3.1238 \(\int \frac{(A+B x) (d+e x)^{9/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=386 \[ \frac{e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac{e \sqrt{d+e x} \left (b^2 c^2 d e (11 A e+15 B d)-b^3 c e^2 (5 A e+19 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4}-\frac{(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac{e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}-\frac{(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}}-\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3} \]

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*A*e) + b^2*c^2*d*e*(15*B*d + 1
1*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2 - 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d +
5*A*e))*(d + e*x)^(3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5/2))/(5*b^
2*c^2) - ((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))/(b^2*c*(b*x + c*x^2)) - (d^(7
/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*
B*e - b*c*(2*B*d - 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.1937, antiderivative size = 386, normalized size of antiderivative = 1., number of steps used = 8, 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} \[ \frac{e (d+e x)^{3/2} \left (b^2 c e (5 A e+12 B d)-3 b c^2 d (2 A e+B d)+6 A c^3 d^2-7 b^3 B e^2\right )}{3 b^2 c^3}+\frac{e \sqrt{d+e x} \left (b^2 c^2 d e (11 A e+15 B d)-b^3 c e^2 (5 A e+19 B d)-b c^3 d^2 (3 A e+B d)+2 A c^4 d^3+7 b^4 B e^3\right )}{b^2 c^4}-\frac{(d+e x)^{7/2} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}+\frac{e (d+e x)^{5/2} \left (-5 b c (A e+B d)+10 A c^2 d+7 b^2 B e\right )}{5 b^2 c^2}-\frac{(c d-b e)^{7/2} \left (-b c (2 B d-5 A e)+4 A c^2 d-7 b^2 B e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}}-\frac{d^{7/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (9 A b e-4 A c d+2 b B d)}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(e*(2*A*c^4*d^3 + 7*b^4*B*e^3 - b*c^3*d^2*(B*d + 3*A*e) - b^3*c*e^2*(19*B*d + 5*A*e) + b^2*c^2*d*e*(15*B*d + 1
1*A*e))*Sqrt[d + e*x])/(b^2*c^4) + (e*(6*A*c^3*d^2 - 7*b^3*B*e^2 - 3*b*c^2*d*(B*d + 2*A*e) + b^2*c*e*(12*B*d +
5*A*e))*(d + e*x)^(3/2))/(3*b^2*c^3) + (e*(10*A*c^2*d + 7*b^2*B*e - 5*b*c*(B*d + A*e))*(d + e*x)^(5/2))/(5*b^
2*c^2) - ((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))/(b^2*c*(b*x + c*x^2)) - (d^(7
/2)*(2*b*B*d - 4*A*c*d + 9*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/b^3 - ((c*d - b*e)^(7/2)*(4*A*c^2*d - 7*b^2*
B*e - b*c*(2*B*d - 5*A*e))*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*c^(9/2))

