3.7.21 \(\int \frac {(1+a x)^2}{(c+d x) \sqrt {1-a^2 x^2}} \, dx\) [621]

Optimal. Leaf size=107 \[ -\frac {\sqrt {1-a^2 x^2}}{d}-\frac {(a c-2 d) \sin ^{-1}(a x)}{d^2}+\frac {(a c-d)^2 \tan ^{-1}\left (\frac {d+a^2 c x}{\sqrt {a^2 c^2-d^2} \sqrt {1-a^2 x^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}} \]

[Out]

-(a*c-2*d)*arcsin(a*x)/d^2+(a*c-d)^2*arctan((a^2*c*x+d)/(a^2*c^2-d^2)^(1/2)/(-a^2*x^2+1)^(1/2))/d^2/(a^2*c^2-d
^2)^(1/2)-(-a^2*x^2+1)^(1/2)/d

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Rubi [A]
time = 0.11, antiderivative size = 107, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 29, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.172, Rules used = {1668, 858, 222, 739, 210} \begin {gather*} \frac {(a c-d)^2 \text {ArcTan}\left (\frac {a^2 c x+d}{\sqrt {1-a^2 x^2} \sqrt {a^2 c^2-d^2}}\right )}{d^2 \sqrt {a^2 c^2-d^2}}-\frac {\sqrt {1-a^2 x^2}}{d}-\frac {\text {ArcSin}(a x) (a c-2 d)}{d^2} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[1 - a^2*x^2]/d) - ((a*c - 2*d)*ArcSin[a*x])/d^2 + ((a*c - d)^2*ArcTan[(d + a^2*c*x)/(Sqrt[a^2*c^2 - d^2
]*Sqrt[1 - a^2*x^2])])/(d^2*Sqrt[a^2*c^2 - d^2])

Rule 210

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(-(Rt[-a, 2]*Rt[-b, 2])^(-1))*ArcTan[Rt[-b, 2]*(x/Rt[-a, 2])
], x] /; FreeQ[{a, b}, x] && PosQ[a/b] && (LtQ[a, 0] || LtQ[b, 0])

Rule 222

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Simp[ArcSin[Rt[-b, 2]*(x/Sqrt[a])]/Rt[-b, 2], x] /; FreeQ[{a, b}
, x] && GtQ[a, 0] && NegQ[b]

Rule 739

Int[1/(((d_) + (e_.)*(x_))*Sqrt[(a_) + (c_.)*(x_)^2]), x_Symbol] :> -Subst[Int[1/(c*d^2 + a*e^2 - x^2), x], x,
(a*e - c*d*x)/Sqrt[a + c*x^2]] /; FreeQ[{a, c, d, e}, x]

Rule 858

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dist[g/e, Int[(d
+ e*x)^(m + 1)*(a + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a,
c, d, e, f, g, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] &&  !IGtQ[m, 0]

Rule 1668

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff
[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x)^(m + q - 1)*((a + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q + 2*p + 1))), x]
+ Dist[1/(c*e^q*(m + q + 2*p + 1)), Int[(d + e*x)^m*(a + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*
f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x)^(q - 2)*(a*e^2*(m + q - 1) - c*d^2*(m + q + 2*p + 1) - 2*c*d*e*(
m + q + p)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, c, d, e, m, p}, x] && PolyQ[Pq
, x] && NeQ[c*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && True) &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[
p] || ILtQ[p + 1/2, 0]))

