3.6.6 \(\int \frac {(d+e x)^4}{(a+c x^2)^2} \, dx\) [506]

Optimal. Leaf size=149 \[ -\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \]

[Out]

-3/2*e^2*(-a*e^2+c*d^2)*x/a/c^2-1/2*d*e^3*x^2/a/c-1/2*(-c*d*x+a*e)*(e*x+d)^3/a/c/(c*x^2+a)+1/2*(-3*a^2*e^4+6*a
*c*d^2*e^2+c^2*d^4)*arctan(x*c^(1/2)/a^(1/2))/a^(3/2)/c^(5/2)+2*d*e^3*ln(c*x^2+a)/c^2

________________________________________________________________________________________

Rubi [A]
time = 0.08, antiderivative size = 149, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 17, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.294, Rules used = {753, 815, 649, 211, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )}{2 a^{3/2} c^{5/2}}-\frac {3 e^2 x \left (c d^2-a e^2\right )}{2 a c^2}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(d+e x)^3 (a e-c d x)}{2 a c \left (a+c x^2\right )} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^4/(a + c*x^2)^2,x]

[Out]

(-3*e^2*(c*d^2 - a*e^2)*x)/(2*a*c^2) - (d*e^3*x^2)/(2*a*c) - ((a*e - c*d*x)*(d + e*x)^3)/(2*a*c*(a + c*x^2)) +
((c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*
x^2])/c^2

Rule 211

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

Rule 266

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 649

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 753

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[(d + e*x)^(m - 1)*(a*e - c*d*x)*((a
+ c*x^2)^(p + 1)/(2*a*c*(p + 1))), x] + Dist[1/((p + 1)*(-2*a*c)), Int[(d + e*x)^(m - 2)*Simp[a*e^2*(m - 1) -
c*d^2*(2*p + 3) - d*c*e*(m + 2*p + 2)*x, x]*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^
2 + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && IntQuadraticQ[a, 0, c, d, e, m, p, x]

Rule 815

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
d + e*x)^m*((f + g*x)/(a + c*x^2)), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

Rubi steps

\begin {align*} \int \frac {(d+e x)^4}{\left (a+c x^2\right )^2} \, dx &=-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {(d+e x)^2 \left (c d^2+3 a e^2-2 c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \left (-3 e^2 \left (d^2-\frac {a e^2}{c}\right )-2 d e^3 x+\frac {c^2 d^4+6 a c d^2 e^2-3 a^2 e^4+8 a c d e^3 x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\int \frac {c^2 d^4+6 a c d^2 e^2-3 a^2 e^4+8 a c d e^3 x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (4 d e^3\right ) \int \frac {x}{a+c x^2} \, dx}{c}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \int \frac {1}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac {3 e^2 \left (c d^2-a e^2\right ) x}{2 a c^2}-\frac {d e^3 x^2}{2 a c}-\frac {(a e-c d x) (d+e x)^3}{2 a c \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2}\\ \end {align*}

________________________________________________________________________________________

Mathematica [A]
time = 0.06, size = 137, normalized size = 0.92 \begin {gather*} \frac {e^4 x}{c^2}+\frac {c^2 d^4 x+a^2 e^3 (4 d+e x)-2 a c d^2 e (2 d+3 e x)}{2 a c^2 \left (a+c x^2\right )}+\frac {\left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{2 a^{3/2} c^{5/2}}+\frac {2 d e^3 \log \left (a+c x^2\right )}{c^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^4/(a + c*x^2)^2,x]

[Out]

(e^4*x)/c^2 + (c^2*d^4*x + a^2*e^3*(4*d + e*x) - 2*a*c*d^2*e*(2*d + 3*e*x))/(2*a*c^2*(a + c*x^2)) + ((c^2*d^4
+ 6*a*c*d^2*e^2 - 3*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(5/2)) + (2*d*e^3*Log[a + c*x^2])/c^2

