3.1339 \(\int \frac{(A+B x) (d+e x)^4}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=220 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )}{c^3}-\frac{3 e^2 x \left (-a A e^2-4 a B d e+A c d^2\right )}{2 a c^2}-\frac{e^3 x^2 (A c d-2 a B e)}{2 a c^2}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

[Out]

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

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Rubi [A]  time = 0.275479, antiderivative size = 220, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.227, Rules used = {819, 801, 635, 205, 260} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )}{c^3}-\frac{3 e^2 x \left (-a A e^2-4 a B d e+A c d^2\right )}{2 a c^2}-\frac{e^3 x^2 (A c d-2 a B e)}{2 a c^2}-\frac{(d+e x)^3 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^2,x]

[Out]

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

Rule 819

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 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 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{(A+B x) (d+e x)^4}{\left (a+c x^2\right )^2} \, dx &=-\frac{(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{(d+e x)^2 \left (A c d^2+a e (4 B d+3 A e)-2 e (A c d-2 a B e) x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\int \left (-\frac{3 e^2 \left (A c d^2-4 a B d e-a A e^2\right )}{c}-\frac{2 e^3 (A c d-2 a B e) x}{c}+\frac{4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+4 a e^2 \left (3 B c d^2+2 A c d e-a B e^2\right ) x}{c \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac{3 e^2 \left (A c d^2-4 a B d e-a A e^2\right ) x}{2 a c^2}-\frac{e^3 (A c d-2 a B e) x^2}{2 a c^2}-\frac{(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+4 a e^2 \left (3 B c d^2+2 A c d e-a B e^2\right ) x}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 e^2 \left (A c d^2-4 a B d e-a A e^2\right ) x}{2 a c^2}-\frac{e^3 (A c d-2 a B e) x^2}{2 a c^2}-\frac{(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\left (2 e^2 \left (3 B c d^2+2 A c d e-a B e^2\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )\right ) \int \frac{1}{a+c x^2} \, dx}{2 a c^2}\\ &=-\frac{3 e^2 \left (A c d^2-4 a B d e-a A e^2\right ) x}{2 a c^2}-\frac{e^3 (A c d-2 a B e) x^2}{2 a c^2}-\frac{(d+e x)^3 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\left (4 a B d e \left (c d^2-3 a e^2\right )+A \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )\right ) \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right )}{2 a^{3/2} c^{5/2}}+\frac{e^2 \left (3 B c d^2+2 A c d e-a B e^2\right ) \log \left (a+c x^2\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.223381, size = 231, normalized size = 1.05 \[ \frac{\frac{a^2 c e^2 (A e (4 d+e x)+2 B d (3 d+2 e x))-a^3 B e^4-a c^2 d^2 (2 A e (2 d+3 e x)+B d (d+4 e x))+A c^3 d^4 x}{a \left (a+c x^2\right )}+\frac{\sqrt{c} \tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (c d^2-3 a e^2\right )\right )}{a^{3/2}}+2 e^2 \log \left (a+c x^2\right ) \left (-a B e^2+2 A c d e+3 B c d^2\right )+2 c e^3 x (A e+4 B d)+B c e^4 x^2}{2 c^3} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^4)/(a + c*x^2)^2,x]

[Out]