Rubi steps

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

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Mathematica [B] Leaf count is larger than twice the leaf count of optimal. \(501\) vs. \(2(107)=214\).
time = 1.14, size = 501, normalized size = 4.68 \begin {gather*} -\frac {a d^3 (a c+d) \sqrt {1-a^2 x^2}-a (a c-d) \sqrt {2 a^2 c^2-d^2-2 a c \sqrt {a^2 c^2-d^2}} \left (a^2 c^2-d^2+a c \sqrt {a^2 c^2-d^2}\right ) \tan ^{-1}\left (\frac {d \left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right )}{\sqrt {2 a^2 c^2-d^2-2 a c \sqrt {a^2 c^2-d^2}}}\right )+a (a c-d) \left (-a^2 c^2+d^2+a c \sqrt {a^2 c^2-d^2}\right ) \sqrt {2 a^2 c^2-d^2+2 a c \sqrt {a^2 c^2-d^2}} \tan ^{-1}\left (\frac {d \left (\sqrt {-a^2} x-\sqrt {1-a^2 x^2}\right )}{\sqrt {2 a^2 c^2-d^2+2 a c \sqrt {a^2 c^2-d^2}}}\right )+\sqrt {-a^2} (a c-d) d^2 \sqrt {-a^2 c^2+d^2} \tan ^{-1}\left (\frac {-\sqrt {-a^2} d^2 x \sqrt {1-a^2 x^2}+a^2 \left (c^2-d^2 x^2\right )}{a c \sqrt {-a^2 c^2+d^2}}\right )+\sqrt {-a^2} (a c-2 d) d^2 (a c+d) \log \left (-\sqrt {-a^2} x+\sqrt {1-a^2 x^2}\right )}{a d^4 (a c+d)} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-((a*d^3*(a*c + d)*Sqrt[1 - a^2*x^2] - a*(a*c - d)*Sqrt[2*a^2*c^2 - d^2 - 2*a*c*Sqrt[a^2*c^2 - d^2]]*(a^2*c^2
- d^2 + a*c*Sqrt[a^2*c^2 - d^2])*ArcTan[(d*(Sqrt[-a^2]*x - Sqrt[1 - a^2*x^2]))/Sqrt[2*a^2*c^2 - d^2 - 2*a*c*Sq
rt[a^2*c^2 - d^2]]] + a*(a*c - d)*(-(a^2*c^2) + d^2 + a*c*Sqrt[a^2*c^2 - d^2])*Sqrt[2*a^2*c^2 - d^2 + 2*a*c*Sq
rt[a^2*c^2 - d^2]]*ArcTan[(d*(Sqrt[-a^2]*x - Sqrt[1 - a^2*x^2]))/Sqrt[2*a^2*c^2 - d^2 + 2*a*c*Sqrt[a^2*c^2 - d
^2]]] + Sqrt[-a^2]*(a*c - d)*d^2*Sqrt[-(a^2*c^2) + d^2]*ArcTan[(-(Sqrt[-a^2]*d^2*x*Sqrt[1 - a^2*x^2]) + a^2*(c
^2 - d^2*x^2))/(a*c*Sqrt[-(a^2*c^2) + d^2])] + Sqrt[-a^2]*(a*c - 2*d)*d^2*(a*c + d)*Log[-(Sqrt[-a^2]*x) + Sqrt
[1 - a^2*x^2]])/(a*d^4*(a*c + d)))

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(241\) vs. \(2(99)=198\).
time = 0.10, size = 242, normalized size = 2.26

method	result	size
default	\(-\frac {a \left (\frac {d \sqrt {-a^{2} x^{2}+1}}{a}+\frac {a c \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}-\frac {2 d \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{\sqrt {a^{2}}}\right )}{d^{2}}-\frac {\left (a^{2} c^{2}-2 a c d +d^{2}\right ) \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d^{3} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\)	\(242\)
risch	\(\frac {a^{2} x^{2}-1}{d \sqrt {-a^{2} x^{2}+1}}-\frac {a^{2} \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right ) c}{d^{2} \sqrt {a^{2}}}+\frac {2 a \arctan \left (\frac {\sqrt {a^{2}}\, x}{\sqrt {-a^{2} x^{2}+1}}\right )}{d \sqrt {a^{2}}}-\frac {\ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right ) a^{2} c^{2}}{d^{3} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}+\frac {2 \ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right ) a c}{d^{2} \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}-\frac {\ln \left (\frac {-\frac {2 \left (a^{2} c^{2}-d^{2}\right )}{d^{2}}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}\, \sqrt {-a^{2} \left (x +\frac {c}{d}\right )^{2}+\frac {2 a^{2} c \left (x +\frac {c}{d}\right )}{d}-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{d \sqrt {-\frac {a^{2} c^{2}-d^{2}}{d^{2}}}}\)	\(532\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-a/d^2*(1/a*d*(-a^2*x^2+1)^(1/2)+a*c/(a^2)^(1/2)*arctan((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2))-2*d/(a^2)^(1/2)*arct
an((a^2)^(1/2)*x/(-a^2*x^2+1)^(1/2)))-(a^2*c^2-2*a*c*d+d^2)/d^3/(-(a^2*c^2-d^2)/d^2)^(1/2)*ln((-2*(a^2*c^2-d^2
)/d^2+2*a^2*c/d*(x+c/d)+2*(-(a^2*c^2-d^2)/d^2)^(1/2)*(-a^2*(x+c/d)^2+2*a^2*c/d*(x+c/d)-(a^2*c^2-d^2)/d^2)^(1/2
))/(x+c/d))

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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((a*x+1)^2/(d*x+c)/(-a^2*x^2+1)^(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(d-a*c>0)', see `assume?` for m
ore details)

