3.471 \(\int \frac{(a+c x^2)^2}{(d+e x)^8} \, dx\)

Optimal. Leaf size=114 \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{5 e^5 (d+e x)^5}+\frac{2 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2}{7 e^5 (d+e x)^7}-\frac{c^2}{3 e^5 (d+e x)^3}+\frac{c^2 d}{e^5 (d+e x)^4} \]

[Out]

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

________________________________________________________________________________________

Rubi [A]  time = 0.0695633, antiderivative size = 114, 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} \[ -\frac{2 c \left (a e^2+3 c d^2\right )}{5 e^5 (d+e x)^5}+\frac{2 c d \left (a e^2+c d^2\right )}{3 e^5 (d+e x)^6}-\frac{\left (a e^2+c d^2\right )^2}{7 e^5 (d+e x)^7}-\frac{c^2}{3 e^5 (d+e x)^3}+\frac{c^2 d}{e^5 (d+e x)^4} \]

Antiderivative was successfully verified.

[In]

Int[(a + c*x^2)^2/(d + e*x)^8,x]

[Out]

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

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)^8} \, dx &=\int \left (\frac{\left (c d^2+a e^2\right )^2}{e^4 (d+e x)^8}-\frac{4 c d \left (c d^2+a e^2\right )}{e^4 (d+e x)^7}+\frac{2 c \left (3 c d^2+a e^2\right )}{e^4 (d+e x)^6}-\frac{4 c^2 d}{e^4 (d+e x)^5}+\frac{c^2}{e^4 (d+e x)^4}\right ) \, dx\\ &=-\frac{\left (c d^2+a e^2\right )^2}{7 e^5 (d+e x)^7}+\frac{2 c d \left (c d^2+a e^2\right )}{3 e^5 (d+e x)^6}-\frac{2 c \left (3 c d^2+a e^2\right )}{5 e^5 (d+e x)^5}+\frac{c^2 d}{e^5 (d+e x)^4}-\frac{c^2}{3 e^5 (d+e x)^3}\\ \end{align*}

Mathematica [A]  time = 0.035664, size = 90, normalized size = 0.79 \[ -\frac{15 a^2 e^4+2 a c e^2 \left (d^2+7 d e x+21 e^2 x^2\right )+c^2 \left (21 d^2 e^2 x^2+7 d^3 e x+d^4+35 d e^3 x^3+35 e^4 x^4\right )}{105 e^5 (d+e x)^7} \]

Antiderivative was successfully verified.

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^8,x]

[Out]

-(15*a^2*e^4 + 2*a*c*e^2*(d^2 + 7*d*e*x + 21*e^2*x^2) + c^2*(d^4 + 7*d^3*e*x + 21*d^2*e^2*x^2 + 35*d*e^3*x^3 +
35*e^4*x^4))/(105*e^5*(d + e*x)^7)

________________________________________________________________________________________

Maple [A]  time = 0.046, size = 119, normalized size = 1. \begin{align*}{\frac{{c}^{2}d}{{e}^{5} \left ( ex+d \right ) ^{4}}}+{\frac{2\,cd \left ( a{e}^{2}+c{d}^{2} \right ) }{3\,{e}^{5} \left ( ex+d \right ) ^{6}}}-{\frac{{c}^{2}}{3\,{e}^{5} \left ( ex+d \right ) ^{3}}}-{\frac{{a}^{2}{e}^{4}+2\,ac{d}^{2}{e}^{2}+{c}^{2}{d}^{4}}{7\,{e}^{5} \left ( ex+d \right ) ^{7}}}-{\frac{2\,c \left ( a{e}^{2}+3\,c{d}^{2} \right ) }{5\,{e}^{5} \left ( ex+d \right ) ^{5}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((c*x^2+a)^2/(e*x+d)^8,x)

[Out]

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

________________________________________________________________________________________

Maxima [A]  time = 1.60838, size = 234, normalized size = 2.05 \begin{align*} -\frac{35 \, c^{2} e^{4} x^{4} + 35 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 21 \,{\left (c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} + 7 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} 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)^8,x, algorithm="maxima")

[Out]

-1/105*(35*c^2*e^4*x^4 + 35*c^2*d*e^3*x^3 + c^2*d^4 + 2*a*c*d^2*e^2 + 15*a^2*e^4 + 21*(c^2*d^2*e^2 + 2*a*c*e^4
)*x^2 + 7*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

