### 3.465 $$\int \frac{(a+c x^2)^2}{(d+e x)^2} \, dx$$

Optimal. Leaf size=94 $\frac{c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac{4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2}$

[Out]

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

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Rubi [A]  time = 0.0756636, antiderivative size = 94, 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} $\frac{c x \left (2 a e^2+3 c d^2\right )}{e^4}-\frac{\left (a e^2+c d^2\right )^2}{e^5 (d+e x)}-\frac{4 c d \left (a e^2+c d^2\right ) \log (d+e x)}{e^5}-\frac{c^2 d x^2}{e^3}+\frac{c^2 x^3}{3 e^2}$

Antiderivative was successfully veriﬁed.

[In]

Int[(a + c*x^2)^2/(d + e*x)^2,x]

[Out]

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

Mathematica [A]  time = 0.0556821, size = 91, normalized size = 0.97 $\frac{3 c e x \left (2 a e^2+3 c d^2\right )-\frac{3 \left (a e^2+c d^2\right )^2}{d+e x}-12 c d \left (a e^2+c d^2\right ) \log (d+e x)-3 c^2 d e^2 x^2+c^2 e^3 x^3}{3 e^5}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

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

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Maple [A]  time = 0.049, size = 126, normalized size = 1.3 \begin{align*}{\frac{{c}^{2}{x}^{3}}{3\,{e}^{2}}}-{\frac{{c}^{2}d{x}^{2}}{{e}^{3}}}+2\,{\frac{acx}{{e}^{2}}}+3\,{\frac{{c}^{2}{d}^{2}x}{{e}^{4}}}-4\,{\frac{cd\ln \left ( ex+d \right ) a}{{e}^{3}}}-4\,{\frac{{c}^{2}{d}^{3}\ln \left ( ex+d \right ) }{{e}^{5}}}-{\frac{{a}^{2}}{e \left ( ex+d \right ) }}-2\,{\frac{ac{d}^{2}}{{e}^{3} \left ( ex+d \right ) }}-{\frac{{c}^{2}{d}^{4}}{{e}^{5} \left ( ex+d \right ) }} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Maxima [A]  time = 1.17403, size = 151, normalized size = 1.61 \begin{align*} -\frac{c^{2} d^{4} + 2 \, a c d^{2} e^{2} + a^{2} e^{4}}{e^{6} x + d e^{5}} + \frac{c^{2} e^{2} x^{3} - 3 \, c^{2} d e x^{2} + 3 \,{\left (3 \, c^{2} d^{2} + 2 \, a c e^{2}\right )} x}{3 \, e^{4}} - \frac{4 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} \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)^2,x, algorithm="maxima")

[Out]

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

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Fricas [A]  time = 1.81917, size = 309, normalized size = 3.29 \begin{align*} \frac{c^{2} e^{4} x^{4} - 2 \, c^{2} d e^{3} x^{3} - 3 \, c^{2} d^{4} - 6 \, a c d^{2} e^{2} - 3 \, a^{2} e^{4} + 6 \,{\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 3 \,{\left (3 \, c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x - 12 \,{\left (c^{2} d^{4} + a c d^{2} e^{2} +{\left (c^{2} d^{3} e + a c d e^{3}\right )} x\right )} \log \left (e x + d\right )}{3 \,{\left (e^{6} x + d 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)^2,x, algorithm="fricas")

[Out]

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

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Sympy [A]  time = 0.720812, size = 105, normalized size = 1.12 \begin{align*} - \frac{c^{2} d x^{2}}{e^{3}} + \frac{c^{2} x^{3}}{3 e^{2}} - \frac{4 c d \left (a e^{2} + c d^{2}\right ) \log{\left (d + e x \right )}}{e^{5}} - \frac{a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4}}{d e^{5} + e^{6} x} + \frac{x \left (2 a c e^{2} + 3 c^{2} d^{2}\right )}{e^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.27953, size = 201, normalized size = 2.14 \begin{align*} \frac{1}{3} \,{\left (c^{2} - \frac{6 \, c^{2} d}{x e + d} + \frac{6 \,{\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}}\right )}{\left (x e + d\right )}^{3} e^{\left (-5\right )} + 4 \,{\left (c^{2} d^{3} + a c d e^{2}\right )} e^{\left (-5\right )} \log \left (\frac{{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) -{\left (\frac{c^{2} d^{4} e^{3}}{x e + d} + \frac{2 \, a c d^{2} e^{5}}{x e + d} + \frac{a^{2} e^{7}}{x e + d}\right )} e^{\left (-8\right )} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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