### 3.514 $$\int \frac{(d+e x)^3}{(a+c x^2)^3} \, dx$$

Optimal. Leaf size=98 $\frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2}$

[Out]

(x*(d + e*x)^3)/(4*a*(a + c*x^2)^2) - (3*d*(a*e - c*d*x)*(d + e*x))/(8*a^2*c*(a + c*x^2)) + (3*d*(c*d^2 + a*e^
2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))

________________________________________________________________________________________

Rubi [A]  time = 0.0380723, antiderivative size = 98, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 3, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.176, Rules used = {729, 723, 205} $\frac{3 d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{3/2}}-\frac{3 d (d+e x) (a e-c d x)}{8 a^2 c \left (a+c x^2\right )}+\frac{x (d+e x)^3}{4 a \left (a+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^3/(a + c*x^2)^3,x]

[Out]

(x*(d + e*x)^3)/(4*a*(a + c*x^2)^2) - (3*d*(a*e - c*d*x)*(d + e*x))/(8*a^2*c*(a + c*x^2)) + (3*d*(c*d^2 + a*e^
2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(3/2))

Rule 729

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> -Simp[((d + e*x)^m*(2*c*x)*(a + c*x^2)^(
p + 1))/(4*a*c*(p + 1)), x] - Dist[(m*(2*c*d))/(4*a*c*(p + 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1), x],
x] /; FreeQ[{a, c, d, e, m, p}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 3, 0] && LtQ[p, -1]

Rule 723

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[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

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]

Rubi steps

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

Mathematica [A]  time = 0.103615, size = 127, normalized size = 1.3 $\frac{\frac{\sqrt{a} \left (-a^2 c e \left (6 d^2+3 d e x+4 e^2 x^2\right )-2 a^3 e^3+a c^2 d x \left (5 d^2+3 e^2 x^2\right )+3 c^3 d^3 x^3\right )}{\left (a+c x^2\right )^2}+3 \sqrt{c} d \left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^2}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^3/(a + c*x^2)^3,x]

[Out]

((Sqrt[a]*(-2*a^3*e^3 + 3*c^3*d^3*x^3 + a*c^2*d*x*(5*d^2 + 3*e^2*x^2) - a^2*c*e*(6*d^2 + 3*d*e*x + 4*e^2*x^2))
)/(a + c*x^2)^2 + 3*Sqrt[c]*d*(c*d^2 + a*e^2)*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^2)

________________________________________________________________________________________

Maple [A]  time = 0.051, size = 133, normalized size = 1.4 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ({\frac{3\,d \left ( a{e}^{2}+c{d}^{2} \right ){x}^{3}}{8\,{a}^{2}}}-{\frac{{e}^{3}{x}^{2}}{2\,c}}-{\frac{d \left ( 3\,a{e}^{2}-5\,c{d}^{2} \right ) x}{8\,ac}}-{\frac{e \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{4\,{c}^{2}}} \right ) }+{\frac{3\,d{e}^{2}}{8\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{3}}{8\,{a}^{2}}\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)^3/(c*x^2+a)^3,x)

[Out]

(3/8*d*(a*e^2+c*d^2)/a^2*x^3-1/2*e^3*x^2/c-1/8*d*(3*a*e^2-5*c*d^2)/a/c*x-1/4*e*(a*e^2+3*c*d^2)/c^2)/(c*x^2+a)^
2+3/8*d/a/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*e^2+3/8*d^3/a^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))

