\(\int \frac {(a+c x^2)^4}{(d+e x)^2} \, dx\) [496]

Optimal result

Integrand size = 17, antiderivative size = 255 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right ) x}{e^8}-\frac {c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^7}+\frac {c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^3}{3 e^6}-\frac {c^3 d \left (c d^2+2 a e^2\right ) x^4}{e^5}+\frac {c^3 \left (3 c d^2+4 a e^2\right ) x^5}{5 e^4}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2}-\frac {\left (c d^2+a e^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^9} \]

[Out]

c*(4*a^3*e^6+18*a^2*c*d^2*e^4+20*a*c^2*d^4*e^2+7*c^3*d^6)*x/e^8-c^2*d*(6*a^2*e^4+8*a*c*d^2*e^2+3*c^2*d^4)*x^2/
e^7+1/3*c^2*(6*a^2*e^4+12*a*c*d^2*e^2+5*c^2*d^4)*x^3/e^6-c^3*d*(2*a*e^2+c*d^2)*x^4/e^5+1/5*c^3*(4*a*e^2+3*c*d^
2)*x^5/e^4-1/3*c^4*d*x^6/e^3+1/7*c^4*x^7/e^2-(a*e^2+c*d^2)^4/e^9/(e*x+d)-8*c*d*(a*e^2+c*d^2)^3*ln(e*x+d)/e^9

Rubi [A] (verified)

Time = 0.17 (sec) , antiderivative size = 255, 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} \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=-\frac {c^2 d x^2 \left (6 a^2 e^4+8 a c d^2 e^2+3 c^2 d^4\right )}{e^7}+\frac {c^2 x^3 \left (6 a^2 e^4+12 a c d^2 e^2+5 c^2 d^4\right )}{3 e^6}+\frac {c x \left (4 a^3 e^6+18 a^2 c d^2 e^4+20 a c^2 d^4 e^2+7 c^3 d^6\right )}{e^8}-\frac {c^3 d x^4 \left (2 a e^2+c d^2\right )}{e^5}+\frac {c^3 x^5 \left (4 a e^2+3 c d^2\right )}{5 e^4}-\frac {\left (a e^2+c d^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (a e^2+c d^2\right )^3 \log (d+e x)}{e^9}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2} \]

[In]

Int[(a + c*x^2)^4/(d + e*x)^2,x]

[Out]

(c*(7*c^3*d^6 + 20*a*c^2*d^4*e^2 + 18*a^2*c*d^2*e^4 + 4*a^3*e^6)*x)/e^8 - (c^2*d*(3*c^2*d^4 + 8*a*c*d^2*e^2 +
6*a^2*e^4)*x^2)/e^7 + (c^2*(5*c^2*d^4 + 12*a*c*d^2*e^2 + 6*a^2*e^4)*x^3)/(3*e^6) - (c^3*d*(c*d^2 + 2*a*e^2)*x^
4)/e^5 + (c^3*(3*c*d^2 + 4*a*e^2)*x^5)/(5*e^4) - (c^4*d*x^6)/(3*e^3) + (c^4*x^7)/(7*e^2) - (c*d^2 + a*e^2)^4/(
e^9*(d + e*x)) - (8*c*d*(c*d^2 + a*e^2)^3*Log[d + e*x])/e^9

