3.1.29 \(\int \frac {(a+c x^2)^2 (A+B x+C x^2)}{d+e x} \, dx\) [29]

Optimal. Leaf size=297 \[ -\frac {\left (a^2 e^4 (C d-B e)+c^2 d^3 \left (C d^2-e (B d-A e)\right )+2 a c d e^2 \left (C d^2-e (B d-A e)\right )\right ) x}{e^6}+\frac {\left (a^2 C e^4+c^2 d^2 \left (C d^2-e (B d-A e)\right )+2 a c e^2 \left (C d^2-e (B d-A e)\right )\right ) x^2}{2 e^5}-\frac {c \left (2 a e^2 (C d-B e)+c d \left (C d^2-e (B d-A e)\right )\right ) x^3}{3 e^4}+\frac {c \left (2 a C e^2+c \left (C d^2-e (B d-A e)\right )\right ) x^4}{4 e^3}-\frac {c^2 (C d-B e) x^5}{5 e^2}+\frac {c^2 C x^6}{6 e}+\frac {\left (c d^2+a e^2\right )^2 \left (C d^2-B d e+A e^2\right ) \log (d+e x)}{e^7} \]

[Out]

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

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Rubi [A]
time = 0.40, antiderivative size = 295, 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 = {1642} \begin {gather*} \frac {x^2 \left (a^2 C e^4+2 a c e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^4-d^2 e (B d-A e)\right )\right )}{2 e^5}-\frac {x \left (a^2 e^4 (C d-B e)+2 a c d e^2 \left (C d^2-e (B d-A e)\right )+c^2 \left (C d^5-d^3 e (B d-A e)\right )\right )}{e^6}-\frac {c x^3 \left (2 a e^2 (C d-B e)-c d e (B d-A e)+c C d^3\right )}{3 e^4}+\frac {\left (a e^2+c d^2\right )^2 \log (d+e x) \left (A e^2-B d e+C d^2\right )}{e^7}+\frac {c x^4 \left (2 a C e^2-c e (B d-A e)+c C d^2\right )}{4 e^3}-\frac {c^2 x^5 (C d-B e)}{5 e^2}+\frac {c^2 C x^6}{6 e} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 1642

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

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Mathematica [A]
time = 0.11, size = 285, normalized size = 0.96 \begin {gather*} \frac {e x \left (30 a^2 e^4 (-2 C d+2 B e+C e x)+10 a c e^2 \left (C \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+2 e \left (3 A e (-2 d+e x)+B \left (6 d^2-3 d e x+2 e^2 x^2\right )\right )\right )+c^2 \left (C \left (-60 d^5+30 d^4 e x-20 d^3 e^2 x^2+15 d^2 e^3 x^3-12 d e^4 x^4+10 e^5 x^5\right )+e \left (5 A e \left (-12 d^3+6 d^2 e x-4 d e^2 x^2+3 e^3 x^3\right )+B \left (60 d^4-30 d^3 e x+20 d^2 e^2 x^2-15 d e^3 x^3+12 e^4 x^4\right )\right )\right )\right )+60 \left (c d^2+a e^2\right )^2 \left (C d^2+e (-B d+A e)\right ) \log (d+e x)}{60 e^7} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

(e*x*(30*a^2*e^4*(-2*C*d + 2*B*e + C*e*x) + 10*a*c*e^2*(C*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + 2*
e*(3*A*e*(-2*d + e*x) + B*(6*d^2 - 3*d*e*x + 2*e^2*x^2))) + c^2*(C*(-60*d^5 + 30*d^4*e*x - 20*d^3*e^2*x^2 + 15
*d^2*e^3*x^3 - 12*d*e^4*x^4 + 10*e^5*x^5) + e*(5*A*e*(-12*d^3 + 6*d^2*e*x - 4*d*e^2*x^2 + 3*e^3*x^3) + B*(60*d
^4 - 30*d^3*e*x + 20*d^2*e^2*x^2 - 15*d*e^3*x^3 + 12*e^4*x^4)))) + 60*(c*d^2 + a*e^2)^2*(C*d^2 + e*(-(B*d) + A
*e))*Log[d + e*x])/(60*e^7)

