3.12.72 \(\int \frac {(A+B x) (d+e x)^5}{(a+c x^2)^3} \, dx\)

Optimal. Leaf size=304 \[ -\frac {e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}-\frac {(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right ) \left (A c d \left (15 a^2 e^4+10 a c d^2 e^2+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac {e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac {(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \]

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Rubi [A]  time = 0.42, antiderivative size = 304, 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, 774, 635, 205, 260} \begin {gather*} -\frac {(d+e x)^2 \left (2 a e \left (2 a A e^2+5 a B d e+A c d^2\right )-x \left (A c d \left (5 a e^2+3 c d^2\right )+5 a B e \left (c d^2-a e^2\right )\right )\right )}{8 a^2 c^2 \left (a+c x^2\right )}-\frac {e^2 x \left (A c d \left (7 a e^2+3 c d^2\right )+5 a B e \left (c d^2-3 a e^2\right )\right )}{8 a^2 c^3}+\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+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{8 a^{5/2} c^{7/2}}+\frac {e^4 \log \left (a+c x^2\right ) (A e+5 B d)}{2 c^3}-\frac {(d+e x)^4 (a (A e+B d)-x (A c d-a B e))}{4 a c \left (a+c x^2\right )^2} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3,x]

[Out]

-(e^2*(5*a*B*e*(c*d^2 - 3*a*e^2) + A*c*d*(3*c*d^2 + 7*a*e^2))*x)/(8*a^2*c^3) - ((d + e*x)^4*(a*(B*d + A*e) - (
A*c*d - a*B*e)*x))/(4*a*c*(a + c*x^2)^2) - ((d + e*x)^2*(2*a*e*(A*c*d^2 + 5*a*B*d*e + 2*a*A*e^2) - (5*a*B*e*(c
*d^2 - a*e^2) + A*c*d*(3*c*d^2 + 5*a*e^2))*x))/(8*a^2*c^2*(a + c*x^2)) + ((5*a*B*e*(c^2*d^4 + 6*a*c*d^2*e^2 -
3*a^2*e^4) + A*c*d*(3*c^2*d^4 + 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/(8*a^(5/2)*c^(7/2))
+ (e^4*(5*B*d + A*e)*Log[a + c*x^2])/(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 774

Int[(((d_.) + (e_.)*(x_))*((f_) + (g_.)*(x_)))/((a_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(e*g*x)/c, x] + Dist[1
/c, Int[(c*d*f - a*e*g + c*(e*f + d*g)*x)/(a + c*x^2), x], x] /; FreeQ[{a, c, d, e, f, g}, x]

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 )^3} \, dx &=-\frac {(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}+\frac {\int \frac {(d+e x)^3 \left (3 A c d^2+a e (5 B d+4 A e)-e (A c d-5 a B e) x\right )}{\left (a+c x^2\right )^2} \, dx}{4 a c}\\ &=-\frac {(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {(d+e x) \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )-e \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x\right )}{a+c x^2} \, dx}{8 a^2 c^2}\\ &=-\frac {e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac {(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\int \frac {a e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right )+c d \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right )+c \left (-d e \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right )+e \left (5 a B d e \left (c d^2+5 a e^2\right )+A \left (3 c^2 d^4+7 a c d^2 e^2+8 a^2 e^4\right )\right )\right ) x}{a+c x^2} \, dx}{8 a^2 c^3}\\ &=-\frac {e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac {(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (e^4 (5 B d+A e)\right ) \int \frac {x}{a+c x^2} \, dx}{c^2}+\frac {\left (5 a B e \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \int \frac {1}{a+c x^2} \, dx}{8 a^2 c^3}\\ &=-\frac {e^2 \left (5 a B e \left (c d^2-3 a e^2\right )+A c d \left (3 c d^2+7 a e^2\right )\right ) x}{8 a^2 c^3}-\frac {(d+e x)^4 (a (B d+A e)-(A c d-a B e) x)}{4 a c \left (a+c x^2\right )^2}-\frac {(d+e x)^2 \left (2 a e \left (A c d^2+5 a B d e+2 a A e^2\right )-\left (5 a B e \left (c d^2-a e^2\right )+A c d \left (3 c d^2+5 a e^2\right )\right ) x\right )}{8 a^2 c^2 \left (a+c x^2\right )}+\frac {\left (5 a B e \left (c^2 d^4+6 a c d^2 e^2-3 a^2 e^4\right )+A c d \left (3 c^2 d^4+10 a c d^2 e^2+15 a^2 e^4\right )\right ) \tan ^{-1}\left (\frac {\sqrt {c} x}{\sqrt {a}}\right )}{8 a^{5/2} c^{7/2}}+\frac {e^4 (5 B d+A e) \log \left (a+c x^2\right )}{2 c^3}\\ \end {align*}

