3.1247 \(\int \frac{(A+B x) (d+e x)^{9/2}}{(b x+c x^2)^3} \, dx\)

Optimal. Leaf size=461 \[ \frac{(d+e x)^{3/2} \left (x \left (b^2 c^2 d e (10 A e+9 B d)+b^3 c e^2 (A e+4 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-5 b^4 B e^3\right )+b c d^2 \left (-b c (13 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{3 e \sqrt{d+e x} \left (b^2 c^2 d e (2 A e+3 B d)+b^3 c e^2 (A e+2 B d)-4 b c^3 d^2 (3 A e+B d)+8 A c^4 d^3-5 b^4 B e^3\right )}{4 b^4 c^3}+\frac{3 (c d-b e)^{5/2} \left (-b^2 c e (8 B d-A e)-4 b c^2 d (2 B d-A e)+16 A c^3 d^2-5 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{7/2}}-\frac{3 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

[Out]

(-3*e*(8*A*c^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(2*B*d + A*e) + b^2*c^2*d*e*(3*B*d + 2*A*e) - 4*b*c^3*d^2*(B*d +
3*A*e))*Sqrt[d + e*x])/(4*b^4*c^3) - ((d + e*x)^(7/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(
2*b^2*c*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 13*A*e)) + (24*A*c
^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(4*B*d + A*e) - 12*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(9*B*d + 10*A*e))*x)
)/(4*b^4*c^2*(b*x + c*x^2)) - (3*d^(5/2)*(16*A*c^2*d^2 + 3*b^2*e*(4*B*d + 7*A*e) - 4*b*c*d*(2*B*d + 9*A*e))*Ar
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*A*c^3*d^2 - 5*b^3*B*e^2 - 4*b*c^2*d*(2*B*d -
A*e) - b^2*c*e*(8*B*d - A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 1.21513, antiderivative size = 461, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 5, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.192, Rules used = {818, 824, 826, 1166, 208} \[ \frac{(d+e x)^{3/2} \left (x \left (b^2 c^2 d e (10 A e+9 B d)+b^3 c e^2 (A e+4 B d)-12 b c^3 d^2 (3 A e+B d)+24 A c^4 d^3-5 b^4 B e^3\right )+b c d^2 \left (-b c (13 A e+6 B d)+12 A c^2 d+2 b^2 B e\right )\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{3 e \sqrt{d+e x} \left (b^2 c^2 d e (2 A e+3 B d)+b^3 c e^2 (A e+2 B d)-4 b c^3 d^2 (3 A e+B d)+8 A c^4 d^3-5 b^4 B e^3\right )}{4 b^4 c^3}+\frac{3 (c d-b e)^{5/2} \left (-b^2 c e (8 B d-A e)-4 b c^2 d (2 B d-A e)+16 A c^3 d^2-5 b^3 B e^2\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{7/2}}-\frac{3 d^{5/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )}{4 b^5}-\frac{(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{2 b^2 c \left (b x+c x^2\right )^2} \]

Antiderivative was successfully verified.

[In]

Int[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x]

[Out]

(-3*e*(8*A*c^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(2*B*d + A*e) + b^2*c^2*d*e*(3*B*d + 2*A*e) - 4*b*c^3*d^2*(B*d +
3*A*e))*Sqrt[d + e*x])/(4*b^4*c^3) - ((d + e*x)^(7/2)*(A*b*c*d + (2*A*c^2*d + b^2*B*e - b*c*(B*d + A*e))*x))/(
2*b^2*c*(b*x + c*x^2)^2) + ((d + e*x)^(3/2)*(b*c*d^2*(12*A*c^2*d + 2*b^2*B*e - b*c*(6*B*d + 13*A*e)) + (24*A*c
^4*d^3 - 5*b^4*B*e^3 + b^3*c*e^2*(4*B*d + A*e) - 12*b*c^3*d^2*(B*d + 3*A*e) + b^2*c^2*d*e*(9*B*d + 10*A*e))*x)
)/(4*b^4*c^2*(b*x + c*x^2)) - (3*d^(5/2)*(16*A*c^2*d^2 + 3*b^2*e*(4*B*d + 7*A*e) - 4*b*c*d*(2*B*d + 9*A*e))*Ar
cTanh[Sqrt[d + e*x]/Sqrt[d]])/(4*b^5) + (3*(c*d - b*e)^(5/2)*(16*A*c^3*d^2 - 5*b^3*B*e^2 - 4*b*c^2*d*(2*B*d -
A*e) - b^2*c*e*(8*B*d - A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(4*b^5*c^(7/2))

