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

Optimal. Leaf size=216 \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )-a \left (3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{2 a^{3/2} c^{5/2}}-\frac {e \log \left (a+c x^2\right ) \left (2 a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^3}-\frac {3 e^2 x (A c d-a (B e+3 C d))}{2 a c^2}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac {e^3 x^2 (A c-2 a C)}{2 a c^2} \]

[Out]

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

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Rubi [A]  time = 0.50, antiderivative size = 216, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1645, 801, 635, 205, 260} \[ \frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )-a \left (3 a e^2 (B e+3 C d)-c d^2 (3 B e+C d)\right )\right )}{2 a^{3/2} c^{5/2}}-\frac {e \log \left (a+c x^2\right ) \left (2 a C e^2-c \left (e (A e+3 B d)+3 C d^2\right )\right )}{2 c^3}-\frac {3 e^2 x (A c d-a (B e+3 C d))}{2 a c^2}-\frac {(d+e x)^3 (a B-x (A c-a C))}{2 a c \left (a+c x^2\right )}-\frac {e^3 x^2 (A c-2 a C)}{2 a c^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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]

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 801

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Int[ExpandIntegrand[(
(d + e*x)^m*(f + g*x))/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x] && NeQ[c*d^2 + a*e^2, 0] && Integer
Q[m]

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]))

Rubi steps

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

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Mathematica [A]  time = 0.20, size = 233, normalized size = 1.08 \[ \frac {\frac {\sqrt {c} \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (3 a e^2+c d^2\right )+a \left (c d^2 (3 B e+C d)-3 a e^2 (B e+3 C d)\right )\right )}{a^{3/2}}+\frac {-a^3 C e^3+a^2 c e (e (A e+3 B d+B e x)+3 C d (d+e x))-a c^2 d \left (3 A e (d+e x)+B d (d+3 e x)+C d^2 x\right )+A c^3 d^3 x}{a \left (a+c x^2\right )}+e \log \left (a+c x^2\right ) \left (-2 a C e^2+c e (A e+3 B d)+3 c C d^2\right )+2 c e^2 x (B e+3 C d)+c C e^3 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 0.92, size = 931, normalized size = 4.31 \[ \left [\frac {2 \, C a^{2} c^{2} e^{3} x^{4} + 2 \, C a^{3} c e^{3} x^{2} - 2 \, B a^{2} c^{2} d^{3} + 6 \, B a^{3} c d e^{2} + 6 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e - 2 \, {\left (C a^{4} - A a^{3} c\right )} e^{3} + 4 \, {\left (3 \, C a^{2} c^{2} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{3} + {\left (3 \, B a^{2} c d^{2} e - 3 \, B a^{3} e^{3} + {\left (C a^{2} c + A a c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} - A a^{2} c\right )} d e^{2} + {\left (3 \, B a c^{2} d^{2} e - 3 \, B a^{2} c e^{3} + {\left (C a c^{2} + A c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} + 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) - 2 \, {\left (3 \, B a^{2} c^{2} d^{2} e - 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + 2 \, {\left (3 \, C a^{3} c d^{2} e + 3 \, B a^{3} c d e^{2} - {\left (2 \, C a^{4} - A a^{3} c\right )} e^{3} + {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - {\left (2 \, C a^{3} c - A a^{2} c^{2}\right )} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{4 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}, \frac {C a^{2} c^{2} e^{3} x^{4} + C a^{3} c e^{3} x^{2} - B a^{2} c^{2} d^{3} + 3 \, B a^{3} c d e^{2} + 3 \, {\left (C a^{3} c - A a^{2} c^{2}\right )} d^{2} e - {\left (C a^{4} - A a^{3} c\right )} e^{3} + 2 \, {\left (3 \, C a^{2} c^{2} d e^{2} + B a^{2} c^{2} e^{3}\right )} x^{3} + {\left (3 \, B a^{2} c d^{2} e - 3 \, B a^{3} e^{3} + {\left (C a^{2} c + A a c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} - A a^{2} c\right )} d e^{2} + {\left (3 \, B a c^{2} d^{2} e - 3 \, B a^{2} c e^{3} + {\left (C a c^{2} + A c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) - {\left (3 \, B a^{2} c^{2} d^{2} e - 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - A a c^{3}\right )} d^{3} - 3 \, {\left (3 \, C a^{3} c - A a^{2} c^{2}\right )} d e^{2}\right )} x + {\left (3 \, C a^{3} c d^{2} e + 3 \, B a^{3} c d e^{2} - {\left (2 \, C a^{4} - A a^{3} c\right )} e^{3} + {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - {\left (2 \, C a^{3} c - A a^{2} c^{2}\right )} e^{3}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{2 \, {\left (a^{2} c^{4} x^{2} + a^{3} c^{3}\right )}}\right ] \]

