3.1338 \(\int \frac{(A+B x) (d+e x)^5}{(a+c x^2)^2} \, dx\)

Optimal. Leaf size=297 \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+5 a B e \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{2 a^{3/2} c^{7/2}}-\frac{e^3 x^2 \left (-a A e^2-5 a B d e+2 A c d^2\right )}{a c^2}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a A e^3-5 a B d e^2+5 A c d^2 e+5 B c d^3\right )}{c^3}-\frac{e^2 x \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right )}{2 a c^3}-\frac{e^4 x^3 (3 A c d-5 a B e)}{6 a c^2}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

[Out]

-(e^2*(3*A*c*d*(2*c*d^2 - 5*a*e^2) - 5*a*B*e*(6*c*d^2 - a*e^2))*x)/(2*a*c^3) - (e^3*(2*A*c*d^2 - 5*a*B*d*e - a
*A*e^2)*x^2)/(a*c^2) - (e^4*(3*A*c*d - 5*a*B*e)*x^3)/(6*a*c^2) - ((d + e*x)^4*(a*(B*d + A*e) - (A*c*d - a*B*e)
*x))/(2*a*c*(a + c*x^2)) + ((A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) + 5*a*B*e*(c^2*d^4 - 6*a*c*d^2*e^2
+ a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(7/2)) + (e^2*(5*B*c*d^3 + 5*A*c*d^2*e - 5*a*B*d*e^2 - a
*A*e^3)*Log[a + c*x^2])/c^3

________________________________________________________________________________________

Rubi [A]  time = 0.338922, antiderivative size = 297, 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} \[ \frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+5 a B e \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{2 a^{3/2} c^{7/2}}-\frac{e^3 x^2 \left (-a A e^2-5 a B d e+2 A c d^2\right )}{a c^2}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a A e^3-5 a B d e^2+5 A c d^2 e+5 B c d^3\right )}{c^3}-\frac{e^2 x \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right )}{2 a c^3}-\frac{e^4 x^3 (3 A c d-5 a B e)}{6 a c^2}-\frac{(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{2 a c \left (a+c x^2\right )} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^2,x]

[Out]

