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

Optimal. Leaf size=292 \[ \frac {x \left (a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 d^2 \left (5 C d^2-e (4 B d-3 A e)\right )\right )}{e^6}-\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^7 (d+e x)}-\frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)+c d \left (6 C d^2-e (5 B d-4 A e)\right )\right )}{e^7}-\frac {c x^2 \left (2 a e^2 (2 C d-B e)+c d \left (4 C d^2-e (3 B d-2 A e)\right )\right )}{2 e^5}+\frac {c x^3 \left (2 a C e^2+c \left (3 C d^2-e (2 B d-A e)\right )\right )}{3 e^4}-\frac {c^2 x^4 (2 C d-B e)}{4 e^3}+\frac {c^2 C x^5}{5 e^2} \]

[Out]

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

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Rubi [A]  time = 0.53, antiderivative size = 289, normalized size of antiderivative = 0.99, number of steps used = 2, number of rules used = 1, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.037, Rules used = {1628} \[ \frac {x \left (a^2 C e^4+2 a c e^2 \left (3 C d^2-e (2 B d-A e)\right )+c^2 \left (5 C d^4-d^2 e (4 B d-3 A e)\right )\right )}{e^6}+\frac {c x^3 \left (2 a C e^2-c e (2 B d-A e)+3 c C d^2\right )}{3 e^4}-\frac {c x^2 \left (2 a e^2 (2 C d-B e)-c d e (3 B d-2 A e)+4 c C d^3\right )}{2 e^5}-\frac {\left (a e^2+c d^2\right )^2 \left (A e^2-B d e+C d^2\right )}{e^7 (d+e x)}-\frac {\left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)-c d e (5 B d-4 A e)+6 c C d^3\right )}{e^7}-\frac {c^2 x^4 (2 C d-B e)}{4 e^3}+\frac {c^2 C x^5}{5 e^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1628

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Int[ExpandIntegra
nd[(d + e*x)^m*Pq*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m}, x] && PolyQ[Pq, x] && IGtQ[p, -2]

Rubi steps

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

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Mathematica [A]  time = 0.28, size = 272, normalized size = 0.93 \[ \frac {60 e x \left (a^2 C e^4+2 a c e^2 \left (e (A e-2 B d)+3 C d^2\right )+c^2 \left (d^2 e (3 A e-4 B d)+5 C d^4\right )\right )-30 c e^2 x^2 \left (-2 a e^2 (B e-2 C d)+c d e (2 A e-3 B d)+4 c C d^3\right )-\frac {60 \left (a e^2+c d^2\right )^2 \left (e (A e-B d)+C d^2\right )}{d+e x}+20 c e^3 x^3 \left (2 a C e^2+c e (A e-2 B d)+3 c C d^2\right )-60 \left (a e^2+c d^2\right ) \log (d+e x) \left (a e^2 (2 C d-B e)+c d e (4 A e-5 B d)+6 c C d^3\right )+15 c^2 e^4 x^4 (B e-2 C d)+12 c^2 C e^5 x^5}{60 e^7} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(60*e*(a^2*C*e^4 + 2*a*c*e^2*(3*C*d^2 + e*(-2*B*d + A*e)) + c^2*(5*C*d^4 + d^2*e*(-4*B*d + 3*A*e)))*x - 30*c*e
^2*(4*c*C*d^3 + c*d*e*(-3*B*d + 2*A*e) - 2*a*e^2*(-2*C*d + B*e))*x^2 + 20*c*e^3*(3*c*C*d^2 + 2*a*C*e^2 + c*e*(
-2*B*d + A*e))*x^3 + 15*c^2*e^4*(-2*C*d + B*e)*x^4 + 12*c^2*C*e^5*x^5 - (60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*
d) + A*e)))/(d + e*x) - 60*(c*d^2 + a*e^2)*(6*c*C*d^3 + c*d*e*(-5*B*d + 4*A*e) + a*e^2*(2*C*d - B*e))*Log[d +
e*x])/(60*e^7)