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)^2,x, algorithm="fricas")

[Out]

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

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giac [A]  time = 0.17, size = 289, normalized size = 1.34 \[ \frac {{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} - 2 \, C a e^{3} + A c e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (C a c d^{3} + A c^{2} d^{3} + 3 \, B a c d^{2} e - 9 \, C a^{2} d e^{2} + 3 \, A a c d e^{2} - 3 \, B a^{2} e^{3}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} + \frac {C c^{2} x^{2} e^{3} + 6 \, C c^{2} d x e^{2} + 2 \, B c^{2} x e^{3}}{2 \, c^{4}} - \frac {B a c^{2} d^{3} - 3 \, C a^{2} c d^{2} e + 3 \, A a c^{2} d^{2} e - 3 \, B a^{2} c d e^{2} + C a^{3} e^{3} - A a^{2} c e^{3} + {\left (C a c^{2} d^{3} - A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 3 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - B a^{2} c e^{3}\right )} x}{2 \, {\left (c x^{2} + a\right )} a c^{3}} \]

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)^2,x, algorithm="giac")

[Out]

1/2*(3*C*c*d^2*e + 3*B*c*d*e^2 - 2*C*a*e^3 + A*c*e^3)*log(c*x^2 + a)/c^3 + 1/2*(C*a*c*d^3 + A*c^2*d^3 + 3*B*a*
c*d^2*e - 9*C*a^2*d*e^2 + 3*A*a*c*d*e^2 - 3*B*a^2*e^3)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a*c^2) + 1/2*(C*c^2*x^
2*e^3 + 6*C*c^2*d*x*e^2 + 2*B*c^2*x*e^3)/c^4 - 1/2*(B*a*c^2*d^3 - 3*C*a^2*c*d^2*e + 3*A*a*c^2*d^2*e - 3*B*a^2*
c*d*e^2 + C*a^3*e^3 - A*a^2*c*e^3 + (C*a*c^2*d^3 - A*c^3*d^3 + 3*B*a*c^2*d^2*e - 3*C*a^2*c*d*e^2 + 3*A*a*c^2*d
*e^2 - B*a^2*c*e^3)*x)/((c*x^2 + a)*a*c^3)

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maple [B]  time = 0.02, size = 484, normalized size = 2.24 \[ \frac {A \,d^{3} x}{2 \left (c \,x^{2}+a \right ) a}+\frac {A \,d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}-\frac {3 A d \,e^{2} x}{2 \left (c \,x^{2}+a \right ) c}+\frac {3 A d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {B a \,e^{3} x}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {3 B a \,e^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}-\frac {3 B \,d^{2} e x}{2 \left (c \,x^{2}+a \right ) c}+\frac {3 B \,d^{2} e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {3 C a d \,e^{2} x}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {9 C a d \,e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}-\frac {C \,d^{3} x}{2 \left (c \,x^{2}+a \right ) c}+\frac {C \,d^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {C \,e^{3} x^{2}}{2 c^{2}}+\frac {A a \,e^{3}}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {3 A \,d^{2} e}{2 \left (c \,x^{2}+a \right ) c}+\frac {A \,e^{3} \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {3 B a d \,e^{2}}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {B \,d^{3}}{2 \left (c \,x^{2}+a \right ) c}+\frac {3 B d \,e^{2} \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {B \,e^{3} x}{c^{2}}-\frac {C \,a^{2} e^{3}}{2 \left (c \,x^{2}+a \right ) c^{3}}+\frac {3 C a \,d^{2} e}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {C a \,e^{3} \ln \left (c \,x^{2}+a \right )}{c^{3}}+\frac {3 C \,d^{2} e \ln \left (c \,x^{2}+a \right )}{2 c^{2}}+\frac {3 C d \,e^{2} x}{c^{2}} \]

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)^2,x)

[Out]