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Mathematica [A]  time = 0.25, size = 341, normalized size = 1.12 \begin {gather*} \frac {\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+3 c^2 d^4\right )+5 a B e \left (-3 a^2 e^4+6 a c d^2 e^2+c^2 d^4\right )\right )}{a^{5/2}}+\frac {2 \sqrt {c} \left (-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\right )}{a \left (a+c x^2\right )^2}+\frac {\sqrt {c} \left (a^3 e^4 (8 A e+40 B d+9 B e x)-5 a^2 c d e^2 (A e (8 d+5 e x)+2 B d (4 d+5 e x))+5 a c^2 d^3 e x (2 A e+B d)+3 A c^3 d^5 x\right )}{a^2 \left (a+c x^2\right )}+4 \sqrt {c} e^4 \log \left (a+c x^2\right ) (A e+5 B d)+8 B \sqrt {c} e^5 x}{8 c^{7/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3,x]

[Out]

(8*B*Sqrt[c]*e^5*x + (2*Sqrt[c]*(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))))/(a*(a + c*x^2)^2) + (Sqrt[c]*(3*A*c^3*d
^5*x + 5*a*c^2*d^3*e*(B*d + 2*A*e)*x + a^3*e^4*(40*B*d + 8*A*e + 9*B*e*x) - 5*a^2*c*d*e^2*(2*B*d*(4*d + 5*e*x)
+ A*e*(8*d + 5*e*x))))/(a^2*(a + c*x^2)) + ((5*a*B*e*(c^2*d^4 + 6*a*c*d^2*e^2 - 3*a^2*e^4) + A*c*d*(3*c^2*d^4
+ 10*a*c*d^2*e^2 + 15*a^2*e^4))*ArcTan[(Sqrt[c]*x)/Sqrt[a]])/a^(5/2) + 4*Sqrt[c]*e^4*(5*B*d + A*e)*Log[a + c*
x^2])/(8*c^(7/2))

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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 )^3} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3,x]

[Out]

IntegrateAlgebraic[((A + B*x)*(d + e*x)^5)/(a + c*x^2)^3, x]

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fricas [B]  time = 0.46, size = 1403, normalized size = 4.62

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^5/(c*x^2+a)^3,x, algorithm="fricas")

[Out]