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fricas [A]  time = 0.97, size = 553, normalized size = 1.89 \[ \frac {12 \, C c^{2} e^{6} x^{6} - 60 \, C c^{2} d^{6} + 60 \, B c^{2} d^{5} e + 120 \, B a c d^{3} e^{3} + 60 \, B a^{2} d e^{5} - 60 \, A a^{2} e^{6} - 60 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} - 60 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} - 3 \, {\left (6 \, C c^{2} d e^{5} - 5 \, B c^{2} e^{6}\right )} x^{5} + 5 \, {\left (6 \, C c^{2} d^{2} e^{4} - 5 \, B c^{2} d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} e^{6}\right )} x^{4} - 10 \, {\left (6 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} - 6 \, B a c e^{6} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d e^{5}\right )} x^{3} + 30 \, {\left (6 \, C c^{2} d^{4} e^{2} - 5 \, B c^{2} d^{3} e^{3} - 6 \, B a c d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{4} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} e^{6}\right )} x^{2} + 60 \, {\left (5 \, C c^{2} d^{5} e - 4 \, B c^{2} d^{4} e^{2} - 4 \, B a c d^{2} e^{4} + 3 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x - 60 \, {\left (6 \, C c^{2} d^{6} - 5 \, B c^{2} d^{5} e - 6 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4} + {\left (6 \, C c^{2} d^{5} e - 5 \, B c^{2} d^{4} e^{2} - 6 \, B a c d^{2} e^{4} - B a^{2} e^{6} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{3} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{5}\right )} x\right )} \log \left (e x + d\right )}{60 \, {\left (e^{8} x + d e^{7}\right )}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*C*c^2*e^6*x^6 - 60*C*c^2*d^6 + 60*B*c^2*d^5*e + 120*B*a*c*d^3*e^3 + 60*B*a^2*d*e^5 - 60*A*a^2*e^6 - 6
0*(2*C*a*c + A*c^2)*d^4*e^2 - 60*(C*a^2 + 2*A*a*c)*d^2*e^4 - 3*(6*C*c^2*d*e^5 - 5*B*c^2*e^6)*x^5 + 5*(6*C*c^2*
d^2*e^4 - 5*B*c^2*d*e^5 + 4*(2*C*a*c + A*c^2)*e^6)*x^4 - 10*(6*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 - 6*B*a*c*e^6 +
4*(2*C*a*c + A*c^2)*d*e^5)*x^3 + 30*(6*C*c^2*d^4*e^2 - 5*B*c^2*d^3*e^3 - 6*B*a*c*d*e^5 + 4*(2*C*a*c + A*c^2)*
d^2*e^4 + 2*(C*a^2 + 2*A*a*c)*e^6)*x^2 + 60*(5*C*c^2*d^5*e - 4*B*c^2*d^4*e^2 - 4*B*a*c*d^2*e^4 + 3*(2*C*a*c +
A*c^2)*d^3*e^3 + (C*a^2 + 2*A*a*c)*d*e^5)*x - 60*(6*C*c^2*d^6 - 5*B*c^2*d^5*e - 6*B*a*c*d^3*e^3 - B*a^2*d*e^5
+ 4*(2*C*a*c + A*c^2)*d^4*e^2 + 2*(C*a^2 + 2*A*a*c)*d^2*e^4 + (6*C*c^2*d^5*e - 5*B*c^2*d^4*e^2 - 6*B*a*c*d^2*e
^4 - B*a^2*e^6 + 4*(2*C*a*c + A*c^2)*d^3*e^3 + 2*(C*a^2 + 2*A*a*c)*d*e^5)*x)*log(e*x + d))/(e^8*x + d*e^7)