Rule 818

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> -Si
mp[((d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*(2*a*c*(e*f + d*g) - b*(c*d*f + a*e*g) - (2*c^2*d*f + b^2*e*g
- c*(b*e*f + b*d*g + 2*a*e*g))*x))/(c*(p + 1)*(b^2 - 4*a*c)), x] - Dist[1/(c*(p + 1)*(b^2 - 4*a*c)), Int[(d +
e*x)^(m - 2)*(a + b*x + c*x^2)^(p + 1)*Simp[2*c^2*d^2*f*(2*p + 3) + b*e*g*(a*e*(m - 1) + b*d*(p + 2)) - c*(2*a
*e*(e*f*(m - 1) + d*g*m) + b*d*(d*g*(2*p + 3) - e*f*(m - 2*p - 4))) + e*(b^2*e*g*(m + p + 1) + 2*c^2*d*f*(m +
2*p + 2) - c*(2*a*e*g*m + b*(e*f + d*g)*(m + 2*p + 2)))*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && Ne
Q[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[p, -1] && GtQ[m, 1] && ((EqQ[m, 2] && EqQ[p, -3] &&
RationalQ[a, b, c, d, e, f, g]) ||  !ILtQ[m + 2*p + 3, 0])

Rule 824

Int[(((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_)))/((a_.) + (b_.)*(x_) + (c_.)*(x_)^2), x_Symbol] :> Simp[(g
*(d + e*x)^m)/(c*m), x] + Dist[1/c, Int[((d + e*x)^(m - 1)*Simp[c*d*f - a*e*g + (g*c*d - b*e*g + c*e*f)*x, x])
/(a + b*x + c*x^2), x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*
e^2, 0] && FractionQ[m] && GtQ[m, 0]

Rule 826

Int[((f_.) + (g_.)*(x_))/(Sqrt[(d_.) + (e_.)*(x_)]*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)), x_Symbol] :> Dist[2,
Subst[Int[(e*f - d*g + g*x^2)/(c*d^2 - b*d*e + a*e^2 - (2*c*d - b*e)*x^2 + c*x^4), x], x, Sqrt[d + e*x]], x] /
; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0]

Rule 1166

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] :> With[{q = Rt[b^2 - 4*a*c, 2]}, Di
st[e/2 + (2*c*d - b*e)/(2*q), Int[1/(b/2 - q/2 + c*x^2), x], x] + Dist[e/2 - (2*c*d - b*e)/(2*q), Int[1/(b/2 +
q/2 + c*x^2), x], x]] /; FreeQ[{a, b, c, d, e}, x] && NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - a*e^2, 0] && PosQ[b^
2 - 4*a*c]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rubi steps

\begin{align*} \int \frac{(A+B x) (d+e x)^{9/2}}{\left (b x+c x^2\right )^3} \, dx &=-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{\int \frac{(d+e x)^{5/2} \left (-\frac{1}{2} d \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\frac{1}{2} e \left (2 A c^2 d+5 b^2 B e-b c (B d+A e)\right ) x\right )}{\left (b x+c x^2\right )^2} \, dx}{2 b^2 c}\\ &=-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{3}{4} c^2 d^2 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )-\frac{3}{4} e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) x\right )}{b x+c x^2} \, dx}{2 b^4 c^2}\\ &=-\frac{3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt{d+e x}}{4 b^4 c^3}-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{\int \frac{\frac{3}{4} c^3 d^3 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )+\frac{3}{4} e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{2 b^4 c^3}\\ &=-\frac{3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt{d+e x}}{4 b^4 c^3}-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}+\frac{\operatorname{Subst}\left (\int \frac{-\frac{3}{4} d e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right )+\frac{3}{4} c^3 d^3 e \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )+\frac{3}{4} e \left (8 A c^5 d^4-5 b^5 B e^4+b^3 c^2 d e^2 (B d+A e)+b^4 c e^3 (7 B d+A e)-4 b c^4 d^3 (B d+4 A e)+b^2 c^3 d^2 e (5 B d+7 A e)\right ) x^2}{c d^2-b d e+(-2 c d+b e) x^2+c x^4} \, dx,x,\sqrt{d+e x}\right )}{b^4 c^3}\\ &=-\frac{3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt{d+e x}}{4 b^4 c^3}-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{\left (3 (c d-b e)^3 \left (16 A c^3 d^2-5 b^3 B e^2-4 b c^2 d (2 B d-A e)-b^2 c e (8 B d-A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5 c^3}+\frac{\left (3 c d^3 \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right )\right ) \operatorname{Subst}\left (\int \frac{1}{-\frac{b e}{2}+\frac{1}{2} (-2 c d+b e)+c x^2} \, dx,x,\sqrt{d+e x}\right )}{4 b^5}\\ &=-\frac{3 e \left (8 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (2 B d+A e)+b^2 c^2 d e (3 B d+2 A e)-4 b c^3 d^2 (B d+3 A e)\right ) \sqrt{d+e x}}{4 b^4 c^3}-\frac{(d+e x)^{7/2} \left (A b c d+\left (2 A c^2 d+b^2 B e-b c (B d+A e)\right ) x\right )}{2 b^2 c \left (b x+c x^2\right )^2}+\frac{(d+e x)^{3/2} \left (b c d^2 \left (12 A c^2 d+2 b^2 B e-b c (6 B d+13 A e)\right )+\left (24 A c^4 d^3-5 b^4 B e^3+b^3 c e^2 (4 B d+A e)-12 b c^3 d^2 (B d+3 A e)+b^2 c^2 d e (9 B d+10 A e)\right ) x\right )}{4 b^4 c^2 \left (b x+c x^2\right )}-\frac{3 d^{5/2} \left (16 A c^2 d^2+3 b^2 e (4 B d+7 A e)-4 b c d (2 B d+9 A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{4 b^5}+\frac{3 (c d-b e)^{5/2} \left (16 A c^3 d^2-5 b^3 B e^2-4 b c^2 d (2 B d-A e)-b^2 c e (8 B d-A e)\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{4 b^5 c^{7/2}}\\ \end{align*}