-(e^2*(3*A*c*d*(2*c*d^2 - 5*a*e^2) - 5*a*B*e*(6*c*d^2 - a*e^2))*x)/(2*a*c^3) - (e^3*(2*A*c*d^2 - 5*a*B*d*e - a
*A*e^2)*x^2)/(a*c^2) - (e^4*(3*A*c*d - 5*a*B*e)*x^3)/(6*a*c^2) - ((d + e*x)^4*(a*(B*d + A*e) - (A*c*d - a*B*e)
*x))/(2*a*c*(a + c*x^2)) + ((A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) + 5*a*B*e*(c^2*d^4 - 6*a*c*d^2*e^2
+ a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(7/2)) + (e^2*(5*B*c*d^3 + 5*A*c*d^2*e - 5*a*B*d*e^2 - a
*A*e^3)*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)^5}{\left (a+c x^2\right )^2} \, dx &=-\frac{(d+e x)^4 (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)^3 \left (A c d^2+a e (5 B d+4 A e)-e (3 A c d-5 a B e) x\right )}{a+c x^2} \, dx}{2 a c}\\ &=-\frac{(d+e x)^4 (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{e^2 \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right )}{c^2}-\frac{4 e^3 \left (2 A c d^2-5 a B d e-a A e^2\right ) x}{c}-\frac{e^4 (3 A c d-5 a B e) x^2}{c}+\frac{A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+5 a B e \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )+4 a c e^2 \left (5 B c d^3+5 A c d^2 e-5 a B d e^2-a A e^3\right ) x}{c^2 \left (a+c x^2\right )}\right ) \, dx}{2 a c}\\ &=-\frac{e^2 \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right ) x}{2 a c^3}-\frac{e^3 \left (2 A c d^2-5 a B d e-a A e^2\right ) x^2}{a c^2}-\frac{e^4 (3 A c d-5 a B e) x^3}{6 a c^2}-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\int \frac{A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+5 a B e \left (c^2 d^4-6 a c d^2 e^2+a^2 e^4\right )+4 a c e^2 \left (5 B c d^3+5 A c d^2 e-5 a B d e^2-a A e^3\right ) x}{a+c x^2} \, dx}{2 a c^3}\\ &=-\frac{e^2 \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right ) x}{2 a c^3}-\frac{e^3 \left (2 A c d^2-5 a B d e-a A e^2\right ) x^2}{a c^2}-\frac{e^4 (3 A c d-5 a B e) x^3}{6 a c^2}-\frac{(d+e x)^4 (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 (5 B c d^3+5 A c d^2 e-5 a B d e^2-a A e^3\right )\right ) \int \frac{x}{a+c x^2} \, dx}{c^2}+\frac{\left (A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+5 a B e \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}{2 a c^3}\\ &=-\frac{e^2 \left (3 A c d \left (2 c d^2-5 a e^2\right )-5 a B e \left (6 c d^2-a e^2\right )\right ) x}{2 a c^3}-\frac{e^3 \left (2 A c d^2-5 a B d e-a A e^2\right ) x^2}{a c^2}-\frac{e^4 (3 A c d-5 a B e) x^3}{6 a c^2}-\frac{(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{2 a c \left (a+c x^2\right )}+\frac{\left (A c d \left (c^2 d^4+10 a c d^2 e^2-15 a^2 e^4\right )+5 a B e \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 )}{2 a^{3/2} c^{7/2}}+\frac{e^2 \left (5 B c d^3+5 A c d^2 e-5 a B d e^2-a A e^3\right ) \log \left (a+c x^2\right )}{c^3}\\ \end{align*}

Mathematica [A]  time = 0.184356, size = 307, normalized size = 1.03 \[ \frac{5 a^2 c d e^2 (A e (2 d+e x)+2 B d (d+e x))-a^3 e^4 (A e+5 B d+B e x)-a c^2 d^3 (5 A e (d+2 e x)+B d (d+5 e x))+A c^3 d^5 x}{2 a c^3 \left (a+c x^2\right )}+\frac{\tan ^{-1}\left (\frac{\sqrt{c} x}{\sqrt{a}}\right ) \left (A c d \left (-15 a^2 e^4+10 a c d^2 e^2+c^2 d^4\right )+5 a B e \left (a^2 e^4-6 a c d^2 e^2+c^2 d^4\right )\right )}{2 a^{3/2} c^{7/2}}+\frac{e^2 \log \left (a+c x^2\right ) \left (-a A e^3-5 a B d e^2+5 A c d^2 e+5 B c d^3\right )}{c^3}+\frac{e^3 x \left (-2 a B e^2+5 A c d e+10 B c d^2\right )}{c^3}+\frac{e^4 x^2 (A e+5 B d)}{2 c^2}+\frac{B e^5 x^3}{3 c^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^2,x]

[Out]

(e^3*(10*B*c*d^2 + 5*A*c*d*e - 2*a*B*e^2)*x)/c^3 + (e^4*(5*B*d + A*e)*x^2)/(2*c^2) + (B*e^5*x^3)/(3*c^2) + (A*
c^3*d^5*x - a^3*e^4*(5*B*d + A*e + B*e*x) + 5*a^2*c*d*e^2*(2*B*d*(d + e*x) + A*e*(2*d + e*x)) - a*c^2*d^3*(5*A
*e*(d + 2*e*x) + B*d*(d + 5*e*x)))/(2*a*c^3*(a + c*x^2)) + ((A*c*d*(c^2*d^4 + 10*a*c*d^2*e^2 - 15*a^2*e^4) + 5
*a*B*e*(c^2*d^4 - 6*a*c*d^2*e^2 + a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(2*a^(3/2)*c^(7/2)) + (e^2*(5*B*c*d^3
+ 5*A*c*d^2*e - 5*a*B*d*e^2 - a*A*e^3)*Log[a + c*x^2])/c^3