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giac [A]  time = 0.18, size = 497, normalized size = 1.70 \[ \frac {1}{60} \, {\left (12 \, C c^{2} - \frac {15 \, {\left (6 \, C c^{2} d e - B c^{2} e^{2}\right )} e^{\left (-1\right )}}{x e + d} + \frac {20 \, {\left (15 \, C c^{2} d^{2} e^{2} - 5 \, B c^{2} d e^{3} + 2 \, C a c e^{4} + A c^{2} e^{4}\right )} e^{\left (-2\right )}}{{\left (x e + d\right )}^{2}} - \frac {60 \, {\left (10 \, C c^{2} d^{3} e^{3} - 5 \, B c^{2} d^{2} e^{4} + 4 \, C a c d e^{5} + 2 \, A c^{2} d e^{5} - B a c e^{6}\right )} e^{\left (-3\right )}}{{\left (x e + d\right )}^{3}} + \frac {60 \, {\left (15 \, C c^{2} d^{4} e^{4} - 10 \, B c^{2} d^{3} e^{5} + 12 \, C a c d^{2} e^{6} + 6 \, A c^{2} d^{2} e^{6} - 6 \, B a c d e^{7} + C a^{2} e^{8} + 2 \, A a c e^{8}\right )} e^{\left (-4\right )}}{{\left (x e + d\right )}^{4}}\right )} {\left (x e + d\right )}^{5} e^{\left (-7\right )} + {\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e + 8 \, C a c d^{3} e^{2} + 4 \, A c^{2} d^{3} e^{2} - 6 \, B a c d^{2} e^{3} + 2 \, C a^{2} d e^{4} + 4 \, A a c d e^{4} - B a^{2} e^{5}\right )} e^{\left (-7\right )} \log \left (\frac {{\left | x e + d \right |} e^{\left (-1\right )}}{{\left (x e + d\right )}^{2}}\right ) - {\left (\frac {C c^{2} d^{6} e^{5}}{x e + d} - \frac {B c^{2} d^{5} e^{6}}{x e + d} + \frac {2 \, C a c d^{4} e^{7}}{x e + d} + \frac {A c^{2} d^{4} e^{7}}{x e + d} - \frac {2 \, B a c d^{3} e^{8}}{x e + d} + \frac {C a^{2} d^{2} e^{9}}{x e + d} + \frac {2 \, A a c d^{2} e^{9}}{x e + d} - \frac {B a^{2} d e^{10}}{x e + d} + \frac {A a^{2} e^{11}}{x e + d}\right )} e^{\left (-12\right )} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/60*(12*C*c^2 - 15*(6*C*c^2*d*e - B*c^2*e^2)*e^(-1)/(x*e + d) + 20*(15*C*c^2*d^2*e^2 - 5*B*c^2*d*e^3 + 2*C*a*
c*e^4 + A*c^2*e^4)*e^(-2)/(x*e + d)^2 - 60*(10*C*c^2*d^3*e^3 - 5*B*c^2*d^2*e^4 + 4*C*a*c*d*e^5 + 2*A*c^2*d*e^5
- B*a*c*e^6)*e^(-3)/(x*e + d)^3 + 60*(15*C*c^2*d^4*e^4 - 10*B*c^2*d^3*e^5 + 12*C*a*c*d^2*e^6 + 6*A*c^2*d^2*e^
6 - 6*B*a*c*d*e^7 + C*a^2*e^8 + 2*A*a*c*e^8)*e^(-4)/(x*e + d)^4)*(x*e + d)^5*e^(-7) + (6*C*c^2*d^5 - 5*B*c^2*d
^4*e + 8*C*a*c*d^3*e^2 + 4*A*c^2*d^3*e^2 - 6*B*a*c*d^2*e^3 + 2*C*a^2*d*e^4 + 4*A*a*c*d*e^4 - B*a^2*e^5)*e^(-7)
*log(abs(x*e + d)*e^(-1)/(x*e + d)^2) - (C*c^2*d^6*e^5/(x*e + d) - B*c^2*d^5*e^6/(x*e + d) + 2*C*a*c*d^4*e^7/(
x*e + d) + A*c^2*d^4*e^7/(x*e + d) - 2*B*a*c*d^3*e^8/(x*e + d) + C*a^2*d^2*e^9/(x*e + d) + 2*A*a*c*d^2*e^9/(x*
e + d) - B*a^2*d*e^10/(x*e + d) + A*a^2*e^11/(x*e + d))*e^(-12)

