3.505 \(\int \frac{(d+e x)^5}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=190 \[ \frac{d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}-\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{a c^2}+\frac{e^3 \left (5 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^3}-\frac{3 d e^2 x \left (2 c d^2-5 a e^2\right )}{2 a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(d+e x)^4 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

[Out]

(-3*d*e^2*(2*c*d^2 - 5*a*e^2)*x)/(2*a*c^2) - (e^3*(2*c*d^2 - a*e^2)*x^2)/(a*c^2) - (d*e^4*x^3)/(2*a*c) - ((a*e
- c*d*x)*(d + e*x)^4)/(2*a*c*(a + c*x^2)) + (d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqr
t[a]])/(2*a^(3/2)*c^(5/2)) + (e^3*(5*c*d^2 - a*e^2)*Log[a + c*x^2])/c^3

________________________________________________________________________________________

Rubi [A]  time = 0.176979, antiderivative size = 190, normalized size of antiderivative = 1., 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 = {739, 801, 635, 205, 260} \[ \frac{d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}-\frac{e^3 x^2 \left (2 c d^2-a e^2\right )}{a c^2}+\frac{e^3 \left (5 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^3}-\frac{3 d e^2 x \left (2 c d^2-5 a e^2\right )}{2 a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(d+e x)^4 (a e-c d x)}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[(d + e*x)^5/(a + c*x^2)^2,x]

[Out]

(-3*d*e^2*(2*c*d^2 - 5*a*e^2)*x)/(2*a*c^2) - (e^3*(2*c*d^2 - a*e^2)*x^2)/(a*c^2) - (d*e^4*x^3)/(2*a*c) - ((a*e
- c*d*x)*(d + e*x)^4)/(2*a*c*(a + c*x^2)) + (d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqr
t[a]])/(2*a^(3/2)*c^(5/2)) + (e^3*(5*c*d^2 - a*e^2)*Log[a + c*x^2])/c^3

Rule 739

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 801

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]

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 205

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

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{(d+e x)^5}{\left (a+c x^2\right )^2} \, dx &=-\frac{(a e-c d x) (d+e x)^4}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{(d+e x)^3 \left (c d^2+4 a e^2-3 c d e x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(a e-c d x) (d+e x)^4}{2 a c \left (a+c x^2\right )}+\frac{\int \left (-3 d e^2 \left (2 d^2-\frac{5 a e^2}{c}\right )-\frac{4 e^3 \left (2 c d^2-a e^2\right ) x}{c}-3 d e^4 x^2+\frac{c^2 d^5+10 a c d^3 e^2-15 a^2 d e^4+4 a e^3 \left (5 c d^2-a e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac{3 d e^2 \left (2 c d^2-5 a e^2\right ) x}{2 a c^2}-\frac{e^3 \left (2 c d^2-a e^2\right ) x^2}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(a e-c d x) (d+e x)^4}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{c^2 d^5+10 a c d^3 e^2-15 a^2 d e^4+4 a e^3 \left (5 c d^2-a e^2\right ) x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 d e^2 \left (2 c d^2-5 a e^2\right ) x}{2 a c^2}-\frac{e^3 \left (2 c d^2-a e^2\right ) x^2}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(a e-c d x) (d+e x)^4}{2 a c \left (a+c x^2\right )}+\frac{\left (2 e^3 \left (5 c d^2-a e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 d e^2 \left (2 c d^2-5 a e^2\right ) x}{2 a c^2}-\frac{e^3 \left (2 c d^2-a e^2\right ) x^2}{a c^2}-\frac{d e^4 x^3}{2 a c}-\frac{(a e-c d x) (d+e x)^4}{2 a c \left (a+c x^2\right )}+\frac{d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{e^3 \left (5 c d^2-a e^2\right ) \log \left (a+c x^2\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.125413, size = 164, normalized size = 0.86 \[ \frac{\frac{5 a^2 c d e^3 (2 d+e x)-a^3 e^5-5 a c^2 d^3 e (d+2 e x)+c^3 d^5 x}{a \left (a+c x^2\right )}+\frac{\sqrt{c} d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{a^{3/2}}+2 \left (5 c d^2 e^3-a e^5\right ) \log \left (a+c x^2\right )+10 c d e^4 x+c e^5 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[(d + e*x)^5/(a + c*x^2)^2,x]

[Out]

(10*c*d*e^4*x + c*e^5*x^2 + (-(a^3*e^5) + c^3*d^5*x + 5*a^2*c*d*e^3*(2*d + e*x) - 5*a*c^2*d^3*e*(d + 2*e*x))/(
a*(a + c*x^2)) + (Sqrt[c]*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(3/2) + 2*(
5*c*d^2*e^3 - a*e^5)*Log[a + c*x^2])/(2*c^3)