________________________________________________________________________________________

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

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.05173, size = 822, normalized size = 8.39 \begin{align*} \left [-\frac{8 \, a^{3} c e^{3} x^{2} + 12 \, a^{3} c d^{2} e + 4 \, a^{4} e^{3} - 6 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} + 3 \,{\left (a^{2} c d^{3} + a^{3} d e^{2} +{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{2} d^{3} + a^{2} c d e^{2}\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 (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{16 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}, -\frac{4 \, a^{3} c e^{3} x^{2} + 6 \, a^{3} c d^{2} e + 2 \, a^{4} e^{3} - 3 \,{\left (a c^{3} d^{3} + a^{2} c^{2} d e^{2}\right )} x^{3} - 3 \,{\left (a^{2} c d^{3} + a^{3} d e^{2} +{\left (c^{3} d^{3} + a c^{2} d e^{2}\right )} x^{4} + 2 \,{\left (a c^{2} d^{3} + a^{2} c d e^{2}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, a^{2} c^{2} d^{3} - 3 \, a^{3} c d e^{2}\right )} x}{8 \,{\left (a^{3} c^{4} x^{4} + 2 \, a^{4} c^{3} x^{2} + a^{5} c^{2}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/16*(8*a^3*c*e^3*x^2 + 12*a^3*c*d^2*e + 4*a^4*e^3 - 6*(a*c^3*d^3 + a^2*c^2*d*e^2)*x^3 + 3*(a^2*c*d^3 + a^3*
d*e^2 + (c^3*d^3 + a*c^2*d*e^2)*x^4 + 2*(a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x
- a)/(c*x^2 + a)) - 2*(5*a^2*c^2*d^3 - 3*a^3*c*d*e^2)*x)/(a^3*c^4*x^4 + 2*a^4*c^3*x^2 + a^5*c^2), -1/8*(4*a^3*
c*e^3*x^2 + 6*a^3*c*d^2*e + 2*a^4*e^3 - 3*(a*c^3*d^3 + a^2*c^2*d*e^2)*x^3 - 3*(a^2*c*d^3 + a^3*d*e^2 + (c^3*d^
3 + a*c^2*d*e^2)*x^4 + 2*(a*c^2*d^3 + a^2*c*d*e^2)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (5*a^2*c^2*d^3 - 3*a
^3*c*d*e^2)*x)/(a^3*c^4*x^4 + 2*a^4*c^3*x^2 + a^5*c^2)]

________________________________________________________________________________________

Sympy [B]  time = 1.72044, size = 272, normalized size = 2.78 \begin{align*} - \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (- \frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{3 d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right ) \log{\left (\frac{3 a^{3} c d \sqrt{- \frac{1}{a^{5} c^{3}}} \left (a e^{2} + c d^{2}\right )}{3 a d e^{2} + 3 c d^{3}} + x \right )}}{16} + \frac{- 2 a^{3} e^{3} - 6 a^{2} c d^{2} e - 4 a^{2} c e^{3} x^{2} + x^{3} \left (3 a c^{2} d e^{2} + 3 c^{3} d^{3}\right ) + x \left (- 3 a^{2} c d e^{2} + 5 a c^{2} d^{3}\right )}{8 a^{4} c^{2} + 16 a^{3} c^{3} x^{2} + 8 a^{2} c^{4} x^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-3*d*sqrt(-1/(a**5*c**3))*(a*e**2 + c*d**2)*log(-3*a**3*c*d*sqrt(-1/(a**5*c**3))*(a*e**2 + c*d**2)/(3*a*d*e**2
+ 3*c*d**3) + x)/16 + 3*d*sqrt(-1/(a**5*c**3))*(a*e**2 + c*d**2)*log(3*a**3*c*d*sqrt(-1/(a**5*c**3))*(a*e**2
+ c*d**2)/(3*a*d*e**2 + 3*c*d**3) + x)/16 + (-2*a**3*e**3 - 6*a**2*c*d**2*e - 4*a**2*c*e**3*x**2 + x**3*(3*a*c
**2*d*e**2 + 3*c**3*d**3) + x*(-3*a**2*c*d*e**2 + 5*a*c**2*d**3))/(8*a**4*c**2 + 16*a**3*c**3*x**2 + 8*a**2*c*
*4*x**4)

________________________________________________________________________________________

Giac [A]  time = 1.33972, size = 167, normalized size = 1.7 \begin{align*} \frac{3 \,{\left (c d^{3} + a d e^{2}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c} + \frac{3 \, c^{3} d^{3} x^{3} + 3 \, a c^{2} d x^{3} e^{2} + 5 \, a c^{2} d^{3} x - 4 \, a^{2} c x^{2} e^{3} - 3 \, a^{2} c d x e^{2} - 6 \, a^{2} c d^{2} e - 2 \, a^{3} e^{3}}{8 \,{\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

3/8*(c*d^3 + a*d*e^2)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c) + 1/8*(3*c^3*d^3*x^3 + 3*a*c^2*d*x^3*e^2 + 5*a*c
^2*d^3*x - 4*a^2*c*x^2*e^3 - 3*a^2*c*d*x*e^2 - 6*a^2*c*d^2*e - 2*a^3*e^3)/((c*x^2 + a)^2*a^2*c^2)