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maple [A]  time = 0.01, size = 527, normalized size = 1.80 \[ \frac {C \,c^{2} x^{5}}{5 e^{2}}+\frac {B \,c^{2} x^{4}}{4 e^{2}}-\frac {C \,c^{2} d \,x^{4}}{2 e^{3}}+\frac {A \,c^{2} x^{3}}{3 e^{2}}-\frac {2 B \,c^{2} d \,x^{3}}{3 e^{3}}+\frac {2 C a c \,x^{3}}{3 e^{2}}+\frac {C \,c^{2} d^{2} x^{3}}{e^{4}}-\frac {A \,c^{2} d \,x^{2}}{e^{3}}+\frac {B a c \,x^{2}}{e^{2}}+\frac {3 B \,c^{2} d^{2} x^{2}}{2 e^{4}}-\frac {2 C a c d \,x^{2}}{e^{3}}-\frac {2 C \,c^{2} d^{3} x^{2}}{e^{5}}-\frac {A \,a^{2}}{\left (e x +d \right ) e}-\frac {2 A a c \,d^{2}}{\left (e x +d \right ) e^{3}}-\frac {4 A a c d \ln \left (e x +d \right )}{e^{3}}+\frac {2 A a c x}{e^{2}}-\frac {A \,c^{2} d^{4}}{\left (e x +d \right ) e^{5}}-\frac {4 A \,c^{2} d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {3 A \,c^{2} d^{2} x}{e^{4}}+\frac {B \,a^{2} d}{\left (e x +d \right ) e^{2}}+\frac {B \,a^{2} \ln \left (e x +d \right )}{e^{2}}+\frac {2 B a c \,d^{3}}{\left (e x +d \right ) e^{4}}+\frac {6 B a c \,d^{2} \ln \left (e x +d \right )}{e^{4}}-\frac {4 B a c d x}{e^{3}}+\frac {B \,c^{2} d^{5}}{\left (e x +d \right ) e^{6}}+\frac {5 B \,c^{2} d^{4} \ln \left (e x +d \right )}{e^{6}}-\frac {4 B \,c^{2} d^{3} x}{e^{5}}-\frac {C \,a^{2} d^{2}}{\left (e x +d \right ) e^{3}}-\frac {2 C \,a^{2} d \ln \left (e x +d \right )}{e^{3}}+\frac {C \,a^{2} x}{e^{2}}-\frac {2 C a c \,d^{4}}{\left (e x +d \right ) e^{5}}-\frac {8 C a c \,d^{3} \ln \left (e x +d \right )}{e^{5}}+\frac {6 C a c \,d^{2} x}{e^{4}}-\frac {C \,c^{2} d^{6}}{\left (e x +d \right ) e^{7}}-\frac {6 C \,c^{2} d^{5} \ln \left (e x +d \right )}{e^{7}}+\frac {5 C \,c^{2} d^{4} x}{e^{6}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-2/e^3*C*x^2*a*c*d-4/e^3*d*a*c*B*x+6/e^4*C*a*c*d^2*x-2/e^3/(e*x+d)*A*a*c*d^2+2/e^4/(e*x+d)*B*a*c*d^3-2/e^5/(e*
x+d)*C*a*c*d^4-4/e^3*ln(e*x+d)*A*a*c*d+6/e^4*ln(e*x+d)*B*a*c*d^2-8/e^5*ln(e*x+d)*C*a*c*d^3+1/5*c^2*C*x^5/e^2-1
/e/(e*x+d)*A*a^2+1/e^2*ln(e*x+d)*B*a^2+1/4/e^2*B*x^4*c^2+1/3/e^2*A*x^3*c^2+1/e^2*a^2*C*x-4/e^5*ln(e*x+d)*A*c^2
*d^3+5/e^6*ln(e*x+d)*B*c^2*d^4-4/e^5*B*c^2*d^3*x+5/e^6*C*c^2*d^4*x-1/e^5/(e*x+d)*A*c^2*d^4+1/e^2/(e*x+d)*B*d*a
^2+1/e^6/(e*x+d)*B*c^2*d^5-1/e^3/(e*x+d)*C*a^2*d^2-1/e^7/(e*x+d)*C*c^2*d^6+2/3/e^2*C*x^3*a*c+1/e^4*C*x^3*c^2*d
^2-1/e^3*A*x^2*c^2*d+1/e^2*B*x^2*a*c+3/2/e^4*B*x^2*c^2*d^2-2/e^5*C*x^2*c^2*d^3+2/e^2*A*a*c*x+3/e^4*A*c^2*d^2*x
-1/2/e^3*C*x^4*c^2*d-2/3/e^3*B*x^3*c^2*d-2/e^3*ln(e*x+d)*C*a^2*d-6/e^7*ln(e*x+d)*C*c^2*d^5