-1/2/c/(c*x^2+a)*B*d^3+1/2/c^2*ln(c*x^2+a)*A*e^3+1/2*e^3/c^2*C*x^2+e^3/c^2*B*x+3/2/c/(a*c)^(1/2)*arctan(1/(a*c
)^(1/2)*c*x)*A*d*e^2-3/2/c^2*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*e^3+3/2/c/(a*c)^(1/2)*arctan(1/(a*c)^(1
/2)*c*x)*B*d^2*e-3/2/c/(c*x^2+a)*A*x*d*e^2+1/2/c^2/(c*x^2+a)*B*x*a*e^3-3/2/c/(c*x^2+a)*B*x*d^2*e+3/2/c^2/(c*x^
2+a)*B*a*d*e^2+3/2/c^2/(c*x^2+a)*C*a*d^2*e+1/2/(c*x^2+a)/a*x*A*d^3+3*e^2/c^2*C*d*x+3/2/c^2/(c*x^2+a)*C*x*a*d*e
^2-9/2/c^2*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*C*d*e^2+3/2/c^2*ln(c*x^2+a)*B*d*e^2-1/c^3*a*ln(c*x^2+a)*C*e
^3+3/2/c^2*ln(c*x^2+a)*C*d^2*e+1/2/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^3+1/2/c/(a*c)^(1/2)*arctan(1/(a
*c)^(1/2)*c*x)*C*d^3-1/2/c/(c*x^2+a)*C*x*d^3+1/2/c^2/(c*x^2+a)*A*a*e^3-3/2/c/(c*x^2+a)*A*d^2*e-1/2/c^3/(c*x^2+
a)*C*a^2*e^3

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maxima [A]  time = 0.98, size = 287, normalized size = 1.33 \[ -\frac {B a c^{2} d^{3} - 3 \, B a^{2} c d e^{2} - 3 \, {\left (C a^{2} c - A a c^{2}\right )} d^{2} e + {\left (C a^{3} - A a^{2} c\right )} e^{3} + {\left (3 \, B a c^{2} d^{2} e - B a^{2} c e^{3} + {\left (C a c^{2} - A c^{3}\right )} d^{3} - 3 \, {\left (C a^{2} c - A a c^{2}\right )} d e^{2}\right )} x}{2 \, {\left (a c^{4} x^{2} + a^{2} c^{3}\right )}} + \frac {C e^{3} x^{2} + 2 \, {\left (3 \, C d e^{2} + B e^{3}\right )} x}{2 \, c^{2}} + \frac {{\left (3 \, C c d^{2} e + 3 \, B c d e^{2} - {\left (2 \, C a - A c\right )} e^{3}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, B a c d^{2} e - 3 \, B a^{2} e^{3} + {\left (C a c + A c^{2}\right )} d^{3} - 3 \, {\left (3 \, C a^{2} - A a c\right )} d e^{2}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \, \sqrt {a c} a c^{2}} \]

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)^2,x, algorithm="maxima")

[Out]

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

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mupad [B]  time = 4.01, size = 303, normalized size = 1.40 \[ \frac {x\,\left (B\,e^3+3\,C\,d\,e^2\right )}{c^2}-\frac {\frac {C\,a^2\,e^3-3\,C\,a\,c\,d^2\,e-3\,B\,a\,c\,d\,e^2-A\,a\,c\,e^3+B\,c^2\,d^3+3\,A\,c^2\,d^2\,e}{2\,c}-\frac {x\,\left (3\,C\,a^2\,d\,e^2+B\,a^2\,e^3-C\,a\,c\,d^3-3\,B\,a\,c\,d^2\,e-3\,A\,a\,c\,d\,e^2+A\,c^2\,d^3\right )}{2\,a}}{c^3\,x^2+a\,c^2}+\frac {C\,e^3\,x^2}{2\,c^2}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-9\,C\,a^2\,d\,e^2-3\,B\,a^2\,e^3+C\,a\,c\,d^3+3\,B\,a\,c\,d^2\,e+3\,A\,a\,c\,d\,e^2+A\,c^2\,d^3\right )}{2\,a^{3/2}\,c^{5/2}}+\frac {\ln \left (c\,x^2+a\right )\,\left (-32\,C\,a^4\,c^3\,e^3+48\,C\,a^3\,c^4\,d^2\,e+48\,B\,a^3\,c^4\,d\,e^2+16\,A\,a^3\,c^4\,e^3\right )}{32\,a^3\,c^6} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x*(B*e^3 + 3*C*d*e^2))/c^2 - ((B*c^2*d^3 + C*a^2*e^3 - A*a*c*e^3 + 3*A*c^2*d^2*e - 3*B*a*c*d*e^2 - 3*C*a*c*d^
2*e)/(2*c) - (x*(A*c^2*d^3 + B*a^2*e^3 - C*a*c*d^3 + 3*C*a^2*d*e^2 - 3*A*a*c*d*e^2 - 3*B*a*c*d^2*e))/(2*a))/(a
*c^2 + c^3*x^2) + (C*e^3*x^2)/(2*c^2) + (atan((c^(1/2)*x)/a^(1/2))*(A*c^2*d^3 - 3*B*a^2*e^3 + C*a*c*d^3 - 9*C*
a^2*d*e^2 + 3*A*a*c*d*e^2 + 3*B*a*c*d^2*e))/(2*a^(3/2)*c^(5/2)) + (log(a + c*x^2)*(16*A*a^3*c^4*e^3 - 32*C*a^4
*c^3*e^3 + 48*B*a^3*c^4*d*e^2 + 48*C*a^3*c^4*d^2*e))/(32*a^3*c^6)

