3.1246 \(\int \frac{A+B x}{(d+e x)^{7/2} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=457 \[ -\frac{e \left (-2 b^2 c^2 d^2 e (6 B d-13 A e)+8 b^3 c d e^2 (B d-3 A e)+b^4 \left (-e^3\right ) (2 B d-7 A e)-b c^3 d^3 (4 A e+B d)+2 A c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{e \left (-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (b^2 (-e) (2 B d-7 A e)-5 b c d (2 A e+B d)+10 A c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{c^{7/2} \left (11 A b c e-4 A c^2 d-9 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)} \]

[Out]

-(e*(10*A*c^2*d^2 - b^2*e*(2*B*d - 7*A*e) - 5*b*c*d*(B*d + 2*A*e)))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))
- (e*(6*A*c^3*d^3 - b^2*c*d*e*(6*B*d - 17*A*e) + b^3*e^2*(2*B*d - 7*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e)))/(3*b^2*
d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*A*c^4*d^4 - 2*b^2*c^2*d^2*e*(6*B*d - 13*A*e) - b^4*e^3*(2*B*d - 7*A
*e) + 8*b^3*c*d*e^2*(B*d - 3*A*e) - b*c^3*d^3*(B*d + 4*A*e)))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*
d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) - ((2*b*B*d - 4*A*
c*d - 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) + (c^(7/2)*(2*b*B*c*d - 4*A*c^2*d - 9*b^2*B*e + 1
1*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(9/2))

________________________________________________________________________________________

Rubi [A]  time = 1.28271, antiderivative size = 457, 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 = {822, 828, 826, 1166, 208} \[ -\frac{e \left (-2 b^2 c^2 d^2 e (6 B d-13 A e)+8 b^3 c d e^2 (B d-3 A e)+b^4 \left (-e^3\right ) (2 B d-7 A e)-b c^3 d^3 (4 A e+B d)+2 A c^4 d^4\right )}{b^2 d^4 \sqrt{d+e x} (c d-b e)^4}-\frac{e \left (-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (3 A e+B d)+6 A c^3 d^3\right )}{3 b^2 d^3 (d+e x)^{3/2} (c d-b e)^3}-\frac{e \left (b^2 (-e) (2 B d-7 A e)-5 b c d (2 A e+B d)+10 A c^2 d^2\right )}{5 b^2 d^2 (d+e x)^{5/2} (c d-b e)^2}+\frac{c^{7/2} \left (11 A b c e-4 A c^2 d-9 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (-7 A b e-4 A c d+2 b B d)}{b^3 d^{9/2}}-\frac{c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) (d+e x)^{5/2} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

-(e*(10*A*c^2*d^2 - b^2*e*(2*B*d - 7*A*e) - 5*b*c*d*(B*d + 2*A*e)))/(5*b^2*d^2*(c*d - b*e)^2*(d + e*x)^(5/2))
- (e*(6*A*c^3*d^3 - b^2*c*d*e*(6*B*d - 17*A*e) + b^3*e^2*(2*B*d - 7*A*e) - 3*b*c^2*d^2*(B*d + 3*A*e)))/(3*b^2*
d^3*(c*d - b*e)^3*(d + e*x)^(3/2)) - (e*(2*A*c^4*d^4 - 2*b^2*c^2*d^2*e*(6*B*d - 13*A*e) - b^4*e^3*(2*B*d - 7*A
*e) + 8*b^3*c*d*e^2*(B*d - 3*A*e) - b*c^3*d^3*(B*d + 4*A*e)))/(b^2*d^4*(c*d - b*e)^4*Sqrt[d + e*x]) - (A*b*(c*
d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x)/(b^2*d*(c*d - b*e)*(d + e*x)^(5/2)*(b*x + c*x^2)) - ((2*b*B*d - 4*A*
c*d - 7*A*b*e)*ArcTanh[Sqrt[d + e*x]/Sqrt[d]])/(b^3*d^(9/2)) + (c^(7/2)*(2*b*B*c*d - 4*A*c^2*d - 9*b^2*B*e + 1
1*A*b*c*e)*ArcTanh[(Sqrt[c]*Sqrt[d + e*x])/Sqrt[c*d - b*e]])/(b^3*(c*d - b*e)^(9/2))