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maxima [A]  time = 0.48, size = 392, normalized size = 1.34 \[ -\frac {C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4}}{e^{8} x + d e^{7}} + \frac {12 \, C c^{2} e^{4} x^{5} - 15 \, {\left (2 \, C c^{2} d e^{3} - B c^{2} e^{4}\right )} x^{4} + 20 \, {\left (3 \, C c^{2} d^{2} e^{2} - 2 \, B c^{2} d e^{3} + {\left (2 \, C a c + A c^{2}\right )} e^{4}\right )} x^{3} - 30 \, {\left (4 \, C c^{2} d^{3} e - 3 \, B c^{2} d^{2} e^{2} - 2 \, B a c e^{4} + 2 \, {\left (2 \, C a c + A c^{2}\right )} d e^{3}\right )} x^{2} + 60 \, {\left (5 \, C c^{2} d^{4} - 4 \, B c^{2} d^{3} e - 4 \, B a c d e^{3} + 3 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} e^{2} + {\left (C a^{2} + 2 \, A a c\right )} e^{4}\right )} x}{60 \, e^{6}} - \frac {{\left (6 \, C c^{2} d^{5} - 5 \, B c^{2} d^{4} e - 6 \, B a c d^{2} e^{3} - B a^{2} e^{5} + 4 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} e^{2} + 2 \, {\left (C a^{2} + 2 \, A a c\right )} d e^{4}\right )} \log \left (e x + d\right )}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-(C*c^2*d^6 - B*c^2*d^5*e - 2*B*a*c*d^3*e^3 - B*a^2*d*e^5 + A*a^2*e^6 + (2*C*a*c + A*c^2)*d^4*e^2 + (C*a^2 + 2
*A*a*c)*d^2*e^4)/(e^8*x + d*e^7) + 1/60*(12*C*c^2*e^4*x^5 - 15*(2*C*c^2*d*e^3 - B*c^2*e^4)*x^4 + 20*(3*C*c^2*d
^2*e^2 - 2*B*c^2*d*e^3 + (2*C*a*c + A*c^2)*e^4)*x^3 - 30*(4*C*c^2*d^3*e - 3*B*c^2*d^2*e^2 - 2*B*a*c*e^4 + 2*(2
*C*a*c + A*c^2)*d*e^3)*x^2 + 60*(5*C*c^2*d^4 - 4*B*c^2*d^3*e - 4*B*a*c*d*e^3 + 3*(2*C*a*c + A*c^2)*d^2*e^2 + (
C*a^2 + 2*A*a*c)*e^4)*x)/e^6 - (6*C*c^2*d^5 - 5*B*c^2*d^4*e - 6*B*a*c*d^2*e^3 - B*a^2*e^5 + 4*(2*C*a*c + A*c^2
)*d^3*e^2 + 2*(C*a^2 + 2*A*a*c)*d*e^4)*log(e*x + d)/e^7