________________________________________________________________________________________

Maple [A]  time = 0.013, size = 553, normalized size = 1.9 \begin{align*} -{\frac{15\,aAd{e}^{4}}{2\,{c}^{2}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-15\,{\frac{aB{d}^{2}{e}^{3}}{{c}^{2}\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{5\,aAdx{e}^{4}}{2\,{c}^{2} \left ( c{x}^{2}+a \right ) }}+5\,{\frac{aBx{d}^{2}{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{5\,B{d}^{4}e}{2\,c}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{5\,Bx{d}^{4}e}{2\,c \left ( c{x}^{2}+a \right ) }}+5\,{\frac{A{d}^{2}a{e}^{3}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}-5\,{\frac{xA{d}^{3}{e}^{2}}{c \left ( c{x}^{2}+a \right ) }}-{\frac{{a}^{2}Bx{e}^{5}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}-5\,{\frac{a\ln \left ( c{x}^{2}+a \right ) Bd{e}^{4}}{{c}^{3}}}+5\,{\frac{A{d}^{3}{e}^{2}}{c\sqrt{ac}}\arctan \left ({\frac{cx}{\sqrt{ac}}} \right ) }+{\frac{5\,B{e}^{5}{a}^{2}}{2\,{c}^{3}}\arctan \left ({cx{\frac{1}{\sqrt{ac}}}} \right ){\frac{1}{\sqrt{ac}}}}-{\frac{5\,B{a}^{2}d{e}^{4}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}+5\,{\frac{aB{d}^{3}{e}^{2}}{{c}^{2} \left ( c{x}^{2}+a \right ) }}+{\frac{{e}^{5}A{x}^{2}}{2\,{c}^{2}}}-{\frac{B{d}^{5}}{2\,c \left ( c{x}^{2}+a \right ) }}+{\frac{B{e}^{5}{x}^{3}}{3\,{c}^{2}}}-{\frac{A{a}^{2}{e}^{5}}{2\,{c}^{3} \left ( c{x}^{2}+a \right ) }}-{\frac{5\,A{d}^{4}e}{2\,c \left ( c{x}^{2}+a \right ) }}-{\frac{a\ln \left ( c{x}^{2}+a \right ) A{e}^{5}}{{c}^{3}}}-2\,{\frac{B{e}^{5}ax}{{c}^{3}}}+10\,{\frac{{e}^{3}B{d}^{2}x}{{c}^{2}}}+{\frac{5\,{e}^{4}B{x}^{2}d}{2\,{c}^{2}}}+5\,{\frac{{e}^{4}Adx}{{c}^{2}}}+5\,{\frac{\ln \left ( c{x}^{2}+a \right ) A{d}^{2}{e}^{3}}{{c}^{2}}}+5\,{\frac{\ln \left ( c{x}^{2}+a \right ) B{d}^{3}{e}^{2}}{{c}^{2}}}+{\frac{xA{d}^{5}}{ \left ( 2\,c{x}^{2}+2\,a \right ) a}}+{\frac{A{d}^{5}}{2\,a}\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((B*x+A)*(e*x+d)^5/(c*x^2+a)^2,x)

[Out]