[1/16*(16*B*a^3*c^3*e^5*x^5 - 4*B*a^3*c^3*d^5 - 20*A*a^3*c^3*d^4*e - 40*B*a^4*c^2*d^3*e^2 - 40*A*a^4*c^2*d^2*e
^3 + 60*B*a^5*c*d*e^4 + 12*A*a^5*c*e^5 + 2*(3*A*a*c^5*d^5 + 5*B*a^2*c^4*d^4*e + 10*A*a^2*c^4*d^3*e^2 - 50*B*a^
3*c^3*d^2*e^3 - 25*A*a^3*c^3*d*e^4 + 25*B*a^4*c^2*e^5)*x^3 - 16*(5*B*a^3*c^3*d^3*e^2 + 5*A*a^3*c^3*d^2*e^3 - 5
*B*a^4*c^2*d*e^4 - A*a^4*c^2*e^5)*x^2 + (3*A*a^2*c^3*d^5 + 5*B*a^3*c^2*d^4*e + 10*A*a^3*c^2*d^3*e^2 + 30*B*a^4
*c*d^2*e^3 + 15*A*a^4*c*d*e^4 - 15*B*a^5*e^5 + (3*A*c^5*d^5 + 5*B*a*c^4*d^4*e + 10*A*a*c^4*d^3*e^2 + 30*B*a^2*
c^3*d^2*e^3 + 15*A*a^2*c^3*d*e^4 - 15*B*a^3*c^2*e^5)*x^4 + 2*(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 - 15*B*a^4*c*e^5)*x^2)*sqrt(-a*c)*log((c*x^2 + 2*sqrt(-a*
c)*x - a)/(c*x^2 + a)) + 10*(A*a^2*c^4*d^5 - B*a^3*c^3*d^4*e - 2*A*a^3*c^3*d^3*e^2 - 6*B*a^4*c^2*d^2*e^3 - 3*A
*a^4*c^2*d*e^4 + 3*B*a^5*c*e^5)*x + 8*(5*B*a^5*c*d*e^4 + A*a^5*c*e^5 + (5*B*a^3*c^3*d*e^4 + A*a^3*c^3*e^5)*x^4
+ 2*(5*B*a^4*c^2*d*e^4 + A*a^4*c^2*e^5)*x^2)*log(c*x^2 + a))/(a^3*c^6*x^4 + 2*a^4*c^5*x^2 + a^5*c^4), 1/8*(8*
B*a^3*c^3*e^5*x^5 - 2*B*a^3*c^3*d^5 - 10*A*a^3*c^3*d^4*e - 20*B*a^4*c^2*d^3*e^2 - 20*A*a^4*c^2*d^2*e^3 + 30*B*
a^5*c*d*e^4 + 6*A*a^5*c*e^5 + (3*A*a*c^5*d^5 + 5*B*a^2*c^4*d^4*e + 10*A*a^2*c^4*d^3*e^2 - 50*B*a^3*c^3*d^2*e^3
- 25*A*a^3*c^3*d*e^4 + 25*B*a^4*c^2*e^5)*x^3 - 8*(5*B*a^3*c^3*d^3*e^2 + 5*A*a^3*c^3*d^2*e^3 - 5*B*a^4*c^2*d*e
^4 - A*a^4*c^2*e^5)*x^2 + (3*A*a^2*c^3*d^5 + 5*B*a^3*c^2*d^4*e + 10*A*a^3*c^2*d^3*e^2 + 30*B*a^4*c*d^2*e^3 + 1
5*A*a^4*c*d*e^4 - 15*B*a^5*e^5 + (3*A*c^5*d^5 + 5*B*a*c^4*d^4*e + 10*A*a*c^4*d^3*e^2 + 30*B*a^2*c^3*d^2*e^3 +
15*A*a^2*c^3*d*e^4 - 15*B*a^3*c^2*e^5)*x^4 + 2*(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 - 15*B*a^4*c*e^5)*x^2)*sqrt(a*c)*arctan(sqrt(a*c)*x/a) + 5*(A*a^2*c^4*d
^5 - B*a^3*c^3*d^4*e - 2*A*a^3*c^3*d^3*e^2 - 6*B*a^4*c^2*d^2*e^3 - 3*A*a^4*c^2*d*e^4 + 3*B*a^5*c*e^5)*x + 4*(5
*B*a^5*c*d*e^4 + A*a^5*c*e^5 + (5*B*a^3*c^3*d*e^4 + A*a^3*c^3*e^5)*x^4 + 2*(5*B*a^4*c^2*d*e^4 + A*a^4*c^2*e^5)
*x^2)*log(c*x^2 + a))/(a^3*c^6*x^4 + 2*a^4*c^5*x^2 + a^5*c^4)]