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 {c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right )}{e^8}-\frac {2 c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x}{e^7}+\frac {c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^6}-\frac {4 c^3 d \left (c d^2+2 a e^2\right ) x^3}{e^5}+\frac {c^3 \left (3 c d^2+4 a e^2\right ) x^4}{e^4}-\frac {2 c^4 d x^5}{e^3}+\frac {c^4 x^6}{e^2}+\frac {\left (c d^2+a e^2\right )^4}{e^8 (d+e x)^2}-\frac {8 c d \left (c d^2+a e^2\right )^3}{e^8 (d+e x)}\right ) \, dx \\ & = \frac {c \left (7 c^3 d^6+20 a c^2 d^4 e^2+18 a^2 c d^2 e^4+4 a^3 e^6\right ) x}{e^8}-\frac {c^2 d \left (3 c^2 d^4+8 a c d^2 e^2+6 a^2 e^4\right ) x^2}{e^7}+\frac {c^2 \left (5 c^2 d^4+12 a c d^2 e^2+6 a^2 e^4\right ) x^3}{3 e^6}-\frac {c^3 d \left (c d^2+2 a e^2\right ) x^4}{e^5}+\frac {c^3 \left (3 c d^2+4 a e^2\right ) x^5}{5 e^4}-\frac {c^4 d x^6}{3 e^3}+\frac {c^4 x^7}{7 e^2}-\frac {\left (c d^2+a e^2\right )^4}{e^9 (d+e x)}-\frac {8 c d \left (c d^2+a e^2\right )^3 \log (d+e x)}{e^9} \\ \end{align*}

Mathematica [A] (verified)

Time = 0.10 (sec) , antiderivative size = 289, normalized size of antiderivative = 1.13 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {-105 a^4 e^8+420 a^3 c e^6 \left (-d^2+d e x+e^2 x^2\right )+210 a^2 c^2 e^4 \left (-3 d^4+9 d^3 e x+6 d^2 e^2 x^2-2 d e^3 x^3+e^4 x^4\right )+42 a c^3 e^2 \left (-10 d^6+50 d^5 e x+30 d^4 e^2 x^2-10 d^3 e^3 x^3+5 d^2 e^4 x^4-3 d e^5 x^5+2 e^6 x^6\right )+c^4 \left (-105 d^8+735 d^7 e x+420 d^6 e^2 x^2-140 d^5 e^3 x^3+70 d^4 e^4 x^4-42 d^3 e^5 x^5+28 d^2 e^6 x^6-20 d e^7 x^7+15 e^8 x^8\right )-840 c d \left (c d^2+a e^2\right )^3 (d+e x) \log (d+e x)}{105 e^9 (d+e x)} \]

[In]

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

[Out]

(-105*a^4*e^8 + 420*a^3*c*e^6*(-d^2 + d*e*x + e^2*x^2) + 210*a^2*c^2*e^4*(-3*d^4 + 9*d^3*e*x + 6*d^2*e^2*x^2 -
2*d*e^3*x^3 + e^4*x^4) + 42*a*c^3*e^2*(-10*d^6 + 50*d^5*e*x + 30*d^4*e^2*x^2 - 10*d^3*e^3*x^3 + 5*d^2*e^4*x^4
- 3*d*e^5*x^5 + 2*e^6*x^6) + c^4*(-105*d^8 + 735*d^7*e*x + 420*d^6*e^2*x^2 - 140*d^5*e^3*x^3 + 70*d^4*e^4*x^4
- 42*d^3*e^5*x^5 + 28*d^2*e^6*x^6 - 20*d*e^7*x^7 + 15*e^8*x^8) - 840*c*d*(c*d^2 + a*e^2)^3*(d + e*x)*Log[d +
e*x])/(105*e^9*(d + e*x))

Maple [A] (verified)

Time = 2.20 (sec) , antiderivative size = 315, normalized size of antiderivative = 1.24