Mathematica [A]  time = 5.80614, size = 589, normalized size = 1.28 \[ \frac{\frac{(b+c x) \left ((b+c x) \left (945 c^{11/2} (c d-b e)^2 \left (\frac{2}{315} \sqrt{d+e x} \left (408 d^2 e^2 x^2+506 d^3 e x+563 d^4+185 d e^3 x^3+35 e^4 x^4\right )-2 d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right ) \left (3 b^2 e (7 A e+4 B d)-4 b c d (9 A e+2 B d)+16 A c^2 d^2\right )-6 c^2 d^2 \left (b^2 c e (A e-8 B d)+4 b c^2 d (A e-2 B d)+16 A c^3 d^2-5 b^3 B e^2\right ) \left (3 (c d-b e) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )+35 c^{9/2} (d+e x)^{9/2}\right )\right )-630 b c^{13/2} (d+e x)^{11/2} \left (-b^2 c d e (26 A e+9 B d)+b^3 e^2 (7 A e+4 B d)+12 b c^2 d^2 (3 A e+B d)-24 A c^3 d^3\right )\right )}{b^4 c^{11/2} d (c d-b e)^2}+\frac{630 c (d+e x)^{11/2} \left (b^2 (-e) (7 A e+4 B d)+b c d (17 A e+6 B d)-12 A c^2 d^2\right )}{b^2 d (b e-c d)}-\frac{630 (d+e x)^{11/2} (7 A b e-8 A c d+4 b B d)}{b d x}-\frac{1260 A (d+e x)^{11/2}}{x^2}}{2520 b d (b+c x)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[((A + B*x)*(d + e*x)^(9/2))/(b*x + c*x^2)^3,x]

[Out]

((630*c*(-12*A*c^2*d^2 - b^2*e*(4*B*d + 7*A*e) + b*c*d*(6*B*d + 17*A*e))*(d + e*x)^(11/2))/(b^2*d*(-(c*d) + b*
e)) - (1260*A*(d + e*x)^(11/2))/x^2 - (630*(4*b*B*d - 8*A*c*d + 7*A*b*e)*(d + e*x)^(11/2))/(b*d*x) + ((b + c*x
)*(-630*b*c^(13/2)*(-24*A*c^3*d^3 + 12*b*c^2*d^2*(B*d + 3*A*e) + b^3*e^2*(4*B*d + 7*A*e) - b^2*c*d*e*(9*B*d +
26*A*e))*(d + e*x)^(11/2) + (b + c*x)*(945*c^(11/2)*(c*d - b*e)^2*(16*A*c^2*d^2 + 3*b^2*e*(4*B*d + 7*A*e) - 4*
b*c*d*(2*B*d + 9*A*e))*((2*Sqrt[d + e*x]*(563*d^4 + 506*d^3*e*x + 408*d^2*e^2*x^2 + 185*d*e^3*x^3 + 35*e^4*x^4
))/315 - 2*d^(9/2)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]]) - 6*c^2*d^2*(16*A*c^3*d^2 - 5*b^3*B*e^2 + b^2*c*e*(-8*B*d +
A*e) + 4*b*c^2*d*(-2*B*d + A*e))*(35*c^(9/2)*(d + e*x)^(9/2) + 3*(c*d - b*e)*(15*c^(7/2)*(d + e*x)^(7/2) + 7*
(c*d - b*e)*(3*c^(5/2)*(d + e*x)^(5/2) + 5*(c*d - b*e)*(Sqrt[c]*Sqrt[d + e*x]*(4*c*d - 3*b*e + c*e*x) - 3*(c*d
- b*e)^(3/2)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))))/(b^4*c^(11/2)*d*(c*d - b*e)^2))/(2520*b*
d*(b + c*x)^2)

