### 3.43 $$\int \frac{(d+e x)^3 (A+B x+C x^2)}{a+c x^2} \, dx$$

Optimal. Leaf size=240 $-\frac{e x^2 \left (a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^2}+\frac{\log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{2 c^3}-\frac{x \left (a e^2 (B e+3 C d)-c d \left (3 e (A e+B d)+C d^2\right )\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^3 (B e+3 C d)}{3 c}+\frac{C e^3 x^4}{4 c}$

[Out]

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

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Rubi [A]  time = 0.471217, antiderivative size = 237, normalized size of antiderivative = 0.99, number of steps used = 5, number of rules used = 4, integrand size = 27, $$\frac{\text{number of rules}}{\text{integrand size}}$$ = 0.148, Rules used = {1629, 635, 205, 260} $\frac{e x^2 \left (-a C e^2+c e (A e+3 B d)+3 c C d^2\right )}{2 c^2}+\frac{\log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{2 c^3}+\frac{x \left (-a e^2 (B e+3 C d)+3 c d e (A e+B d)+c C d^3\right )}{c^2}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^3 (B e+3 C d)}{3 c}+\frac{C e^3 x^4}{4 c}$

Antiderivative was successfully veriﬁed.

[In]

Int[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2),x]

[Out]

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

Rule 1629

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

Rule 635

Int[((d_) + (e_.)*(x_))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Dist[d, Int[1/(a + c*x^2), x], x] + Dist[e, Int[x/
(a + c*x^2), x], x] /; FreeQ[{a, c, d, e}, x] &&  !NiceSqrtQ[-(a*c)]

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]

Rule 260

Int[(x_)^(m_.)/((a_) + (b_.)*(x_)^(n_)), x_Symbol] :> Simp[Log[RemoveContent[a + b*x^n, x]]/(b*n), x] /; FreeQ
[{a, b, m, n}, x] && EqQ[m, n - 1]

Rubi steps

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

Mathematica [A]  time = 0.224381, size = 223, normalized size = 0.93 $\frac{c x \left (-6 a e^2 (2 B e+6 C d+C e x)+2 c e \left (3 A e (6 d+e x)+B \left (18 d^2+9 d e x+2 e^2 x^2\right )\right )+3 c C \left (6 d^2 e x+4 d^3+4 d e^2 x^2+e^3 x^3\right )\right )+6 \log \left (a+c x^2\right ) \left (e (A c-a C) \left (3 c d^2-a e^2\right )+B c d \left (c d^2-3 a e^2\right )\right )}{12 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (c d^2-3 a e^2\right )+a \left (a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{\sqrt{a} c^{5/2}}$

Antiderivative was successfully veriﬁed.

[In]

Integrate[((d + e*x)^3*(A + B*x + C*x^2))/(a + c*x^2),x]

[Out]

((A*c*d*(c*d^2 - 3*a*e^2) + a*(a*e^2*(3*C*d + B*e) - c*d^2*(C*d + 3*B*e)))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[
a]*c^(5/2)) + (c*x*(-6*a*e^2*(6*C*d + 2*B*e + C*e*x) + 3*c*C*(4*d^3 + 6*d^2*e*x + 4*d*e^2*x^2 + e^3*x^3) + 2*c
*e*(3*A*e*(6*d + e*x) + B*(18*d^2 + 9*d*e*x + 2*e^2*x^2))) + 6*(B*c*d*(c*d^2 - 3*a*e^2) + (A*c - a*C)*e*(3*c*d
^2 - a*e^2))*Log[a + c*x^2])/(12*c^3)