method	result	size
norman	\(\frac {-\frac {a^{4} e^{8}+8 a^{3} c \,d^{2} e^{6}+24 a^{2} c^{2} d^{4} e^{4}+24 a \,c^{3} d^{6} e^{2}+8 c^{4} d^{8}}{e^{9}}+\frac {c^{4} x^{8}}{7 e}+\frac {4 c \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) x^{2}}{e^{7}}+\frac {2 c^{2} \left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{4}}{3 e^{5}}+\frac {4 c^{3} \left (3 e^{2} a +c \,d^{2}\right ) x^{6}}{15 e^{3}}-\frac {4 d \,c^{4} x^{7}}{21 e^{2}}-\frac {4 d \,c^{2} \left (3 a^{2} e^{4}+3 a c \,d^{2} e^{2}+c^{2} d^{4}\right ) x^{3}}{3 e^{6}}-\frac {2 d \,c^{3} \left (3 e^{2} a +c \,d^{2}\right ) x^{5}}{5 e^{4}}}{e x +d}-\frac {8 c d \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(315\)
default	\(\frac {c \left (\frac {1}{7} x^{7} c^{3} e^{6}-\frac {1}{3} x^{6} c^{3} d \,e^{5}+\frac {4}{5} x^{5} a \,c^{2} e^{6}+\frac {3}{5} x^{5} c^{3} d^{2} e^{4}-2 x^{4} a \,c^{2} d \,e^{5}-x^{4} c^{3} d^{3} e^{3}+2 x^{3} a^{2} c \,e^{6}+4 x^{3} a \,c^{2} d^{2} e^{4}+\frac {5}{3} x^{3} c^{3} d^{4} e^{2}-6 x^{2} a^{2} c d \,e^{5}-8 x^{2} a \,c^{2} d^{3} e^{3}-3 x^{2} c^{3} d^{5} e +4 e^{6} a^{3} x +18 d^{2} e^{4} a^{2} c x +20 d^{4} e^{2} c^{2} a x +7 c^{3} d^{6} x \right )}{e^{8}}-\frac {a^{4} e^{8}+4 a^{3} c \,d^{2} e^{6}+6 a^{2} c^{2} d^{4} e^{4}+4 a \,c^{3} d^{6} e^{2}+c^{4} d^{8}}{e^{9} \left (e x +d \right )}-\frac {8 c d \left (e^{6} a^{3}+3 d^{2} e^{4} a^{2} c +3 d^{4} e^{2} c^{2} a +c^{3} d^{6}\right ) \ln \left (e x +d \right )}{e^{9}}\)	\(326\)
risch	\(-\frac {3 c^{4} x^{2} d^{5}}{e^{7}}+\frac {4 c \,a^{3} x}{e^{2}}+\frac {7 c^{4} d^{6} x}{e^{8}}-\frac {c^{4} d^{8}}{e^{9} \left (e x +d \right )}+\frac {4 c^{3} x^{5} a}{5 e^{2}}+\frac {3 c^{4} x^{5} d^{2}}{5 e^{4}}-\frac {c^{4} x^{4} d^{3}}{e^{5}}+\frac {2 c^{2} x^{3} a^{2}}{e^{2}}+\frac {5 c^{4} x^{3} d^{4}}{3 e^{6}}-\frac {a^{4}}{e \left (e x +d \right )}-\frac {6 a^{2} c^{2} d^{4}}{e^{5} \left (e x +d \right )}-\frac {4 a \,c^{3} d^{6}}{e^{7} \left (e x +d \right )}-\frac {2 c^{3} x^{4} a d}{e^{3}}+\frac {c^{4} x^{7}}{7 e^{2}}-\frac {8 c d \ln \left (e x +d \right ) a^{3}}{e^{3}}+\frac {4 c^{3} x^{3} a \,d^{2}}{e^{4}}-\frac {24 c^{2} d^{3} \ln \left (e x +d \right ) a^{2}}{e^{5}}-\frac {24 c^{3} d^{5} \ln \left (e x +d \right ) a}{e^{7}}-\frac {8 c^{4} d^{7} \ln \left (e x +d \right )}{e^{9}}-\frac {6 c^{2} x^{2} a^{2} d}{e^{3}}-\frac {8 c^{3} x^{2} a \,d^{3}}{e^{5}}+\frac {18 c^{2} d^{2} a^{2} x}{e^{4}}+\frac {20 c^{3} d^{4} a x}{e^{6}}-\frac {4 a^{3} c \,d^{2}}{e^{3} \left (e x +d \right )}-\frac {c^{4} d \,x^{6}}{3 e^{3}}\)	\(378\)
parallelrisch	\(-\frac {2520 \ln \left (e x +d \right ) x a \,c^{3} d^{5} e^{3}+2520 \ln \left (e x +d \right ) x \,a^{2} c^{2} d^{3} e^{5}+42 x^{5} c^{4} d^{3} e^{5}-210 x^{4} a^{2} c^{2} e^{8}-70 x^{4} c^{4} d^{4} e^{4}+140 x^{3} c^{4} d^{5} e^{3}-420 x^{2} a^{3} c \,e^{8}-420 x^{2} c^{4} d^{6} e^{2}+20 d \,c^{4} x^{7} e^{7}-84 x^{6} a \,c^{3} e^{8}-28 x^{6} c^{4} d^{2} e^{6}+840 \ln \left (e x +d \right ) x \,a^{3} c d \,e^{7}+840 a^{3} c \,d^{2} e^{6}+2520 a^{2} c^{2} d^{4} e^{4}+2520 a \,c^{3} d^{6} e^{2}+840 \ln \left (e x +d \right ) x \,c^{4} d^{7} e +420 x^{3} a \,c^{3} d^{3} e^{5}-1260 x^{2} a^{2} c^{2} d^{2} e^{6}-1260 x^{2} a \,c^{3} d^{4} e^{4}+840 \ln \left (e x +d \right ) a^{3} c \,d^{2} e^{6}+2520 \ln \left (e x +d \right ) a^{2} c^{2} d^{4} e^{4}+2520 \ln \left (e x +d \right ) a \,c^{3} d^{6} e^{2}+840 c^{4} d^{8}+105 a^{4} e^{8}+840 \ln \left (e x +d \right ) c^{4} d^{8}-15 x^{8} c^{4} e^{8}+420 x^{3} a^{2} c^{2} d \,e^{7}+126 x^{5} a \,c^{3} d \,e^{7}-210 x^{4} a \,c^{3} d^{2} e^{6}}{105 e^{9} \left (e x +d \right )}\)	\(432\)