________________________________________________________________________________________

Maple [A]
time = 0.48, size = 137, normalized size = 0.92

method	result	size
default	\(\frac {e^{4} x}{c^{2}}-\frac {\frac {-\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}-2 d e \left (e^{2} a -c \,d^{2}\right )}{c \,x^{2}+a}+\frac {-4 a d \,e^{3} \ln \left (c \,x^{2}+a \right )+\frac {\left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{2 a}}{c^{2}}\)	\(137\)
risch	\(\frac {e^{4} x}{c^{2}}+\frac {\frac {\left (a^{2} e^{4}-6 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x}{2 a}+2 d e \left (e^{2} a -c \,d^{2}\right )}{c^{2} \left (c \,x^{2}+a \right )}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}+\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a -\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}+\frac {2 \ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) d \,e^{3}}{c^{2}}-\frac {\ln \left (-3 e^{4} a^{3}+6 d^{2} e^{2} a^{2} c +d^{4} c^{2} a +\sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}\, x \right ) \sqrt {-a c \left (3 a^{2} e^{4}-6 a c \,d^{2} e^{2}-c^{2} d^{4}\right )^{2}}}{4 c^{3} a^{2}}\)	\(445\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4/(c*x^2+a)^2,x,method=_RETURNVERBOSE)

[Out]

e^4*x/c^2-1/c^2*((-1/2*(a^2*e^4-6*a*c*d^2*e^2+c^2*d^4)/a*x-2*d*e*(a*e^2-c*d^2))/(c*x^2+a)+1/2/a*(-4*a*d*e^3*ln
(c*x^2+a)+(3*a^2*e^4-6*a*c*d^2*e^2-c^2*d^4)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2))))

________________________________________________________________________________________

Maxima [A]
time = 0.55, size = 134, normalized size = 0.90 \begin {gather*} \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (a c^{3} x^{2} + a^{2} c^{2}\right )}} + \frac {x e^{4}}{c^{2}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

2*d*e^3*log(c*x^2 + a)/c^2 - 1/2*(4*a*c*d^3*e - 4*a^2*d*e^3 - (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*x)/(a*c^3*x^
2 + a^2*c^2) + x*e^4/c^2 + 1/2*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2)

________________________________________________________________________________________

Fricas [A]
time = 1.38, size = 414, normalized size = 2.78 \begin {gather*} \left [\frac {2 \, a c^{3} d^{4} x - 12 \, a^{2} c^{2} d^{2} x e^{2} - 8 \, a^{2} c^{2} d^{3} e + 8 \, a^{3} c d e^{3} + 8 \, {\left (a^{2} c^{2} d x^{2} + a^{3} c d\right )} e^{3} \log \left (c x^{2} + a\right ) + {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} - 3 \, {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 6 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (2 \, a^{2} c^{2} x^{3} + 3 \, a^{3} c x\right )} e^{4}}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {a c^{3} d^{4} x - 6 \, a^{2} c^{2} d^{2} x e^{2} - 4 \, a^{2} c^{2} d^{3} e + 4 \, a^{3} c d e^{3} + 4 \, {\left (a^{2} c^{2} d x^{2} + a^{3} c d\right )} e^{3} \log \left (c x^{2} + a\right ) + {\left (c^{3} d^{4} x^{2} + a c^{2} d^{4} - 3 \, {\left (a^{2} c x^{2} + a^{3}\right )} e^{4} + 6 \, {\left (a c^{2} d^{2} x^{2} + a^{2} c d^{2}\right )} e^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (2 \, a^{2} c^{2} x^{3} + 3 \, a^{3} c x\right )} e^{4}}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*a*c^3*d^4*x - 12*a^2*c^2*d^2*x*e^2 - 8*a^2*c^2*d^3*e + 8*a^3*c*d*e^3 + 8*(a^2*c^2*d*x^2 + a^3*c*d)*e^3
*log(c*x^2 + a) + (c^3*d^4*x^2 + a*c^2*d^4 - 3*(a^2*c*x^2 + a^3)*e^4 + 6*(a*c^2*d^2*x^2 + a^2*c*d^2)*e^2)*sqrt
(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(2*a^2*c^2*x^3 + 3*a^3*c*x)*e^4)/(a^2*c^4*x^2 + a^3*c
^3), 1/2*(a*c^3*d^4*x - 6*a^2*c^2*d^2*x*e^2 - 4*a^2*c^2*d^3*e + 4*a^3*c*d*e^3 + 4*(a^2*c^2*d*x^2 + a^3*c*d)*e^
3*log(c*x^2 + a) + (c^3*d^4*x^2 + a*c^2*d^4 - 3*(a^2*c*x^2 + a^3)*e^4 + 6*(a*c^2*d^2*x^2 + a^2*c*d^2)*e^2)*sqr
t(a*c)*arctan(sqrt(a*c)*x/a) + (2*a^2*c^2*x^3 + 3*a^3*c*x)*e^4)/(a^2*c^4*x^2 + a^3*c^3)]