________________________________________________________________________________________

Maple [A]  time = 0.056, size = 248, normalized size = 1.3 \begin{align*}{\frac{{e}^{5}{x}^{2}}{2\,{c}^{2}}}+5\,{\frac{{e}^{4}xd}{{c}^{2}}}+{\frac{5\,adx{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-5\,{\frac{{d}^{3}x{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}+{\frac{{d}^{5}x}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}-{\frac{{a}^{2}{e}^{5}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+5\,{\frac{{e}^{3}a{d}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{5\,e{d}^{4}}{2\,c \left ( c{x}^{2}+a \right ) }}-{\frac{a\ln \left ( c{x}^{2}+a \right ){e}^{5}}{{c}^{3}}}+5\,{\frac{\ln \left ( c{x}^{2}+a \right ){d}^{2}{e}^{3}}{{c}^{2}}}-{\frac{15\,ad{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+5\,{\frac{{d}^{3}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{{d}^{5}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^5/(c*x^2+a)^2,x)

[Out]

1/2*e^5*x^2/c^2+5*e^4/c^2*x*d+5/2/c^2/(c*x^2+a)*d*a*x*e^4-5/c/(c*x^2+a)*d^3*x*e^2+1/2/(c*x^2+a)*d^5/a*x-1/2/c^
3/(c*x^2+a)*e^5*a^2+5/c^2/(c*x^2+a)*e^3*a*d^2-5/2/c/(c*x^2+a)*e*d^4-1/c^3*a*ln(c*x^2+a)*e^5+5/c^2*ln(c*x^2+a)*
d^2*e^3-15/2/c^2*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d*e^4+5/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^3*e^2+1
/2/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*d^5

________________________________________________________________________________________

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((e*x+d)^5/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [A]  time = 1.97456, size = 1142, normalized size = 6.01 \begin{align*} \left [\frac{2 \, a^{2} c^{2} e^{5} x^{4} + 20 \, a^{2} c^{2} d e^{4} x^{3} + 2 \, a^{3} c e^{5} x^{2} - 10 \, a^{2} c^{2} d^{4} e + 20 \, a^{3} c d^{2} e^{3} - 2 \, a^{4} e^{5} +{\left (a c^{2} d^{5} + 10 \, a^{2} c d^{3} e^{2} - 15 \, a^{3} d e^{4} +{\left (c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4}\right )} x^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} + 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 2 \,{\left (a c^{3} d^{5} - 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x + 4 \,{\left (5 \, a^{3} c d^{2} e^{3} - a^{4} e^{5} +{\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{4 \,{\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac{a^{2} c^{2} e^{5} x^{4} + 10 \, a^{2} c^{2} d e^{4} x^{3} + a^{3} c e^{5} x^{2} - 5 \, a^{2} c^{2} d^{4} e + 10 \, a^{3} c d^{2} e^{3} - a^{4} e^{5} +{\left (a c^{2} d^{5} + 10 \, a^{2} c d^{3} e^{2} - 15 \, a^{3} d e^{4} +{\left (c^{3} d^{5} + 10 \, a c^{2} d^{3} e^{2} - 15 \, a^{2} c d e^{4}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) +{\left (a c^{3} d^{5} - 10 \, a^{2} c^{2} d^{3} e^{2} + 15 \, a^{3} c d e^{4}\right )} x + 2 \,{\left (5 \, a^{3} c d^{2} e^{3} - a^{4} e^{5} +{\left (5 \, a^{2} c^{2} d^{2} e^{3} - a^{3} c e^{5}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{2 \,{\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*a^2*c^2*e^5*x^4 + 20*a^2*c^2*d*e^4*x^3 + 2*a^3*c*e^5*x^2 - 10*a^2*c^2*d^4*e + 20*a^3*c*d^2*e^3 - 2*a^4
*e^5 + (a*c^2*d^5 + 10*a^2*c*d^3*e^2 - 15*a^3*d*e^4 + (c^3*d^5 + 10*a*c^2*d^3*e^2 - 15*a^2*c*d*e^4)*x^2)*sqrt(
-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(a*c^3*d^5 - 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x +
4*(5*a^3*c*d^2*e^3 - a^4*e^5 + (5*a^2*c^2*d^2*e^3 - a^3*c*e^5)*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3), 1
/2*(a^2*c^2*e^5*x^4 + 10*a^2*c^2*d*e^4*x^3 + a^3*c*e^5*x^2 - 5*a^2*c^2*d^4*e + 10*a^3*c*d^2*e^3 - a^4*e^5 + (a
*c^2*d^5 + 10*a^2*c*d^3*e^2 - 15*a^3*d*e^4 + (c^3*d^5 + 10*a*c^2*d^3*e^2 - 15*a^2*c*d*e^4)*x^2)*sqrt(a*c)*arct
an(sqrt(a*c)*x/a) + (a*c^3*d^5 - 10*a^2*c^2*d^3*e^2 + 15*a^3*c*d*e^4)*x + 2*(5*a^3*c*d^2*e^3 - a^4*e^5 + (5*a^
2*c^2*d^2*e^3 - a^3*c*e^5)*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3)]

