3.28.64 \(\int \frac {1}{(d+c x)^2 \sqrt {a x+\sqrt {b^2+a^2 x^2}}} \, dx\)

Optimal. Leaf size=261 \[ -\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {a \tan ^{-1}\left (\frac {\sqrt {c} \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{\sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}\right )}{\sqrt {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}-\frac {1}{c \sqrt {\sqrt {a^2 x^2+b^2}+a x} (c x+d)} \]

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Rubi [A]  time = 0.61, antiderivative size = 310, normalized size of antiderivative = 1.19, number of steps used = 7, number of rules used = 6, integrand size = 31, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.194, Rules used = {2119, 1648, 12, 707, 1093, 205} \begin {gather*} -\frac {a \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 {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {a d-\sqrt {a^2 d^2+b^2 c^2}}}+\frac {a \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 {c} \sqrt {a^2 d^2+b^2 c^2} \sqrt {\sqrt {a^2 d^2+b^2 c^2}+a d}}+\frac {2 a \sqrt {\sqrt {a^2 x^2+b^2}+a x}}{c \left (-c \left (\sqrt {a^2 x^2+b^2}+a x\right )^2-2 a d \left (\sqrt {a^2 x^2+b^2}+a x\right )+b^2 c\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[1/((d + c*x)^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

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

Rule 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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 707

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2*e, Subst[Int[1/(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}, x] && NeQ[
b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && NeQ[2*c*d - b*e, 0]

Rule 1093

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

Rule 1648

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = Polynomi
alQuotient[Pq, a + b*x + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + b*x + c*x^2, x], x, 0], g = Coeff[Po
lynomialRemainder[Pq, a + b*x + c*x^2, x], x, 1]}, Simp[((d + e*x)^(m + 1)*(a + b*x + c*x^2)^(p + 1)*(f*(b*c*d
- b^2*e + 2*a*c*e) - a*g*(2*c*d - b*e) + c*(f*(2*c*d - b*e) - g*(b*d - 2*a*e))*x))/((p + 1)*(b^2 - 4*a*c)*(c*
d^2 - b*d*e + a*e^2)), x] + Dist[1/((p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^m*(a + b*x +
c*x^2)^(p + 1)*ExpandToSum[(p + 1)*(b^2 - 4*a*c)*(c*d^2 - b*d*e + a*e^2)*Q + f*(b*c*d*e*(2*p - m + 2) + b^2*e
^2*(p + m + 2) - 2*c^2*d^2*(2*p + 3) - 2*a*c*e^2*(m + 2*p + 3)) - g*(a*e*(b*e - 2*c*d*m + b*e*m) - b*d*(3*c*d
- b*e + 2*c*d*p - b*e*p)) + c*e*(g*(b*d - 2*a*e) - f*(2*c*d - b*e))*(m + 2*p + 4)*x, x], x], x]] /; FreeQ[{a,
b, c, d, e, m}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] &&  !
(IGtQ[m, 0] && RationalQ[a, b, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

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]

Rubi steps

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

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Mathematica [A]  time = 0.89, size = 273, normalized size = 1.05 \begin {gather*} -\frac {a \left (-\frac {\sqrt {-\sqrt {a^2 d^2+b^2 c^2}-a d} \tan ^{-1}\left (\frac {b \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 )}{b \sqrt {a^2 d^2+b^2 c^2}}+\frac {\sqrt {\sqrt {a^2 d^2+b^2 c^2}-a d} \tan ^{-1}\left (\frac {b \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 )}{b \sqrt {a^2 d^2+b^2 c^2}}+\frac {\sqrt {c}}{\sqrt {\sqrt {a^2 x^2+b^2}+a x} (a c x+a d)}\right )}{c^{3/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[1/((d + c*x)^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

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

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IntegrateAlgebraic [A]  time = 1.04, size = 261, normalized size = 1.00 \begin {gather*} -\frac {1}{c (d+c x) \sqrt {a x+\sqrt {b^2+a^2 x^2}}}-\frac {a \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 {c} \sqrt {b^2 c^2+a^2 d^2} \sqrt {a d-\sqrt {b^2 c^2+a^2 d^2}}}+\frac {a \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 {c} \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[1/((d + c*x)^2*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]]),x]

[Out]

-(1/(c*(d + c*x)*Sqrt[a*x + Sqrt[b^2 + a^2*x^2]])) - (a*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[c]*Sqrt[b^2*c^2 + a^2*d^2]*Sqrt[a*d - Sqrt[b^2*c^2 + a^2*d^2]]) + (a*Ar
cTan[(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[c]*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 [B]  time = 0.54, size = 1174, normalized size = 4.50 \begin {gather*} \frac {{\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} - {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d + {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} + {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) - {\left (b^{2} c^{2} x + b^{2} c d\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}} \log \left (\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} a^{2} - {\left (a b^{2} c^{2} + a^{3} d^{2} + {\left (b^{4} c^{5} d + a^{2} b^{2} c^{3} d^{3}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}\right )} \sqrt {\frac {a^{3} d - {\left (b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}\right )} \sqrt {\frac {a^{4}}{b^{6} c^{8} + a^{2} b^{4} c^{6} d^{2}}}}{b^{4} c^{5} + a^{2} b^{2} c^{3} d^{2}}}\right ) + 2 \, \sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (a x - \sqrt {a^{2} x^{2} + b^{2}}\right )}}{2 \, {\left (b^{2} c^{2} x + b^{2} c d\right )}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*((b^2*c^2*x + b^2*c*d)*sqrt((a^3*d + (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b
^4*c^5 + a^2*b^2*c^3*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 + (a*b^2*c^2 + a^3*d^2 - (b^4*c^5*d + a^2*b
^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*d + (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8
+ a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))) - (b^2*c^2*x + b^2*c*d)*sqrt((a^3*d + (b^4*c^5 + a^2*b^2*c
^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2)
)*a^2 - (a*b^2*c^2 + a^3*d^2 - (b^4*c^5*d + a^2*b^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*
d + (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))) + (b^2*c^
2*x + b^2*c*d)*sqrt((a^3*d - (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2
*b^2*c^3*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 + (a*b^2*c^2 + a^3*d^2 + (b^4*c^5*d + a^2*b^2*c^3*d^3)*
sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*d - (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c
^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))) - (b^2*c^2*x + b^2*c*d)*sqrt((a^3*d - (b^4*c^5 + a^2*b^2*c^3*d^2)*sqrt
(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))*log(sqrt(a*x + sqrt(a^2*x^2 + b^2))*a^2 - (a*b
^2*c^2 + a^3*d^2 + (b^4*c^5*d + a^2*b^2*c^3*d^3)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))*sqrt((a^3*d - (b^4*c^5
+ a^2*b^2*c^3*d^2)*sqrt(a^4/(b^6*c^8 + a^2*b^4*c^6*d^2)))/(b^4*c^5 + a^2*b^2*c^3*d^2))) + 2*sqrt(a*x + sqrt(a
^2*x^2 + b^2))*(a*x - sqrt(a^2*x^2 + b^2)))/(b^2*c^2*x + b^2*c*d)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maple [F]  time = 180.00, size = 0, normalized size = 0.00 \[\int \frac {1}{\left (c x +d \right )^{2} \sqrt {a x +\sqrt {a^{2} x^{2}+b^{2}}}}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} {\left (c x + d\right )}^{2}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {1}{\sqrt {a x + \sqrt {a^{2} x^{2} + b^{2}}} \left (c x + d\right )^{2}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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