3.1.57 \(\int \frac {(d+e x)^3 (A+B x+C x^2)}{(a+c x^2)^3} \, dx\) [57]

Optimal. Leaf size=209 \[ -\frac {(a B-(A c-a C) x) (d+e x)^3}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x) \left (a e (3 A c d+5 a C d+3 a B e)-\left (3 A c^2 d^2-a \left (4 a C e^2-c d (C d+3 B e)\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (3 a e^2 (A c d+3 a C d+a B e)+c d^2 (3 A c d+a C d+3 a B e)\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{5/2}}+\frac {C e^3 \log \left (a+c x^2\right )}{2 c^3} \]

[Out]

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

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Rubi [A]
time = 0.18, antiderivative size = 209, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 27, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {1659, 833, 649, 211, 266} \begin {gather*} \frac {\text {ArcTan}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (c d^2 (3 a B e+a C d+3 A c d)+3 a e^2 (a B e+3 a C d+A c d)\right )}{8 a^{5/2} c^{5/2}}-\frac {(d+e x) \left (a e (3 a B e+5 a C d+3 A c d)-x \left (3 A c^2 d^2-a \left (4 a C e^2-c d (3 B e+C d)\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {(d+e x)^3 (a B-x (A c-a C))}{4 a c \left (a+c x^2\right )^2}+\frac {C e^3 \log \left (a+c x^2\right )}{2 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

-1/4*((a*B - (A*c - a*C)*x)*(d + e*x)^3)/(a*c*(a + c*x^2)^2) - ((d + e*x)*(a*e*(3*A*c*d + 5*a*C*d + 3*a*B*e) -
(3*A*c^2*d^2 - a*(4*a*C*e^2 - c*d*(C*d + 3*B*e)))*x))/(8*a^2*c^2*(a + c*x^2)) + ((3*a*e^2*(A*c*d + 3*a*C*d +
a*B*e) + c*d^2*(3*A*c*d + a*C*d + 3*a*B*e))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(5/2)) + (C*e^3*Log[a +
c*x^2])/(2*c^3)

Rule 211

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

Rule 266

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 649

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 833

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

Rule 1659

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

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Mathematica [A]
time = 0.17, size = 281, normalized size = 1.34 \begin {gather*} \frac {\frac {-2 a^3 C e^3+2 A c^3 d^3 x-2 a c^2 d \left (C d^2 x+3 A e (d+e x)+B d (d+3 e x)\right )+2 a^2 c e (3 C d (d+e x)+e (3 B d+A e+B e x))}{a \left (a+c x^2\right )^2}+\frac {8 a^3 C e^3+3 A c^3 d^3 x+a c^2 d \left (C d^2+3 e (B d+A e)\right ) x-a^2 c e (3 C d (4 d+5 e x)+e (12 B d+4 A e+5 B e x))}{a^2 \left (a+c x^2\right )}+\frac {\sqrt {c} \left (3 A c d \left (c d^2+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 )}{a^{5/2}}+4 C e^3 \log \left (a+c x^2\right )}{8 c^3} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

((-2*a^3*C*e^3 + 2*A*c^3*d^3*x - 2*a*c^2*d*(C*d^2*x + 3*A*e*(d + e*x) + B*d*(d + 3*e*x)) + 2*a^2*c*e*(3*C*d*(d
+ e*x) + e*(3*B*d + A*e + B*e*x)))/(a*(a + c*x^2)^2) + (8*a^3*C*e^3 + 3*A*c^3*d^3*x + a*c^2*d*(C*d^2 + 3*e*(B
*d + A*e))*x - a^2*c*e*(3*C*d*(4*d + 5*e*x) + e*(12*B*d + 4*A*e + 5*B*e*x)))/(a^2*(a + c*x^2)) + (Sqrt[c]*(3*A
*c*d*(c*d^2 + 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^(5/2) +
4*C*e^3*Log[a + c*x^2])/(8*c^3)

