3.64 $$\int \frac{(d+e x)^4 (A+B x+C x^2)}{(a+c x^2)^4} \, dx$$

Optimal. Leaf size=234 $\frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^{7/2} c^{7/2}}-\frac{(d+e x) (a e-c d x) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^3 c^3 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}$

[Out]

-((a*B - (A*c - a*C)*x)*(d + e*x)^4)/(6*a*c*(a + c*x^2)^3) - ((d + e*x)^3*(a*(A*c + 5*a*C)*e - c*(5*A*c*d + a*
C*d + 4*a*B*e)*x))/(24*a^2*c^2*(a + c*x^2)^2) - ((a*(A*c + 5*a*C)*e^2 + c*d*(5*A*c*d + a*C*d + 4*a*B*e))*(a*e
- c*d*x)*(d + e*x))/(16*a^3*c^3*(a + c*x^2)) + ((c*d^2 + a*e^2)*(a*(A*c + 5*a*C)*e^2 + c*d*(5*A*c*d + a*C*d +
4*a*B*e))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(7/2))

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Rubi [A]  time = 0.291445, antiderivative size = 234, normalized size of antiderivative = 1., number of steps used = 4, number of rules used = 4, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.148, Rules used = {1645, 805, 723, 205} $\frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^{7/2} c^{7/2}}-\frac{(d+e x) (a e-c d x) \left (c d (4 a B e+a C d+5 A c d)+a e^2 (5 a C+A c)\right )}{16 a^3 c^3 \left (a+c x^2\right )}-\frac{(d+e x)^3 (a e (5 a C+A c)-c x (4 a B e+a C d+5 A c d))}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{(d+e x)^4 (a B-x (A c-a C))}{6 a c \left (a+c x^2\right )^3}$

Antiderivative was successfully veriﬁed.

[In]

Int[((d + e*x)^4*(A + B*x + C*x^2))/(a + c*x^2)^4,x]

[Out]

-((a*B - (A*c - a*C)*x)*(d + e*x)^4)/(6*a*c*(a + c*x^2)^3) - ((d + e*x)^3*(a*(A*c + 5*a*C)*e - c*(5*A*c*d + a*
C*d + 4*a*B*e)*x))/(24*a^2*c^2*(a + c*x^2)^2) - ((a*(A*c + 5*a*C)*e^2 + c*d*(5*A*c*d + a*C*d + 4*a*B*e))*(a*e
- c*d*x)*(d + e*x))/(16*a^3*c^3*(a + c*x^2)) + ((c*d^2 + a*e^2)*(a*(A*c + 5*a*C)*e^2 + c*d*(5*A*c*d + a*C*d +
4*a*B*e))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(7/2))

Rule 1645

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> With[{Q = PolynomialQuotient[Pq,
a + c*x^2, x], f = Coeff[PolynomialRemainder[Pq, a + c*x^2, x], x, 0], g = Coeff[PolynomialRemainder[Pq, a + c
*x^2, x], x, 1]}, Simp[((d + e*x)^m*(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] + Dist[1/(2*a*c*(p
+ 1)), Int[(d + e*x)^(m - 1)*(a + c*x^2)^(p + 1)*ExpandToSum[2*a*c*(p + 1)*(d + e*x)*Q - a*e*g*m + c*d*f*(2*p
+ 3) + c*e*f*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, c, d, e}, x] && PolyQ[Pq, x] && NeQ[c*d^2 + a*e^2, 0] &
& LtQ[p, -1] && GtQ[m, 0] &&  !(IGtQ[m, 0] && RationalQ[a, c, d, e] && (IntegerQ[p] || ILtQ[p + 1/2, 0]))

Rule 805

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^m*
(a + c*x^2)^(p + 1)*(a*g - c*f*x))/(2*a*c*(p + 1)), x] - Dist[(m*(c*d*f + a*e*g))/(2*a*c*(p + 1)), Int[(d + e*
x)^(m - 1)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[Simplif
y[m + 2*p + 3], 0] && LtQ[p, -1]

