2 x^2}}\right )}{c}\\ &=-\frac {b^2}{a \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {4 d \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{c}+\frac {\left (a x+\sqrt {b^2+a^2 x^2}\right )^{3/2}}{3 a}+\frac {4 d \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2}}+\frac {4 d \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}} \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2}}\\ \end {align*}

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Mathematica [A]  time = 0.55, size = 388, normalized size = 0.52 \begin {gather*} \frac {-\frac {4 a d \left (a d \left (a d-\sqrt {a^2 d^2+b^2 c^2}\right )+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{c^{3/2} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {4 a d \left (a d \left (\sqrt {a^2 d^2+b^2 c^2}+a d\right )+b^2 c^2\right ) \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{c^{3/2} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}-\frac {4 a d \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{c}-\frac {b^2}{\sqrt {\sqrt {a^2 x^2+b^2}+a x}}+\frac {1}{3} \left (\sqrt {a^2 x^2+b^2}+a x\right )^{3/2}}{a} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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IntegrateAlgebraic [A]  time = 1.95, size = 747, normalized size = 1.00 \begin {gather*} \frac {2 (-6 a d+a c x) \sqrt {b^2+a^2 x^2}+2 \left (-b^2 c-6 a^2 d x+a^2 c x^2\right )}{3 a c \sqrt {a x+\sqrt {b^2+a^2 x^2}}}+\frac {4 a d^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}-\frac {4 b^2 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}-\frac {4 a^2 d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 a d^2 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 b^2 \sqrt {c} d \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{\sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}+\frac {4 a^2 d^3 \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {a x+\sqrt {b^2+a^2 x^2}}}{\sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}}\right )}{c^{3/2} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d+\sqrt {b^2 c^2+a^2 d^2}}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(2*(-6*a*d + a*c*x)*Sqrt[b^2 + a^2*x^2] + 2*(-(b^2*c) - 6*a^2*d*x + a^2*c*x^2))/(3*a*c*Sqrt[a*x + Sqrt[b^2 + a
^2*x^2]]) + (4*a*d^2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(c
^(3/2)*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) - (4*b^2*Sqrt[c]*d*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]
)/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) - (4*a^2
*d^3*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[b^2*
c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (4*a*d^2*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]
)/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]) + (4*b^2*Sqrt[c]*d*ArcTa
n[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]]])/(Sqrt[b^2*c^2 + a^2*d^2]*Sqr
t[a*d + Sqrt[b^2*c^2 + a^2*d^2]]) + (4*a^2*d^3*ArcTan[(Sqrt[c]*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])/Sqrt[a*d + Sqr
t[b^2*c^2 + a^2*d^2]]])/(c^(3/2)*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d + Sqrt[b^2*c^2 + a^2*d^2]])

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fricas [A]  time = 0.50, size = 511, normalized size = 0.68 \begin {gather*} \frac {2 \, {\left (3 \, a c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d + 4 \, c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) - 3 \, a c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d - 4 \, c \sqrt {-\frac {a d^{3} + c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) + 3 \, a c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d + 4 \, c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) - 3 \, a c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}} \log \left (4 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} d - 4 \, c \sqrt {-\frac {a d^{3} - c^{3} \sqrt {\frac {b^{2} c^{2} d^{4} + a^{2} d^{6}}{c^{6}}}}{c^{3}}}\right ) + {\left (2 \, a c x - 6 \, a d - \sqrt {a^{2} x^{2} + b^{2}} c\right )} \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}}\right )}}{3 \, a c} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2/3*(3*a*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d
+ 4*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)) - 3*a*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4
+ a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d - 4*c*sqrt(-(a*d^3 + c^3*sqrt((b^2*c^2*d^4 + a^
2*d^6)/c^6))/c^3)) + 3*a*c*sqrt(-(a*d^3 - c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^
2*x^2 + b^2))*d + 4*c*sqrt(-(a*d^3 - c^3*sqrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)) - 3*a*c*sqrt(-(a*d^3 - c^3*s
qrt((b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)*log(4*sqrt(a*x + sqrt(a^2*x^2 + b^2))*d - 4*c*sqrt(-(a*d^3 - c^3*sqrt((
b^2*c^2*d^4 + a^2*d^6)/c^6))/c^3)) + (2*a*c*x - 6*a*d - sqrt(a^2*x^2 + b^2)*c)*sqrt(a*x + sqrt(a^2*x^2 + b^2))
)/(a*c)

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {\left (c x -d \right ) \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}{c x +d}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [F]  time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int -\frac {\sqrt {a\,x+\sqrt {a^2\,x^2+b^2}}\,\left (d-c\,x\right )}{d+c\,x} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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