-15/2/c^2*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d*e^4-15/c^2*a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d^2*e^3
+5/2/c^2/(c*x^2+a)*a*x*A*d*e^4+5/c^2/(c*x^2+a)*a*x*B*d^2*e^3+5/2/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*B*d^4*e
-5/2/c/(c*x^2+a)*x*B*d^4*e+5/c^2/(c*x^2+a)*A*d^2*a*e^3-5/c/(c*x^2+a)*x*A*d^3*e^2-1/2/c^3/(c*x^2+a)*a^2*x*B*e^5
-5/c^3*a*ln(c*x^2+a)*B*d*e^4+5/c/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^3*e^2+5/2/c^3*a^2/(a*c)^(1/2)*arctan(
x*c/(a*c)^(1/2))*B*e^5-5/2/c^3/(c*x^2+a)*B*a^2*d*e^4+5/c^2/(c*x^2+a)*a*B*d^3*e^2+1/2*e^5/c^2*A*x^2-1/2/c/(c*x^
2+a)*B*d^5+1/3*e^5/c^2*B*x^3-1/2/c^3/(c*x^2+a)*A*a^2*e^5-5/2/c/(c*x^2+a)*A*d^4*e-1/c^3*a*ln(c*x^2+a)*A*e^5-2*e
^5/c^3*a*B*x+10*e^3/c^2*B*d^2*x+5/2*e^4/c^2*B*x^2*d+5*e^4/c^2*A*d*x+5/c^2*ln(c*x^2+a)*A*d^2*e^3+5/c^2*ln(c*x^2
+a)*B*d^3*e^2+1/2/(c*x^2+a)/a*x*A*d^5+1/2/a/(a*c)^(1/2)*arctan(x*c/(a*c)^(1/2))*A*d^5

________________________________________________________________________________________

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)^5/(c*x^2+a)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [B]  time = 2.31656, size = 2456, normalized size = 8.27 \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)^5/(c*x^2+a)^2,x, algorithm="fricas")

[Out]

[1/12*(4*B*a^2*c^3*e^5*x^5 - 6*B*a^2*c^3*d^5 - 30*A*a^2*c^3*d^4*e + 60*B*a^3*c^2*d^3*e^2 + 60*A*a^3*c^2*d^2*e^
3 - 30*B*a^4*c*d*e^4 - 6*A*a^4*c*e^5 + 6*(5*B*a^2*c^3*d*e^4 + A*a^2*c^3*e^5)*x^4 + 20*(6*B*a^2*c^3*d^2*e^3 + 3
*A*a^2*c^3*d*e^4 - B*a^3*c^2*e^5)*x^3 + 6*(5*B*a^3*c^2*d*e^4 + A*a^3*c^2*e^5)*x^2 - 3*(A*a*c^3*d^5 + 5*B*a^2*c
^2*d^4*e + 10*A*a^2*c^2*d^3*e^2 - 30*B*a^3*c*d^2*e^3 - 15*A*a^3*c*d*e^4 + 5*B*a^4*e^5 + (A*c^4*d^5 + 5*B*a*c^3
*d^4*e + 10*A*a*c^3*d^3*e^2 - 30*B*a^2*c^2*d^2*e^3 - 15*A*a^2*c^2*d*e^4 + 5*B*a^3*c*e^5)*x^2)*sqrt(-a*c)*log((
c*x^2 - 2*sqrt(-a*c)*x - a)/(c*x^2 + a)) + 6*(A*a*c^4*d^5 - 5*B*a^2*c^3*d^4*e - 10*A*a^2*c^3*d^3*e^2 + 30*B*a^
3*c^2*d^2*e^3 + 15*A*a^3*c^2*d*e^4 - 5*B*a^4*c*e^5)*x + 12*(5*B*a^3*c^2*d^3*e^2 + 5*A*a^3*c^2*d^2*e^3 - 5*B*a^
4*c*d*e^4 - A*a^4*c*e^5 + (5*B*a^2*c^3*d^3*e^2 + 5*A*a^2*c^3*d^2*e^3 - 5*B*a^3*c^2*d*e^4 - A*a^3*c^2*e^5)*x^2)
*log(c*x^2 + a))/(a^2*c^5*x^2 + a^3*c^4), 1/6*(2*B*a^2*c^3*e^5*x^5 - 3*B*a^2*c^3*d^5 - 15*A*a^2*c^3*d^4*e + 30
*B*a^3*c^2*d^3*e^2 + 30*A*a^3*c^2*d^2*e^3 - 15*B*a^4*c*d*e^4 - 3*A*a^4*c*e^5 + 3*(5*B*a^2*c^3*d*e^4 + A*a^2*c^
3*e^5)*x^4 + 10*(6*B*a^2*c^3*d^2*e^3 + 3*A*a^2*c^3*d*e^4 - B*a^3*c^2*e^5)*x^3 + 3*(5*B*a^3*c^2*d*e^4 + A*a^3*c
^2*e^5)*x^2 + 3*(A*a*c^3*d^5 + 5*B*a^2*c^2*d^4*e + 10*A*a^2*c^2*d^3*e^2 - 30*B*a^3*c*d^2*e^3 - 15*A*a^3*c*d*e^
4 + 5*B*a^4*e^5 + (A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 30*B*a^2*c^2*d^2*e^3 - 15*A*a^2*c^2*d*e^
4 + 5*B*a^3*c*e^5)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + 3*(A*a*c^4*d^5 - 5*B*a^2*c^3*d^4*e - 10*A*a^2*c^3*d^
3*e^2 + 30*B*a^3*c^2*d^2*e^3 + 15*A*a^3*c^2*d*e^4 - 5*B*a^4*c*e^5)*x + 6*(5*B*a^3*c^2*d^3*e^2 + 5*A*a^3*c^2*d^
2*e^3 - 5*B*a^4*c*d*e^4 - A*a^4*c*e^5 + (5*B*a^2*c^3*d^3*e^2 + 5*A*a^2*c^3*d^2*e^3 - 5*B*a^3*c^2*d*e^4 - A*a^3
*c^2*e^5)*x^2)*log(c*x^2 + a))/(a^2*c^5*x^2 + a^3*c^4)]

