### 3.467 $$\int \frac{(a+c x^2)^2}{(d+e x)^4} \, dx$$

Optimal. Leaf size=101 $-\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4}$

[Out]

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

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Rubi [A]  time = 0.0746482, antiderivative size = 101, 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} $-\frac{2 c \left (a e^2+3 c d^2\right )}{e^5 (d+e x)}+\frac{2 c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^2}-\frac{\left (a e^2+c d^2\right )^2}{3 e^5 (d+e x)^3}-\frac{4 c^2 d \log (d+e x)}{e^5}+\frac{c^2 x}{e^4}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^4,x]

[Out]

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

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

Mathematica [A]  time = 0.0539365, size = 110, normalized size = 1.09 $-\frac{a^2 e^4+2 a c e^2 \left (d^2+3 d e x+3 e^2 x^2\right )+c^2 \left (9 d^2 e^2 x^2+27 d^3 e x+13 d^4-9 d e^3 x^3-3 e^4 x^4\right )+12 c^2 d (d+e x)^3 \log (d+e x)}{3 e^5 (d+e x)^3}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

-(a^2*e^4 + 2*a*c*e^2*(d^2 + 3*d*e*x + 3*e^2*x^2) + c^2*(13*d^4 + 27*d^3*e*x + 9*d^2*e^2*x^2 - 9*d*e^3*x^3 - 3
*e^4*x^4) + 12*c^2*d*(d + e*x)^3*Log[d + e*x])/(3*e^5*(d + e*x)^3)

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Maple [A]  time = 0.048, size = 140, normalized size = 1.4 \begin{align*}{\frac{{c}^{2}x}{{e}^{4}}}-{\frac{{a}^{2}}{3\,e \left ( ex+d \right ) ^{3}}}-{\frac{2\,ac{d}^{2}}{3\,{e}^{3} \left ( ex+d \right ) ^{3}}}-{\frac{{c}^{2}{d}^{4}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-4\,{\frac{{c}^{2}d\ln \left ( ex+d \right ) }{{e}^{5}}}-2\,{\frac{ac}{{e}^{3} \left ( ex+d \right ) }}-6\,{\frac{{c}^{2}{d}^{2}}{{e}^{5} \left ( ex+d \right ) }}+2\,{\frac{acd}{{e}^{3} \left ( ex+d \right ) ^{2}}}+2\,{\frac{{c}^{2}{d}^{3}}{{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)^4,x)

[Out]

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

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Maxima [A]  time = 1.29147, size = 176, normalized size = 1.74 \begin{align*} -\frac{13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} e^{5}\right )}} + \frac{c^{2} x}{e^{4}} - \frac{4 \, c^{2} d \log \left (e x + d\right )}{e^{5}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-1/3*(13*c^2*d^4 + 2*a*c*d^2*e^2 + a^2*e^4 + 6*(3*c^2*d^2*e^2 + a*c*e^4)*x^2 + 6*(5*c^2*d^3*e + a*c*d*e^3)*x)/
(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5) + c^2*x/e^4 - 4*c^2*d*log(e*x + d)/e^5

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Fricas [A]  time = 1.94676, size = 373, normalized size = 3.69 \begin{align*} \frac{3 \, c^{2} e^{4} x^{4} + 9 \, c^{2} d e^{3} x^{3} - 13 \, c^{2} d^{4} - 2 \, a c d^{2} e^{2} - a^{2} e^{4} - 3 \,{\left (3 \, c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} - 3 \,{\left (9 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \,{\left (c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{2} e^{2} x^{2} + 3 \, c^{2} d^{3} e x + c^{2} d^{4}\right )} \log \left (e x + d\right )}{3 \,{\left (e^{8} x^{3} + 3 \, d e^{7} x^{2} + 3 \, d^{2} e^{6} x + d^{3} 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)^4,x, algorithm="fricas")

[Out]

1/3*(3*c^2*e^4*x^4 + 9*c^2*d*e^3*x^3 - 13*c^2*d^4 - 2*a*c*d^2*e^2 - a^2*e^4 - 3*(3*c^2*d^2*e^2 + 2*a*c*e^4)*x^
2 - 3*(9*c^2*d^3*e + 2*a*c*d*e^3)*x - 12*(c^2*d*e^3*x^3 + 3*c^2*d^2*e^2*x^2 + 3*c^2*d^3*e*x + c^2*d^4)*log(e*x
+ d))/(e^8*x^3 + 3*d*e^7*x^2 + 3*d^2*e^6*x + d^3*e^5)

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Sympy [A]  time = 1.43739, size = 134, normalized size = 1.33 \begin{align*} - \frac{4 c^{2} d \log{\left (d + e x \right )}}{e^{5}} + \frac{c^{2} x}{e^{4}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} + 13 c^{2} d^{4} + x^{2} \left (6 a c e^{4} + 18 c^{2} d^{2} e^{2}\right ) + x \left (6 a c d e^{3} + 30 c^{2} d^{3} e\right )}{3 d^{3} e^{5} + 9 d^{2} e^{6} x + 9 d e^{7} x^{2} + 3 e^{8} x^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c**2*d*log(d + e*x)/e**5 + c**2*x/e**4 - (a**2*e**4 + 2*a*c*d**2*e**2 + 13*c**2*d**4 + x**2*(6*a*c*e**4 + 1
8*c**2*d**2*e**2) + x*(6*a*c*d*e**3 + 30*c**2*d**3*e))/(3*d**3*e**5 + 9*d**2*e**6*x + 9*d*e**7*x**2 + 3*e**8*x
**3)

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Giac [A]  time = 1.2176, size = 136, normalized size = 1.35 \begin{align*} -4 \, c^{2} d e^{\left (-5\right )} \log \left ({\left | x e + d \right |}\right ) + c^{2} x e^{\left (-4\right )} - \frac{{\left (13 \, c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + a^{2} e^{4} + 6 \,{\left (5 \, c^{2} d^{3} e + a c d e^{3}\right )} x\right )} e^{\left (-5\right )}}{3 \,{\left (x e + d\right )}^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-4*c^2*d*e^(-5)*log(abs(x*e + d)) + c^2*x*e^(-4) - 1/3*(13*c^2*d^4 + 2*a*c*d^2*e^2 + 6*(3*c^2*d^2*e^2 + a*c*e^
4)*x^2 + a^2*e^4 + 6*(5*c^2*d^3*e + a*c*d*e^3)*x)*e^(-5)/(x*e + d)^3