[In]

int((c*x^2+a)^4/(e*x+d)^2,x,method=_RETURNVERBOSE)

[Out]

(-(a^4*e^8+8*a^3*c*d^2*e^6+24*a^2*c^2*d^4*e^4+24*a*c^3*d^6*e^2+8*c^4*d^8)/e^9+1/7/e*c^4*x^8+4*c/e^7*(a^3*e^6+3
*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2+c^3*d^6)*x^2+2/3*c^2*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4)/e^5*x^4+4/15*c^3*(3*a*e^
2+c*d^2)/e^3*x^6-4/21*d/e^2*c^4*x^7-4/3*d*c^2*(3*a^2*e^4+3*a*c*d^2*e^2+c^2*d^4)/e^6*x^3-2/5*d*c^3*(3*a*e^2+c*d
^2)/e^4*x^5)/(e*x+d)-8*c*d/e^9*(a^3*e^6+3*a^2*c*d^2*e^4+3*a*c^2*d^4*e^2+c^3*d^6)*ln(e*x+d)

Fricas [A] (verification not implemented)

none

Time = 0.54 (sec) , antiderivative size = 421, normalized size of antiderivative = 1.65 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {15 \, c^{4} e^{8} x^{8} - 20 \, c^{4} d e^{7} x^{7} - 105 \, c^{4} d^{8} - 420 \, a c^{3} d^{6} e^{2} - 630 \, a^{2} c^{2} d^{4} e^{4} - 420 \, a^{3} c d^{2} e^{6} - 105 \, a^{4} e^{8} + 28 \, {\left (c^{4} d^{2} e^{6} + 3 \, a c^{3} e^{8}\right )} x^{6} - 42 \, {\left (c^{4} d^{3} e^{5} + 3 \, a c^{3} d e^{7}\right )} x^{5} + 70 \, {\left (c^{4} d^{4} e^{4} + 3 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )} x^{4} - 140 \, {\left (c^{4} d^{5} e^{3} + 3 \, a c^{3} d^{3} e^{5} + 3 \, a^{2} c^{2} d e^{7}\right )} x^{3} + 420 \, {\left (c^{4} d^{6} e^{2} + 3 \, a c^{3} d^{4} e^{4} + 3 \, a^{2} c^{2} d^{2} e^{6} + a^{3} c e^{8}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{7} e + 20 \, a c^{3} d^{5} e^{3} + 18 \, a^{2} c^{2} d^{3} e^{5} + 4 \, a^{3} c d e^{7}\right )} x - 840 \, {\left (c^{4} d^{8} + 3 \, a c^{3} d^{6} e^{2} + 3 \, a^{2} c^{2} d^{4} e^{4} + a^{3} c d^{2} e^{6} + {\left (c^{4} d^{7} e + 3 \, a c^{3} d^{5} e^{3} + 3 \, a^{2} c^{2} d^{3} e^{5} + a^{3} c d e^{7}\right )} x\right )} \log \left (e x + d\right )}{105 \, {\left (e^{10} x + d e^{9}\right )}} \]

[In]

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

[Out]

1/105*(15*c^4*e^8*x^8 - 20*c^4*d*e^7*x^7 - 105*c^4*d^8 - 420*a*c^3*d^6*e^2 - 630*a^2*c^2*d^4*e^4 - 420*a^3*c*d
^2*e^6 - 105*a^4*e^8 + 28*(c^4*d^2*e^6 + 3*a*c^3*e^8)*x^6 - 42*(c^4*d^3*e^5 + 3*a*c^3*d*e^7)*x^5 + 70*(c^4*d^4
*e^4 + 3*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)*x^4 - 140*(c^4*d^5*e^3 + 3*a*c^3*d^3*e^5 + 3*a^2*c^2*d*e^7)*x^3 + 420*
(c^4*d^6*e^2 + 3*a*c^3*d^4*e^4 + 3*a^2*c^2*d^2*e^6 + a^3*c*e^8)*x^2 + 105*(7*c^4*d^7*e + 20*a*c^3*d^5*e^3 + 18
*a^2*c^2*d^3*e^5 + 4*a^3*c*d*e^7)*x - 840*(c^4*d^8 + 3*a*c^3*d^6*e^2 + 3*a^2*c^2*d^4*e^4 + a^3*c*d^2*e^6 + (c^
4*d^7*e + 3*a*c^3*d^5*e^3 + 3*a^2*c^2*d^3*e^5 + a^3*c*d*e^7)*x)*log(e*x + d))/(e^10*x + d*e^9)

Sympy [A] (verification not implemented)

Time = 0.56 (sec) , antiderivative size = 314, normalized size of antiderivative = 1.23 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=- \frac {c^{4} d x^{6}}{3 e^{3}} + \frac {c^{4} x^{7}}{7 e^{2}} - \frac {8 c d \left (a e^{2} + c d^{2}\right )^{3} \log {\left (d + e x \right )}}{e^{9}} + x^{5} \cdot \left (\frac {4 a c^{3}}{5 e^{2}} + \frac {3 c^{4} d^{2}}{5 e^{4}}\right ) + x^{4} \left (- \frac {2 a c^{3} d}{e^{3}} - \frac {c^{4} d^{3}}{e^{5}}\right ) + x^{3} \cdot \left (\frac {2 a^{2} c^{2}}{e^{2}} + \frac {4 a c^{3} d^{2}}{e^{4}} + \frac {5 c^{4} d^{4}}{3 e^{6}}\right ) + x^{2} \left (- \frac {6 a^{2} c^{2} d}{e^{3}} - \frac {8 a c^{3} d^{3}}{e^{5}} - \frac {3 c^{4} d^{5}}{e^{7}}\right ) + x \left (\frac {4 a^{3} c}{e^{2}} + \frac {18 a^{2} c^{2} d^{2}}{e^{4}} + \frac {20 a c^{3} d^{4}}{e^{6}} + \frac {7 c^{4} d^{6}}{e^{8}}\right ) + \frac {- a^{4} e^{8} - 4 a^{3} c d^{2} e^{6} - 6 a^{2} c^{2} d^{4} e^{4} - 4 a c^{3} d^{6} e^{2} - c^{4} d^{8}}{d e^{9} + e^{10} x} \]