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Maple [A]
time = 0.15, size = 333, normalized size = 1.59

method	result	size
default	\(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,a^{2} e^{3}+3 B a c \,d^{2} e -15 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x^{3}}{8 a^{2} c}-\frac {e \left (A c \,e^{2}+3 B c d e -2 a C \,e^{2}+3 C c \,d^{2}\right ) x^{2}}{2 c^{2}}-\frac {\left (3 A a c d \,e^{2}-5 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x}{8 c^{2} a}-\frac {A a c \,e^{3}+3 A \,c^{2} d^{2} e +3 B a c d \,e^{2}+B \,c^{2} d^{3}-3 C \,a^{2} e^{3}+3 C a c \,d^{2} e}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\frac {4 C \,a^{2} e^{3} \ln \left (c \,x^{2}+a \right )}{c}+\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}+3 B \,a^{2} e^{3}+3 B a c \,d^{2} e +9 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}}}{8 a^{2} c^{2}}\)	\(333\)
risch	\(\frac {\frac {\left (3 A a c d \,e^{2}+3 A \,c^{2} d^{3}-5 B \,a^{2} e^{3}+3 B a c \,d^{2} e -15 C \,a^{2} d \,e^{2}+C a c \,d^{3}\right ) x^{3}}{8 a^{2} c}-\frac {e \left (A c \,e^{2}+3 B c d e -2 a C \,e^{2}+3 C c \,d^{2}\right ) x^{2}}{2 c^{2}}-\frac {\left (3 A a c d \,e^{2}-5 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 ) x}{8 c^{2} a}-\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}-3 C \,a^{2} e^{3}+3 C a c \,d^{2} e}{4 c^{3}}}{\left (c \,x^{2}+a \right )^{2}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}-\sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}\, x \right ) C \,e^{3}}{2 c^{3}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}-\sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}\, x \right ) \sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}}{16 a^{3} c^{3}}+\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}+\sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}\, x \right ) C \,e^{3}}{2 c^{3}}-\frac {\ln \left (3 A \,a^{2} c d \,e^{2}+3 A a \,c^{2} d^{3}+3 B \,a^{3} e^{3}+3 B \,a^{2} c \,d^{2} e +9 C \,a^{3} d \,e^{2}+C \,a^{2} c \,d^{3}+\sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}\, x \right ) \sqrt {-a c \left (3 A a c d \,e^{2}+3 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 )^{2}}}{16 a^{3} c^{3}}\)	\(899\)

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)^3,x,method=_RETURNVERBOSE)

[Out]

(1/8*(3*A*a*c*d*e^2+3*A*c^2*d^3-5*B*a^2*e^3+3*B*a*c*d^2*e-15*C*a^2*d*e^2+C*a*c*d^3)/a^2/c*x^3-1/2*e*(A*c*e^2+3
*B*c*d*e-2*C*a*e^2+3*C*c*d^2)/c^2*x^2-1/8*(3*A*a*c*d*e^2-5*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)/c^2/a*x-1/4*(A*a*c*e^3+3*A*c^2*d^2*e+3*B*a*c*d*e^2+B*c^2*d^3-3*C*a^2*e^3+3*C*a*c*d^2*e)/c^3)/(c*x^2+
a)^2+1/8/a^2/c^2*(4*C*a^2*e^3/c*ln(c*x^2+a)+(3*A*a*c*d*e^2+3*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)/(a*c)^(1/2)*arctan(c*x/(a*c)^(1/2)))

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Maxima [A]
time = 0.52, size = 378, normalized size = 1.81 \begin {gather*} -\frac {2 \, B a^{2} c^{2} d^{3} + 6 \, B a^{3} c d e^{2} - 6 \, C a^{4} e^{3} + 2 \, A a^{3} c e^{3} - {\left (3 \, B a c^{3} d^{2} e - 5 \, B a^{2} c^{2} e^{3} + {\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} - 3 \, {\left (5 \, C a^{2} c^{2} e^{2} - A a c^{3} e^{2}\right )} d\right )} x^{3} + 6 \, {\left (C a^{3} c e + A a^{2} c^{2} e\right )} d^{2} + 4 \, {\left (3 \, C a^{2} c^{2} d^{2} e + 3 \, B a^{2} c^{2} d e^{2} - 2 \, C a^{3} c e^{3} + A a^{2} c^{2} e^{3}\right )} x^{2} + {\left (3 \, B a^{2} c^{2} d^{2} e + 3 \, B a^{3} c e^{3} + {\left (C a^{2} c^{2} - 5 \, A a c^{3}\right )} d^{3} + 3 \, {\left (3 \, C a^{3} c e^{2} + A a^{2} c^{2} e^{2}\right )} d\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {C e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, B a c d^{2} e + {\left (C a c + 3 \, A c^{2}\right )} d^{3} + 3 \, B a^{2} e^{3} + 3 \, {\left (3 \, C a^{2} e^{2} + A a c e^{2}\right )} d\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{2}} \end {gather*}