________________________________________________________________________________________

Sympy [B]  time = 25.8674, size = 1083, normalized size = 3.65 \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)**5/(c*x**2+a)**2,x)

[Out]

B*e**5*x**3/(3*c**2) + (-e**2*(A*a*e**3 - 5*A*c*d**2*e + 5*B*a*d*e**2 - 5*B*c*d**3)/c**3 - sqrt(-a**3*c**7)*(-
15*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 + 5*B*a*c**2*
d**4*e)/(4*a**3*c**7))*log(x + (4*A*a**3*e**5 - 20*A*a**2*c*d**2*e**3 + 20*B*a**3*d*e**4 - 20*B*a**2*c*d**3*e*
*2 + 4*a**2*c**3*(-e**2*(A*a*e**3 - 5*A*c*d**2*e + 5*B*a*d*e**2 - 5*B*c*d**3)/c**3 - sqrt(-a**3*c**7)*(-15*A*a
**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 + 5*B*a*c**2*d**4*e
)/(4*a**3*c**7)))/(-15*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a**2*c*d**
2*e**3 + 5*B*a*c**2*d**4*e)) + (-e**2*(A*a*e**3 - 5*A*c*d**2*e + 5*B*a*d*e**2 - 5*B*c*d**3)/c**3 + sqrt(-a**3*
c**7)*(-15*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 + 5*B
*a*c**2*d**4*e)/(4*a**3*c**7))*log(x + (4*A*a**3*e**5 - 20*A*a**2*c*d**2*e**3 + 20*B*a**3*d*e**4 - 20*B*a**2*c
*d**3*e**2 + 4*a**2*c**3*(-e**2*(A*a*e**3 - 5*A*c*d**2*e + 5*B*a*d*e**2 - 5*B*c*d**3)/c**3 + sqrt(-a**3*c**7)*
(-15*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 + 5*B*a*c**
2*d**4*e)/(4*a**3*c**7)))/(-15*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 + A*c**3*d**5 + 5*B*a**3*e**5 - 30*B*a*
*2*c*d**2*e**3 + 5*B*a*c**2*d**4*e)) - (A*a**3*e**5 - 10*A*a**2*c*d**2*e**3 + 5*A*a*c**2*d**4*e + 5*B*a**3*d*e
**4 - 10*B*a**2*c*d**3*e**2 + B*a*c**2*d**5 + x*(-5*A*a**2*c*d*e**4 + 10*A*a*c**2*d**3*e**2 - A*c**3*d**5 + B*
a**3*e**5 - 10*B*a**2*c*d**2*e**3 + 5*B*a*c**2*d**4*e))/(2*a**2*c**3 + 2*a*c**4*x**2) + x**2*(A*e**5 + 5*B*d*e
**4)/(2*c**2) - x*(-5*A*c*d*e**4 + 2*B*a*e**5 - 10*B*c*d**2*e**3)/c**3