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Maple [A]
time = 0.09, size = 441, normalized size = 1.48

method	result	size
norman	\(\frac {\left (2 A a c \,e^{4}+A \,c^{2} d^{2} e^{2}-2 B a c d \,e^{3}-B \,c^{2} d^{3} e +a^{2} C \,e^{4}+2 C a c \,d^{2} e^{2}+C \,c^{2} d^{4}\right ) x^{2}}{2 e^{5}}-\frac {\left (2 A a c d \,e^{4}+A \,c^{2} d^{3} e^{2}-B \,a^{2} e^{5}-2 B a c \,d^{2} e^{3}-B \,c^{2} d^{4} e +C \,a^{2} d \,e^{4}+2 C a c \,d^{3} e^{2}+C \,c^{2} d^{5}\right ) x}{e^{6}}-\frac {c \left (d \,e^{2} c A -2 B a \,e^{3}-B c \,d^{2} e +2 C a d \,e^{2}+C c \,d^{3}\right ) x^{3}}{3 e^{4}}+\frac {c \left (A c \,e^{2}-B c d e +2 a C \,e^{2}+C c \,d^{2}\right ) x^{4}}{4 e^{3}}+\frac {c^{2} C \,x^{6}}{6 e}+\frac {c^{2} \left (B e -C d \right ) x^{5}}{5 e^{2}}+\frac {\left (A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}\)	\(375\)
default	\(\frac {\left (A \,a^{2} e^{6}+2 A a c \,d^{2} e^{4}+A \,c^{2} d^{4} e^{2}-B \,a^{2} d \,e^{5}-2 B a c \,d^{3} e^{3}-B \,c^{2} d^{5} e +C \,a^{2} d^{2} e^{4}+2 C a c \,d^{4} e^{2}+C \,c^{2} d^{6}\right ) \ln \left (e x +d \right )}{e^{7}}-\frac {B a c d \,e^{4} x^{2}+\frac {1}{3} A \,c^{2} d \,e^{4} x^{3}-\frac {2}{3} B a c \,e^{5} x^{3}-\frac {1}{3} B \,c^{2} d^{2} e^{3} x^{3}+\frac {1}{3} C \,c^{2} d^{3} e^{2} x^{3}-A a c \,e^{5} x^{2}-\frac {1}{2} A \,c^{2} d^{2} e^{3} x^{2}+\frac {1}{2} B \,c^{2} d^{3} e^{2} x^{2}-\frac {1}{2} C \,c^{2} d^{4} e \,x^{2}+\frac {1}{4} B \,c^{2} d \,e^{4} x^{4}-\frac {1}{2} C a c \,e^{5} x^{4}-\frac {1}{4} C \,c^{2} d^{2} e^{3} x^{4}+\frac {1}{5} C \,c^{2} d \,e^{4} x^{5}-B \,c^{2} d^{4} e x +C \,a^{2} d \,e^{4} x +A \,c^{2} d^{3} e^{2} x -\frac {1}{6} c^{2} C \,x^{6} e^{5}-\frac {1}{5} B \,c^{2} e^{5} x^{5}-\frac {1}{4} A \,c^{2} e^{5} x^{4}-\frac {1}{2} C \,a^{2} e^{5} x^{2}-B \,a^{2} e^{5} x +C \,c^{2} d^{5} x +\frac {2}{3} C a c d \,e^{4} x^{3}-C a c \,d^{2} e^{3} x^{2}+2 A a c d \,e^{4} x -2 B a c \,d^{2} e^{3} x +2 C a c \,d^{3} e^{2} x}{e^{6}}\)	\(441\)
risch	\(\frac {B \,c^{2} x^{5}}{5 e}+\frac {A \,c^{2} x^{4}}{4 e}+\frac {C \,a^{2} x^{2}}{2 e}+\frac {B \,a^{2} x}{e}+\frac {\ln \left (e x +d \right ) A \,a^{2}}{e}-\frac {2 \ln \left (e x +d \right ) B a c \,d^{3}}{e^{4}}+\frac {2 \ln \left (e x +d \right ) C a c \,d^{4}}{e^{5}}-\frac {2 C a c d \,x^{3}}{3 e^{2}}+\frac {C a c \,d^{2} x^{2}}{e^{3}}-\frac {2 A a c d x}{e^{2}}+\frac {2 B a c \,d^{2} x}{e^{3}}-\frac {2 C a c \,d^{3} x}{e^{4}}+\frac {2 \ln \left (e x +d \right ) A a c \,d^{2}}{e^{3}}-\frac {B a c d \,x^{2}}{e^{2}}+\frac {c^{2} C \,x^{6}}{6 e}+\frac {B \,c^{2} d^{4} x}{e^{5}}-\frac {C \,a^{2} d x}{e^{2}}-\frac {A \,c^{2} d^{3} x}{e^{4}}-\frac {C \,c^{2} d^{5} x}{e^{6}}+\frac {\ln \left (e x +d \right ) A \,c^{2} d^{4}}{e^{5}}-\frac {\ln \left (e x +d \right ) B \,a^{2} d}{e^{2}}-\frac {\ln \left (e x +d \right ) B \,c^{2} d^{5}}{e^{6}}+\frac {\ln \left (e x +d \right ) C \,a^{2} d^{2}}{e^{3}}+\frac {\ln \left (e x +d \right ) C \,c^{2} d^{6}}{e^{7}}-\frac {A \,c^{2} d \,x^{3}}{3 e^{2}}+\frac {2 B a c \,x^{3}}{3 e}+\frac {B \,c^{2} d^{2} x^{3}}{3 e^{3}}-\frac {C \,c^{2} d^{3} x^{3}}{3 e^{4}}+\frac {A a c \,x^{2}}{e}+\frac {A \,c^{2} d^{2} x^{2}}{2 e^{3}}-\frac {B \,c^{2} d^{3} x^{2}}{2 e^{4}}+\frac {C \,c^{2} d^{4} x^{2}}{2 e^{5}}-\frac {B \,c^{2} d \,x^{4}}{4 e^{2}}+\frac {C a c \,x^{4}}{2 e}+\frac {C \,c^{2} d^{2} x^{4}}{4 e^{3}}-\frac {C \,c^{2} d \,x^{5}}{5 e^{2}}\)	\(490\)

