### 3.513 $$\int \frac{(d+e x)^4}{(a+c x^2)^3} \, dx$$

Optimal. Leaf size=120 $-\frac{3 (d+e x) \left (a e^2+c d^2\right ) (a e-c d x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}$

[Out]

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

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Rubi [A]  time = 0.0479793, antiderivative size = 120, normalized size of antiderivative = 1., number of steps used = 3, number of rules used = 2, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.118, Rules used = {723, 205} $-\frac{3 (d+e x) \left (a e^2+c d^2\right ) (a e-c d x)}{8 a^2 c^2 \left (a+c x^2\right )}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}-\frac{(d+e x)^3 (a e-c d x)}{4 a c \left (a+c x^2\right )^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(d + e*x)^4/(a + c*x^2)^3,x]

[Out]

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

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

Mathematica [A]  time = 0.101979, size = 148, normalized size = 1.23 $\frac{-a^2 c e \left (6 d^2 e x+8 d^3+16 d e^2 x^2+5 e^3 x^3\right )-a^3 e^3 (8 d+3 e x)+a c^2 d^2 x \left (5 d^2+6 e^2 x^2\right )+3 c^3 d^4 x^3}{8 a^2 c^2 \left (a+c x^2\right )^2}+\frac{3 \left (a e^2+c d^2\right )^2 \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{8 a^{5/2} c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(d + e*x)^4/(a + c*x^2)^3,x]

[Out]

(3*c^3*d^4*x^3 - a^3*e^3*(8*d + 3*e*x) + a*c^2*d^2*x*(5*d^2 + 6*e^2*x^2) - a^2*c*e*(8*d^3 + 6*d^2*e*x + 16*d*e
^2*x^2 + 5*e^3*x^3))/(8*a^2*c^2*(a + c*x^2)^2) + (3*(c*d^2 + a*e^2)^2*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*
c^(5/2))

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Maple [A]  time = 0.051, size = 189, normalized size = 1.6 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{2}} \left ( -{\frac{ \left ( 5\,{a}^{2}{e}^{4}-6\,ac{d}^{2}{e}^{2}-3\,{c}^{2}{d}^{4} \right ){x}^{3}}{8\,{a}^{2}c}}-2\,{\frac{d{e}^{3}{x}^{2}}{c}}-{\frac{ \left ( 3\,{a}^{2}{e}^{4}+6\,ac{d}^{2}{e}^{2}-5\,{c}^{2}{d}^{4} \right ) x}{8\,a{c}^{2}}}-{\frac{de \left ( a{e}^{2}+c{d}^{2} \right ) }{{c}^{2}}} \right ) }+{\frac{3\,{e}^{4}}{8\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{2}{e}^{2}}{4\,ac}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,{d}^{4}}{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)^4/(c*x^2+a)^3,x)

[Out]

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

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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((e*x+d)^4/(c*x^2+a)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.90785, size = 1104, normalized size = 9.2 \begin{align*} \left [-\frac{32 \, a^{3} c^{2} d e^{3} x^{2} + 16 \, a^{3} c^{2} d^{3} e + 16 \, a^{4} c d e^{3} - 2 \,{\left (3 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} - 5 \, a^{3} c^{2} e^{4}\right )} x^{3} + 3 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c 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 (5 \, a^{2} c^{3} d^{4} - 6 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x}{16 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac{16 \, a^{3} c^{2} d e^{3} x^{2} + 8 \, a^{3} c^{2} d^{3} e + 8 \, a^{4} c d e^{3} -{\left (3 \, a c^{4} d^{4} + 6 \, a^{2} c^{3} d^{2} e^{2} - 5 \, a^{3} c^{2} e^{4}\right )} x^{3} - 3 \,{\left (a^{2} c^{2} d^{4} + 2 \, a^{3} c d^{2} e^{2} + a^{4} e^{4} +{\left (c^{4} d^{4} + 2 \, a c^{3} d^{2} e^{2} + a^{2} c^{2} e^{4}\right )} x^{4} + 2 \,{\left (a c^{3} d^{4} + 2 \, a^{2} c^{2} d^{2} e^{2} + a^{3} c e^{4}\right )} x^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) -{\left (5 \, a^{2} c^{3} d^{4} - 6 \, a^{3} c^{2} d^{2} e^{2} - 3 \, a^{4} c e^{4}\right )} x}{8 \,{\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^4/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[-1/16*(32*a^3*c^2*d*e^3*x^2 + 16*a^3*c^2*d^3*e + 16*a^4*c*d*e^3 - 2*(3*a*c^4*d^4 + 6*a^2*c^3*d^2*e^2 - 5*a^3*
c^2*e^4)*x^3 + 3*(a^2*c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4 + 2*
(a*c^3*d^4 + 2*a^2*c^2*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) - 2*
(5*a^2*c^3*d^4 - 6*a^3*c^2*d^2*e^2 - 3*a^4*c*e^4)*x)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3), -1/8*(16*a^3*c^2
*d*e^3*x^2 + 8*a^3*c^2*d^3*e + 8*a^4*c*d*e^3 - (3*a*c^4*d^4 + 6*a^2*c^3*d^2*e^2 - 5*a^3*c^2*e^4)*x^3 - 3*(a^2*
c^2*d^4 + 2*a^3*c*d^2*e^2 + a^4*e^4 + (c^4*d^4 + 2*a*c^3*d^2*e^2 + a^2*c^2*e^4)*x^4 + 2*(a*c^3*d^4 + 2*a^2*c^2
*d^2*e^2 + a^3*c*e^4)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) - (5*a^2*c^3*d^4 - 6*a^3*c^2*d^2*e^2 - 3*a^4*c*e^4)
*x)/(a^3*c^5*x^4 + 2*a^4*c^4*x^2 + a^5*c^3)]

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Sympy [B]  time = 2.63037, size = 328, normalized size = 2.73 \begin{align*} - \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (- \frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} + \frac{3 \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2} \log{\left (\frac{3 a^{3} c^{2} \sqrt{- \frac{1}{a^{5} c^{5}}} \left (a e^{2} + c d^{2}\right )^{2}}{3 a^{2} e^{4} + 6 a c d^{2} e^{2} + 3 c^{2} d^{4}} + x \right )}}{16} - \frac{8 a^{3} d e^{3} + 8 a^{2} c d^{3} e + 16 a^{2} c d e^{3} x^{2} + x^{3} \left (5 a^{2} c e^{4} - 6 a c^{2} d^{2} e^{2} - 3 c^{3} d^{4}\right ) + x \left (3 a^{3} e^{4} + 6 a^{2} c d^{2} e^{2} - 5 a c^{2} d^{4}\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)**4/(c*x**2+a)**3,x)

[Out]

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

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Giac [A]  time = 1.33361, size = 217, normalized size = 1.81 \begin{align*} \frac{3 \,{\left (c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{8 \, \sqrt{a c} a^{2} c^{2}} + \frac{3 \, c^{3} d^{4} x^{3} + 6 \, a c^{2} d^{2} x^{3} e^{2} + 5 \, a c^{2} d^{4} x - 5 \, a^{2} c x^{3} e^{4} - 16 \, a^{2} c d x^{2} e^{3} - 6 \, a^{2} c d^{2} x e^{2} - 8 \, a^{2} c d^{3} e - 3 \, a^{3} x e^{4} - 8 \, a^{3} d 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)^4/(c*x^2+a)^3,x, algorithm="giac")

[Out]

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