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giac [A]  time = 0.16, size = 403, normalized size = 1.33 \begin {gather*} \frac {B x e^{5}}{c^{3}} + \frac {{\left (5 \, B d e^{4} + A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, 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} - 15 \, B a^{3} e^{5}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} c^{3}} - \frac {2 \, B a^{2} c^{2} d^{5} + 10 \, A a^{2} c^{2} d^{4} e + 20 \, B a^{3} c d^{3} e^{2} + 20 \, A a^{3} c d^{2} e^{3} - 30 \, B a^{4} d e^{4} - 6 \, A a^{4} e^{5} - {\left (3 \, A c^{4} d^{5} + 5 \, B a c^{3} d^{4} e + 10 \, A a c^{3} d^{3} e^{2} - 50 \, B a^{2} c^{2} d^{2} e^{3} - 25 \, A a^{2} c^{2} d e^{4} + 9 \, B a^{3} c e^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{2} c^{2} d^{3} e^{2} + 5 \, A a^{2} c^{2} d^{2} e^{3} - 5 \, B a^{3} c d e^{4} - A a^{3} c e^{5}\right )} x^{2} - {\left (5 \, 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} + 7 \, B a^{4} e^{5}\right )} x}{8 \, {\left (c x^{2} + a\right )}^{2} a^{2} 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)^3,x, algorithm="giac")

[Out]

B*x*e^5/c^3 + 1/2*(5*B*d*e^4 + A*e^5)*log(c*x^2 + a)/c^3 + 1/8*(3*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 - 15*B*a^3*e^5)*arctan(c*x/sqrt(a*c))/(sqrt(a*c)*a^2*c^3) - 1/8*(
2*B*a^2*c^2*d^5 + 10*A*a^2*c^2*d^4*e + 20*B*a^3*c*d^3*e^2 + 20*A*a^3*c*d^2*e^3 - 30*B*a^4*d*e^4 - 6*A*a^4*e^5
- (3*A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 50*B*a^2*c^2*d^2*e^3 - 25*A*a^2*c^2*d*e^4 + 9*B*a^3*c*
e^5)*x^3 + 8*(5*B*a^2*c^2*d^3*e^2 + 5*A*a^2*c^2*d^2*e^3 - 5*B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^2 - (5*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 + 7*B*a^4*e^5)*x)/((c*x^2 +
a)^2*a^2*c^3)

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maple [B]  time = 0.06, size = 678, normalized size = 2.23 \begin {gather*} \frac {5 A \,d^{3} e^{2} x^{3}}{4 \left (c \,x^{2}+a \right )^{2} a}+\frac {3 A c \,d^{5} x^{3}}{8 \left (c \,x^{2}+a \right )^{2} a^{2}}-\frac {25 A d \,e^{4} x^{3}}{8 \left (c \,x^{2}+a \right )^{2} c}+\frac {9 B a \,e^{5} x^{3}}{8 \left (c \,x^{2}+a \right )^{2} c^{2}}+\frac {5 B \,d^{4} e \,x^{3}}{8 \left (c \,x^{2}+a \right )^{2} a}-\frac {25 B \,d^{2} e^{3} x^{3}}{4 \left (c \,x^{2}+a \right )^{2} c}+\frac {A a \,e^{5} x^{2}}{\left (c \,x^{2}+a \right )^{2} c^{2}}-\frac {5 A \,d^{2} e^{3} x^{2}}{\left (c \,x^{2}+a \right )^{2} c}+\frac {5 B a d \,e^{4} x^{2}}{\left (c \,x^{2}+a \right )^{2} c^{2}}-\frac {5 B \,d^{3} e^{2} x^{2}}{\left (c \,x^{2}+a \right )^{2} c}-\frac {15 A a d \,e^{4} x}{8 \left (c \,x^{2}+a \right )^{2} c^{2}}+\frac {5 A \,d^{5} x}{8 \left (c \,x^{2}+a \right )^{2} a}-\frac {5 A \,d^{3} e^{2} x}{4 \left (c \,x^{2}+a \right )^{2} c}+\frac {7 B \,a^{2} e^{5} x}{8 \left (c \,x^{2}+a \right )^{2} c^{3}}-\frac {15 B a \,d^{2} e^{3} x}{4 \left (c \,x^{2}+a \right )^{2} c^{2}}-\frac {5 B \,d^{4} e x}{8 \left (c \,x^{2}+a \right )^{2} c}+\frac {3 A \,a^{2} e^{5}}{4 \left (c \,x^{2}+a \right )^{2} c^{3}}-\frac {5 A a \,d^{2} e^{3}}{2 \left (c \,x^{2}+a \right )^{2} c^{2}}+\frac {5 A \,d^{3} e^{2} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, a c}+\frac {3 A \,d^{5} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a^{2}}-\frac {5 A \,d^{4} e}{4 \left (c \,x^{2}+a \right )^{2} c}+\frac {15 A d \,e^{4} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, c^{2}}+\frac {15 B \,a^{2} d \,e^{4}}{4 \left (c \,x^{2}+a \right )^{2} c^{3}}-\frac {5 B a \,d^{3} e^{2}}{2 \left (c \,x^{2}+a \right )^{2} c^{2}}-\frac {15 B a \,e^{5} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, c^{3}}+\frac {5 B \,d^{4} e \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \sqrt {a c}\, a c}-\frac {B \,d^{5}}{4 \left (c \,x^{2}+a \right )^{2} c}+\frac {15 B \,d^{2} e^{3} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{4 \sqrt {a c}\, c^{2}}+\frac {A \,e^{5} \ln \left (c \,x^{2}+a \right )}{2 c^{3}}+\frac {5 B d \,e^{4} \ln \left (c \,x^{2}+a \right )}{2 c^{3}}+\frac {B \,e^{5} x}{c^{3}} \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)^3,x)