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),x,method=_RETURNVERBOSE)

[Out]

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

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Maxima [A]
time = 0.33, size = 368, normalized size = 1.24 \begin {gather*} {\left (C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} + {\left (2 \, C a c e^{2} + A c^{2} e^{2}\right )} d^{4} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (C a^{2} e^{4} + 2 \, A a c e^{4}\right )} d^{2}\right )} e^{\left (-7\right )} \log \left (x e + d\right ) + \frac {1}{60} \, {\left (10 \, C c^{2} x^{6} e^{5} - 12 \, {\left (C c^{2} d e^{4} - B c^{2} e^{5}\right )} x^{5} + 15 \, {\left (C c^{2} d^{2} e^{3} - B c^{2} d e^{4} + 2 \, C a c e^{5} + A c^{2} e^{5}\right )} x^{4} - 20 \, {\left (C c^{2} d^{3} e^{2} - B c^{2} d^{2} e^{3} - 2 \, B a c e^{5} + {\left (2 \, C a c e^{4} + A c^{2} e^{4}\right )} d\right )} x^{3} + 30 \, {\left (C c^{2} d^{4} e - B c^{2} d^{3} e^{2} - 2 \, B a c d e^{4} + C a^{2} e^{5} + 2 \, A a c e^{5} + {\left (2 \, C a c e^{3} + A c^{2} e^{3}\right )} d^{2}\right )} x^{2} - 60 \, {\left (C c^{2} d^{5} - B c^{2} d^{4} e - 2 \, B a c d^{2} e^{3} + {\left (2 \, C a c e^{2} + A c^{2} e^{2}\right )} d^{3} - B a^{2} e^{5} + {\left (C a^{2} e^{4} + 2 \, A a c e^{4}\right )} d\right )} x\right )} e^{\left (-6\right )} \end {gather*}