Rule 723

Int[((d_) + (e_.)*(x_))^(m_)*((a_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp[((d + e*x)^(m - 1)*(a*e - c*d*x)*(a
+ c*x^2)^(p + 1))/(2*a*c*(p + 1)), x] + Dist[((2*p + 3)*(c*d^2 + a*e^2))/(2*a*c*(p + 1)), Int[(d + e*x)^(m -
2)*(a + c*x^2)^(p + 1), x], x] /; FreeQ[{a, c, d, e}, x] && NeQ[c*d^2 + a*e^2, 0] && EqQ[m + 2*p + 2, 0] && Lt
Q[p, -1]

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{(d+e x)^4 \left (A+B x+C x^2\right )}{\left (a+c x^2\right )^4} \, dx &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{\int \frac{(d+e x)^3 (-5 A c d-a C d-4 a B e-(A c+5 a C) e x)}{\left (a+c x^2\right )^3} \, dx}{6 a c}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}+\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) \int \frac{(d+e x)^2}{\left (a+c x^2\right )^2} \, dx}{8 a^2 c^2}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) (a e-c d x) (d+e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{\left (\left (c d^2+a e^2\right ) \left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right )\right ) \int \frac{1}{a+c x^2} \, dx}{16 a^3 c^3}\\ &=-\frac{(a B-(A c-a C) x) (d+e x)^4}{6 a c \left (a+c x^2\right )^3}-\frac{(d+e x)^3 (a (A c+5 a C) e-c (5 A c d+a C d+4 a B e) x)}{24 a^2 c^2 \left (a+c x^2\right )^2}-\frac{\left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) (a e-c d x) (d+e x)}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{\left (c d^2+a e^2\right ) \left (a (A c+5 a C) e^2+c d (5 A c d+a C d+4 a B e)\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{16 a^{7/2} c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.305285, size = 437, normalized size = 1.87 $\frac{a^2 c e^2 x \left (e (A e+4 B d)+6 C d^2\right )-a^3 e^3 (8 B e+32 C d+11 C e x)+a c^2 d^2 x \left (6 A e^2+4 B d e+C d^2\right )+5 A c^3 d^4 x}{16 a^3 c^3 \left (a+c x^2\right )}+\frac{a^2 c e \left (e (A e (4 d+e x)+2 B d (3 d+2 e x))+2 C d^2 (2 d+3 e x)\right )-a^3 e^3 (B e+4 C d+C e x)-a c^2 d^2 \left (4 A d e+6 A e^2 x+B d (d+4 e x)+C d^2 x\right )+A c^3 d^4 x}{6 a c^3 \left (a+c x^2\right )^3}+\frac{-a^2 c e \left (e (A e (24 d+7 e x)+4 B d (9 d+7 e x))+6 C d^2 (4 d+7 e x)\right )+a^3 e^3 (12 B e+48 C d+13 C e x)+a c^2 d^2 x \left (6 A e^2+4 B d e+C d^2\right )+5 A c^3 d^4 x}{24 a^2 c^3 \left (a+c x^2\right )^2}+\frac{\left (a e^2+c d^2\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c \left (a e^2+5 c d^2\right )+a \left (5 a C e^2+c d (4 B e+C d)\right )\right )}{16 a^{7/2} c^{7/2}}$

Antiderivative was successfully veriﬁed.

[In]

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

[Out]

(5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^2*c*e^2*(6*C*d^2 + e*(4*B*d + A*e))*x - a^3*e^3*(
32*C*d + 8*B*e + 11*C*e*x))/(16*a^3*c^3*(a + c*x^2)) + (A*c^3*d^4*x - a^3*e^3*(4*C*d + B*e + C*e*x) - a*c^2*d^
2*(4*A*d*e + C*d^2*x + 6*A*e^2*x + B*d*(d + 4*e*x)) + a^2*c*e*(2*C*d^2*(2*d + 3*e*x) + e*(A*e*(4*d + e*x) + 2*
B*d*(3*d + 2*e*x))))/(6*a*c^3*(a + c*x^2)^3) + (5*A*c^3*d^4*x + a*c^2*d^2*(C*d^2 + 4*B*d*e + 6*A*e^2)*x + a^3*
e^3*(48*C*d + 12*B*e + 13*C*e*x) - a^2*c*e*(6*C*d^2*(4*d + 7*e*x) + e*(4*B*d*(9*d + 7*e*x) + A*e*(24*d + 7*e*x
))))/(24*a^2*c^3*(a + c*x^2)^2) + ((c*d^2 + a*e^2)*(A*c*(5*c*d^2 + a*e^2) + a*(5*a*C*e^2 + c*d*(C*d + 4*B*e)))
*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(16*a^(7/2)*c^(7/2))