[Out]

-1/4/c/(c*x^2+a)^2*B*d^5+1/2/c^3*ln(c*x^2+a)*A*e^5+5/4/c/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^3*e^2+5/8
/c/a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*d^4*e-15/4/c^2/(c*x^2+a)^2*a*x*B*d^2*e^3-15/8/c^2/(c*x^2+a)^2*a*x
*A*d*e^4+5/c^2/(c*x^2+a)^2*B*x^2*a*d*e^4+3/8/a^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d^5+5/2/c^3*ln(c*x^2+
a)*B*d*e^4+3/4/c^3/(c*x^2+a)^2*A*a^2*e^5-5/4/c/(c*x^2+a)^2*A*d^4*e+5/8/(c*x^2+a)^2/a*x*A*d^5+5/8/(c*x^2+a)^2/a
*x^3*B*d^4*e+5/4/(c*x^2+a)^2/a*x^3*A*d^3*e^2+15/8/c^2/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*A*d*e^4+15/4/c^2/(
a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)*B*d^2*e^3+1/c^2/(c*x^2+a)^2*A*x^2*a*e^5+9/8/c^2/(c*x^2+a)^2*a*x^3*B*e^5-2
5/8/c/(c*x^2+a)^2*x^3*A*d*e^4+3/8*c/(c*x^2+a)^2/a^2*x^3*A*d^5-15/8/c^3*a/(a*c)^(1/2)*arctan(1/(a*c)^(1/2)*c*x)
*B*e^5+B/c^3*e^5*x-5/c/(c*x^2+a)^2*A*x^2*d^2*e^3-5/c/(c*x^2+a)^2*B*x^2*d^3*e^2-5/4/c/(c*x^2+a)^2*x*A*d^3*e^2-5
/8/c/(c*x^2+a)^2*x*B*d^4*e-5/2/c^2/(c*x^2+a)^2*A*d^2*a*e^3-5/2/c^2/(c*x^2+a)^2*B*d^3*a*e^2+15/4/c^3/(c*x^2+a)^
2*B*a^2*d*e^4+7/8/c^3/(c*x^2+a)^2*a^2*x*B*e^5-25/4/c/(c*x^2+a)^2*x^3*B*d^2*e^3