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Fricas [A]
time = 4.41, size = 318, normalized size = 2.97 \begin {gather*} \left [-\frac {{\left (a c - d\right )} \sqrt {-\frac {a c - d}{a c + d}} \log \left (\frac {a^{2} c d x + d^{2} - {\left (a^{2} c^{2} - d^{2}\right )} \sqrt {-a^{2} x^{2} + 1} - {\left (a c d + d^{2} + {\left (a^{3} c^{2} + a^{2} c d\right )} x + \sqrt {-a^{2} x^{2} + 1} {\left (a c d + d^{2}\right )}\right )} \sqrt {-\frac {a c - d}{a c + d}}}{d x + c}\right ) - 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) + \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}, \frac {2 \, {\left (a c - d\right )} \sqrt {\frac {a c - d}{a c + d}} \arctan \left (\frac {{\left (d x - \sqrt {-a^{2} x^{2} + 1} c + c\right )} \sqrt {\frac {a c - d}{a c + d}}}{{\left (a c - d\right )} x}\right ) + 2 \, {\left (a c - 2 \, d\right )} \arctan \left (\frac {\sqrt {-a^{2} x^{2} + 1} - 1}{a x}\right ) - \sqrt {-a^{2} x^{2} + 1} d}{d^{2}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-((a*c - d)*sqrt(-(a*c - d)/(a*c + d))*log((a^2*c*d*x + d^2 - (a^2*c^2 - d^2)*sqrt(-a^2*x^2 + 1) - (a*c*d + d
^2 + (a^3*c^2 + a^2*c*d)*x + sqrt(-a^2*x^2 + 1)*(a*c*d + d^2))*sqrt(-(a*c - d)/(a*c + d)))/(d*x + c)) - 2*(a*c
- 2*d)*arctan((sqrt(-a^2*x^2 + 1) - 1)/(a*x)) + sqrt(-a^2*x^2 + 1)*d)/d^2, (2*(a*c - d)*sqrt((a*c - d)/(a*c +
d))*arctan((d*x - sqrt(-a^2*x^2 + 1)*c + c)*sqrt((a*c - d)/(a*c + d))/((a*c - d)*x)) + 2*(a*c - 2*d)*arctan((
sqrt(-a^2*x^2 + 1) - 1)/(a*x)) - sqrt(-a^2*x^2 + 1)*d)/d^2]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]
time = 3.90, size = 131, normalized size = 1.22 \begin {gather*} -\frac {{\left (a^{2} c - 2 \, a d\right )} \arcsin \left (a x\right ) \mathrm {sgn}\left (a\right )}{d^{2} {\left | a \right |}} - \frac {\sqrt {-a^{2} x^{2} + 1}}{d} - \frac {2 \, {\left (a^{3} c^{2} - 2 \, a^{2} c d + a d^{2}\right )} \arctan \left (\frac {d + \frac {{\left (\sqrt {-a^{2} x^{2} + 1} {\left | a \right |} + a\right )} c}{a x}}{\sqrt {a^{2} c^{2} - d^{2}}}\right )}{\sqrt {a^{2} c^{2} - d^{2}} d^{2} {\left | a \right |}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(a^2*c - 2*a*d)*arcsin(a*x)*sgn(a)/(d^2*abs(a)) - sqrt(-a^2*x^2 + 1)/d - 2*(a^3*c^2 - 2*a^2*c*d + a*d^2)*arct
an((d + (sqrt(-a^2*x^2 + 1)*abs(a) + a)*c/(a*x))/sqrt(a^2*c^2 - d^2))/(sqrt(a^2*c^2 - d^2)*d^2*abs(a))

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Mupad [B]
time = 0.12, size = 148, normalized size = 1.38 \begin {gather*} -\frac {\sqrt {1-a^2\,x^2}}{d}-\frac {\mathrm {asinh}\left (x\,\sqrt {-a^2}\right )\,\left (2\,a\,\sqrt {-a^2}-\frac {a^2\,c\,\sqrt {-a^2}}{d}\right )}{a^2\,d}-\frac {\left (\ln \left (\sqrt {1-\frac {a^2\,c^2}{d^2}}\,\sqrt {1-a^2\,x^2}+\frac {a^2\,c\,x}{d}+1\right )-\ln \left (c+d\,x\right )\right )\,\left (a^2\,c^2-2\,a\,c\,d+d^2\right )}{d^3\,\sqrt {1-\frac {a^2\,c^2}{d^2}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

- (1 - a^2*x^2)^(1/2)/d - (asinh(x*(-a^2)^(1/2))*(2*a*(-a^2)^(1/2) - (a^2*c*(-a^2)^(1/2))/d))/(a^2*d) - ((log(
(1 - (a^2*c^2)/d^2)^(1/2)*(1 - a^2*x^2)^(1/2) + (a^2*c*x)/d + 1) - log(c + d*x))*(d^2 + a^2*c^2 - 2*a*c*d))/(d
^3*(1 - (a^2*c^2)/d^2)^(1/2))

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