________________________________________________________________________________________

Giac [A]  time = 1.16959, size = 468, normalized size = 1.58 \begin{align*} \frac{{\left (5 \, B c d^{3} e^{2} + 5 \, A c d^{2} e^{3} - 5 \, B a d e^{4} - A a e^{5}\right )} \log \left (c x^{2} + a\right )}{c^{3}} + \frac{{\left (A c^{3} d^{5} + 5 \, B a c^{2} d^{4} e + 10 \, A a c^{2} d^{3} e^{2} - 30 \, B a^{2} c d^{2} e^{3} - 15 \, A a^{2} c d e^{4} + 5 \, B a^{3} e^{5}\right )} \arctan \left (\frac{c x}{\sqrt{a c}}\right )}{2 \, \sqrt{a c} a c^{3}} - \frac{B a c^{2} d^{5} + 5 \, A a c^{2} d^{4} e - 10 \, B a^{2} c d^{3} e^{2} - 10 \, A a^{2} c d^{2} e^{3} + 5 \, B a^{3} d e^{4} + A a^{3} e^{5} -{\left (A c^{3} d^{5} - 5 \, B a c^{2} d^{4} e - 10 \, A a c^{2} d^{3} e^{2} + 10 \, B a^{2} c d^{2} e^{3} + 5 \, A a^{2} c d e^{4} - B a^{3} e^{5}\right )} x}{2 \,{\left (c x^{2} + a\right )} a c^{3}} + \frac{2 \, B c^{4} x^{3} e^{5} + 15 \, B c^{4} d x^{2} e^{4} + 60 \, B c^{4} d^{2} x e^{3} + 3 \, A c^{4} x^{2} e^{5} + 30 \, A c^{4} d x e^{4} - 12 \, B a c^{3} x e^{5}}{6 \, c^{6}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+a)^2,x, algorithm="giac")

[Out]

(5*B*c*d^3*e^2 + 5*A*c*d^2*e^3 - 5*B*a*d*e^4 - A*a*e^5)*log(c*x^2 + a)/c^3 + 1/2*(A*c^3*d^5 + 5*B*a*c^2*d^4*e
+ 10*A*a*c^2*d^3*e^2 - 30*B*a^2*c*d^2*e^3 - 15*A*a^2*c*d*e^4 + 5*B*a^3*e^5)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a
*c^3) - 1/2*(B*a*c^2*d^5 + 5*A*a*c^2*d^4*e - 10*B*a^2*c*d^3*e^2 - 10*A*a^2*c*d^2*e^3 + 5*B*a^3*d*e^4 + A*a^3*e
^5 - (A*c^3*d^5 - 5*B*a*c^2*d^4*e - 10*A*a*c^2*d^3*e^2 + 10*B*a^2*c*d^2*e^3 + 5*A*a^2*c*d*e^4 - B*a^3*e^5)*x)/
((c*x^2 + a)*a*c^3) + 1/6*(2*B*c^4*x^3*e^5 + 15*B*c^4*d*x^2*e^4 + 60*B*c^4*d^2*x*e^3 + 3*A*c^4*x^2*e^5 + 30*A*
c^4*d*x*e^4 - 12*B*a*c^3*x*e^5)/c^6