3.1330 \(\int \frac{(A+B x) (d+e x)^4}{a+c x^2} \, dx\)

Optimal. Leaf size=240 \[ \frac{\log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )}{2 c^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (4 a B d e \left (c d^2-a e^2\right )-A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^2 \left (-a B e^2+4 A c d e+6 B c d^2\right )}{2 c^2}+\frac{e x \left (-a A e^3-4 a B d e^2+6 A c d^2 e+4 B c d^3\right )}{c^2}+\frac{e^3 x^3 (A e+4 B d)}{3 c}+\frac{B e^4 x^4}{4 c} \]

[Out]

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

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Rubi [A]  time = 0.261257, antiderivative size = 240, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 22, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.182, Rules used = {801, 635, 205, 260} \[ \frac{\log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )}{2 c^3}-\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (4 a B d e \left (c d^2-a e^2\right )-A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{\sqrt{a} c^{5/2}}+\frac{e^2 x^2 \left (-a B e^2+4 A c d e+6 B c d^2\right )}{2 c^2}+\frac{e x \left (-a A e^3-4 a B d e^2+6 A c d^2 e+4 B c d^3\right )}{c^2}+\frac{e^3 x^3 (A e+4 B d)}{3 c}+\frac{B e^4 x^4}{4 c} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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

Mathematica [A]  time = 0.193508, size = 217, normalized size = 0.9 \[ \frac{6 \log \left (a+c x^2\right ) \left (B \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 A c d e \left (c d^2-a e^2\right )\right )+c e x \left (-12 a A e^3-6 a B e^2 (8 d+e x)+4 A c e \left (18 d^2+6 d e x+e^2 x^2\right )+B c \left (36 d^2 e x+48 d^3+16 d e^2 x^2+3 e^3 x^3\right )\right )}{12 c^3}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )+4 a B d e \left (a e^2-c d^2\right )\right )}{\sqrt{a} c^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((4*a*B*d*e*(-(c*d^2) + a*e^2) + A*(c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(Sqrt[a]*
c^(5/2)) + (c*e*x*(-12*a*A*e^3 - 6*a*B*e^2*(8*d + e*x) + 4*A*c*e*(18*d^2 + 6*d*e*x + e^2*x^2) + B*c*(48*d^3 +
36*d^2*e*x + 16*d*e^2*x^2 + 3*e^3*x^3)) + 6*(4*A*c*d*e*(c*d^2 - a*e^2) + B*(c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4)
)*Log[a + c*x^2])/(12*c^3)

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Maple [A]  time = 0.007, size = 345, normalized size = 1.4 \begin{align*}{\frac{B{e}^{4}{x}^{4}}{4\,c}}+{\frac{A{e}^{4}{x}^{3}}{3\,c}}+{\frac{4\,{e}^{3}B{x}^{3}d}{3\,c}}+2\,{\frac{A{e}^{3}{x}^{2}d}{c}}-{\frac{B{e}^{4}{x}^{2}a}{2\,{c}^{2}}}+3\,{\frac{{e}^{2}B{x}^{2}{d}^{2}}{c}}-{\frac{Aa{e}^{4}x}{{c}^{2}}}+6\,{\frac{A{e}^{2}{d}^{2}x}{c}}-4\,{\frac{Bad{e}^{3}x}{{c}^{2}}}+4\,{\frac{Be{d}^{3}x}{c}}-2\,{\frac{\ln \left ( c{x}^{2}+a \right ) Ada{e}^{3}}{{c}^{2}}}+2\,{\frac{\ln \left ( c{x}^{2}+a \right ) A{d}^{3}e}{c}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{e}^{4}{a}^{2}}{2\,{c}^{3}}}-3\,{\frac{\ln \left ( c{x}^{2}+a \right ) Ba{d}^{2}{e}^{2}}{{c}^{2}}}+{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{4}}{2\,c}}+{\frac{A{a}^{2}{e}^{4}}{{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-6\,{\frac{aA{d}^{2}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{A{d}^{4}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}+4\,{\frac{B{a}^{2}d{e}^{3}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }-4\,{\frac{aB{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),x)

