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

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

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Rubi [A]  time = 0.34, antiderivative size = 297, normalized size of antiderivative = 1.00, 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} \begin {gather*} \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 )} \end {gather*}

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 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 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])

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*}

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Mathematica [A]  time = 0.18, size = 307, normalized size = 1.03 \begin {gather*} \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 {-a^3 e^4 (A e+5 B d+B e x)+5 a^2 c d e^2 (A e (2 d+e x)+2 B d (d+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 {e^3 x \left (-2 a B e^2+5 A c d e+10 B c d^2\right )}{c^3}+\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^4 x^2 (A e+5 B d)}{2 c^2}+\frac {B e^5 x^3}{3 c^2} \end {gather*}

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

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IntegrateAlgebraic [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {(A+B x) (d+e x)^5}{\left (a+c x^2\right )^2} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

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

[Out]

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

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fricas [B]  time = 0.43, size = 1190, normalized size = 4.01 \begin {gather*} \left [\frac {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 \, {\left (5 \, B a^{2} c^{3} d e^{4} + A a^{2} c^{3} e^{5}\right )} x^{4} + 20 \, {\left (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}\right )} x^{3} + 6 \, {\left (5 \, B a^{3} c^{2} d e^{4} + A a^{3} c^{2} e^{5}\right )} x^{2} - 3 \, {\left (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} + {\left (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}\right )} x^{2}\right )} \sqrt {-a c} \log \left (\frac {c x^{2} - 2 \, \sqrt {-a c} x - a}{c x^{2} + a}\right ) + 6 \, {\left (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}\right )} x + 12 \, {\left (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} + {\left (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}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{12 \, {\left (a^{2} c^{5} x^{2} + a^{3} c^{4}\right )}}, \frac {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 \, {\left (5 \, B a^{2} c^{3} d e^{4} + A a^{2} c^{3} e^{5}\right )} x^{4} + 10 \, {\left (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}\right )} x^{3} + 3 \, {\left (5 \, B a^{3} c^{2} d e^{4} + A a^{3} c^{2} e^{5}\right )} x^{2} + 3 \, {\left (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} + {\left (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}\right )} x^{2}\right )} \sqrt {a c} \arctan \left (\frac {\sqrt {a c} x}{a}\right ) + 3 \, {\left (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}\right )} x + 6 \, {\left (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} + {\left (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}\right )} x^{2}\right )} \log \left (c x^{2} + a\right )}{6 \, {\left (a^{2} c^{5} x^{2} + a^{3} c^{4}\right )}}\right ] \end {gather*}

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)]

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giac [A]  time = 0.16, size = 347, normalized size = 1.17 \begin {gather*} \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 {gather*}

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

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maple [A]  time = 0.06, size = 553, normalized size = 1.86 \begin {gather*} \frac {B \,e^{5} x^{3}}{3 c^{2}}+\frac {5 A a d \,e^{4} x}{2 \left (c \,x^{2}+a \right ) c^{2}}-\frac {15 A a d \,e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{2}}+\frac {A \,d^{5} x}{2 \left (c \,x^{2}+a \right ) a}+\frac {A \,d^{5} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, a}-\frac {5 A \,d^{3} e^{2} x}{\left (c \,x^{2}+a \right ) c}+\frac {5 A \,d^{3} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c}+\frac {A \,e^{5} x^{2}}{2 c^{2}}-\frac {B \,a^{2} e^{5} x}{2 \left (c \,x^{2}+a \right ) c^{3}}+\frac {5 B \,a^{2} e^{5} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c^{3}}+\frac {5 B a \,d^{2} e^{3} x}{\left (c \,x^{2}+a \right ) c^{2}}-\frac {15 B a \,d^{2} e^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{\sqrt {a c}\, c^{2}}-\frac {5 B \,d^{4} e x}{2 \left (c \,x^{2}+a \right ) c}+\frac {5 B \,d^{4} e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{2 \sqrt {a c}\, c}+\frac {5 B d \,e^{4} x^{2}}{2 c^{2}}-\frac {A \,a^{2} e^{5}}{2 \left (c \,x^{2}+a \right ) c^{3}}+\frac {5 A a \,d^{2} e^{3}}{\left (c \,x^{2}+a \right ) c^{2}}-\frac {A a \,e^{5} \ln \left (c \,x^{2}+a \right )}{c^{3}}-\frac {5 A \,d^{4} e}{2 \left (c \,x^{2}+a \right ) c}+\frac {5 A \,d^{2} e^{3} \ln \left (c \,x^{2}+a \right )}{c^{2}}+\frac {5 A d \,e^{4} x}{c^{2}}-\frac {5 B \,a^{2} d \,e^{4}}{2 \left (c \,x^{2}+a \right ) c^{3}}+\frac {5 B a \,d^{3} e^{2}}{\left (c \,x^{2}+a \right ) c^{2}}-\frac {5 B a d \,e^{4} \ln \left (c \,x^{2}+a \right )}{c^{3}}-\frac {2 B a \,e^{5} x}{c^{3}}-\frac {B \,d^{5}}{2 \left (c \,x^{2}+a \right ) c}+\frac {5 B \,d^{3} e^{2} \ln \left (c \,x^{2}+a \right )}{c^{2}}+\frac {10 B \,d^{2} e^{3} x}{c^{2}} \end {gather*}

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]

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

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maxima [A]  time = 1.34, size = 356, normalized size = 1.20 \begin {gather*} -\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 (a c^{4} x^{2} + a^{2} c^{3}\right )}} + \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 {2 \, B c e^{5} x^{3} + 3 \, {\left (5 \, B c d e^{4} + A c e^{5}\right )} x^{2} + 6 \, {\left (10 \, B c d^{2} e^{3} + 5 \, A c d e^{4} - 2 \, B a e^{5}\right )} x}{6 \, 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}} \end {gather*}