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Maple [A]  time = 0.052, size = 399, normalized size = 1.7 \begin{align*}{\frac{C{e}^{3}{x}^{4}}{4\,c}}+{\frac{B{x}^{3}{e}^{3}}{3\,c}}+{\frac{C{x}^{3}d{e}^{2}}{c}}+{\frac{A{x}^{2}{e}^{3}}{2\,c}}+{\frac{3\,B{x}^{2}d{e}^{2}}{2\,c}}-{\frac{C{x}^{2}a{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,C{x}^{2}{d}^{2}e}{2\,c}}+3\,{\frac{Ad{e}^{2}x}{c}}-{\frac{Ba{e}^{3}x}{{c}^{2}}}+3\,{\frac{B{d}^{2}ex}{c}}-3\,{\frac{Cad{e}^{2}x}{{c}^{2}}}+{\frac{C{d}^{3}x}{c}}-{\frac{\ln \left ( c{x}^{2}+a \right ) aA{e}^{3}}{2\,{c}^{2}}}+{\frac{3\,\ln \left ( c{x}^{2}+a \right ) A{d}^{2}e}{2\,c}}-{\frac{3\,\ln \left ( c{x}^{2}+a \right ) aBd{e}^{2}}{2\,{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{3}}{2\,c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) C{a}^{2}{e}^{3}}{2\,{c}^{3}}}-{\frac{3\,\ln \left ( c{x}^{2}+a \right ) Ca{d}^{2}e}{2\,{c}^{2}}}-3\,{\frac{Aad{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{A{d}^{3}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+{\frac{{a}^{2}B{e}^{3}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-3\,{\frac{Ba{d}^{2}e}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+3\,{\frac{C{a}^{2}d{e}^{2}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }-{\frac{{d}^{3}aC}{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)^3*(C*x^2+B*x+A)/(c*x^2+a),x)

[Out]

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

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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)^3*(C*x^2+B*x+A)/(c*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.88152, size = 1239, normalized size = 5.16 \begin{align*} \left [\frac{3 \, C a c^{2} e^{3} x^{4} + 4 \,{\left (3 \, C a c^{2} d e^{2} + B a c^{2} e^{3}\right )} x^{3} + 6 \,{\left (3 \, C a c^{2} d^{2} e + 3 \, B a c^{2} d e^{2} -{\left (C a^{2} c - A a c^{2}\right )} e^{3}\right )} x^{2} + 6 \,{\left (3 \, B a c d^{2} e - B a^{2} e^{3} +{\left (C a c - A c^{2}\right )} d^{3} - 3 \,{\left (C a^{2} - A a c\right )} d e^{2}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 12 \,{\left (C a c^{2} d^{3} + 3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x + 6 \,{\left (B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d^{2} e +{\left (C a^{3} - A a^{2} c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}, \frac{3 \, C a c^{2} e^{3} x^{4} + 4 \,{\left (3 \, C a c^{2} d e^{2} + B a c^{2} e^{3}\right )} x^{3} + 6 \,{\left (3 \, C a c^{2} d^{2} e + 3 \, B a c^{2} d e^{2} -{\left (C a^{2} c - A a c^{2}\right )} e^{3}\right )} x^{2} - 12 \,{\left (3 \, B a c d^{2} e - B a^{2} e^{3} +{\left (C a c - A c^{2}\right )} d^{3} - 3 \,{\left (C a^{2} - A a c\right )} d e^{2}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 12 \,{\left (C a c^{2} d^{3} + 3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x + 6 \,{\left (B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \,{\left (C a^{2} c - A a c^{2}\right )} d^{2} e +{\left (C a^{3} - A a^{2} c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}\right ] \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(C*x^2+B*x+A)/(c*x^2+a),x, algorithm="fricas")

[Out]

[1/12*(3*C*a*c^2*e^3*x^4 + 4*(3*C*a*c^2*d*e^2 + B*a*c^2*e^3)*x^3 + 6*(3*C*a*c^2*d^2*e + 3*B*a*c^2*d*e^2 - (C*a
^2*c - A*a*c^2)*e^3)*x^2 + 6*(3*B*a*c*d^2*e - B*a^2*e^3 + (C*a*c - A*c^2)*d^3 - 3*(C*a^2 - A*a*c)*d*e^2)*sqrt(
-a*c)*log((c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 12*(C*a*c^2*d^3 + 3*B*a*c^2*d^2*e - B*a^2*c*e^3 - 3*(C*a
^2*c - A*a*c^2)*d*e^2)*x + 6*(B*a*c^2*d^3 - 3*B*a^2*c*d*e^2 - 3*(C*a^2*c - A*a*c^2)*d^2*e + (C*a^3 - A*a^2*c)*
e^3)*log(c*x^2 + a))/(a*c^3), 1/12*(3*C*a*c^2*e^3*x^4 + 4*(3*C*a*c^2*d*e^2 + B*a*c^2*e^3)*x^3 + 6*(3*C*a*c^2*d
^2*e + 3*B*a*c^2*d*e^2 - (C*a^2*c - A*a*c^2)*e^3)*x^2 - 12*(3*B*a*c*d^2*e - B*a^2*e^3 + (C*a*c - A*c^2)*d^3 -
3*(C*a^2 - A*a*c)*d*e^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + 12*(C*a*c^2*d^3 + 3*B*a*c^2*d^2*e - B*a^2*c*e^3 - 3
*(C*a^2*c - A*a*c^2)*d*e^2)*x + 6*(B*a*c^2*d^3 - 3*B*a^2*c*d*e^2 - 3*(C*a^2*c - A*a*c^2)*d^2*e + (C*a^3 - A*a^
2*c)*e^3)*log(c*x^2 + a))/(a*c^3)]