[In]

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

[Out]

-c**4*d*x**6/(3*e**3) + c**4*x**7/(7*e**2) - 8*c*d*(a*e**2 + c*d**2)**3*log(d + e*x)/e**9 + x**5*(4*a*c**3/(5*
e**2) + 3*c**4*d**2/(5*e**4)) + x**4*(-2*a*c**3*d/e**3 - c**4*d**3/e**5) + x**3*(2*a**2*c**2/e**2 + 4*a*c**3*d
**2/e**4 + 5*c**4*d**4/(3*e**6)) + x**2*(-6*a**2*c**2*d/e**3 - 8*a*c**3*d**3/e**5 - 3*c**4*d**5/e**7) + x*(4*a
**3*c/e**2 + 18*a**2*c**2*d**2/e**4 + 20*a*c**3*d**4/e**6 + 7*c**4*d**6/e**8) + (-a**4*e**8 - 4*a**3*c*d**2*e*
*6 - 6*a**2*c**2*d**4*e**4 - 4*a*c**3*d**6*e**2 - c**4*d**8)/(d*e**9 + e**10*x)

Maxima [A] (verification not implemented)

none

Time = 0.19 (sec) , antiderivative size = 330, normalized size of antiderivative = 1.29 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=-\frac {c^{4} d^{8} + 4 \, a c^{3} d^{6} e^{2} + 6 \, a^{2} c^{2} d^{4} e^{4} + 4 \, a^{3} c d^{2} e^{6} + a^{4} e^{8}}{e^{10} x + d e^{9}} + \frac {15 \, c^{4} e^{6} x^{7} - 35 \, c^{4} d e^{5} x^{6} + 21 \, {\left (3 \, c^{4} d^{2} e^{4} + 4 \, a c^{3} e^{6}\right )} x^{5} - 105 \, {\left (c^{4} d^{3} e^{3} + 2 \, a c^{3} d e^{5}\right )} x^{4} + 35 \, {\left (5 \, c^{4} d^{4} e^{2} + 12 \, a c^{3} d^{2} e^{4} + 6 \, a^{2} c^{2} e^{6}\right )} x^{3} - 105 \, {\left (3 \, c^{4} d^{5} e + 8 \, a c^{3} d^{3} e^{3} + 6 \, a^{2} c^{2} d e^{5}\right )} x^{2} + 105 \, {\left (7 \, c^{4} d^{6} + 20 \, a c^{3} d^{4} e^{2} + 18 \, a^{2} c^{2} d^{2} e^{4} + 4 \, a^{3} c e^{6}\right )} x}{105 \, e^{8}} - \frac {8 \, {\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} \log \left (e x + d\right )}{e^{9}} \]

[In]

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

[Out]

-(c^4*d^8 + 4*a*c^3*d^6*e^2 + 6*a^2*c^2*d^4*e^4 + 4*a^3*c*d^2*e^6 + a^4*e^8)/(e^10*x + d*e^9) + 1/105*(15*c^4*
e^6*x^7 - 35*c^4*d*e^5*x^6 + 21*(3*c^4*d^2*e^4 + 4*a*c^3*e^6)*x^5 - 105*(c^4*d^3*e^3 + 2*a*c^3*d*e^5)*x^4 + 35
*(5*c^4*d^4*e^2 + 12*a*c^3*d^2*e^4 + 6*a^2*c^2*e^6)*x^3 - 105*(3*c^4*d^5*e + 8*a*c^3*d^3*e^3 + 6*a^2*c^2*d*e^5
)*x^2 + 105*(7*c^4*d^6 + 20*a*c^3*d^4*e^2 + 18*a^2*c^2*d^2*e^4 + 4*a^3*c*e^6)*x)/e^8 - 8*(c^4*d^7 + 3*a*c^3*d^
5*e^2 + 3*a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*log(e*x + d)/e^9