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

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Maple [B]  time = 0.012, size = 414, normalized size = 1.9 \begin{align*}{\frac{B{e}^{4}{x}^{2}}{2\,{c}^{2}}}+{\frac{{e}^{4}Ax}{{c}^{2}}}+4\,{\frac{{e}^{3}Bdx}{{c}^{2}}}+{\frac{aAx{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}-3\,{\frac{xA{d}^{2}{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}+{\frac{xA{d}^{4}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+2\,{\frac{aBxd{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{Bx{d}^{3}e}{c \left ( c{x}^{2}+a \right ) }}+2\,{\frac{Ada{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-2\,{\frac{A{d}^{3}e}{c \left ( c{x}^{2}+a \right ) }}-{\frac{B{e}^{4}{a}^{2}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+3\,{\frac{Ba{d}^{2}{e}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-{\frac{B{d}^{4}}{2\,c \left ( c{x}^{2}+a \right ) }}+2\,{\frac{\ln \left ( c{x}^{2}+a \right ) Ad{e}^{3}}{{c}^{2}}}-{\frac{a\ln \left ( c{x}^{2}+a \right ) B{e}^{4}}{{c}^{3}}}+3\,{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{2}{e}^{2}}{{c}^{2}}}-{\frac{3\,Aa{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+3\,{\frac{A{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{A{d}^{4}}{2\,a}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{Bad{e}^{3}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+2\,{\frac{B{d}^{3}e}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^4/(c*x^2+a)^2,x)

[Out]

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

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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((B*x+A)*(e*x+d)^4/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.08065, size = 1758, normalized size = 7.99 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/4*(2*B*a^2*c^2*e^4*x^4 + 2*B*a^3*c*e^4*x^2 - 2*B*a^2*c^2*d^4 - 8*A*a^2*c^2*d^3*e + 12*B*a^3*c*d^2*e^2 + 8*A
*a^3*c*d*e^3 - 2*B*a^4*e^4 + 4*(4*B*a^2*c^2*d*e^3 + A*a^2*c^2*e^4)*x^3 + (A*a*c^2*d^4 + 4*B*a^2*c*d^3*e + 6*A*
a^2*c*d^2*e^2 - 12*B*a^3*d*e^3 - 3*A*a^3*e^4 + (A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a*c^2*d^2*e^2 - 12*B*a^2*c*d
*e^3 - 3*A*a^2*c*e^4)*x^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 2*(A*a*c^3*d^4 - 4*B*a^2
*c^2*d^3*e - 6*A*a^2*c^2*d^2*e^2 + 12*B*a^3*c*d*e^3 + 3*A*a^3*c*e^4)*x + 4*(3*B*a^3*c*d^2*e^2 + 2*A*a^3*c*d*e^
3 - B*a^4*e^4 + (3*B*a^2*c^2*d^2*e^2 + 2*A*a^2*c^2*d*e^3 - B*a^3*c*e^4)*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^
3*c^3), 1/2*(B*a^2*c^2*e^4*x^4 + B*a^3*c*e^4*x^2 - B*a^2*c^2*d^4 - 4*A*a^2*c^2*d^3*e + 6*B*a^3*c*d^2*e^2 + 4*A
*a^3*c*d*e^3 - B*a^4*e^4 + 2*(4*B*a^2*c^2*d*e^3 + A*a^2*c^2*e^4)*x^3 + (A*a*c^2*d^4 + 4*B*a^2*c*d^3*e + 6*A*a^
2*c*d^2*e^2 - 12*B*a^3*d*e^3 - 3*A*a^3*e^4 + (A*c^3*d^4 + 4*B*a*c^2*d^3*e + 6*A*a*c^2*d^2*e^2 - 12*B*a^2*c*d*e
^3 - 3*A*a^2*c*e^4)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + (A*a*c^3*d^4 - 4*B*a^2*c^2*d^3*e - 6*A*a^2*c^2*d^2*
e^2 + 12*B*a^3*c*d*e^3 + 3*A*a^3*c*e^4)*x + 2*(3*B*a^3*c*d^2*e^2 + 2*A*a^3*c*d*e^3 - B*a^4*e^4 + (3*B*a^2*c^2*
d^2*e^2 + 2*A*a^2*c^2*d*e^3 - B*a^3*c*e^4)*x^2)*log(c*x^2 + a))/(a^2*c^4*x^2 + a^3*c^3)]