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Maple [B]  time = 0.056, size = 647, normalized size = 2.8 \begin{align*}{\frac{1}{ \left ( c{x}^{2}+a \right ) ^{3}} \left ({\frac{ \left ( A{e}^{4}{a}^{2}c+6\,Aa{c}^{2}{d}^{2}{e}^{2}+5\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c+4\,Ba{c}^{2}{d}^{3}e-11\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}+Ca{c}^{2}{d}^{4} \right ){x}^{5}}{16\,{a}^{3}c}}-{\frac{{e}^{3} \left ( Be+4\,Cd \right ){x}^{4}}{2\,c}}-{\frac{ \left ( A{e}^{4}{a}^{2}c-6\,Aa{c}^{2}{d}^{2}{e}^{2}-5\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c-4\,Ba{c}^{2}{d}^{3}e+5\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}-Ca{c}^{2}{d}^{4} \right ){x}^{3}}{6\,{a}^{2}{c}^{2}}}-{\frac{e \left ( 2\,Acd{e}^{2}+Ba{e}^{3}+3\,Bc{d}^{2}e+4\,Cad{e}^{2}+2\,Cc{d}^{3} \right ){x}^{2}}{2\,{c}^{2}}}-{\frac{ \left ( A{e}^{4}{a}^{2}c+6\,Aa{c}^{2}{d}^{2}{e}^{2}-11\,A{c}^{3}{d}^{4}+4\,Bd{a}^{2}{e}^{3}c+4\,Ba{c}^{2}{d}^{3}e+5\,{a}^{3}C{e}^{4}+6\,C{a}^{2}c{d}^{2}{e}^{2}+Ca{c}^{2}{d}^{4} \right ) x}{16\,a{c}^{3}}}-{\frac{2\,Aacd{e}^{3}+4\,A{c}^{2}{d}^{3}e+B{a}^{2}{e}^{4}+3\,Bac{d}^{2}{e}^{2}+B{c}^{2}{d}^{4}+4\,C{a}^{2}d{e}^{3}+2\,Cac{d}^{3}e}{6\,{c}^{3}}} \right ) }+{\frac{A{e}^{4}}{16\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,A{d}^{2}{e}^{2}}{8\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,A{d}^{4}}{16\,{a}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{Bd{e}^{3}}{4\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{B{d}^{3}e}{4\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{5\,C{e}^{4}}{16\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{3\,C{d}^{2}{e}^{2}}{8\,a{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{C{d}^{4}}{16\,{a}^{2}c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

int((e*x+d)^4*(C*x^2+B*x+A)/(c*x^2+a)^4,x)

[Out]

(1/16*(A*a^2*c*e^4+6*A*a*c^2*d^2*e^2+5*A*c^3*d^4+4*B*a^2*c*d*e^3+4*B*a*c^2*d^3*e-11*C*a^3*e^4+6*C*a^2*c*d^2*e^
2+C*a*c^2*d^4)/a^3/c*x^5-1/2*e^3*(B*e+4*C*d)/c*x^4-1/6*(A*a^2*c*e^4-6*A*a*c^2*d^2*e^2-5*A*c^3*d^4+4*B*a^2*c*d*
e^3-4*B*a*c^2*d^3*e+5*C*a^3*e^4+6*C*a^2*c*d^2*e^2-C*a*c^2*d^4)/a^2/c^2*x^3-1/2*e*(2*A*c*d*e^2+B*a*e^3+3*B*c*d^
2*e+4*C*a*d*e^2+2*C*c*d^3)/c^2*x^2-1/16*(A*a^2*c*e^4+6*A*a*c^2*d^2*e^2-11*A*c^3*d^4+4*B*a^2*c*d*e^3+4*B*a*c^2*
d^3*e+5*C*a^3*e^4+6*C*a^2*c*d^2*e^2+C*a*c^2*d^4)/a/c^3*x-1/6*(2*A*a*c*d*e^3+4*A*c^2*d^3*e+B*a^2*e^4+3*B*a*c*d^
2*e^2+B*c^2*d^4+4*C*a^2*d*e^3+2*C*a*c*d^3*e)/c^3)/(c*x^2+a)^3+1/16/a/c^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A
*e^4+3/8/a^2/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^2*e^2+5/16/a^3/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^
4+1/4/a/c^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d*e^3+1/4/a^2/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d^3*e+
5/16/c^3/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*e^4+3/8/a/c^2/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d^2*e^2+1/1
6/a^2/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*C*d^4