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),x, algorithm="maxima")

[Out]

(C*c^2*d^6 - B*c^2*d^5*e - 2*B*a*c*d^3*e^3 + (2*C*a*c*e^2 + A*c^2*e^2)*d^4 - B*a^2*d*e^5 + A*a^2*e^6 + (C*a^2*
e^4 + 2*A*a*c*e^4)*d^2)*e^(-7)*log(x*e + d) + 1/60*(10*C*c^2*x^6*e^5 - 12*(C*c^2*d*e^4 - B*c^2*e^5)*x^5 + 15*(
C*c^2*d^2*e^3 - B*c^2*d*e^4 + 2*C*a*c*e^5 + A*c^2*e^5)*x^4 - 20*(C*c^2*d^3*e^2 - B*c^2*d^2*e^3 - 2*B*a*c*e^5 +
(2*C*a*c*e^4 + A*c^2*e^4)*d)*x^3 + 30*(C*c^2*d^4*e - B*c^2*d^3*e^2 - 2*B*a*c*d*e^4 + C*a^2*e^5 + 2*A*a*c*e^5
+ (2*C*a*c*e^3 + A*c^2*e^3)*d^2)*x^2 - 60*(C*c^2*d^5 - B*c^2*d^4*e - 2*B*a*c*d^2*e^3 + (2*C*a*c*e^2 + A*c^2*e^
2)*d^3 - B*a^2*e^5 + (C*a^2*e^4 + 2*A*a*c*e^4)*d)*x)*e^(-6)

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Fricas [A]
time = 0.34, size = 370, normalized size = 1.25 \begin {gather*} -\frac {1}{60} \, {\left (60 \, C c^{2} d^{5} x e - {\left (10 \, C c^{2} x^{6} + 12 \, B c^{2} x^{5} + 40 \, B a c x^{3} + 15 \, {\left (2 \, C a c + A c^{2}\right )} x^{4} + 60 \, B a^{2} x + 30 \, {\left (C a^{2} + 2 \, A a c\right )} x^{2}\right )} e^{6} + {\left (12 \, C c^{2} d x^{5} + 15 \, B c^{2} d x^{4} + 60 \, B a c d x^{2} + 20 \, {\left (2 \, C a c + A c^{2}\right )} d x^{3} + 60 \, {\left (C a^{2} + 2 \, A a c\right )} d x\right )} e^{5} - 5 \, {\left (3 \, C c^{2} d^{2} x^{4} + 4 \, B c^{2} d^{2} x^{3} + 24 \, B a c d^{2} x + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{2} x^{2}\right )} e^{4} + 10 \, {\left (2 \, C c^{2} d^{3} x^{3} + 3 \, B c^{2} d^{3} x^{2} + 6 \, {\left (2 \, C a c + A c^{2}\right )} d^{3} x\right )} e^{3} - 30 \, {\left (C c^{2} d^{4} x^{2} + 2 \, B c^{2} d^{4} x\right )} e^{2} - 60 \, {\left (C c^{2} d^{6} - B c^{2} d^{5} e - 2 \, B a c d^{3} e^{3} + {\left (2 \, C a c + A c^{2}\right )} d^{4} e^{2} - B a^{2} d e^{5} + A a^{2} e^{6} + {\left (C a^{2} + 2 \, A a c\right )} d^{2} e^{4}\right )} \log \left (x e + d\right )\right )} e^{\left (-7\right )} \end {gather*}

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),x, algorithm="fricas")

[Out]