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maxima [A]  time = 1.15, size = 438, normalized size = 1.44 \begin {gather*} \frac {B e^{5} x}{c^{3}} - \frac {2 \, B a^{2} c^{2} d^{5} + 10 \, A a^{2} c^{2} d^{4} e + 20 \, B a^{3} c d^{3} e^{2} + 20 \, A a^{3} c d^{2} e^{3} - 30 \, B a^{4} d e^{4} - 6 \, A a^{4} e^{5} - {\left (3 \, A c^{4} d^{5} + 5 \, B a c^{3} d^{4} e + 10 \, A a c^{3} d^{3} e^{2} - 50 \, B a^{2} c^{2} d^{2} e^{3} - 25 \, A a^{2} c^{2} d e^{4} + 9 \, B a^{3} c e^{5}\right )} x^{3} + 8 \, {\left (5 \, B a^{2} c^{2} d^{3} e^{2} + 5 \, A a^{2} c^{2} d^{2} e^{3} - 5 \, B a^{3} c d e^{4} - A a^{3} c e^{5}\right )} x^{2} - {\left (5 \, 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} + 7 \, B a^{4} e^{5}\right )} x}{8 \, {\left (a^{2} c^{5} x^{4} + 2 \, a^{3} c^{4} x^{2} + a^{4} c^{3}\right )}} + \frac {{\left (5 \, B d e^{4} + A e^{5}\right )} \log \left (c x^{2} + a\right )}{2 \, c^{3}} + \frac {{\left (3 \, 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} - 15 \, B a^{3} e^{5}\right )} \arctan \left (\frac {c x}{\sqrt {a c}}\right )}{8 \, \sqrt {a c} a^{2} 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)^3,x, algorithm="maxima")

[Out]

B*e^5*x/c^3 - 1/8*(2*B*a^2*c^2*d^5 + 10*A*a^2*c^2*d^4*e + 20*B*a^3*c*d^3*e^2 + 20*A*a^3*c*d^2*e^3 - 30*B*a^4*d
*e^4 - 6*A*a^4*e^5 - (3*A*c^4*d^5 + 5*B*a*c^3*d^4*e + 10*A*a*c^3*d^3*e^2 - 50*B*a^2*c^2*d^2*e^3 - 25*A*a^2*c^2
*d*e^4 + 9*B*a^3*c*e^5)*x^3 + 8*(5*B*a^2*c^2*d^3*e^2 + 5*A*a^2*c^2*d^2*e^3 - 5*B*a^3*c*d*e^4 - A*a^3*c*e^5)*x^
2 - (5*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 + 7*B*a^
4*e^5)*x)/(a^2*c^5*x^4 + 2*a^3*c^4*x^2 + a^4*c^3) + 1/2*(5*B*d*e^4 + A*e^5)*log(c*x^2 + a)/c^3 + 1/8*(3*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 - 15*B*a^3*e^5)*arctan(c*x/
sqrt(a*c))/(sqrt(a*c)*a^2*c^3)

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mupad [B]  time = 0.29, size = 424, normalized size = 1.39 \begin {gather*} \frac {\ln \left (c\,x^2+a\right )\,\left (256\,A\,a^5\,c^4\,e^5+1280\,B\,d\,a^5\,c^4\,e^4\right )}{512\,a^5\,c^7}-\frac {\frac {B\,c^2\,d^5}{4}-\frac {3\,A\,a^2\,e^5}{4}-x^2\,\left (-5\,B\,c^2\,d^3\,e^2-5\,A\,c^2\,d^2\,e^3+5\,B\,a\,c\,d\,e^4+A\,a\,c\,e^5\right )-\frac {x^3\,\left (9\,B\,a^3\,c\,e^5-50\,B\,a^2\,c^2\,d^2\,e^3-25\,A\,a^2\,c^2\,d\,e^4+5\,B\,a\,c^3\,d^4\,e+10\,A\,a\,c^3\,d^3\,e^2+3\,A\,c^4\,d^5\right )}{8\,a^2}+\frac {x\,\left (-7\,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^2-5\,A\,c^3\,d^5\right )}{8\,a}-\frac {15\,B\,a^2\,d\,e^4}{4}+\frac {5\,A\,c^2\,d^4\,e}{4}+\frac {5\,A\,a\,c\,d^2\,e^3}{2}+\frac {5\,B\,a\,c\,d^3\,e^2}{2}}{a^2\,c^3+2\,a\,c^4\,x^2+c^5\,x^4}+\frac {B\,e^5\,x}{c^3}+\frac {\mathrm {atan}\left (\frac {\sqrt {c}\,x}{\sqrt {a}}\right )\,\left (-15\,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+3\,A\,c^3\,d^5\right )}{8\,a^{5/2}\,c^{7/2}} \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)^3,x)