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Sympy [B]  time = 14.6221, size = 833, normalized size = 3.79 \begin{align*} \frac{B e^{4} x^{2}}{2 c^{2}} + \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} - \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right ) \log{\left (x + \frac{8 A a^{2} c d e^{3} - 4 B a^{3} e^{4} + 12 B a^{2} c d^{2} e^{2} - 4 a^{2} c^{3} \left (- \frac{e^{2} \left (- 2 A c d e + B a e^{2} - 3 B c d^{2}\right )}{c^{3}} + \frac{\sqrt{- a^{3} c^{7}} \left (3 A a^{2} e^{4} - 6 A a c d^{2} e^{2} - A c^{2} d^{4} + 12 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{4 a^{3} c^{6}}\right )}{3 A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} - A c^{3} d^{4} + 12 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e} \right )} + \frac{4 A a^{2} c d e^{3} - 4 A a c^{2} d^{3} e - B a^{3} e^{4} + 6 B a^{2} c d^{2} e^{2} - B a c^{2} d^{4} + x \left (A a^{2} c e^{4} - 6 A a c^{2} d^{2} e^{2} + A c^{3} d^{4} + 4 B a^{2} c d e^{3} - 4 B a c^{2} d^{3} e\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} + \frac{x \left (A e^{4} + 4 B d e^{3}\right )}{c^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**4/(c*x**2+a)**2,x)

[Out]

B*e**4*x**2/(2*c**2) + (-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 - sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*
A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6))*log(x + (8*A*a**2*c*d*e**3 -
4*B*a**3*e**4 + 12*B*a**2*c*d**2*e**2 - 4*a**2*c**3*(-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 - sqrt(-
a**3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6)
))/(3*A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e)) + (-e**2*(
-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A*a*c*d**2*e**2 - A*c**2*d**4 +
12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6))*log(x + (8*A*a**2*c*d*e**3 - 4*B*a**3*e**4 + 12*B*a**2*c*d*
*2*e**2 - 4*a**2*c**3*(-e**2*(-2*A*c*d*e + B*a*e**2 - 3*B*c*d**2)/c**3 + sqrt(-a**3*c**7)*(3*A*a**2*e**4 - 6*A
*a*c*d**2*e**2 - A*c**2*d**4 + 12*B*a**2*d*e**3 - 4*B*a*c*d**3*e)/(4*a**3*c**6)))/(3*A*a**2*c*e**4 - 6*A*a*c**
2*d**2*e**2 - A*c**3*d**4 + 12*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e)) + (4*A*a**2*c*d*e**3 - 4*A*a*c**2*d**3*e
- B*a**3*e**4 + 6*B*a**2*c*d**2*e**2 - B*a*c**2*d**4 + x*(A*a**2*c*e**4 - 6*A*a*c**2*d**2*e**2 + A*c**3*d**4 +
4*B*a**2*c*d*e**3 - 4*B*a*c**2*d**3*e))/(2*a**2*c**3 + 2*a*c**4*x**2) + x*(A*e**4 + 4*B*d*e**3)/c**2

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Giac [A]  time = 1.15537, size = 352, normalized size = 1.6 \begin{align*} \frac{{\left (3 \, B c d^{2} e^{2} + 2 \, A c d e^{3} - B a e^{4}\right )} \log \left (c x^{2} + a\right )}{c^{3}} + \frac{{\left (A c^{2} d^{4} + 4 \, B a c d^{3} e + 6 \, A a c d^{2} e^{2} - 12 \, B a^{2} d e^{3} - 3 \, A a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{2}} + \frac{B c^{2} x^{2} e^{4} + 8 \, B c^{2} d x e^{3} + 2 \, A c^{2} x e^{4}}{2 \, c^{4}} - \frac{B a c^{2} d^{4} + 4 \, A a c^{2} d^{3} e - 6 \, B a^{2} c d^{2} e^{2} - 4 \, A a^{2} c d e^{3} + B a^{3} e^{4} -{\left (A c^{3} d^{4} - 4 \, B a c^{2} d^{3} e - 6 \, A a c^{2} d^{2} e^{2} + 4 \, B a^{2} c d e^{3} + A a^{2} c e^{4}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^4/(c*x^2+a)^2,x, algorithm="giac")

[Out]

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