### 3.469 $$\int \frac{(a+c x^2)^2}{(d+e x)^6} \, dx$$

Optimal. Leaf size=110 $-\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2}$

[Out]

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

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Rubi [A]  time = 0.0685907, antiderivative size = 110, normalized size of antiderivative = 1., number of steps used = 2, number of rules used = 1, integrand size = 17, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.059, Rules used = {697} $-\frac{2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac{c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac{\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac{c^2}{e^5 (d+e x)}+\frac{2 c^2 d}{e^5 (d+e x)^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

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

Rule 697

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(a + c*
x^2)^p, x], x] /; FreeQ[{a, c, d, e, m}, x] && NeQ[c*d^2 + a*e^2, 0] && IGtQ[p, 0]

Rubi steps

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

Mathematica [A]  time = 0.0374355, size = 90, normalized size = 0.82 $-\frac{3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (10 d^2 e^2 x^2+5 d^3 e x+d^4+10 d e^3 x^3+5 e^4 x^4\right )}{15 e^5 (d+e x)^5}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

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

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Maple [A]  time = 0.048, size = 119, normalized size = 1.1 \begin{align*}{\frac{cd \left ( a{e}^{2}+c{d}^{2} \right ) }{{e}^{5} \left ( ex+d \right ) ^{4}}}-{\frac{2\,c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{5\,{e}^{5} \left ( ex+d \right ) ^{5}}}-{\frac{{c}^{2}}{{e}^{5} \left ( ex+d \right ) }}+2\,{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{2}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^6,x)

[Out]

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

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Maxima [A]  time = 1.33863, size = 204, normalized size = 1.85 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="maxima")

[Out]

-1/15*(15*c^2*e^4*x^4 + 30*c^2*d*e^3*x^3 + 3*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4 + 10*(3*c^2*d^2*e^2 + a*c*e^4)*
x^2 + 5*(3*c^2*d^3*e + a*c*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x +
d^5*e^5)

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Fricas [A]  time = 1.80105, size = 312, normalized size = 2.84 \begin{align*} -\frac{15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \,{\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \,{\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="fricas")

[Out]

-1/15*(15*c^2*e^4*x^4 + 30*c^2*d*e^3*x^3 + 3*c^2*d^4 + a*c*d^2*e^2 + 3*a^2*e^4 + 10*(3*c^2*d^2*e^2 + a*c*e^4)*
x^2 + 5*(3*c^2*d^3*e + a*c*d*e^3)*x)/(e^10*x^5 + 5*d*e^9*x^4 + 10*d^2*e^8*x^3 + 10*d^3*e^7*x^2 + 5*d^4*e^6*x +
d^5*e^5)

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Sympy [A]  time = 2.3188, size = 160, normalized size = 1.45 \begin{align*} - \frac{3 a^{2} e^{4} + a c d^{2} e^{2} + 3 c^{2} d^{4} + 30 c^{2} d e^{3} x^{3} + 15 c^{2} e^{4} x^{4} + x^{2} \left (10 a c e^{4} + 30 c^{2} d^{2} e^{2}\right ) + x \left (5 a c d e^{3} + 15 c^{2} d^{3} e\right )}{15 d^{5} e^{5} + 75 d^{4} e^{6} x + 150 d^{3} e^{7} x^{2} + 150 d^{2} e^{8} x^{3} + 75 d e^{9} x^{4} + 15 e^{10} x^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**6,x)

[Out]

-(3*a**2*e**4 + a*c*d**2*e**2 + 3*c**2*d**4 + 30*c**2*d*e**3*x**3 + 15*c**2*e**4*x**4 + x**2*(10*a*c*e**4 + 30
*c**2*d**2*e**2) + x*(5*a*c*d*e**3 + 15*c**2*d**3*e))/(15*d**5*e**5 + 75*d**4*e**6*x + 150*d**3*e**7*x**2 + 15
0*d**2*e**8*x**3 + 75*d*e**9*x**4 + 15*e**10*x**5)

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Giac [A]  time = 1.29556, size = 132, normalized size = 1.2 \begin{align*} -\frac{{\left (15 \, c^{2} x^{4} e^{4} + 30 \, c^{2} d x^{3} e^{3} + 30 \, c^{2} d^{2} x^{2} e^{2} + 15 \, c^{2} d^{3} x e + 3 \, c^{2} d^{4} + 10 \, a c x^{2} e^{4} + 5 \, a c d x e^{3} + a c d^{2} e^{2} + 3 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{15 \,{\left (x e + d\right )}^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^6,x, algorithm="giac")

[Out]

-1/15*(15*c^2*x^4*e^4 + 30*c^2*d*x^3*e^3 + 30*c^2*d^2*x^2*e^2 + 15*c^2*d^3*x*e + 3*c^2*d^4 + 10*a*c*x^2*e^4 +
5*a*c*d*x*e^3 + a*c*d^2*e^2 + 3*a^2*e^4)*e^(-5)/(x*e + d)^5