-1/60*(60*C*c^2*d^5*x*e - (10*C*c^2*x^6 + 12*B*c^2*x^5 + 40*B*a*c*x^3 + 15*(2*C*a*c + A*c^2)*x^4 + 60*B*a^2*x
+ 30*(C*a^2 + 2*A*a*c)*x^2)*e^6 + (12*C*c^2*d*x^5 + 15*B*c^2*d*x^4 + 60*B*a*c*d*x^2 + 20*(2*C*a*c + A*c^2)*d*x
^3 + 60*(C*a^2 + 2*A*a*c)*d*x)*e^5 - 5*(3*C*c^2*d^2*x^4 + 4*B*c^2*d^2*x^3 + 24*B*a*c*d^2*x + 6*(2*C*a*c + A*c^
2)*d^2*x^2)*e^4 + 10*(2*C*c^2*d^3*x^3 + 3*B*c^2*d^3*x^2 + 6*(2*C*a*c + A*c^2)*d^3*x)*e^3 - 30*(C*c^2*d^4*x^2 +
2*B*c^2*d^4*x)*e^2 - 60*(C*c^2*d^6 - B*c^2*d^5*e - 2*B*a*c*d^3*e^3 + (2*C*a*c + A*c^2)*d^4*e^2 - B*a^2*d*e^5
+ A*a^2*e^6 + (C*a^2 + 2*A*a*c)*d^2*e^4)*log(x*e + d))*e^(-7)

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Sympy [A]
time = 0.47, size = 359, normalized size = 1.21 \begin {gather*} \frac {C c^{2} x^{6}}{6 e} + x^{5} \left (\frac {B c^{2}}{5 e} - \frac {C c^{2} d}{5 e^{2}}\right ) + x^{4} \left (\frac {A c^{2}}{4 e} - \frac {B c^{2} d}{4 e^{2}} + \frac {C a c}{2 e} + \frac {C c^{2} d^{2}}{4 e^{3}}\right ) + x^{3} \left (- \frac {A c^{2} d}{3 e^{2}} + \frac {2 B a c}{3 e} + \frac {B c^{2} d^{2}}{3 e^{3}} - \frac {2 C a c d}{3 e^{2}} - \frac {C c^{2} d^{3}}{3 e^{4}}\right ) + x^{2} \left (\frac {A a c}{e} + \frac {A c^{2} d^{2}}{2 e^{3}} - \frac {B a c d}{e^{2}} - \frac {B c^{2} d^{3}}{2 e^{4}} + \frac {C a^{2}}{2 e} + \frac {C a c d^{2}}{e^{3}} + \frac {C c^{2} d^{4}}{2 e^{5}}\right ) + x \left (- \frac {2 A a c d}{e^{2}} - \frac {A c^{2} d^{3}}{e^{4}} + \frac {B a^{2}}{e} + \frac {2 B a c d^{2}}{e^{3}} + \frac {B c^{2} d^{4}}{e^{5}} - \frac {C a^{2} d}{e^{2}} - \frac {2 C a c d^{3}}{e^{4}} - \frac {C c^{2} d^{5}}{e^{6}}\right ) + \frac {\left (a e^{2} + c d^{2}\right )^{2} \left (A e^{2} - B d e + C d^{2}\right ) \log {\left (d + e x \right )}}{e^{7}} \end {gather*}

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),x)

[Out]

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

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Giac [A]
time = 4.54, size = 416, normalized size = 1.40 \begin {gather*} {\left (C c^{2} d^{6} - B c^{2} d^{5} e + 2 \, C a c d^{4} e^{2} + A c^{2} d^{4} e^{2} - 2 \, B a c d^{3} e^{3} + C a^{2} d^{2} e^{4} + 2 \, A a c d^{2} e^{4} - B a^{2} d e^{5} + A a^{2} e^{6}\right )} e^{\left (-7\right )} \log \left ({\left | x e + d \right |}\right ) + \frac {1}{60} \, {\left (10 \, C c^{2} x^{6} e^{5} - 12 \, C c^{2} d x^{5} e^{4} + 15 \, C c^{2} d^{2} x^{4} e^{3} - 20 \, C c^{2} d^{3} x^{3} e^{2} + 30 \, C c^{2} d^{4} x^{2} e - 60 \, C c^{2} d^{5} x + 12 \, B c^{2} x^{5} e^{5} - 15 \, B c^{2} d x^{4} e^{4} + 20 \, B c^{2} d^{2} x^{3} e^{3} - 30 \, B c^{2} d^{3} x^{2} e^{2} + 60 \, B c^{2} d^{4} x e + 30 \, C a c x^{4} e^{5} + 15 \, A c^{2} x^{4} e^{5} - 40 \, C a c d x^{3} e^{4} - 20 \, A c^{2} d x^{3} e^{4} + 60 \, C a c d^{2} x^{2} e^{3} + 30 \, A c^{2} d^{2} x^{2} e^{3} - 120 \, C a c d^{3} x e^{2} - 60 \, A c^{2} d^{3} x e^{2} + 40 \, B a c x^{3} e^{5} - 60 \, B a c d x^{2} e^{4} + 120 \, B a c d^{2} x e^{3} + 30 \, C a^{2} x^{2} e^{5} + 60 \, A a c x^{2} e^{5} - 60 \, C a^{2} d x e^{4} - 120 \, A a c d x e^{4} + 60 \, B a^{2} x e^{5}\right )} e^{\left (-6\right )} \end {gather*}