________________________________________________________________________________________

Maple [B]  time = 0.039, size = 1421, normalized size = 3.1 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x)

[Out]

15/2*e^4/b/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^2-3/4*e^6/c^2*b/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)-1/e*d^4/b^3/x^2*(e*x+
d)^(3/2)*B+7/4*e^6/c^3*b^2/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)-15/4*e^5/c^3*b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/
2)*c/((b*e-c*d)*c)^(1/2))*B+9/4*e^5/c^2*b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B+27*e*d^(7/2)/b^4*arctanh((e*x+d)^(1/2)
/d^(1/2))*A*c+15/4*e^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^3+3/4*e^4/b/(c*
e*x+b*e)^2*(e*x+d)^(3/2)*A*d-19/4*e^4/c/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d+3/4*e^3/b/(c*e*x+b*e)^2*(e*x+d)^(3/2)*
B*d^2+1/e*d^5/b^3/x^2*(e*x+d)^(1/2)*B+21/4*e^3/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1
/2))*A*d^2+5/2*e^4/c/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^2+5*e^3/b/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^3+21/4*e^4/c^2/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d+6*c^2/b^4/((b*e-c*d)*c)^(1/2)*arctan((e*x+
d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^5-12*c^3/b^5/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2
))*A*d^5+2*e^4*B/c^3*(e*x+d)^(1/2)+15/4*d^4/b^3/x^2*(e*x+d)^(1/2)*A-17/4*d^3/b^3/x^2*(e*x+d)^(3/2)*A+6*d^(9/2)
/b^4*arctanh((e*x+d)^(1/2)/d^(1/2))*B*c-12*d^(9/2)/b^5*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c^2-5/4*e^5/c/(c*e*x+b
*e)^2*(e*x+d)^(3/2)*A+3/4*e^5/c^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A-63/4*e^2*d
^(5/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A-9*e*d^(7/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*B+3/4*e^3/c/b/((b*e
-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^2-12*e*c/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)
^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^4+21/4*e^3*c/b^2/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d^2-31/4*e^2*c^2/b^3/(c*e*x+b
*e)^2*(e*x+d)^(3/2)*A*d^3+3*e*c^3/b^4/(c*e*x+b*e)^2*(e*x+d)^(3/2)*A*d^4+15/4*e^2*c/b^2/(c*e*x+b*e)^2*(e*x+d)^(
3/2)*B*d^3-2*e*c^2/b^3/(c*e*x+b*e)^2*(e*x+d)^(3/2)*B*d^4-15*e^3*c/b^2/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^3+45/4*e
^2*c^2/b^3/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^4-3*e*c^3/b^4/(c*e*x+b*e)^2*A*(e*x+d)^(1/2)*d^5-5*e^5/c^2*b/(c*e*x+
b*e)^2*B*(e*x+d)^(1/2)*d+3/e*d^4/b^4/x^2*(e*x+d)^(3/2)*A*c-3/e*d^5/b^4/x^2*(e*x+d)^(1/2)*A*c-25/4*e^2*c/b^2/(c
*e*x+b*e)^2*B*(e*x+d)^(1/2)*d^4+3/4*e^4/c/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*
d-111/4*e^2*c/b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^3+33*e*c^2/b^4/((b*e-c*d
)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^4+2*e*c^2/b^3/(c*e*x+b*e)^2*B*(e*x+d)^(1/2)*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)^(9/2)/(c*x^2+b*x)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

________________________________________________________________________________________

Fricas [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(9/2)/(c*x^2+b*x)^3,x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(9/2)/(c*x**2+b*x)**3,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.51915, size = 1681, 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)^(9/2)/(c*x^2+b*x)^3,x, algorithm="giac")

[Out]

