3.79 \(\int \frac{x^2}{c+(a+b x)^2} \, dx\)

Optimal. Leaf size=50 \[ \frac{\left (a^2-c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^3 \sqrt{c}}-\frac{a \log \left ((a+b x)^2+c\right )}{b^3}+\frac{x}{b^2} \]

[Out]

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

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Rubi [A]  time = 0.0387091, antiderivative size = 50, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 15, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {371, 702, 635, 203, 260} \[ \frac{\left (a^2-c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^3 \sqrt{c}}-\frac{a \log \left ((a+b x)^2+c\right )}{b^3}+\frac{x}{b^2} \]

Antiderivative was successfully verified.

[In]

Int[x^2/(c + (a + b*x)^2),x]

[Out]

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

Rule 371

Int[((a_) + (b_.)*(v_)^(n_))^(p_.)*(x_)^(m_.), x_Symbol] :> With[{c = Coefficient[v, x, 0], d = Coefficient[v,
x, 1]}, Dist[1/d^(m + 1), Subst[Int[SimplifyIntegrand[(x - c)^m*(a + b*x^n)^p, x], x], x, v], x] /; NeQ[c, 0]
] /; FreeQ[{a, b, n, p}, x] && LinearQ[v, x] && IntegerQ[m]

Rule 702

Int[((d_) + (e_.)*(x_))^(m_)/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[PolynomialDivide[(d + e*x)^m, a + c*x^2,
x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[m, 1] && (NeQ[d, 0] || GtQ[m, 2])

Rule 635

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

Rule 203

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

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.0276808, size = 54, normalized size = 1.08 \[ \frac{-a \log \left (a^2+2 a b x+b^2 x^2+c\right )+\frac{\left (a^2-c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}+b x}{b^3} \]

Antiderivative was successfully verified.

[In]

Integrate[x^2/(c + (a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.002, size = 89, normalized size = 1.8 \begin{align*}{\frac{x}{{b}^{2}}}-{\frac{a\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) }{{b}^{3}}}+{\frac{{a}^{2}}{{b}^{3}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}-{\frac{1}{{b}^{3}}\sqrt{c}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.77995, size = 377, normalized size = 7.54 \begin{align*} \left [\frac{2 \, b c x - 2 \, a c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) +{\left (a^{2} - c\right )} \sqrt{-c} \log \left (\frac{b^{2} x^{2} + 2 \, a b x + a^{2} + 2 \,{\left (b x + a\right )} \sqrt{-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right )}{2 \, b^{3} c}, \frac{b c x - a c \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right ) +{\left (a^{2} - c\right )} \sqrt{c} \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b^{3} c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Sympy [B]  time = 0.577375, size = 153, normalized size = 3.06 \begin{align*} \left (- \frac{a}{b^{3}} - \frac{\sqrt{- c} \left (a^{2} - c\right )}{2 b^{3} c}\right ) \log{\left (x + \frac{a^{3} + a c + 2 b^{3} c \left (- \frac{a}{b^{3}} - \frac{\sqrt{- c} \left (a^{2} - c\right )}{2 b^{3} c}\right )}{a^{2} b - b c} \right )} + \left (- \frac{a}{b^{3}} + \frac{\sqrt{- c} \left (a^{2} - c\right )}{2 b^{3} c}\right ) \log{\left (x + \frac{a^{3} + a c + 2 b^{3} c \left (- \frac{a}{b^{3}} + \frac{\sqrt{- c} \left (a^{2} - c\right )}{2 b^{3} c}\right )}{a^{2} b - b c} \right )} + \frac{x}{b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(-a/b**3 - sqrt(-c)*(a**2 - c)/(2*b**3*c))*log(x + (a**3 + a*c + 2*b**3*c*(-a/b**3 - sqrt(-c)*(a**2 - c)/(2*b*
*3*c)))/(a**2*b - b*c)) + (-a/b**3 + sqrt(-c)*(a**2 - c)/(2*b**3*c))*log(x + (a**3 + a*c + 2*b**3*c*(-a/b**3 +
sqrt(-c)*(a**2 - c)/(2*b**3*c)))/(a**2*b - b*c)) + x/b**2

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Giac [A]  time = 1.17294, size = 73, normalized size = 1.46 \begin{align*} \frac{x}{b^{2}} - \frac{a \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{b^{3}} + \frac{{\left (a^{2} - c\right )} \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b^{3} \sqrt{c}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

x/b^2 - a*log(b^2*x^2 + 2*a*b*x + a^2 + c)/b^3 + (a^2 - c)*arctan((b*x + a)/sqrt(c))/(b^3*sqrt(c))