Giac [A] (verification not implemented)

none

Time = 0.29 (sec) , antiderivative size = 414, normalized size of antiderivative = 1.62 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=\frac {{\left (15 \, c^{4} - \frac {140 \, c^{4} d}{e x + d} + \frac {84 \, {\left (7 \, c^{4} d^{2} e^{2} + a c^{3} e^{4}\right )}}{{\left (e x + d\right )}^{2} e^{2}} - \frac {210 \, {\left (7 \, c^{4} d^{3} e^{3} + 3 \, a c^{3} d e^{5}\right )}}{{\left (e x + d\right )}^{3} e^{3}} + \frac {70 \, {\left (35 \, c^{4} d^{4} e^{4} + 30 \, a c^{3} d^{2} e^{6} + 3 \, a^{2} c^{2} e^{8}\right )}}{{\left (e x + d\right )}^{4} e^{4}} - \frac {420 \, {\left (7 \, c^{4} d^{5} e^{5} + 10 \, a c^{3} d^{3} e^{7} + 3 \, a^{2} c^{2} d e^{9}\right )}}{{\left (e x + d\right )}^{5} e^{5}} + \frac {420 \, {\left (7 \, c^{4} d^{6} e^{6} + 15 \, a c^{3} d^{4} e^{8} + 9 \, a^{2} c^{2} d^{2} e^{10} + a^{3} c e^{12}\right )}}{{\left (e x + d\right )}^{6} e^{6}}\right )} {\left (e x + d\right )}^{7}}{105 \, e^{9}} + \frac {8 \, {\left (c^{4} d^{7} + 3 \, a c^{3} d^{5} e^{2} + 3 \, a^{2} c^{2} d^{3} e^{4} + a^{3} c d e^{6}\right )} \log \left (\frac {{\left | e x + d \right |}}{{\left (e x + d\right )}^{2} {\left | e \right |}}\right )}{e^{9}} - \frac {\frac {c^{4} d^{8} e^{7}}{e x + d} + \frac {4 \, a c^{3} d^{6} e^{9}}{e x + d} + \frac {6 \, a^{2} c^{2} d^{4} e^{11}}{e x + d} + \frac {4 \, a^{3} c d^{2} e^{13}}{e x + d} + \frac {a^{4} e^{15}}{e x + d}}{e^{16}} \]

[In]

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

[Out]

1/105*(15*c^4 - 140*c^4*d/(e*x + d) + 84*(7*c^4*d^2*e^2 + a*c^3*e^4)/((e*x + d)^2*e^2) - 210*(7*c^4*d^3*e^3 +
3*a*c^3*d*e^5)/((e*x + d)^3*e^3) + 70*(35*c^4*d^4*e^4 + 30*a*c^3*d^2*e^6 + 3*a^2*c^2*e^8)/((e*x + d)^4*e^4) -
420*(7*c^4*d^5*e^5 + 10*a*c^3*d^3*e^7 + 3*a^2*c^2*d*e^9)/((e*x + d)^5*e^5) + 420*(7*c^4*d^6*e^6 + 15*a*c^3*d^4
*e^8 + 9*a^2*c^2*d^2*e^10 + a^3*c*e^12)/((e*x + d)^6*e^6))*(e*x + d)^7/e^9 + 8*(c^4*d^7 + 3*a*c^3*d^5*e^2 + 3*
a^2*c^2*d^3*e^4 + a^3*c*d*e^6)*log(abs(e*x + d)/((e*x + d)^2*abs(e)))/e^9 - (c^4*d^8*e^7/(e*x + d) + 4*a*c^3*d
^6*e^9/(e*x + d) + 6*a^2*c^2*d^4*e^11/(e*x + d) + 4*a^3*c*d^2*e^13/(e*x + d) + a^4*e^15/(e*x + d))/e^16