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)^3,x, algorithm="maxima")

[Out]

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

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 545 vs. \(2 (199) = 398\).
time = 0.36, size = 1111, normalized size = 5.32 \begin {gather*} \left [-\frac {4 \, B a^{3} c^{2} d^{3} - 2 \, {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{3} x^{3} + 2 \, {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{3} x - 8 \, {\left (C a^{3} c^{2} x^{4} + 2 \, C a^{4} c x^{2} + C a^{5}\right )} e^{3} \log \left (c x^{2} + a\right ) + {\left ({\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} x^{4} + 2 \, {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{3} x^{2} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{3} + 3 \, {\left (B a^{2} c^{2} x^{4} + 2 \, B a^{3} c x^{2} + B a^{4}\right )} e^{3} + 3 \, {\left ({\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} d x^{4} + 2 \, {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} d x^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} d\right )} e^{2} + 3 \, {\left (B a c^{3} d^{2} x^{4} + 2 \, B a^{2} c^{2} d^{2} x^{2} + B a^{3} c d^{2}\right )} e\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 2 \, {\left (5 \, B a^{3} c^{2} x^{3} + 3 \, B a^{4} c x - 6 \, C a^{5} + 2 \, A a^{4} c - 4 \, {\left (2 \, C a^{4} c - A a^{3} c^{2}\right )} x^{2}\right )} e^{3} + 6 \, {\left (4 \, B a^{3} c^{2} d x^{2} + 2 \, B a^{4} c d + {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} d x^{3} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} d x\right )} e^{2} - 6 \, {\left (B a^{2} c^{3} d^{2} x^{3} - 4 \, C a^{3} c^{2} d^{2} x^{2} - B a^{3} c^{2} d^{2} x - 2 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d^{2}\right )} e}{16 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}, -\frac {2 \, B a^{3} c^{2} d^{3} - {\left (C a^{2} c^{3} + 3 \, A a c^{4}\right )} d^{3} x^{3} + {\left (C a^{3} c^{2} - 5 \, A a^{2} c^{3}\right )} d^{3} x - 4 \, {\left (C a^{3} c^{2} x^{4} + 2 \, C a^{4} c x^{2} + C a^{5}\right )} e^{3} \log \left (c x^{2} + a\right ) - {\left ({\left (C a c^{3} + 3 \, A c^{4}\right )} d^{3} x^{4} + 2 \, {\left (C a^{2} c^{2} + 3 \, A a c^{3}\right )} d^{3} x^{2} + {\left (C a^{3} c + 3 \, A a^{2} c^{2}\right )} d^{3} + 3 \, {\left (B a^{2} c^{2} x^{4} + 2 \, B a^{3} c x^{2} + B a^{4}\right )} e^{3} + 3 \, {\left ({\left (3 \, C a^{2} c^{2} + A a c^{3}\right )} d x^{4} + 2 \, {\left (3 \, C a^{3} c + A a^{2} c^{2}\right )} d x^{2} + {\left (3 \, C a^{4} + A a^{3} c\right )} d\right )} e^{2} + 3 \, {\left (B a c^{3} d^{2} x^{4} + 2 \, B a^{2} c^{2} d^{2} x^{2} + B a^{3} c d^{2}\right )} e\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + {\left (5 \, B a^{3} c^{2} x^{3} + 3 \, B a^{4} c x - 6 \, C a^{5} + 2 \, A a^{4} c - 4 \, {\left (2 \, C a^{4} c - A a^{3} c^{2}\right )} x^{2}\right )} e^{3} + 3 \, {\left (4 \, B a^{3} c^{2} d x^{2} + 2 \, B a^{4} c d + {\left (5 \, C a^{3} c^{2} - A a^{2} c^{3}\right )} d x^{3} + {\left (3 \, C a^{4} c + A a^{3} c^{2}\right )} d x\right )} e^{2} - 3 \, {\left (B a^{2} c^{3} d^{2} x^{3} - 4 \, C a^{3} c^{2} d^{2} x^{2} - B a^{3} c^{2} d^{2} x - 2 \, {\left (C a^{4} c + A a^{3} c^{2}\right )} d^{2}\right )} e}{8 \, {\left (a^{3} c^{5} x^{4} + 2 \, a^{4} c^{4} x^{2} + a^{5} c^{3}\right )}}\right ] \end {gather*}