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mupad [B]  time = 0.12, size = 575, normalized size = 1.97 \[ x^4\,\left (\frac {B\,c^2}{4\,e^2}-\frac {C\,c^2\,d}{2\,e^3}\right )+x\,\left (\frac {C\,a^2+2\,A\,c\,a}{e^2}+\frac {d^2\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e^2}-\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e^2}+\frac {2\,B\,a\,c}{e^2}\right )}{e}\right )-x^3\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{3\,e}-\frac {A\,c^2+2\,C\,a\,c}{3\,e^2}+\frac {C\,c^2\,d^2}{3\,e^4}\right )+x^2\,\left (\frac {d\,\left (\frac {2\,d\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{e}-\frac {A\,c^2+2\,C\,a\,c}{e^2}+\frac {C\,c^2\,d^2}{e^4}\right )}{e}-\frac {d^2\,\left (\frac {B\,c^2}{e^2}-\frac {2\,C\,c^2\,d}{e^3}\right )}{2\,e^2}+\frac {B\,a\,c}{e^2}\right )-\frac {C\,a^2\,d^2\,e^4-B\,a^2\,d\,e^5+A\,a^2\,e^6+2\,C\,a\,c\,d^4\,e^2-2\,B\,a\,c\,d^3\,e^3+2\,A\,a\,c\,d^2\,e^4+C\,c^2\,d^6-B\,c^2\,d^5\,e+A\,c^2\,d^4\,e^2}{e\,\left (x\,e^7+d\,e^6\right )}-\frac {\ln \left (d+e\,x\right )\,\left (2\,C\,a^2\,d\,e^4-B\,a^2\,e^5+8\,C\,a\,c\,d^3\,e^2-6\,B\,a\,c\,d^2\,e^3+4\,A\,a\,c\,d\,e^4+6\,C\,c^2\,d^5-5\,B\,c^2\,d^4\,e+4\,A\,c^2\,d^3\,e^2\right )}{e^7}+\frac {C\,c^2\,x^5}{5\,e^2} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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sympy [A]  time = 2.79, size = 416, normalized size = 1.42 \[ \frac {C c^{2} x^{5}}{5 e^{2}} + x^{4} \left (\frac {B c^{2}}{4 e^{2}} - \frac {C c^{2} d}{2 e^{3}}\right ) + x^{3} \left (\frac {A c^{2}}{3 e^{2}} - \frac {2 B c^{2} d}{3 e^{3}} + \frac {2 C a c}{3 e^{2}} + \frac {C c^{2} d^{2}}{e^{4}}\right ) + x^{2} \left (- \frac {A c^{2} d}{e^{3}} + \frac {B a c}{e^{2}} + \frac {3 B c^{2} d^{2}}{2 e^{4}} - \frac {2 C a c d}{e^{3}} - \frac {2 C c^{2} d^{3}}{e^{5}}\right ) + x \left (\frac {2 A a c}{e^{2}} + \frac {3 A c^{2} d^{2}}{e^{4}} - \frac {4 B a c d}{e^{3}} - \frac {4 B c^{2} d^{3}}{e^{5}} + \frac {C a^{2}}{e^{2}} + \frac {6 C a c d^{2}}{e^{4}} + \frac {5 C c^{2} d^{4}}{e^{6}}\right ) + \frac {- A a^{2} e^{6} - 2 A a c d^{2} e^{4} - A c^{2} d^{4} e^{2} + B a^{2} d e^{5} + 2 B a c d^{3} e^{3} + B c^{2} d^{5} e - C a^{2} d^{2} e^{4} - 2 C a c d^{4} e^{2} - C c^{2} d^{6}}{d e^{7} + e^{8} x} - \frac {\left (a e^{2} + c d^{2}\right ) \left (4 A c d e^{2} - B a e^{3} - 5 B c d^{2} e + 2 C a d e^{2} + 6 C c d^{3}\right ) \log {\left (d + e x \right )}}{e^{7}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

C*c**2*x**5/(5*e**2) + x**4*(B*c**2/(4*e**2) - C*c**2*d/(2*e**3)) + x**3*(A*c**2/(3*e**2) - 2*B*c**2*d/(3*e**3
) + 2*C*a*c/(3*e**2) + C*c**2*d**2/e**4) + x**2*(-A*c**2*d/e**3 + B*a*c/e**2 + 3*B*c**2*d**2/(2*e**4) - 2*C*a*
c*d/e**3 - 2*C*c**2*d**3/e**5) + x*(2*A*a*c/e**2 + 3*A*c**2*d**2/e**4 - 4*B*a*c*d/e**3 - 4*B*c**2*d**3/e**5 +
C*a**2/e**2 + 6*C*a*c*d**2/e**4 + 5*C*c**2*d**4/e**6) + (-A*a**2*e**6 - 2*A*a*c*d**2*e**4 - A*c**2*d**4*e**2 +
B*a**2*d*e**5 + 2*B*a*c*d**3*e**3 + B*c**2*d**5*e - C*a**2*d**2*e**4 - 2*C*a*c*d**4*e**2 - C*c**2*d**6)/(d*e*
*7 + e**8*x) - (a*e**2 + c*d**2)*(4*A*c*d*e**2 - B*a*e**3 - 5*B*c*d**2*e + 2*C*a*d*e**2 + 6*C*c*d**3)*log(d +
e*x)/e**7

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