\(\int \frac {(a+c x^2)^2}{(d+e x)^7} \, dx\) [470]

Optimal result

Integrand size = 17, antiderivative size = 117 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\left (c d^2+a e^2\right )^2}{6 e^5 (d+e x)^6}+\frac {4 c d \left (c d^2+a e^2\right )}{5 e^5 (d+e x)^5}-\frac {c \left (3 c d^2+a e^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c^2 d}{3 e^5 (d+e x)^3}-\frac {c^2}{2 e^5 (d+e x)^2} \]

[Out]

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

Rubi [A] (verified)

Time = 0.05 (sec) , antiderivative size = 117, 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} \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {c \left (a e^2+3 c d^2\right )}{2 e^5 (d+e x)^4}+\frac {4 c d \left (a e^2+c d^2\right )}{5 e^5 (d+e x)^5}-\frac {\left (a e^2+c d^2\right )^2}{6 e^5 (d+e x)^6}-\frac {c^2}{2 e^5 (d+e x)^2}+\frac {4 c^2 d}{3 e^5 (d+e x)^3} \]

[In]

Int[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

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

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

Mathematica [A] (verified)

Time = 0.03 (sec) , antiderivative size = 89, normalized size of antiderivative = 0.76 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {5 a^2 e^4+a c e^2 \left (d^2+6 d e x+15 e^2 x^2\right )+c^2 \left (d^4+6 d^3 e x+15 d^2 e^2 x^2+20 d e^3 x^3+15 e^4 x^4\right )}{30 e^5 (d+e x)^6} \]

[In]

Integrate[(a + c*x^2)^2/(d + e*x)^7,x]

[Out]

-1/30*(5*a^2*e^4 + a*c*e^2*(d^2 + 6*d*e*x + 15*e^2*x^2) + c^2*(d^4 + 6*d^3*e*x + 15*d^2*e^2*x^2 + 20*d*e^3*x^3
+ 15*e^4*x^4))/(e^5*(d + e*x)^6)

Maple [A] (verified)

Time = 2.14 (sec) , antiderivative size = 102, normalized size of antiderivative = 0.87

method	result	size
risch	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}-\frac {c \left (e^{2} a +c \,d^{2}\right ) x^{2}}{2 e^{3}}-\frac {d c \left (e^{2} a +c \,d^{2}\right ) x}{5 e^{4}}-\frac {5 a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{5}}}{\left (e x +d \right )^{6}}\)	\(102\)
gosper	\(-\frac {15 c^{2} x^{4} e^{4}+20 x^{3} c^{2} d \,e^{3}+15 x^{2} a c \,e^{4}+15 x^{2} c^{2} d^{2} e^{2}+6 x a c d \,e^{3}+6 x \,c^{2} d^{3} e +5 a^{2} e^{4}+a c \,d^{2} e^{2}+c^{2} d^{4}}{30 e^{5} \left (e x +d \right )^{6}}\)	\(104\)
norman	\(\frac {-\frac {c^{2} x^{4}}{2 e}-\frac {2 c^{2} d \,x^{3}}{3 e^{2}}-\frac {\left (e^{3} a c +d^{2} e \,c^{2}\right ) x^{2}}{2 e^{4}}-\frac {d \left (e^{3} a c +d^{2} e \,c^{2}\right ) x}{5 e^{5}}-\frac {5 a^{2} e^{5}+a c \,d^{2} e^{3}+c^{2} d^{4} e}{30 e^{6}}}{\left (e x +d \right )^{6}}\)	\(109\)
parallelrisch	\(\frac {-15 x^{4} c^{2} e^{5}-20 c^{2} d \,x^{3} e^{4}-15 a c \,e^{5} x^{2}-15 c^{2} d^{2} e^{3} x^{2}-6 a c d \,e^{4} x -6 c^{2} d^{3} e^{2} x -5 a^{2} e^{5}-a c \,d^{2} e^{3}-c^{2} d^{4} e}{30 e^{6} \left (e x +d \right )^{6}}\)	\(109\)
default	\(-\frac {a^{2} e^{4}+2 a c \,d^{2} e^{2}+c^{2} d^{4}}{6 e^{5} \left (e x +d \right )^{6}}+\frac {4 c^{2} d}{3 e^{5} \left (e x +d \right )^{3}}-\frac {c \left (e^{2} a +3 c \,d^{2}\right )}{2 e^{5} \left (e x +d \right )^{4}}-\frac {c^{2}}{2 e^{5} \left (e x +d \right )^{2}}+\frac {4 c d \left (e^{2} a +c \,d^{2}\right )}{5 e^{5} \left (e x +d \right )^{5}}\)	\(120\)

[In]

int((c*x^2+a)^2/(e*x+d)^7,x,method=_RETURNVERBOSE)

[Out]

(-1/2*c^2*x^4/e-2/3*c^2*d*x^3/e^2-1/2*c/e^3*(a*e^2+c*d^2)*x^2-1/5*d*c/e^4*(a*e^2+c*d^2)*x-1/30/e^5*(5*a^2*e^4+
a*c*d^2*e^2+c^2*d^4))/(e*x+d)^6

Fricas [A] (verification not implemented)

none

Time = 0.25 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

[In]

integrate((c*x^2+a)^2/(e*x+d)^7,x, algorithm="fricas")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 15*(c^2*d^2*e^2 + a*c*e^4)*x^2
+ 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6
*d^5*e^6*x + d^6*e^5)