[Out]

(log(a + c*x^2)*(256*A*a^5*c^4*e^5 + 1280*B*a^5*c^4*d*e^4))/(512*a^5*c^7) - ((B*c^2*d^5)/4 - (3*A*a^2*e^5)/4 -
x^2*(A*a*c*e^5 - 5*A*c^2*d^2*e^3 - 5*B*c^2*d^3*e^2 + 5*B*a*c*d*e^4) - (x^3*(3*A*c^4*d^5 + 9*B*a^3*c*e^5 + 10*
A*a*c^3*d^3*e^2 - 25*A*a^2*c^2*d*e^4 - 50*B*a^2*c^2*d^2*e^3 + 5*B*a*c^3*d^4*e))/(8*a^2) + (x*(10*A*a*c^2*d^3*e
^2 - 7*B*a^3*e^5 - 5*A*c^3*d^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))/(8*a) - (15*B*a^2*d
*e^4)/4 + (5*A*c^2*d^4*e)/4 + (5*A*a*c*d^2*e^3)/2 + (5*B*a*c*d^3*e^2)/2)/(a^2*c^3 + c^5*x^4 + 2*a*c^4*x^2) + (
B*e^5*x)/c^3 + (atan((c^(1/2)*x)/a^(1/2))*(3*A*c^3*d^5 - 15*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))/(8*a^(5/2)*c^(7/2))

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sympy [B]  time = 97.93, size = 1044, normalized size = 3.43 \begin {gather*} \frac {B e^{5} x}{c^{3}} + \left (\frac {e^{4} \left (A e + 5 B d\right )}{2 c^{3}} - \frac {\sqrt {- a^{5} c^{7}} \left (- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 B a^{3} e^{5} - 30 B a^{2} c d^{2} e^{3} - 5 B a c^{2} d^{4} e\right )}{16 a^{5} c^{7}}\right ) \log {\left (x + \frac {8 A a^{3} e^{5} + 40 B a^{3} d e^{4} - 16 a^{3} c^{3} \left (\frac {e^{4} \left (A e + 5 B d\right )}{2 c^{3}} - \frac {\sqrt {- a^{5} c^{7}} \left (- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 B a^{3} e^{5} - 30 B a^{2} c d^{2} e^{3} - 5 B a c^{2} d^{4} e\right )}{16 a^{5} c^{7}}\right )}{- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 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^{4} \left (A e + 5 B d\right )}{2 c^{3}} + \frac {\sqrt {- a^{5} c^{7}} \left (- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 B a^{3} e^{5} - 30 B a^{2} c d^{2} e^{3} - 5 B a c^{2} d^{4} e\right )}{16 a^{5} c^{7}}\right ) \log {\left (x + \frac {8 A a^{3} e^{5} + 40 B a^{3} d e^{4} - 16 a^{3} c^{3} \left (\frac {e^{4} \left (A e + 5 B d\right )}{2 c^{3}} + \frac {\sqrt {- a^{5} c^{7}} \left (- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 B a^{3} e^{5} - 30 B a^{2} c d^{2} e^{3} - 5 B a c^{2} d^{4} e\right )}{16 a^{5} c^{7}}\right )}{- 15 A a^{2} c d e^{4} - 10 A a c^{2} d^{3} e^{2} - 3 A c^{3} d^{5} + 15 B a^{3} e^{5} - 30 B a^{2} c d^{2} e^{3} - 5 B a c^{2} d^{4} e} \right )} + \frac {6 A a^{4} e^{5} - 20 A a^{3} c d^{2} e^{3} - 10 A a^{2} c^{2} d^{4} e + 30 B a^{4} d e^{4} - 20 B a^{3} c d^{3} e^{2} - 2 B a^{2} c^{2} d^{5} + x^{3} \left (- 25 A a^{2} c^{2} d e^{4} + 10 A a c^{3} d^{3} e^{2} + 3 A c^{4} d^{5} + 9 B a^{3} c e^{5} - 50 B a^{2} c^{2} d^{2} e^{3} + 5 B a c^{3} d^{4} e\right ) + x^{2} \left (8 A a^{3} c e^{5} - 40 A a^{2} c^{2} d^{2} e^{3} + 40 B a^{3} c d e^{4} - 40 B a^{2} c^{2} d^{3} e^{2}\right ) + x \left (- 15 A a^{3} c d e^{4} - 10 A a^{2} c^{2} d^{3} e^{2} + 5 A a c^{3} d^{5} + 7 B a^{4} e^{5} - 30 B a^{3} c d^{2} e^{3} - 5 B a^{2} c^{2} d^{4} e\right )}{8 a^{4} c^{3} + 16 a^{3} c^{4} x^{2} + 8 a^{2} c^{5} x^{4}} \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)**3,x)