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)^3,x, algorithm="fricas")

[Out]

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

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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)**3,x)

[Out]

Timed out

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Giac [A]
time = 4.42, size = 348, normalized size = 1.67 \begin {gather*} \frac {C e^{3} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (C a c d^{3} + 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 )}{8 \, \sqrt {a c} a^{2} c^{2}} + \frac {{\left (C a c^{2} d^{3} + 3 \, A c^{3} d^{3} + 3 \, B a c^{2} d^{2} e - 15 \, C a^{2} c d e^{2} + 3 \, A a c^{2} d e^{2} - 5 \, B a^{2} c e^{3}\right )} x^{3} - 4 \, {\left (3 \, C a^{2} c d^{2} e + 3 \, B a^{2} c d e^{2} - 2 \, C a^{3} e^{3} + A a^{2} c e^{3}\right )} x^{2} - {\left (C a^{2} c d^{3} - 5 \, A a c^{2} d^{3} + 3 \, B a^{2} c d^{2} e + 9 \, C a^{3} d e^{2} + 3 \, A a^{2} c d e^{2} + 3 \, B a^{3} e^{3}\right )} x - \frac {2 \, {\left (B a^{2} c^{2} d^{3} + 3 \, C a^{3} c d^{2} e + 3 \, A a^{2} c^{2} d^{2} e + 3 \, B a^{3} c d e^{2} - 3 \, C a^{4} e^{3} + A a^{3} c e^{3}\right )}}{c}}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} c^{2}} \end {gather*}

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)^3,x, algorithm="giac")

[Out]

1/2*C*e^3*log(c*x^2 + a)/c^3 + 1/8*(C*a*c*d^3 + 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^2*c^2) + 1/8*((C*a*c^2*d^3 + 3*A*c^3*d^3 + 3*B*a*c^2*d^2*e - 1
5*C*a^2*c*d*e^2 + 3*A*a*c^2*d*e^2 - 5*B*a^2*c*e^3)*x^3 - 4*(3*C*a^2*c*d^2*e + 3*B*a^2*c*d*e^2 - 2*C*a^3*e^3 +
A*a^2*c*e^3)*x^2 - (C*a^2*c*d^3 - 5*A*a*c^2*d^3 + 3*B*a^2*c*d^2*e + 9*C*a^3*d*e^2 + 3*A*a^2*c*d*e^2 + 3*B*a^3*
e^3)*x - 2*(B*a^2*c^2*d^3 + 3*C*a^3*c*d^2*e + 3*A*a^2*c^2*d^2*e + 3*B*a^3*c*d*e^2 - 3*C*a^4*e^3 + A*a^3*c*e^3)
/c)/((c*x^2 + a)^2*a^2*c^2)