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),x, algorithm="giac")

[Out]

(C*c^2*d^6 - B*c^2*d^5*e + 2*C*a*c*d^4*e^2 + A*c^2*d^4*e^2 - 2*B*a*c*d^3*e^3 + C*a^2*d^2*e^4 + 2*A*a*c*d^2*e^4
- B*a^2*d*e^5 + A*a^2*e^6)*e^(-7)*log(abs(x*e + d)) + 1/60*(10*C*c^2*x^6*e^5 - 12*C*c^2*d*x^5*e^4 + 15*C*c^2*
d^2*x^4*e^3 - 20*C*c^2*d^3*x^3*e^2 + 30*C*c^2*d^4*x^2*e - 60*C*c^2*d^5*x + 12*B*c^2*x^5*e^5 - 15*B*c^2*d*x^4*e
^4 + 20*B*c^2*d^2*x^3*e^3 - 30*B*c^2*d^3*x^2*e^2 + 60*B*c^2*d^4*x*e + 30*C*a*c*x^4*e^5 + 15*A*c^2*x^4*e^5 - 40
*C*a*c*d*x^3*e^4 - 20*A*c^2*d*x^3*e^4 + 60*C*a*c*d^2*x^2*e^3 + 30*A*c^2*d^2*x^2*e^3 - 120*C*a*c*d^3*x*e^2 - 60
*A*c^2*d^3*x*e^2 + 40*B*a*c*x^3*e^5 - 60*B*a*c*d*x^2*e^4 + 120*B*a*c*d^2*x*e^3 + 30*C*a^2*x^2*e^5 + 60*A*a*c*x
^2*e^5 - 60*C*a^2*d*x*e^4 - 120*A*a*c*d*x*e^4 + 60*B*a^2*x*e^5)*e^(-6)

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Mupad [B]
time = 3.68, size = 422, normalized size = 1.42 \begin {gather*} x^5\,\left (\frac {B\,c^2}{5\,e}-\frac {C\,c^2\,d}{5\,e^2}\right )-x\,\left (\frac {d\,\left (\frac {C\,a^2+2\,A\,c\,a}{e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{e}\right )}{e}-\frac {B\,a^2}{e}\right )+x^4\,\left (\frac {A\,c^2+2\,C\,a\,c}{4\,e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{4\,e}\right )-x^3\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{3\,e}-\frac {2\,B\,a\,c}{3\,e}\right )+x^2\,\left (\frac {C\,a^2+2\,A\,c\,a}{2\,e}+\frac {d\,\left (\frac {d\,\left (\frac {A\,c^2+2\,C\,a\,c}{e}-\frac {d\,\left (\frac {B\,c^2}{e}-\frac {C\,c^2\,d}{e^2}\right )}{e}\right )}{e}-\frac {2\,B\,a\,c}{e}\right )}{2\,e}\right )+\frac {\ln \left (d+e\,x\right )\,\left (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\right )}{e^7}+\frac {C\,c^2\,x^6}{6\,e} \end {gather*}

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),x)

[Out]

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

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