2*sqrt(x*e + d)*B*e^4/c^3 - 3/4*(8*B*b*c*d^5 - 16*A*c^2*d^5 - 12*B*b^2*d^4*e + 36*A*b*c*d^4*e - 21*A*b^2*d^3*e
^2)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^5*sqrt(-d)) + 3/4*(8*B*b*c^5*d^5 - 16*A*c^6*d^5 - 16*B*b^2*c^4*d^4*e + 4
4*A*b*c^5*d^4*e + 5*B*b^3*c^3*d^3*e^2 - 37*A*b^2*c^4*d^3*e^2 + B*b^4*c^2*d^2*e^3 + 7*A*b^3*c^3*d^2*e^3 + 7*B*b
^5*c*d*e^4 + A*b^4*c^2*d*e^4 - 5*B*b^6*e^5 + A*b^5*c*e^5)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/(sqrt(-
c^2*d + b*c*e)*b^5*c^3) - 1/4*(12*(x*e + d)^(7/2)*B*b*c^5*d^4*e - 24*(x*e + d)^(7/2)*A*c^6*d^4*e - 36*(x*e + d
)^(5/2)*B*b*c^5*d^5*e + 72*(x*e + d)^(5/2)*A*c^6*d^5*e + 36*(x*e + d)^(3/2)*B*b*c^5*d^6*e - 72*(x*e + d)^(3/2)
*A*c^6*d^6*e - 12*sqrt(x*e + d)*B*b*c^5*d^7*e + 24*sqrt(x*e + d)*A*c^6*d^7*e - 15*(x*e + d)^(7/2)*B*b^2*c^4*d^
3*e^2 + 48*(x*e + d)^(7/2)*A*b*c^5*d^3*e^2 + 63*(x*e + d)^(5/2)*B*b^2*c^4*d^4*e^2 - 180*(x*e + d)^(5/2)*A*b*c^
5*d^4*e^2 - 81*(x*e + d)^(3/2)*B*b^2*c^4*d^5*e^2 + 216*(x*e + d)^(3/2)*A*b*c^5*d^5*e^2 + 33*sqrt(x*e + d)*B*b^
2*c^4*d^6*e^2 - 84*sqrt(x*e + d)*A*b*c^5*d^6*e^2 - 3*(x*e + d)^(7/2)*B*b^3*c^3*d^2*e^3 - 21*(x*e + d)^(7/2)*A*
b^2*c^4*d^2*e^3 - 14*(x*e + d)^(5/2)*B*b^3*c^3*d^3*e^3 + 136*(x*e + d)^(5/2)*A*b^2*c^4*d^3*e^3 + 41*(x*e + d)^
(3/2)*B*b^3*c^3*d^4*e^3 - 217*(x*e + d)^(3/2)*A*b^2*c^4*d^4*e^3 - 24*sqrt(x*e + d)*B*b^3*c^3*d^5*e^3 + 102*sqr
t(x*e + d)*A*b^2*c^4*d^5*e^3 + 19*(x*e + d)^(7/2)*B*b^4*c^2*d*e^4 - 3*(x*e + d)^(7/2)*A*b^3*c^3*d*e^4 - 48*(x*
e + d)^(5/2)*B*b^4*c^2*d^2*e^4 - 24*(x*e + d)^(5/2)*A*b^3*c^3*d^2*e^4 + 39*(x*e + d)^(3/2)*B*b^4*c^2*d^3*e^4 +
74*(x*e + d)^(3/2)*A*b^3*c^3*d^3*e^4 - 10*sqrt(x*e + d)*B*b^4*c^2*d^4*e^4 - 45*sqrt(x*e + d)*A*b^3*c^3*d^4*e^
4 - 9*(x*e + d)^(7/2)*B*b^5*c*e^5 + 5*(x*e + d)^(7/2)*A*b^4*c^2*e^5 + 38*(x*e + d)^(5/2)*B*b^5*c*d*e^5 - 10*(x
*e + d)^(5/2)*A*b^4*c^2*d*e^5 - 49*(x*e + d)^(3/2)*B*b^5*c*d^2*e^5 + 5*(x*e + d)^(3/2)*A*b^4*c^2*d^2*e^5 + 20*
sqrt(x*e + d)*B*b^5*c*d^3*e^5 - 7*(x*e + d)^(5/2)*B*b^6*e^6 + 3*(x*e + d)^(5/2)*A*b^5*c*e^6 + 14*(x*e + d)^(3/
2)*B*b^6*d*e^6 - 6*(x*e + d)^(3/2)*A*b^5*c*d*e^6 - 7*sqrt(x*e + d)*B*b^6*d^2*e^6 + 3*sqrt(x*e + d)*A*b^5*c*d^2
*e^6)/(((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e + d)*b*e - b*d*e)^2*b^4*c^3)