[Out]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.95071, size = 1180, normalized size = 4.92 \begin{align*} \left [\frac{3 \, B a c^{2} e^{4} x^{4} + 4 \,{\left (4 \, B a c^{2} d e^{3} + A a c^{2} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B a c^{2} d^{2} e^{2} + 4 \, A a c^{2} d e^{3} - B a^{2} c e^{4}\right )} x^{2} - 6 \,{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \sqrt{-a c} \log \left (\frac{c x^{2} - 2 \, \sqrt{-a c} x - a}{c x^{2} + a}\right ) + 12 \,{\left (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 + 6 \,{\left (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}\right )} \log \left (c x^{2} + a\right )}{12 \, a c^{3}}, \frac{3 \, B a c^{2} e^{4} x^{4} + 4 \,{\left (4 \, B a c^{2} d e^{3} + A a c^{2} e^{4}\right )} x^{3} + 6 \,{\left (6 \, B a c^{2} d^{2} e^{2} + 4 \, A a c^{2} d e^{3} - B a^{2} c e^{4}\right )} x^{2} + 12 \,{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \sqrt{a c} \arctan \left (\frac{\sqrt{a c} x}{a}\right ) + 12 \,{\left (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 + 6 \,{\left (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}\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((B*x+A)*(e*x+d)^4/(c*x^2+a),x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 6.19954, size = 901, normalized size = 3.75 \begin{align*} \frac{B e^{4} x^{4}}{4 c} + \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right ) \log{\left (x + \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} + 2 a c^{3} \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} - \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right )}{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 )} + \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right ) \log{\left (x + \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} + 2 a c^{3} \left (\frac{- 4 A a c d e^{3} + 4 A c^{2} d^{3} e + B a^{2} e^{4} - 6 B a c d^{2} e^{2} + B c^{2} d^{4}}{2 c^{3}} + \frac{\sqrt{- a c^{7}} \left (A a^{2} e^{4} - 6 A a c d^{2} e^{2} + A c^{2} d^{4} + 4 B a^{2} d e^{3} - 4 B a c d^{3} e\right )}{2 a c^{6}}\right )}{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 )} + \frac{x^{3} \left (A e^{4} + 4 B d e^{3}\right )}{3 c} - \frac{x^{2} \left (- 4 A c d e^{3} + B a e^{4} - 6 B c d^{2} e^{2}\right )}{2 c^{2}} - \frac{x \left (A a e^{4} - 6 A c d^{2} e^{2} + 4 B a d e^{3} - 4 B c d^{3} e\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),x)

[Out]

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

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Giac [A]  time = 1.18022, size = 331, normalized size = 1.38 \begin{align*} \frac{{\left (A c^{2} d^{4} - 4 \, B a c d^{3} e - 6 \, A a c d^{2} e^{2} + 4 \, B a^{2} d e^{3} + A a^{2} e^{4}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{\sqrt{a c} c^{2}} + \frac{{\left (B c^{2} d^{4} + 4 \, A c^{2} d^{3} e - 6 \, B a c d^{2} e^{2} - 4 \, A a c d e^{3} + B a^{2} e^{4}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac{3 \, B c^{3} x^{4} e^{4} + 16 \, B c^{3} d x^{3} e^{3} + 36 \, B c^{3} d^{2} x^{2} e^{2} + 48 \, B c^{3} d^{3} x e + 4 \, A c^{3} x^{3} e^{4} + 24 \, A c^{3} d x^{2} e^{3} + 72 \, A c^{3} d^{2} x e^{2} - 6 \, B a c^{2} x^{2} e^{4} - 48 \, B a c^{2} d x e^{3} - 12 \, A a c^{2} x e^{4}}{12 \, c^{4}} \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),x, algorithm="giac")

[Out]

(A*c^2*d^4 - 4*B*a*c*d^3*e - 6*A*a*c*d^2*e^2 + 4*B*a^2*d*e^3 + A*a^2*e^4)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*c^2
) + 1/2*(B*c^2*d^4 + 4*A*c^2*d^3*e - 6*B*a*c*d^2*e^2 - 4*A*a*c*d*e^3 + B*a^2*e^4)*log(c*x^2 + a)/c^3 + 1/12*(3
*B*c^3*x^4*e^4 + 16*B*c^3*d*x^3*e^3 + 36*B*c^3*d^2*x^2*e^2 + 48*B*c^3*d^3*x*e + 4*A*c^3*x^3*e^4 + 24*A*c^3*d*x
^2*e^3 + 72*A*c^3*d^2*x*e^2 - 6*B*a*c^2*x^2*e^4 - 48*B*a*c^2*d*x*e^3 - 12*A*a*c^2*x*e^4)/c^4