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Mupad [B]
time = 1.77, size = 920, normalized size = 4.40 \begin {gather*} \frac {5\,A\,d^3\,x}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {B\,d^3}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,C\,a^2\,e^3}{4\,\left (a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4\right )}-\frac {3\,A\,d^2\,e}{4\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,d^3\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {C\,d^3\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {A\,a\,e^3}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {A\,e^3\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {5\,B\,e^3\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,e^3\,\ln \left (c\,x^2+a\right )}{2\,c^3}-\frac {3\,B\,a\,d\,e^2}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {3\,C\,a\,d^2\,e}{4\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}+\frac {3\,A\,c\,d^3\,x^3}{8\,\left (a^4+2\,a^3\,c\,x^2+a^2\,c^2\,x^4\right )}-\frac {3\,B\,a\,e^3\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )}-\frac {3\,B\,d\,e^2\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,C\,d^2\,e\,x^2}{2\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {15\,C\,d\,e^2\,x^3}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {C\,a\,e^3\,x^2}{a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4}+\frac {3\,A\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{5/2}\,\sqrt {c}}+\frac {3\,B\,e^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}+\frac {C\,d^3\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {3\,A\,d\,e^2\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}+\frac {3\,B\,d^2\,e\,x^3}{8\,\left (a^3+2\,a^2\,c\,x^2+a\,c^2\,x^4\right )}-\frac {3\,A\,d\,e^2\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}-\frac {3\,B\,d^2\,e\,x}{8\,\left (a^2\,c+2\,a\,c^2\,x^2+c^3\,x^4\right )}+\frac {3\,A\,d\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {3\,B\,d^2\,e\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,a^{3/2}\,c^{3/2}}+\frac {9\,C\,d\,e^2\,\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )}{8\,\sqrt {a}\,c^{5/2}}-\frac {9\,C\,a\,d\,e^2\,x}{8\,\left (a^2\,c^2+2\,a\,c^3\,x^2+c^4\,x^4\right )} \end {gather*}

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)^3,x)

[Out]

(5*A*d^3*x)/(8*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (B*d^3)/(4*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) + (3*C*a^2*e^3)/
(4*(a^2*c^3 + c^5*x^4 + 2*a*c^4*x^2)) - (3*A*d^2*e)/(4*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) + (C*d^3*x^3)/(8*(a^3
+ 2*a^2*c*x^2 + a*c^2*x^4)) - (C*d^3*x)/(8*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (A*a*e^3)/(4*(a^2*c^2 + c^4*x^4
+ 2*a*c^3*x^2)) - (A*e^3*x^2)/(2*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (5*B*e^3*x^3)/(8*(a^2*c + c^3*x^4 + 2*a*c^
2*x^2)) + (C*e^3*log(a + c*x^2))/(2*c^3) - (3*B*a*d*e^2)/(4*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) - (3*C*a*d^2*e)
/(4*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) + (3*A*c*d^3*x^3)/(8*(a^4 + 2*a^3*c*x^2 + a^2*c^2*x^4)) - (3*B*a*e^3*x)
/(8*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2)) - (3*B*d*e^2*x^2)/(2*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (3*C*d^2*e*x^2)
/(2*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (15*C*d*e^2*x^3)/(8*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) + (C*a*e^3*x^2)/(a
^2*c^2 + c^4*x^4 + 2*a*c^3*x^2) + (3*A*d^3*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(5/2)*c^(1/2)) + (3*B*e^3*atan((c^(
1/2)*x)/a^(1/2)))/(8*a^(1/2)*c^(5/2)) + (C*d^3*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(3/2)*c^(3/2)) + (3*A*d*e^2*x^3
)/(8*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) + (3*B*d^2*e*x^3)/(8*(a^3 + 2*a^2*c*x^2 + a*c^2*x^4)) - (3*A*d*e^2*x)/(8
*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) - (3*B*d^2*e*x)/(8*(a^2*c + c^3*x^4 + 2*a*c^2*x^2)) + (3*A*d*e^2*atan((c^(1/
2)*x)/a^(1/2)))/(8*a^(3/2)*c^(3/2)) + (3*B*d^2*e*atan((c^(1/2)*x)/a^(1/2)))/(8*a^(3/2)*c^(3/2)) + (9*C*d*e^2*a
tan((c^(1/2)*x)/a^(1/2)))/(8*a^(1/2)*c^(5/2)) - (9*C*a*d*e^2*x)/(8*(a^2*c^2 + c^4*x^4 + 2*a*c^3*x^2))

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