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

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Rubi [A]  time = 0.06, antiderivative size = 78, normalized size of antiderivative = 1.00, 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 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]

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 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 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])

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*}

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Mathematica [A]  time = 0.07, size = 73, normalized size = 0.94 \[ \frac {-\frac {2 \left (a^3-3 a c\right ) \tan ^{-1}\left (\frac {a+b x}{\sqrt {c}}\right )}{\sqrt {c}}+\left (3 a^2-c\right ) \log \left (a^2+2 a b x+b^2 x^2+c\right )+b x (b x-4 a)}{2 b^4} \]

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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fricas [A]  time = 0.87, size = 198, normalized size = 2.54 \[ \left [\frac {b^{2} c x^{2} - 4 \, a b c x + {\left (a^{3} - 3 \, a c\right )} \sqrt {-c} \log \left (\frac {b^{2} x^{2} + 2 \, a b x + a^{2} - 2 \, {\left (b x + a\right )} \sqrt {-c} - c}{b^{2} x^{2} + 2 \, a b x + a^{2} + c}\right ) + {\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}, \frac {b^{2} c x^{2} - 4 \, a b c x - 2 \, {\left (a^{3} - 3 \, a c\right )} \sqrt {c} \arctan \left (\frac {b x + a}{\sqrt {c}}\right ) + {\left (3 \, a^{2} c - c^{2}\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4} c}\right ] \]

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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giac [A]  time = 0.34, size = 77, normalized size = 0.99 \[ \frac {{\left (3 \, a^{2} - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4}} - \frac {{\left (a^{3} - 3 \, a c\right )} \arctan \left (\frac {b x + a}{\sqrt {c}}\right )}{b^{4} \sqrt {c}} + \frac {b^{2} x^{2} - 4 \, a b x}{2 \, b^{4}} \]

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

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maple [A]  time = 0.01, size = 127, normalized size = 1.63 \[ \frac {x^{2}}{2 b^{2}}-\frac {a^{3} \arctan \left (\frac {2 b^{2} x +2 b a}{2 b \sqrt {c}}\right )}{b^{4} \sqrt {c}}+\frac {3 a^{2} \ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b^{4}}-\frac {2 a x}{b^{3}}+\frac {3 a \sqrt {c}\, \arctan \left (\frac {2 b^{2} x +2 b a}{2 b \sqrt {c}}\right )}{b^{4}}-\frac {c \ln \left (b^{2} x^{2}+2 a b x +a^{2}+c \right )}{2 b^{4}} \]

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 [A]  time = 1.46, size = 81, normalized size = 1.04 \[ \frac {b x^{2} - 4 \, a x}{2 \, b^{3}} + \frac {{\left (3 \, a^{2} - c\right )} \log \left (b^{2} x^{2} + 2 \, a b x + a^{2} + c\right )}{2 \, b^{4}} - \frac {{\left (a^{3} - 3 \, a c\right )} \arctan \left (\frac {b^{2} x + a b}{b \sqrt {c}}\right )}{b^{4} \sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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mupad [B]  time = 2.27, size = 87, normalized size = 1.12 \[ \frac {x^2}{2\,b^2}-\frac {2\,a\,x}{b^3}-\frac {\ln \left (a^2+2\,a\,b\,x+b^2\,x^2+c\right )\,\left (4\,b^4\,c^2-12\,a^2\,b^4\,c\right )}{8\,b^8\,c}+\frac {a\,\mathrm {atan}\left (\frac {a+b\,x}{\sqrt {c}}\right )\,\left (3\,c-a^2\right )}{b^4\,\sqrt {c}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^3/(c + (a + b*x)^2),x)

[Out]

x^2/(2*b^2) - (2*a*x)/b^3 - (log(c + a^2 + b^2*x^2 + 2*a*b*x)*(4*b^4*c^2 - 12*a^2*b^4*c))/(8*b^8*c) + (a*atan(
(a + b*x)/c^(1/2))*(3*c - a^2))/(b^4*c^(1/2))

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sympy [B]  time = 0.69, size = 209, normalized size = 2.68 \[ - \frac {2 a x}{b^{3}} + \left (- \frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) \log {\left (x + \frac {a^{4} - 2 b^{4} c \left (- \frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \left (\frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) \log {\left (x + \frac {a^{4} - 2 b^{4} c \left (\frac {a \sqrt {- c} \left (a^{2} - 3 c\right )}{2 b^{4} c} + \frac {3 a^{2} - c}{2 b^{4}}\right ) - c^{2}}{a^{3} b - 3 a b c} \right )} + \frac {x^{2}}{2 b^{2}} \]

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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