Sympy [A] (verification not implemented)

Time = 1.26 (sec) , antiderivative size = 172, normalized size of antiderivative = 1.47 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=\frac {- 5 a^{2} e^{4} - a c d^{2} e^{2} - c^{2} d^{4} - 20 c^{2} d e^{3} x^{3} - 15 c^{2} e^{4} x^{4} + x^{2} \left (- 15 a c e^{4} - 15 c^{2} d^{2} e^{2}\right ) + x \left (- 6 a c d e^{3} - 6 c^{2} d^{3} e\right )}{30 d^{6} e^{5} + 180 d^{5} e^{6} x + 450 d^{4} e^{7} x^{2} + 600 d^{3} e^{8} x^{3} + 450 d^{2} e^{9} x^{4} + 180 d e^{10} x^{5} + 30 e^{11} x^{6}} \]

[In]

integrate((c*x**2+a)**2/(e*x+d)**7,x)

[Out]

(-5*a**2*e**4 - a*c*d**2*e**2 - c**2*d**4 - 20*c**2*d*e**3*x**3 - 15*c**2*e**4*x**4 + x**2*(-15*a*c*e**4 - 15*
c**2*d**2*e**2) + x*(-6*a*c*d*e**3 - 6*c**2*d**3*e))/(30*d**6*e**5 + 180*d**5*e**6*x + 450*d**4*e**7*x**2 + 60
0*d**3*e**8*x**3 + 450*d**2*e**9*x**4 + 180*d*e**10*x**5 + 30*e**11*x**6)

Maxima [A] (verification not implemented)

none

Time = 0.21 (sec) , antiderivative size = 159, normalized size of antiderivative = 1.36 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4} + 15 \, {\left (c^{2} d^{2} e^{2} + a c e^{4}\right )} x^{2} + 6 \, {\left (c^{2} d^{3} e + a c d e^{3}\right )} x}{30 \, {\left (e^{11} x^{6} + 6 \, d e^{10} x^{5} + 15 \, d^{2} e^{9} x^{4} + 20 \, d^{3} e^{8} x^{3} + 15 \, d^{4} e^{7} x^{2} + 6 \, d^{5} e^{6} x + d^{6} e^{5}\right )}} \]

[In]

integrate((c*x^2+a)^2/(e*x+d)^7,x, algorithm="maxima")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4 + 15*(c^2*d^2*e^2 + a*c*e^4)*x^2
+ 6*(c^2*d^3*e + a*c*d*e^3)*x)/(e^11*x^6 + 6*d*e^10*x^5 + 15*d^2*e^9*x^4 + 20*d^3*e^8*x^3 + 15*d^4*e^7*x^2 + 6
*d^5*e^6*x + d^6*e^5)

Giac [A] (verification not implemented)

none

Time = 0.28 (sec) , antiderivative size = 103, normalized size of antiderivative = 0.88 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {15 \, c^{2} e^{4} x^{4} + 20 \, c^{2} d e^{3} x^{3} + 15 \, c^{2} d^{2} e^{2} x^{2} + 15 \, a c e^{4} x^{2} + 6 \, c^{2} d^{3} e x + 6 \, a c d e^{3} x + c^{2} d^{4} + a c d^{2} e^{2} + 5 \, a^{2} e^{4}}{30 \, {\left (e x + d\right )}^{6} e^{5}} \]

[In]

integrate((c*x^2+a)^2/(e*x+d)^7,x, algorithm="giac")

[Out]

-1/30*(15*c^2*e^4*x^4 + 20*c^2*d*e^3*x^3 + 15*c^2*d^2*e^2*x^2 + 15*a*c*e^4*x^2 + 6*c^2*d^3*e*x + 6*a*c*d*e^3*x
+ c^2*d^4 + a*c*d^2*e^2 + 5*a^2*e^4)/((e*x + d)^6*e^5)

Mupad [B] (verification not implemented)

Time = 9.42 (sec) , antiderivative size = 157, normalized size of antiderivative = 1.34 \[ \int \frac {\left (a+c x^2\right )^2}{(d+e x)^7} \, dx=-\frac {\frac {5\,a^2\,e^4+a\,c\,d^2\,e^2+c^2\,d^4}{30\,e^5}+\frac {c^2\,x^4}{2\,e}+\frac {2\,c^2\,d\,x^3}{3\,e^2}+\frac {c\,x^2\,\left (c\,d^2+a\,e^2\right )}{2\,e^3}+\frac {c\,d\,x\,\left (c\,d^2+a\,e^2\right )}{5\,e^4}}{d^6+6\,d^5\,e\,x+15\,d^4\,e^2\,x^2+20\,d^3\,e^3\,x^3+15\,d^2\,e^4\,x^4+6\,d\,e^5\,x^5+e^6\,x^6} \]

[In]

int((a + c*x^2)^2/(d + e*x)^7,x)

[Out]

-((5*a^2*e^4 + c^2*d^4 + a*c*d^2*e^2)/(30*e^5) + (c^2*x^4)/(2*e) + (2*c^2*d*x^3)/(3*e^2) + (c*x^2*(a*e^2 + c*d
^2))/(2*e^3) + (c*d*x*(a*e^2 + c*d^2))/(5*e^4))/(d^6 + e^6*x^6 + 6*d*e^5*x^5 + 15*d^4*e^2*x^2 + 20*d^3*e^3*x^3
+ 15*d^2*e^4*x^4 + 6*d^5*e*x)