Mupad [B] (verification not implemented)

Time = 9.40 (sec) , antiderivative size = 701, normalized size of antiderivative = 2.75 \[ \int \frac {\left (a+c x^2\right )^4}{(d+e x)^2} \, dx=x^4\,\left (\frac {c^4\,d^3}{2\,e^5}-\frac {d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{2\,e}\right )+x^5\,\left (\frac {4\,a\,c^3}{5\,e^2}+\frac {3\,c^4\,d^2}{5\,e^4}\right )+x^2\,\left (\frac {d\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{2\,e^2}\right )-x^3\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{3\,e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{3\,e}-\frac {2\,a^2\,c^2}{e^2}\right )+x\,\left (\frac {4\,a^3\,c}{e^2}+\frac {d^2\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {d^2\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e^2}+\frac {2\,d\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e}-\frac {6\,a^2\,c^2}{e^2}\right )}{e}-\frac {d^2\,\left (\frac {2\,c^4\,d^3}{e^5}-\frac {2\,d\,\left (\frac {4\,a\,c^3}{e^2}+\frac {3\,c^4\,d^2}{e^4}\right )}{e}\right )}{e^2}\right )}{e}\right )-\frac {a^4\,e^8+4\,a^3\,c\,d^2\,e^6+6\,a^2\,c^2\,d^4\,e^4+4\,a\,c^3\,d^6\,e^2+c^4\,d^8}{e\,\left (x\,e^9+d\,e^8\right )}+\frac {c^4\,x^7}{7\,e^2}-\frac {\ln \left (d+e\,x\right )\,\left (8\,a^3\,c\,d\,e^6+24\,a^2\,c^2\,d^3\,e^4+24\,a\,c^3\,d^5\,e^2+8\,c^4\,d^7\right )}{e^9}-\frac {c^4\,d\,x^6}{3\,e^3} \]

[In]

int((a + c*x^2)^4/(d + e*x)^2,x)

[Out]

x^4*((c^4*d^3)/(2*e^5) - (d*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))/(2*e)) + x^5*((4*a*c^3)/(5*e^2) + (3*c^4*d^2)/(
5*e^4)) + x^2*((d*((d^2*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))/e^2 + (2*d*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^3)/e^2 +
(3*c^4*d^2)/e^4))/e))/e - (6*a^2*c^2)/e^2))/e - (d^2*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4
))/e))/(2*e^2)) - x^3*((d^2*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))/(3*e^2) + (2*d*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^
3)/e^2 + (3*c^4*d^2)/e^4))/e))/(3*e) - (2*a^2*c^2)/e^2) + x*((4*a^3*c)/e^2 + (d^2*((d^2*((4*a*c^3)/e^2 + (3*c^
4*d^2)/e^4))/e^2 + (2*d*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))/e))/e - (6*a^2*c^2)/e^2))/e
^2 - (2*d*((2*d*((d^2*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))/e^2 + (2*d*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^3)/e^2 + (
3*c^4*d^2)/e^4))/e))/e - (6*a^2*c^2)/e^2))/e - (d^2*((2*c^4*d^3)/e^5 - (2*d*((4*a*c^3)/e^2 + (3*c^4*d^2)/e^4))
/e))/e^2))/e) - (a^4*e^8 + c^4*d^8 + 4*a*c^3*d^6*e^2 + 4*a^3*c*d^2*e^6 + 6*a^2*c^2*d^4*e^4)/(e*(d*e^8 + e^9*x)
) + (c^4*x^7)/(7*e^2) - (log(d + e*x)*(8*c^4*d^7 + 24*a*c^3*d^5*e^2 + 24*a^2*c^2*d^3*e^4 + 8*a^3*c*d*e^6))/e^9
- (c^4*d*x^6)/(3*e^3)