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [B]  time = 1.71336, size = 3753, normalized size = 16.04 \begin{align*} \text{result too large to display} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(16*B*a^4*c^3*d^4 + 48*B*a^5*c^2*d^2*e^2 + 16*B*a^6*c*e^4 - 6*(4*B*a^2*c^5*d^3*e + 4*B*a^3*c^4*d*e^3 +
(C*a^2*c^5 + 5*A*a*c^6)*d^4 + 6*(C*a^3*c^4 + A*a^2*c^5)*d^2*e^2 - (11*C*a^4*c^3 - A*a^3*c^4)*e^4)*x^5 + 32*(C*
a^5*c^2 + 2*A*a^4*c^3)*d^3*e + 32*(2*C*a^6*c + A*a^5*c^2)*d*e^3 + 48*(4*C*a^4*c^3*d*e^3 + B*a^4*c^3*e^4)*x^4 -
16*(4*B*a^3*c^4*d^3*e - 4*B*a^4*c^3*d*e^3 + (C*a^3*c^4 + 5*A*a^2*c^5)*d^4 - 6*(C*a^4*c^3 - A*a^3*c^4)*d^2*e^2
- (5*C*a^5*c^2 + A*a^4*c^3)*e^4)*x^3 + 48*(2*C*a^4*c^3*d^3*e + 3*B*a^4*c^3*d^2*e^2 + B*a^5*c^2*e^4 + 2*(2*C*a
^5*c^2 + A*a^4*c^3)*d*e^3)*x^2 + 3*(4*B*a^4*c^2*d^3*e + 4*B*a^5*c*d*e^3 + (4*B*a*c^5*d^3*e + 4*B*a^2*c^4*d*e^3
+ (C*a*c^5 + 5*A*c^6)*d^4 + 6*(C*a^2*c^4 + A*a*c^5)*d^2*e^2 + (5*C*a^3*c^3 + A*a^2*c^4)*e^4)*x^6 + (C*a^4*c^2
+ 5*A*a^3*c^3)*d^4 + 6*(C*a^5*c + A*a^4*c^2)*d^2*e^2 + (5*C*a^6 + A*a^5*c)*e^4 + 3*(4*B*a^2*c^4*d^3*e + 4*B*a
^3*c^3*d*e^3 + (C*a^2*c^4 + 5*A*a*c^5)*d^4 + 6*(C*a^3*c^3 + A*a^2*c^4)*d^2*e^2 + (5*C*a^4*c^2 + A*a^3*c^3)*e^4
)*x^4 + 3*(4*B*a^3*c^3*d^3*e + 4*B*a^4*c^2*d*e^3 + (C*a^3*c^3 + 5*A*a^2*c^4)*d^4 + 6*(C*a^4*c^2 + A*a^3*c^3)*d
^2*e^2 + (5*C*a^5*c + A*a^4*c^2)*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 6*(4*B*a
^4*c^3*d^3*e + 4*B*a^5*c^2*d*e^3 + (C*a^4*c^3 - 11*A*a^3*c^4)*d^4 + 6*(C*a^5*c^2 + A*a^4*c^3)*d^2*e^2 + (5*C*a