________________________________________________________________________________________

Fricas [A]  time = 1.9446, size = 363, normalized size = 3.18 \begin{align*} -\frac{35 \, c^{2} e^{4} x^{4} + 35 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4} + 21 \,{\left (c^{2} d^{2} e^{2} + 2 \, a c e^{4}\right )} x^{2} + 7 \,{\left (c^{2} d^{3} e + 2 \, a c d e^{3}\right )} x}{105 \,{\left (e^{12} x^{7} + 7 \, d e^{11} x^{6} + 21 \, d^{2} e^{10} x^{5} + 35 \, d^{3} e^{9} x^{4} + 35 \, d^{4} e^{8} x^{3} + 21 \, d^{5} e^{7} x^{2} + 7 \, d^{6} e^{6} x + d^{7} 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)^8,x, algorithm="fricas")

[Out]

-1/105*(35*c^2*e^4*x^4 + 35*c^2*d*e^3*x^3 + c^2*d^4 + 2*a*c*d^2*e^2 + 15*a^2*e^4 + 21*(c^2*d^2*e^2 + 2*a*c*e^4
)*x^2 + 7*(c^2*d^3*e + 2*a*c*d*e^3)*x)/(e^12*x^7 + 7*d*e^11*x^6 + 21*d^2*e^10*x^5 + 35*d^3*e^9*x^4 + 35*d^4*e^
8*x^3 + 21*d^5*e^7*x^2 + 7*d^6*e^6*x + d^7*e^5)

________________________________________________________________________________________

Sympy [A]  time = 3.76125, size = 184, normalized size = 1.61 \begin{align*} - \frac{15 a^{2} e^{4} + 2 a c d^{2} e^{2} + c^{2} d^{4} + 35 c^{2} d e^{3} x^{3} + 35 c^{2} e^{4} x^{4} + x^{2} \left (42 a c e^{4} + 21 c^{2} d^{2} e^{2}\right ) + x \left (14 a c d e^{3} + 7 c^{2} d^{3} e\right )}{105 d^{7} e^{5} + 735 d^{6} e^{6} x + 2205 d^{5} e^{7} x^{2} + 3675 d^{4} e^{8} x^{3} + 3675 d^{3} e^{9} x^{4} + 2205 d^{2} e^{10} x^{5} + 735 d e^{11} x^{6} + 105 e^{12} x^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x**2+a)**2/(e*x+d)**8,x)

[Out]

-(15*a**2*e**4 + 2*a*c*d**2*e**2 + c**2*d**4 + 35*c**2*d*e**3*x**3 + 35*c**2*e**4*x**4 + x**2*(42*a*c*e**4 + 2
1*c**2*d**2*e**2) + x*(14*a*c*d*e**3 + 7*c**2*d**3*e))/(105*d**7*e**5 + 735*d**6*e**6*x + 2205*d**5*e**7*x**2
+ 3675*d**4*e**8*x**3 + 3675*d**3*e**9*x**4 + 2205*d**2*e**10*x**5 + 735*d*e**11*x**6 + 105*e**12*x**7)

________________________________________________________________________________________

Giac [A]  time = 1.34554, size = 132, normalized size = 1.16 \begin{align*} -\frac{{\left (35 \, c^{2} x^{4} e^{4} + 35 \, c^{2} d x^{3} e^{3} + 21 \, c^{2} d^{2} x^{2} e^{2} + 7 \, c^{2} d^{3} x e + c^{2} d^{4} + 42 \, a c x^{2} e^{4} + 14 \, a c d x e^{3} + 2 \, a c d^{2} e^{2} + 15 \, a^{2} e^{4}\right )} e^{\left (-5\right )}}{105 \,{\left (x e + d\right )}^{7}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((c*x^2+a)^2/(e*x+d)^8,x, algorithm="giac")

[Out]

-1/105*(35*c^2*x^4*e^4 + 35*c^2*d*x^3*e^3 + 21*c^2*d^2*x^2*e^2 + 7*c^2*d^3*x*e + c^2*d^4 + 42*a*c*x^2*e^4 + 14
*a*c*d*x*e^3 + 2*a*c*d^2*e^2 + 15*a^2*e^4)*e^(-5)/(x*e + d)^7