[Out]

B*e**5*x/c**3 + (e**4*(A*e + 5*B*d)/(2*c**3) - sqrt(-a**5*c**7)*(-15*A*a**2*c*d*e**4 - 10*A*a*c**2*d**3*e**2 -
3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 - 5*B*a*c**2*d**4*e)/(16*a**5*c**7))*log(x + (8*A*a**3
*e**5 + 40*B*a**3*d*e**4 - 16*a**3*c**3*(e**4*(A*e + 5*B*d)/(2*c**3) - sqrt(-a**5*c**7)*(-15*A*a**2*c*d*e**4 -
10*A*a*c**2*d**3*e**2 - 3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 - 5*B*a*c**2*d**4*e)/(16*a**5*
c**7)))/(-15*A*a**2*c*d*e**4 - 10*A*a*c**2*d**3*e**2 - 3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3
- 5*B*a*c**2*d**4*e)) + (e**4*(A*e + 5*B*d)/(2*c**3) + sqrt(-a**5*c**7)*(-15*A*a**2*c*d*e**4 - 10*A*a*c**2*d**
3*e**2 - 3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 - 5*B*a*c**2*d**4*e)/(16*a**5*c**7))*log(x + (
8*A*a**3*e**5 + 40*B*a**3*d*e**4 - 16*a**3*c**3*(e**4*(A*e + 5*B*d)/(2*c**3) + sqrt(-a**5*c**7)*(-15*A*a**2*c*
d*e**4 - 10*A*a*c**2*d**3*e**2 - 3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d**2*e**3 - 5*B*a*c**2*d**4*e)/(
16*a**5*c**7)))/(-15*A*a**2*c*d*e**4 - 10*A*a*c**2*d**3*e**2 - 3*A*c**3*d**5 + 15*B*a**3*e**5 - 30*B*a**2*c*d*
*2*e**3 - 5*B*a*c**2*d**4*e)) + (6*A*a**4*e**5 - 20*A*a**3*c*d**2*e**3 - 10*A*a**2*c**2*d**4*e + 30*B*a**4*d*e
**4 - 20*B*a**3*c*d**3*e**2 - 2*B*a**2*c**2*d**5 + x**3*(-25*A*a**2*c**2*d*e**4 + 10*A*a*c**3*d**3*e**2 + 3*A*
c**4*d**5 + 9*B*a**3*c*e**5 - 50*B*a**2*c**2*d**2*e**3 + 5*B*a*c**3*d**4*e) + x**2*(8*A*a**3*c*e**5 - 40*A*a**
2*c**2*d**2*e**3 + 40*B*a**3*c*d*e**4 - 40*B*a**2*c**2*d**3*e**2) + x*(-15*A*a**3*c*d*e**4 - 10*A*a**2*c**2*d*
*3*e**2 + 5*A*a*c**3*d**5 + 7*B*a**4*e**5 - 30*B*a**3*c*d**2*e**3 - 5*B*a**2*c**2*d**4*e))/(8*a**4*c**3 + 16*a
**3*c**4*x**2 + 8*a**2*c**5*x**4)

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