^6*c + A*a^5*c^2)*e^4)*x)/(a^4*c^7*x^6 + 3*a^5*c^6*x^4 + 3*a^6*c^5*x^2 + a^7*c^4), -1/48*(8*B*a^4*c^3*d^4 + 24
*B*a^5*c^2*d^2*e^2 + 8*B*a^6*c*e^4 - 3*(4*B*a^2*c^5*d^3*e + 4*B*a^3*c^4*d*e^3 + (C*a^2*c^5 + 5*A*a*c^6)*d^4 +
6*(C*a^3*c^4 + A*a^2*c^5)*d^2*e^2 - (11*C*a^4*c^3 - A*a^3*c^4)*e^4)*x^5 + 16*(C*a^5*c^2 + 2*A*a^4*c^3)*d^3*e +
16*(2*C*a^6*c + A*a^5*c^2)*d*e^3 + 24*(4*C*a^4*c^3*d*e^3 + B*a^4*c^3*e^4)*x^4 - 8*(4*B*a^3*c^4*d^3*e - 4*B*a^
4*c^3*d*e^3 + (C*a^3*c^4 + 5*A*a^2*c^5)*d^4 - 6*(C*a^4*c^3 - A*a^3*c^4)*d^2*e^2 - (5*C*a^5*c^2 + A*a^4*c^3)*e^
4)*x^3 + 24*(2*C*a^4*c^3*d^3*e + 3*B*a^4*c^3*d^2*e^2 + B*a^5*c^2*e^4 + 2*(2*C*a^5*c^2 + A*a^4*c^3)*d*e^3)*x^2
- 3*(4*B*a^4*c^2*d^3*e + 4*B*a^5*c*d*e^3 + (4*B*a*c^5*d^3*e + 4*B*a^2*c^4*d*e^3 + (C*a*c^5 + 5*A*c^6)*d^4 + 6*
(C*a^2*c^4 + A*a*c^5)*d^2*e^2 + (5*C*a^3*c^3 + A*a^2*c^4)*e^4)*x^6 + (C*a^4*c^2 + 5*A*a^3*c^3)*d^4 + 6*(C*a^5*
c + A*a^4*c^2)*d^2*e^2 + (5*C*a^6 + A*a^5*c)*e^4 + 3*(4*B*a^2*c^4*d^3*e + 4*B*a^3*c^3*d*e^3 + (C*a^2*c^4 + 5*A
*a*c^5)*d^4 + 6*(C*a^3*c^3 + A*a^2*c^4)*d^2*e^2 + (5*C*a^4*c^2 + A*a^3*c^3)*e^4)*x^4 + 3*(4*B*a^3*c^3*d^3*e +
4*B*a^4*c^2*d*e^3 + (C*a^3*c^3 + 5*A*a^2*c^4)*d^4 + 6*(C*a^4*c^2 + A*a^3*c^3)*d^2*e^2 + (5*C*a^5*c + A*a^4*c^2
)*e^4)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + 3*(4*B*a^4*c^3*d^3*e + 4*B*a^5*c^2*d*e^3 + (C*a^4*c^3 - 11*A*a^3
*c^4)*d^4 + 6*(C*a^5*c^2 + A*a^4*c^3)*d^2*e^2 + (5*C*a^6*c + A*a^5*c^2)*e^4)*x)/(a^4*c^7*x^6 + 3*a^5*c^6*x^4 +
3*a^6*c^5*x^2 + a^7*c^4)]