________________________________________________________________________________________

Sympy [B]  time = 3.54667, size = 515, normalized size = 2.71 \begin{align*} \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} - \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{- 4 a^{3} e^{5} - 4 a^{2} c^{3} \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} - \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) + 20 a^{2} c d^{2} e^{3}}{15 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} - c^{3} d^{5}} \right )} + \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} + \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{- 4 a^{3} e^{5} - 4 a^{2} c^{3} \left (- \frac{e^{3} \left (a e^{2} - 5 c d^{2}\right )}{c^{3}} + \frac{d \sqrt{- a^{3} c^{7}} \left (15 a^{2} e^{4} - 10 a c d^{2} e^{2} - c^{2} d^{4}\right )}{4 a^{3} c^{6}}\right ) + 20 a^{2} c d^{2} e^{3}}{15 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} - c^{3} d^{5}} \right )} + \frac{- a^{3} e^{5} + 10 a^{2} c d^{2} e^{3} - 5 a c^{2} d^{4} e + x \left (5 a^{2} c d e^{4} - 10 a c^{2} d^{3} e^{2} + c^{3} d^{5}\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} + \frac{5 d e^{4} x}{c^{2}} + \frac{e^{5} x^{2}}{2 c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**5/(c*x**2+a)**2,x)

[Out]

(-e**3*(a*e**2 - 5*c*d**2)/c**3 - d*sqrt(-a**3*c**7)*(15*a**2*e**4 - 10*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**
6))*log(x + (-4*a**3*e**5 - 4*a**2*c**3*(-e**3*(a*e**2 - 5*c*d**2)/c**3 - d*sqrt(-a**3*c**7)*(15*a**2*e**4 - 1
0*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**6)) + 20*a**2*c*d**2*e**3)/(15*a**2*c*d*e**4 - 10*a*c**2*d**3*e**2 - c
**3*d**5)) + (-e**3*(a*e**2 - 5*c*d**2)/c**3 + d*sqrt(-a**3*c**7)*(15*a**2*e**4 - 10*a*c*d**2*e**2 - c**2*d**4
)/(4*a**3*c**6))*log(x + (-4*a**3*e**5 - 4*a**2*c**3*(-e**3*(a*e**2 - 5*c*d**2)/c**3 + d*sqrt(-a**3*c**7)*(15*
a**2*e**4 - 10*a*c*d**2*e**2 - c**2*d**4)/(4*a**3*c**6)) + 20*a**2*c*d**2*e**3)/(15*a**2*c*d*e**4 - 10*a*c**2*
d**3*e**2 - c**3*d**5)) + (-a**3*e**5 + 10*a**2*c*d**2*e**3 - 5*a*c**2*d**4*e + x*(5*a**2*c*d*e**4 - 10*a*c**2
*d**3*e**2 + c**3*d**5))/(2*a**2*c**3 + 2*a*c**4*x**2) + 5*d*e**4*x/c**2 + e**5*x**2/(2*c**2)

________________________________________________________________________________________

Giac [A]  time = 1.31711, size = 236, normalized size = 1.24 \begin{align*} \frac{{\left (5 \, c d^{2} e^{3} - a e^{5}\right )} \log \left (c x^{2} + a\right )}{c^{3}} + \frac{{\left (c^{2} d^{5} + 10 \, a c d^{3} e^{2} - 15 \, a^{2} d e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} + \frac{c^{2} x^{2} e^{5} + 10 \, c^{2} d x e^{4}}{2 \, c^{4}} - \frac{5 \, a c^{2} d^{4} e - 10 \, a^{2} c d^{2} e^{3} + a^{3} e^{5} -{\left (c^{3} d^{5} - 10 \, a c^{2} d^{3} e^{2} + 5 \, a^{2} c d e^{4}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^5/(c*x^2+a)^2,x, algorithm="giac")

[Out]

(5*c*d^2*e^3 - a*e^5)*log(c*x^2 + a)/c^3 + 1/2*(c^2*d^5 + 10*a*c*d^3*e^2 - 15*a^2*d*e^4)*arctan(c*x/sqrt(a*c))
/(sqrt(a*c)*a*c^2) + 1/2*(c^2*x^2*e^5 + 10*c^2*d*x*e^4)/c^4 - 1/2*(5*a*c^2*d^4*e - 10*a^2*c*d^2*e^3 + a^3*e^5
- (c^3*d^5 - 10*a*c^2*d^3*e^2 + 5*a^2*c*d*e^4)*x)/((c*x^2 + a)*a*c^3)