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sympy [B]  time = 34.46, size = 952, normalized size = 4.41 \[ \frac {C e^{3} x^{2}}{2 c^{2}} + x \left (\frac {B e^{3}}{c^{2}} + \frac {3 C d e^{2}}{c^{2}}\right ) + \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right ) \log {\left (x + \frac {2 A a^{2} c e^{3} + 6 B a^{2} c d e^{2} - 4 C a^{3} e^{3} + 6 C a^{2} c d^{2} e - 4 a^{2} c^{3} \left (- \frac {e \left (- A c e^{2} - 3 B c d e + 2 C a e^{2} - 3 C c d^{2}\right )}{2 c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 3 A a c d e^{2} - A c^{2} d^{3} + 3 B a^{2} e^{3} - 3 B a c d^{2} e + 9 C a^{2} d e^{2} - C a c d^{3}\right )}{4 a^{3} c^{6}}\right )}{- 3 A a c^{2} d e^{2} - A c^{3} d^{3} + 3 B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 9 C a^{2} c d e^{2} - C a c^{2} d^{3}} \right )} + \frac {A a^{2} c e^{3} - 3 A a c^{2} d^{2} e + 3 B a^{2} c d e^{2} - B a c^{2} d^{3} - C a^{3} e^{3} + 3 C a^{2} c d^{2} e + x \left (- 3 A a c^{2} d e^{2} + A c^{3} d^{3} + B a^{2} c e^{3} - 3 B a c^{2} d^{2} e + 3 C a^{2} c d e^{2} - C a c^{2} d^{3}\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} \]

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)**2,x)

[Out]

C*e**3*x**2/(2*c**2) + x*(B*e**3/c**2 + 3*C*d*e**2/c**2) + (-e*(-A*c*e**2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d**
2)/(2*c**3) - sqrt(-a**3*c**7)*(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e*
*2 - C*a*c*d**3)/(4*a**3*c**6))*log(x + (2*A*a**2*c*e**3 + 6*B*a**2*c*d*e**2 - 4*C*a**3*e**3 + 6*C*a**2*c*d**2
*e - 4*a**2*c**3*(-e*(-A*c*e**2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) - sqrt(-a**3*c**7)*(-3*A*a*c*d
*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e**2 - C*a*c*d**3)/(4*a**3*c**6)))/(-3*A*a*c
**2*d*e**2 - A*c**3*d**3 + 3*B*a**2*c*e**3 - 3*B*a*c**2*d**2*e + 9*C*a**2*c*d*e**2 - C*a*c**2*d**3)) + (-e*(-A
*c*e**2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d**2)/(2*c**3) + sqrt(-a**3*c**7)*(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*
B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*e**2 - C*a*c*d**3)/(4*a**3*c**6))*log(x + (2*A*a**2*c*e**3 + 6*B*a**
2*c*d*e**2 - 4*C*a**3*e**3 + 6*C*a**2*c*d**2*e - 4*a**2*c**3*(-e*(-A*c*e**2 - 3*B*c*d*e + 2*C*a*e**2 - 3*C*c*d
**2)/(2*c**3) + sqrt(-a**3*c**7)*(-3*A*a*c*d*e**2 - A*c**2*d**3 + 3*B*a**2*e**3 - 3*B*a*c*d**2*e + 9*C*a**2*d*
e**2 - C*a*c*d**3)/(4*a**3*c**6)))/(-3*A*a*c**2*d*e**2 - A*c**3*d**3 + 3*B*a**2*c*e**3 - 3*B*a*c**2*d**2*e + 9
*C*a**2*c*d*e**2 - C*a*c**2*d**3)) + (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 + x*(-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))/(2*a**2*c**3 + 2*a*c**4*x**2)

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