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 )^2} \, 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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{(d+e x)^{5/2} \left (\frac{1}{2} c d (2 b B d-4 A c d+9 A b e)+\frac{1}{2} e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c}\\ &=\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{(d+e x)^{3/2} \left (\frac{1}{2} c^2 d^2 (2 b B d-4 A c d+9 A b e)+\frac{1}{2} e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^2}\\ &=\frac{e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{\sqrt{d+e x} \left (\frac{1}{2} c^3 d^3 (2 b B d-4 A c d+9 A b e)+\frac{1}{2} e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) x\right )}{b x+c x^2} \, dx}{b^2 c^3}\\ &=\frac{e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt{d+e x}}{b^2 c^4}+\frac{e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\int \frac{\frac{1}{2} c^4 d^4 (2 b B d-4 A c d+9 A b e)-\frac{1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 c^4}\\ &=\frac{e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt{d+e x}}{b^2 c^4}+\frac{e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}+\frac{2 \operatorname{Subst}\left (\int \frac{\frac{1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac{1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )-\frac{1}{2} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 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^2 c^4}\\ &=\frac{e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt{d+e x}}{b^2 c^4}+\frac{e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}+\frac{\left (c d^4 (2 b B d-4 A c d+9 A b e)\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 )}{b^3}-\frac{\left (2 \left (\frac{1}{4} e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+\frac{\frac{1}{2} e (-2 c d+b e) \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )+2 c \left (\frac{1}{2} c^4 d^4 e (2 b B d-4 A c d+9 A b e)+\frac{1}{2} d e \left (2 A c^5 d^4+7 b^5 B e^4-b c^4 d^3 (B d+4 A e)-b^4 c e^3 (26 B d+5 A e)-2 b^2 c^3 d^2 e (8 B d+7 A e)+2 b^3 c^2 d e^2 (17 B d+8 A e)\right )\right )}{2 b 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 )}{b^2 c^4}\\ &=\frac{e \left (2 A c^4 d^3+7 b^4 B e^3-b c^3 d^2 (B d+3 A e)-b^3 c e^2 (19 B d+5 A e)+b^2 c^2 d e (15 B d+11 A e)\right ) \sqrt{d+e x}}{b^2 c^4}+\frac{e \left (6 A c^3 d^2-7 b^3 B e^2-3 b c^2 d (B d+2 A e)+b^2 c e (12 B d+5 A e)\right ) (d+e x)^{3/2}}{3 b^2 c^3}+\frac{e \left (10 A c^2 d+7 b^2 B e-5 b c (B d+A e)\right ) (d+e x)^{5/2}}{5 b^2 c^2}-\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 )}{b^2 c \left (b x+c x^2\right )}-\frac{d^{7/2} (2 b B d-4 A c d+9 A b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3}+\frac{(c d-b e)^{7/2} \left (2 b B c d-4 A c^2 d+7 b^2 B e-5 A b c e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 c^{9/2}}\\ \end{align*}

Mathematica [A]  time = 2.61326, size = 376, normalized size = 0.97 \[ -\frac{-\frac{315 \left (\frac{2}{315} \sqrt{d+e x} \left (408 d^2 e^2 x^2+506 d^3 e x+563 d^4+185 d e^3 x^3+35 e^4 x^4\right )-2 d^{9/2} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right ) (9 A b e-4 A c d+2 b B d)-\frac{2 d \left (b c (2 B d-5 A e)-4 A c^2 d+7 b^2 B e\right ) \left (3 (c d-b e) \left (7 (c d-b e) \left (5 (c d-b e) \left (\sqrt{c} \sqrt{d+e x} (-3 b e+4 c d+c e x)-3 (c d-b e)^{3/2} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )+3 c^{5/2} (d+e x)^{5/2}\right )+15 c^{7/2} (d+e x)^{7/2}\right )+35 c^{9/2} (d+e x)^{9/2}\right )}{c^{9/2} (c d-b e)}}{630 b^2}+\frac{c (d+e x)^{11/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}+\frac{A (d+e x)^{11/2}}{x (b+c x)}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(((c*(b*B*d - 2*A*c*d + A*b*e)*(d + e*x)^(11/2))/(b*(-(c*d) + b*e)*(b + c*x)) + (A*(d + e*x)^(11/2))/(x*(b +
c*x)) - (315*(2*b*B*d - 4*A*c*d + 9*A*b*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]]) - (2*d*(-4*A*c^2*d + 7*b^2*B*e + b*c*(2
*B*d - 5*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)*ArcTa
nh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])))))/(c^(9/2)*(c*d - b*e)))/(630*b^2))/(b*d))

________________________________________________________________________________________

Maple [B]  time = 0.035, size = 1075, normalized size = 2.8 \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)^2,x)

[Out]