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Sympy [B]  time = 6.76913, size = 1000, normalized size = 4.17 \begin{align*} \frac{C e^{3} x^{4}}{4 c} + \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + 2 a c^{3} \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right )}{- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right ) \log{\left (x + \frac{A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + 2 a c^{3} \left (\frac{- A a c e^{3} + 3 A c^{2} d^{2} e - 3 B a c d e^{2} + B c^{2} d^{3} + C a^{2} e^{3} - 3 C a c d^{2} e}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (- 3 A a c d e^{2} + A c^{2} d^{3} + B a^{2} e^{3} - 3 B a c d^{2} e + 3 C a^{2} d e^{2} - C a c d^{3}\right )}{2 a c^{6}}\right )}{- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \frac{x^{3} \left (B e^{3} + 3 C d e^{2}\right )}{3 c} - \frac{x^{2} \left (- A c e^{3} - 3 B c d e^{2} + C a e^{3} - 3 C c d^{2} e\right )}{2 c^{2}} - \frac{x \left (- 3 A c d e^{2} + B a e^{3} - 3 B c d^{2} e + 3 C a d e^{2} - C c d^{3}\right )}{c^{2}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)**3*(C*x**2+B*x+A)/(c*x**2+a),x)

[Out]

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

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Giac [A]  time = 1.14414, size = 377, normalized size = 1.57 \begin{align*} -\frac{{\left (C a c d^{3} - A c^{2} d^{3} + 3 \, B a c d^{2} e - 3 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} - B a^{2} e^{3}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{{\left (B c^{2} d^{3} - 3 \, C a c d^{2} e + 3 \, A c^{2} d^{2} e - 3 \, B a c d e^{2} + C a^{2} e^{3} - A a c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, C c^{3} x^{4} e^{3} + 12 \, C c^{3} d x^{3} e^{2} + 18 \, C c^{3} d^{2} x^{2} e + 12 \, C c^{3} d^{3} x + 4 \, B c^{3} x^{3} e^{3} + 18 \, B c^{3} d x^{2} e^{2} + 36 \, B c^{3} d^{2} x e - 6 \, C a c^{2} x^{2} e^{3} + 6 \, A c^{3} x^{2} e^{3} - 36 \, C a c^{2} d x e^{2} + 36 \, A c^{3} d x e^{2} - 12 \, B a c^{2} x e^{3}}{12 \, c^{4}} \end{align*}

Veriﬁcation of antiderivative is not currently implemented for this CAS.

[In]

integrate((e*x+d)^3*(C*x^2+B*x+A)/(c*x^2+a),x, algorithm="giac")

[Out]

-(C*a*c*d^3 - A*c^2*d^3 + 3*B*a*c*d^2*e - 3*C*a^2*d*e^2 + 3*A*a*c*d*e^2 - B*a^2*e^3)*arctan(c*x/sqrt(a*c))/(sq
rt(a*c)*c^2) + 1/2*(B*c^2*d^3 - 3*C*a*c*d^2*e + 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 + C*a^2*e^3 - A*a*c*e^3)*log(c*x
^2 + a)/c^3 + 1/12*(3*C*c^3*x^4*e^3 + 12*C*c^3*d*x^3*e^2 + 18*C*c^3*d^2*x^2*e + 12*C*c^3*d^3*x + 4*B*c^3*x^3*e
^3 + 18*B*c^3*d*x^2*e^2 + 36*B*c^3*d^2*x*e - 6*C*a*c^2*x^2*e^3 + 6*A*c^3*x^2*e^3 - 36*C*a*c^2*d*x*e^2 + 36*A*c
^3*d*x*e^2 - 12*B*a*c^2*x*e^3)/c^4