________________________________________________________________________________________

Sympy [B] Leaf count of result is larger than twice the leaf count of optimal. 403 vs. \(2 (138) = 276\).
time = 0.78, size = 403, normalized size = 2.70 \begin {gather*} \left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} - \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) \log {\left (x + \frac {- 4 a^{2} c^{2} \cdot \left (\frac {2 d e^{3}}{c^{2}} + \frac {\sqrt {- a^{3} c^{5}} \cdot \left (3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{5}}\right ) + 8 a^{2} d e^{3}}{3 a^{2} e^{4} - 6 a c d^{2} e^{2} - c^{2} d^{4}} \right )} + \frac {4 a^{2} d e^{3} - 4 a c d^{3} e + x \left (a^{2} e^{4} - 6 a c d^{2} e^{2} + c^{2} d^{4}\right )}{2 a^{2} c^{2} + 2 a c^{3} x^{2}} + \frac {e^{4} x}{c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**4/(c*x**2+a)**2,x)

[Out]

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

________________________________________________________________________________________

Giac [A]
time = 1.58, size = 131, normalized size = 0.88 \begin {gather*} \frac {2 \, d e^{3} \log \left (c x^{2} + a\right )}{c^{2}} + \frac {x e^{4}}{c^{2}} + \frac {{\left (c^{2} d^{4} + 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} - \frac {4 \, a c d^{3} e - 4 \, a^{2} d e^{3} - {\left (c^{2} d^{4} - 6 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^2,x, algorithm="giac")

[Out]

2*d*e^3*log(c*x^2 + a)/c^2 + x*e^4/c^2 + 1/2*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt
(a*c)*a*c^2) - 1/2*(4*a*c*d^3*e - 4*a^2*d*e^3 - (c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)*x)/((c*x^2 + a)*a*c^2)

________________________________________________________________________________________

Mupad [B]
time = 0.34, size = 131, normalized size = 0.88 \begin {gather*} \frac {\frac {x\,\left (a^2\,e^4-6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a}+2\,a\,d\,e^3-2\,c\,d^3\,e}{c^3\,x^2+a\,c^2}+\frac {e^4\,x}{c^2}+\frac {2\,d\,e^3\,\ln \left (c\,x^2+a\right )}{c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-3\,a^2\,e^4+6\,a\,c\,d^2\,e^2+c^2\,d^4\right )}{2\,a^{3/2}\,c^{5/2}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((d + e*x)^4/(a + c*x^2)^2,x)

[Out]

((x*(a^2*e^4 + c^2*d^4 - 6*a*c*d^2*e^2))/(2*a) + 2*a*d*e^3 - 2*c*d^3*e)/(a*c^2 + c^3*x^2) + (e^4*x)/c^2 + (2*d
*e^3*log(a + c*x^2))/c^2 + (atan((c^(1/2)*x)/a^(1/2))*(c^2*d^4 - 3*a^2*e^4 + 6*a*c*d^2*e^2))/(2*a^(3/2)*c^(5/2
))

________________________________________________________________________________________