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]

-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)/(a*c^4*
x^2 + a^2*c^3) + (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 + 1/6*(2*B*c*e^5*x
^3 + 3*(5*B*c*d*e^4 + A*c*e^5)*x^2 + 6*(10*B*c*d^2*e^3 + 5*A*c*d*e^4 - 2*B*a*e^5)*x)/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)

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mupad [B]  time = 0.32, size = 370, normalized size = 1.25 \begin {gather*} \frac {x^2\,\left (A\,e^5+5\,B\,d\,e^4\right )}{2\,c^2}-\frac {\frac {A\,a^2\,e^5}{2}+\frac {B\,c^2\,d^5}{2}-\frac {x\,\left (-B\,a^3\,e^5+10\,B\,a^2\,c\,d^2\,e^3+5\,A\,a^2\,c\,d\,e^4-5\,B\,a\,c^2\,d^4\,e-10\,A\,a\,c^2\,d^3\,e^2+A\,c^3\,d^5\right )}{2\,a}+\frac {5\,B\,a^2\,d\,e^4}{2}+\frac {5\,A\,c^2\,d^4\,e}{2}-5\,A\,a\,c\,d^2\,e^3-5\,B\,a\,c\,d^3\,e^2}{c^4\,x^2+a\,c^3}-x\,\left (\frac {2\,B\,a\,e^5}{c^3}-\frac {5\,d\,e^3\,\left (A\,e+2\,B\,d\right )}{c^2}\right )-\frac {\ln \left (c\,x^2+a\right )\,\left (160\,B\,a^4\,c^4\,d\,e^4+32\,A\,a^4\,c^4\,e^5-160\,B\,a^3\,c^5\,d^3\,e^2-160\,A\,a^3\,c^5\,d^2\,e^3\right )}{32\,a^3\,c^7}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (5\,B\,a^3\,e^5-30\,B\,a^2\,c\,d^2\,e^3-15\,A\,a^2\,c\,d\,e^4+5\,B\,a\,c^2\,d^4\,e+10\,A\,a\,c^2\,d^3\,e^2+A\,c^3\,d^5\right )}{2\,a^{3/2}\,c^{7/2}}+\frac {B\,e^5\,x^3}{3\,c^2} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(x^2*(A*e^5 + 5*B*d*e^4))/(2*c^2) - ((A*a^2*e^5)/2 + (B*c^2*d^5)/2 - (x*(A*c^3*d^5 - B*a^3*e^5 - 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 - 5*B*a*c^2*d^4*e))/(2*a) + (5*B*a^2*d*e^4)/2 + (5*A*c^2*d^4*e)/2
- 5*A*a*c*d^2*e^3 - 5*B*a*c*d^3*e^2)/(a*c^3 + c^4*x^2) - x*((2*B*a*e^5)/c^3 - (5*d*e^3*(A*e + 2*B*d))/c^2) -
(log(a + c*x^2)*(32*A*a^4*c^4*e^5 + 160*B*a^4*c^4*d*e^4 - 160*A*a^3*c^5*d^2*e^3 - 160*B*a^3*c^5*d^3*e^2))/(32*
a^3*c^7) + (atan((c^(1/2)*x)/a^(1/2))*(A*c^3*d^5 + 5*B*a^3*e^5 + 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*c^2*d^4*e))/(2*a^(3/2)*c^(7/2)) + (B*e^5*x^3)/(3*c^2)

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sympy [B]  time = 14.77, size = 1091, normalized size = 3.67 \begin {gather*} \frac {B e^{5} x^{3}}{3 c^{2}} + x^{2} \left (\frac {A e^{5}}{2 c^{2}} + \frac {5 B d e^{4}}{2 c^{2}}\right ) + x \left (\frac {5 A d e^{4}}{c^{2}} - \frac {2 B a e^{5}}{c^{3}} + \frac {10 B d^{2} e^{3}}{c^{2}}\right ) + \left (- \frac {e^{2} \left (A a e^{3} - 5 A c d^{2} e + 5 B a d e^{2} - 5 B c d^{3}\right )}{c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 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\right )}{4 a^{3} c^{7}}\right ) \log {\left (x + \frac {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} \left (- \frac {e^{2} \left (A a e^{3} - 5 A c d^{2} e + 5 B a d e^{2} - 5 B c d^{3}\right )}{c^{3}} - \frac {\sqrt {- a^{3} c^{7}} \left (- 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\right )}{4 a^{3} c^{7}}\right )}{- 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} \right )} + \left (- \frac {e^{2} \left (A a e^{3} - 5 A c d^{2} e + 5 B a d e^{2} - 5 B c d^{3}\right )}{c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 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\right )}{4 a^{3} c^{7}}\right ) \log {\left (x + \frac {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} \left (- \frac {e^{2} \left (A a e^{3} - 5 A c d^{2} e + 5 B a d e^{2} - 5 B c d^{3}\right )}{c^{3}} + \frac {\sqrt {- a^{3} c^{7}} \left (- 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\right )}{4 a^{3} c^{7}}\right )}{- 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} \right )} + \frac {- 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 \left (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\right )}{2 a^{2} c^{3} + 2 a c^{4} x^{2}} \end {gather*}

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) + x**2*(A*e**5/(2*c**2) + 5*B*d*e**4/(2*c**2)) + x*(5*A*d*e**4/c**2 - 2*B*a*e**5/c**3 + 1
0*B*d**2*e**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)*(-1
5*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)

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