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Timed out

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Giac [B]  time = 1.16659, size = 859, normalized size = 3.67 \begin{align*} \frac{{\left (C a c^{2} d^{4} + 5 \, A c^{3} d^{4} + 4 \, B a c^{2} d^{3} e + 6 \, C a^{2} c d^{2} e^{2} + 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + 5 \, C a^{3} e^{4} + A a^{2} c e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{16 \, \sqrt{a c} a^{3} c^{3}} + \frac{3 \, C a c^{4} d^{4} x^{5} + 15 \, A c^{5} d^{4} x^{5} + 12 \, B a c^{4} d^{3} x^{5} e + 18 \, C a^{2} c^{3} d^{2} x^{5} e^{2} + 18 \, A a c^{4} d^{2} x^{5} e^{2} + 8 \, C a^{2} c^{3} d^{4} x^{3} + 40 \, A a c^{4} d^{4} x^{3} + 12 \, B a^{2} c^{3} d x^{5} e^{3} + 32 \, B a^{2} c^{3} d^{3} x^{3} e - 33 \, C a^{3} c^{2} x^{5} e^{4} + 3 \, A a^{2} c^{3} x^{5} e^{4} - 96 \, C a^{3} c^{2} d x^{4} e^{3} - 48 \, C a^{3} c^{2} d^{2} x^{3} e^{2} + 48 \, A a^{2} c^{3} d^{2} x^{3} e^{2} - 48 \, C a^{3} c^{2} d^{3} x^{2} e - 3 \, C a^{3} c^{2} d^{4} x + 33 \, A a^{2} c^{3} d^{4} x - 24 \, B a^{3} c^{2} x^{4} e^{4} - 32 \, B a^{3} c^{2} d x^{3} e^{3} - 72 \, B a^{3} c^{2} d^{2} x^{2} e^{2} - 12 \, B a^{3} c^{2} d^{3} x e - 8 \, B a^{3} c^{2} d^{4} - 40 \, C a^{4} c x^{3} e^{4} - 8 \, A a^{3} c^{2} x^{3} e^{4} - 96 \, C a^{4} c d x^{2} e^{3} - 48 \, A a^{3} c^{2} d x^{2} e^{3} - 18 \, C a^{4} c d^{2} x e^{2} - 18 \, A a^{3} c^{2} d^{2} x e^{2} - 16 \, C a^{4} c d^{3} e - 32 \, A a^{3} c^{2} d^{3} e - 24 \, B a^{4} c x^{2} e^{4} - 12 \, B a^{4} c d x e^{3} - 24 \, B a^{4} c d^{2} e^{2} - 15 \, C a^{5} x e^{4} - 3 \, A a^{4} c x e^{4} - 32 \, C a^{5} d e^{3} - 16 \, A a^{4} c d e^{3} - 8 \, B a^{5} e^{4}}{48 \,{\left (c x^{2} + a\right )}^{3} a^{3} c^{3}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/16*(C*a*c^2*d^4 + 5*A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*C*a^2*c*d^2*e^2 + 6*A*a*c^2*d^2*e^2 + 4*B*a^2*c*d*e^3 +
5*C*a^3*e^4 + A*a^2*c*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^3*c^3) + 1/48*(3*C*a*c^4*d^4*x^5 + 15*A*c^5*d^4*
x^5 + 12*B*a*c^4*d^3*x^5*e + 18*C*a^2*c^3*d^2*x^5*e^2 + 18*A*a*c^4*d^2*x^5*e^2 + 8*C*a^2*c^3*d^4*x^3 + 40*A*a*
c^4*d^4*x^3 + 12*B*a^2*c^3*d*x^5*e^3 + 32*B*a^2*c^3*d^3*x^3*e - 33*C*a^3*c^2*x^5*e^4 + 3*A*a^2*c^3*x^5*e^4 - 9
6*C*a^3*c^2*d*x^4*e^3 - 48*C*a^3*c^2*d^2*x^3*e^2 + 48*A*a^2*c^3*d^2*x^3*e^2 - 48*C*a^3*c^2*d^3*x^2*e - 3*C*a^3
*c^2*d^4*x + 33*A*a^2*c^3*d^4*x - 24*B*a^3*c^2*x^4*e^4 - 32*B*a^3*c^2*d*x^3*e^3 - 72*B*a^3*c^2*d^2*x^2*e^2 - 1
2*B*a^3*c^2*d^3*x*e - 8*B*a^3*c^2*d^4 - 40*C*a^4*c*x^3*e^4 - 8*A*a^3*c^2*x^3*e^4 - 96*C*a^4*c*d*x^2*e^3 - 48*A
*a^3*c^2*d*x^2*e^3 - 18*C*a^4*c*d^2*x*e^2 - 18*A*a^3*c^2*d^2*x*e^2 - 16*C*a^4*c*d^3*e - 32*A*a^3*c^2*d^3*e - 2
4*B*a^4*c*x^2*e^4 - 12*B*a^4*c*d*x*e^3 - 24*B*a^4*c*d^2*e^2 - 15*C*a^5*x*e^4 - 3*A*a^4*c*x*e^4 - 32*C*a^5*d*e^
3 - 16*A*a^4*c*d*e^3 - 8*B*a^5*e^4)/((c*x^2 + a)^3*a^3*c^3)