\(\int \frac {(a+c x^2)^2}{(d+e x)^6} \, dx\) [469]

Optimal result

Integrand size = 17, antiderivative size = 110 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\left (c d^2+a e^2\right )^2}{5 e^5 (d+e x)^5}+\frac {c d \left (c d^2+a e^2\right )}{e^5 (d+e x)^4}-\frac {2 c \left (3 c d^2+a e^2\right )}{3 e^5 (d+e x)^3}+\frac {2 c^2 d}{e^5 (d+e x)^2}-\frac {c^2}{e^5 (d+e x)} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 110, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 1, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.059, Rules used = {711} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {2 c \left (a e^2+3 c d^2\right )}{3 e^5 (d+e x)^3}+\frac {c d \left (a e^2+c d^2\right )}{e^5 (d+e x)^4}-\frac {\left (a e^2+c d^2\right )^2}{5 e^5 (d+e x)^5}-\frac {c^2}{e^5 (d+e x)}+\frac {2 c^2 d}{e^5 (d+e x)^2} \]

[In]

Int[(a + c*x^2)^2/(d + e*x)^6,x]

[Out]

-1/5*(c*d^2 + a*e^2)^2/(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 711

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*} \text {integral}& = \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] (verified)

Time = 0.04 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.82 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {3 a^2 e^4+a c e^2 \left (d^2+5 d e x+10 e^2 x^2\right )+3 c^2 \left (d^4+5 d^3 e x+10 d^2 e^2 x^2+10 d e^3 x^3+5 e^4 x^4\right )}{15 e^5 (d+e x)^5} \]

[In]

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

[Out]

-1/15*(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))/(e^5*(d + e*x)^5)

Maple [A] (verified)

Time = 2.13 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.95

method	result	size
gosper	\(-\frac {15 c^{2} x^{4} e^{4}+30 x^{3} c^{2} d \,e^{3}+10 x^{2} a c \,e^{4}+30 x^{2} c^{2} d^{2} e^{2}+5 x a c d \,e^{3}+15 x \,c^{2} d^{3} e +3 a^{2} e^{4}+a c \,d^{2} e^{2}+3 c^{2} d^{4}}{15 \left (e x +d \right )^{5} e^{5}}\)	\(105\)
risch	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {2 c^{2} d \,x^{3}}{e^{2}}-\frac {2 c \left (e^{2} a +3 c \,d^{2}\right ) x^{2}}{3 e^{3}}-\frac {d c \left (e^{2} a +3 c \,d^{2}\right ) x}{3 e^{4}}-\frac {3 a^{2} e^{4}+a c \,d^{2} e^{2}+3 c^{2} d^{4}}{15 e^{5}}}{\left (e x +d \right )^{5}}\)	\(105\)
parallelrisch	\(\frac {-15 c^{2} x^{4} e^{4}-30 x^{3} c^{2} d \,e^{3}-10 x^{2} a c \,e^{4}-30 x^{2} c^{2} d^{2} e^{2}-5 x a c d \,e^{3}-15 x \,c^{2} d^{3} e -3 a^{2} e^{4}-a c \,d^{2} e^{2}-3 c^{2} d^{4}}{15 e^{5} \left (e x +d \right )^{5}}\)	\(106\)
norman	\(\frac {-\frac {c^{2} x^{4}}{e}-\frac {2 c^{2} d \,x^{3}}{e^{2}}-\frac {2 \left (a c \,e^{2}+3 c^{2} d^{2}\right ) x^{2}}{3 e^{3}}-\frac {d \left (a c \,e^{2}+3 c^{2} d^{2}\right ) x}{3 e^{4}}-\frac {3 a^{2} e^{4}+a c \,d^{2} e^{2}+3 c^{2} d^{4}}{15 e^{5}}}{\left (e x +d \right )^{5}}\)	\(109\)
default	\(-\frac {c^{2}}{e^{5} \left (e x +d \right )}-\frac {2 c \left (e^{2} a +3 c \,d^{2}\right )}{3 e^{5} \left (e x +d \right )^{3}}+\frac {c d \left (e^{2} a +c \,d^{2}\right )}{e^{5} \left (e x +d \right )^{4}}+\frac {2 c^{2} d}{e^{5} \left (e x +d \right )^{2}}-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{5 e^{5} \left (e x +d \right )^{5}}\)	\(119\)

[In]

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

[Out]

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

Fricas [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[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)

Sympy [A] (verification not implemented)

Time = 0.99 (sec) , antiderivative size = 162, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=\frac {- 3 a^{2} e^{4} - a c d^{2} e^{2} - 3 c^{2} d^{4} - 30 c^{2} d e^{3} x^{3} - 15 c^{2} e^{4} x^{4} + x^{2} \left (- 10 a c e^{4} - 30 c^{2} d^{2} e^{2}\right ) + x \left (- 5 a c d e^{3} - 15 c^{2} d^{3} e\right )}{15 d^{5} e^{5} + 75 d^{4} e^{6} x + 150 d^{3} e^{7} x^{2} + 150 d^{2} e^{8} x^{3} + 75 d e^{9} x^{4} + 15 e^{10} x^{5}} \]

[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 - 3
0*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 +
150*d**2*e**8*x**3 + 75*d*e**9*x**4 + 15*e**10*x**5)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.37 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4} + 10 \, {\left (3 \, c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 5 \, {\left (3 \, c^{2} d^{3} e + a c d e^{3}\right )} x}{15 \, {\left (e^{10} x^{5} + 5 \, d e^{9} x^{4} + 10 \, d^{2} e^{8} x^{3} + 10 \, d^{3} e^{7} x^{2} + 5 \, d^{4} e^{6} x + d^{5} e^{5}\right )}} \]

[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)

Giac [A] (verification not implemented)

none

Time = 0.27 (sec) , antiderivative size = 104, normalized size of antiderivative = 0.95 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 30 \, c^{2} d e^{3} x^{3} + 30 \, c^{2} d^{2} e^{2} x^{2} + 10 \, a c e^{4} x^{2} + 15 \, c^{2} d^{3} e x + 5 \, a c d e^{3} x + 3 \, c^{2} d^{4} + a c d^{2} e^{2} + 3 \, a^{2} e^{4}}{15 \, {\left (e x + d\right )}^{5} e^{5}} \]

[In]

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

[Out]

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

Mupad [B] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 148, normalized size of antiderivative = 1.35 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^6} \, dx=-\frac {\frac {3\,a^2\,e^4+a\,c\,d^2\,e^2+3\,c^2\,d^4}{15\,e^5}+\frac {c^2\,x^4}{e}+\frac {2\,c^2\,d\,x^3}{e^2}+\frac {2\,c\,x^2\,\left (3\,c\,d^2+a\,e^2\right )}{3\,e^3}+\frac {c\,d\,x\,\left (3\,c\,d^2+a\,e^2\right )}{3\,e^4}}{d^5+5\,d^4\,e\,x+10\,d^3\,e^2\,x^2+10\,d^2\,e^3\,x^3+5\,d\,e^4\,x^4+e^5\,x^5} \]

[In]

int((a + c*x^2)^2/(d + e*x)^6,x)

[Out]

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