-e^5/c^3*b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e^5/c^4*b^3*(e*x+d)^(1/2)/(c*e*x+b*e)*B+5*e^5/c^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-7*e^5/c^4*b^3/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((
b*e-c*d)*c)^(1/2))*B-16*e^3/c^3*B*b*d*(e*x+d)^(1/2)-6*e^3/c*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d^2+4*e^2/b*(e*x+d)^(1
/2)/(c*e*x+b*e)*A*d^3-4*e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^3-2*c/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^5+4*c^2/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^5+
e/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^4+14*e^3/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A
*d^2+4*e^2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d^3+16*e^2/c/((b*e-c*d)*c)^(1/2
)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B*d^3+e/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*
c)^(1/2))*B*d^4+2/3*e^3/c^2*A*(e*x+d)^(3/2)+2/5*e^2/c^2*B*(e*x+d)^(5/2)-2*d^(9/2)/b^2*arctanh((e*x+d)^(1/2)/d^
(1/2))*B-d^4/b^2*A*(e*x+d)^(1/2)/x+2*e^2/c^2*B*(e*x+d)^(3/2)*d+12*e^2/c^2*B*d^2*(e*x+d)^(1/2)-4/3*e^3/c^3*B*(e
*x+d)^(3/2)*b-4*e^4/c^3*A*b*(e*x+d)^(1/2)+8*e^3/c^2*A*d*(e*x+d)^(1/2)+6*e^4/c^4*B*b^2*(e*x+d)^(1/2)-9*e*d^(7/2
)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4*d^(9/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c+26*e^4/c^3*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-34*e^3/c^2*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+4*e^4/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)*A*d-e*c/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*
A*d^4-4*e^4/c^3*b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d+6*e^3/c^2*b*(e*x+d)^(1/2)/(c*e*x+b*e)*B*d^2-16*e^4/c^2*b/((b
*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A*d-11*e*c/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^4

________________________________________________________________________________________

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)^2,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)^2,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)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.44233, size = 1142, normalized size = 2.96 \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)^2,x, algorithm="giac")

[Out]

(2*B*b*d^5 - 4*A*c*d^5 + 9*A*b*d^4*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)) - (2*B*b*c^5*d^5 - 4*A*c^6
*d^5 - B*b^2*c^4*d^4*e + 11*A*b*c^5*d^4*e - 16*B*b^3*c^3*d^3*e^2 - 4*A*b^2*c^4*d^3*e^2 + 34*B*b^4*c^2*d^2*e^3
- 14*A*b^3*c^3*d^2*e^3 - 26*B*b^5*c*d*e^4 + 16*A*b^4*c^2*d*e^4 + 7*B*b^6*e^5 - 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^3*c^4) + ((x*e + d)^(3/2)*B*b*c^4*d^4*e - 2*(x*e + d)^(3/
2)*A*c^5*d^4*e - sqrt(x*e + d)*B*b*c^4*d^5*e + 2*sqrt(x*e + d)*A*c^5*d^5*e - 4*(x*e + d)^(3/2)*B*b^2*c^3*d^3*e
^2 + 4*(x*e + d)^(3/2)*A*b*c^4*d^3*e^2 + 4*sqrt(x*e + d)*B*b^2*c^3*d^4*e^2 - 5*sqrt(x*e + d)*A*b*c^4*d^4*e^2 +
6*(x*e + d)^(3/2)*B*b^3*c^2*d^2*e^3 - 6*(x*e + d)^(3/2)*A*b^2*c^3*d^2*e^3 - 6*sqrt(x*e + d)*B*b^3*c^2*d^3*e^3
+ 6*sqrt(x*e + d)*A*b^2*c^3*d^3*e^3 - 4*(x*e + d)^(3/2)*B*b^4*c*d*e^4 + 4*(x*e + d)^(3/2)*A*b^3*c^2*d*e^4 + 4
*sqrt(x*e + d)*B*b^4*c*d^2*e^4 - 4*sqrt(x*e + d)*A*b^3*c^2*d^2*e^4 + (x*e + d)^(3/2)*B*b^5*e^5 - (x*e + d)^(3/
2)*A*b^4*c*e^5 - sqrt(x*e + d)*B*b^5*d*e^5 + sqrt(x*e + d)*A*b^4*c*d*e^5)/(((x*e + d)^2*c - 2*(x*e + d)*c*d +
c*d^2 + (x*e + d)*b*e - b*d*e)*b^2*c^4) + 2/15*(3*(x*e + d)^(5/2)*B*c^8*e^2 + 15*(x*e + d)^(3/2)*B*c^8*d*e^2 +
90*sqrt(x*e + d)*B*c^8*d^2*e^2 - 10*(x*e + d)^(3/2)*B*b*c^7*e^3 + 5*(x*e + d)^(3/2)*A*c^8*e^3 - 120*sqrt(x*e
+ d)*B*b*c^7*d*e^3 + 60*sqrt(x*e + d)*A*c^8*d*e^3 + 45*sqrt(x*e + d)*B*b^2*c^6*e^4 - 30*sqrt(x*e + d)*A*b*c^7*
e^4)/c^10