3.78 \(\int \frac{x^3}{c+(a+b x)^2} \, dx\)

Optimal. Leaf size=78 \[ \frac{\left (3 a^2-c\right ) \log \left ((a+b x)^2+c\right )}{2 b^4}-\frac{a \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^4 \sqrt{c}}+\frac{(a+b x)^2}{2 b^4}-\frac{3 a x}{b^3} \]

[Out]

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

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Rubi [A]  time = 0.0630097, antiderivative size = 78, 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} \[ \frac{\left (3 a^2-c\right ) \log \left ((a+b x)^2+c\right )}{2 b^4}-\frac{a \left (a^2-3 c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{b^4 \sqrt{c}}+\frac{(a+b x)^2}{2 b^4}-\frac{3 a x}{b^3} \]

Antiderivative was successfully verified.

[In]

Int[x^3/(c + (a + b*x)^2),x]

[Out]

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

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

Mathematica [A]  time = 0.0496587, size = 73, normalized size = 0.94 \[ \frac{\left (3 a^2-c\right ) \log \left (a^2+2 a b x+b^2 x^2+c\right )-\frac{2 \left (a^3-3 a c\right ) \tan ^{-1}\left (\frac{a+b x}{\sqrt{c}}\right )}{\sqrt{c}}+b x (b x-4 a)}{2 b^4} \]

Antiderivative was successfully verified.

[In]

Integrate[x^3/(c + (a + b*x)^2),x]

[Out]

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

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Maple [A]  time = 0.005, size = 127, normalized size = 1.6 \begin{align*}{\frac{{x}^{2}}{2\,{b}^{2}}}-2\,{\frac{ax}{{b}^{3}}}+{\frac{3\,\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ){a}^{2}}{2\,{b}^{4}}}-{\frac{\ln \left ({b}^{2}{x}^{2}+2\,abx+{a}^{2}+c \right ) c}{2\,{b}^{4}}}-{\frac{{a}^{3}}{{b}^{4}}\arctan \left ({\frac{2\,{b}^{2}x+2\,ab}{2\,b}{\frac{1}{\sqrt{c}}}} \right ){\frac{1}{\sqrt{c}}}}+3\,{\frac{\sqrt{c}a}{{b}^{4}}\arctan \left ( 1/2\,{\frac{2\,{b}^{2}x+2\,ab}{b\sqrt{c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c+(b*x+a)^2),x)

[Out]

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

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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^3/(c+(b*x+a)^2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.8337, size = 466, normalized size = 5.97 \begin{align*} \left [\frac{b^{2} c x^{2} - 4 \, a b c x +{\left (a^{3} - 3 \, a 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 ) +{\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}, \frac{b^{2} c x^{2} - 4 \, a b c x - 2 \,{\left (a^{3} - 3 \, a c\right )} \sqrt{c} \arctan \left (\frac{b x + a}{\sqrt{c}}\right ) +{\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c+(b*x+a)^2),x, algorithm="fricas")

[Out]

[1/2*(b^2*c*x^2 - 4*a*b*c*x + (a^3 - 3*a*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)) + (3*a^2*c - c^2)*log(b^2*x^2 + 2*a*b*x + a^2 + c))/(b^4*c), 1/2*(b^2*c*x^2 - 4
*a*b*c*x - 2*(a^3 - 3*a*c)*sqrt(c)*arctan((b*x + a)/sqrt(c)) + (3*a^2*c - c^2)*log(b^2*x^2 + 2*a*b*x + a^2 + c
))/(b^4*c)]

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Sympy [B]  time = 0.760163, size = 209, normalized size = 2.68 \begin{align*} - \frac{2 a x}{b^{3}} + \left (- \frac{a \sqrt{- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac{3 a^{2} - c}{2 b^{4}}\right ) \log{\left (x + \frac{a^{4} - 2 b^{4} c \left (- \frac{a \sqrt{- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac{3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \left (\frac{a \sqrt{- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac{3 a^{2} - c}{2 b^{4}}\right ) \log{\left (x + \frac{a^{4} - 2 b^{4} c \left (\frac{a \sqrt{- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac{3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \frac{x^{2}}{2 b^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**3/(c+(b*x+a)**2),x)

[Out]

-2*a*x/b**3 + (-a*sqrt(-c)*(a**2 - 3*c)/(2*b**4*c) + (3*a**2 - c)/(2*b**4))*log(x + (a**4 - 2*b**4*c*(-a*sqrt(
-c)*(a**2 - 3*c)/(2*b**4*c) + (3*a**2 - c)/(2*b**4)) - c**2)/(a**3*b - 3*a*b*c)) + (a*sqrt(-c)*(a**2 - 3*c)/(2
*b**4*c) + (3*a**2 - c)/(2*b**4))*log(x + (a**4 - 2*b**4*c*(a*sqrt(-c)*(a**2 - 3*c)/(2*b**4*c) + (3*a**2 - c)/
(2*b**4)) - c**2)/(a**3*b - 3*a*b*c)) + x**2/(2*b**2)

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Giac [A]  time = 1.11854, size = 104, normalized size = 1.33 \begin{align*} \frac{{\left (3 \, a^{2} - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4}} - \frac{{\left (a^{3} - 3 \, a c\right )} \arctan \left (\frac{b x + a}{\sqrt{c}}\right )}{b^{4} \sqrt{c}} + \frac{b^{2} x^{2} - 4 \, a b x}{2 \, b^{4}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^3/(c+(b*x+a)^2),x, algorithm="giac")

[Out]

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