Rule 822

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

Rule 828

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

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)^{7/2} \left (b x+c x^2\right )^2} \, dx &=-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{2} (c d-b e) (2 b B d-4 A c d-7 A b e)-\frac{7}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{7/2} \left (b x+c x^2\right )} \, dx}{b^2 d (c d-b e)}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{2} (c d-b e)^2 (2 b B d-4 A c d-7 A b e)+\frac{1}{2} c e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right ) x}{(d+e x)^{5/2} \left (b x+c x^2\right )} \, dx}{b^2 d^2 (c d-b e)^2}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{2} (c d-b e)^3 (2 b B d-4 A c d-7 A b e)+\frac{1}{2} c e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{3/2} \left (b x+c x^2\right )} \, dx}{b^2 d^3 (c d-b e)^3}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{\int \frac{-\frac{1}{2} (c d-b e)^4 (2 b B d-4 A c d-7 A b e)+\frac{1}{2} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right ) x}{\sqrt{d+e x} \left (b x+c x^2\right )} \, dx}{b^2 d^4 (c d-b e)^4}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{2 \operatorname{Subst}\left (\int \frac{-\frac{1}{2} e (c d-b e)^4 (2 b B d-4 A c d-7 A b e)-\frac{1}{2} c d e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )+\frac{1}{2} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 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 d^4 (c d-b e)^4}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}+\frac{(c (2 b B d-4 A c d-7 A b e)) \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 d^4}-\frac{\left (2 \left (\frac{1}{4} c e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )-\frac{-\frac{1}{2} c e (-2 c d+b e) \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )+2 c \left (-\frac{1}{2} e (c d-b e)^4 (2 b B d-4 A c d-7 A b e)-\frac{1}{2} c d e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 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 d^4 (c d-b e)^4}\\ &=-\frac{e \left (10 A c^2 d^2-b^2 e (2 B d-7 A e)-5 b c d (B d+2 A e)\right )}{5 b^2 d^2 (c d-b e)^2 (d+e x)^{5/2}}-\frac{e \left (6 A c^3 d^3-b^2 c d e (6 B d-17 A e)+b^3 e^2 (2 B d-7 A e)-3 b c^2 d^2 (B d+3 A e)\right )}{3 b^2 d^3 (c d-b e)^3 (d+e x)^{3/2}}-\frac{e \left (2 A c^4 d^4-2 b^2 c^2 d^2 e (6 B d-13 A e)-b^4 e^3 (2 B d-7 A e)+8 b^3 c d e^2 (B d-3 A e)-b c^3 d^3 (B d+4 A e)\right )}{b^2 d^4 (c d-b e)^4 \sqrt{d+e x}}-\frac{A b (c d-b e)+c (2 A c d-b (B d+A e)) x}{b^2 d (c d-b e) (d+e x)^{5/2} \left (b x+c x^2\right )}-\frac{(2 b B d-4 A c d-7 A b e) \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )}{b^3 d^{9/2}}+\frac{c^{7/2} \left (2 b B c d-4 A c^2 d-9 b^2 B e+11 A b c e\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 (c d-b e)^{9/2}}\\ \end{align*}

Mathematica [C]  time = 0.190481, size = 194, normalized size = 0.42 \[ \frac{-x (b+c x) \left (c d^2 \left (b c (11 A e+2 B d)-4 A c^2 d-9 b^2 B e\right ) \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{c (d+e x)}{c d-b e}\right )+(c d-b e)^2 \, _2F_1\left (-\frac{5}{2},1;-\frac{3}{2};\frac{e x}{d}+1\right ) (7 A b e+4 A c d-2 b B d)\right )-5 A b^2 d (c d-b e)^2-5 b c d x (b e-c d) (A b e-2 A c d+b B d)}{5 b^3 d^2 x (b+c x) (d+e x)^{5/2} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(7/2)*(b*x + c*x^2)^2),x]

[Out]

(-5*A*b^2*d*(c*d - b*e)^2 - 5*b*c*d*(-(c*d) + b*e)*(b*B*d - 2*A*c*d + A*b*e)*x - x*(b + c*x)*(c*d^2*(-4*A*c^2*
d - 9*b^2*B*e + b*c*(2*B*d + 11*A*e))*Hypergeometric2F1[-5/2, 1, -3/2, (c*(d + e*x))/(c*d - b*e)] + (c*d - b*e
)^2*(-2*b*B*d + 4*A*c*d + 7*A*b*e)*Hypergeometric2F1[-5/2, 1, -3/2, 1 + (e*x)/d]))/(5*b^3*d^2*(c*d - b*e)^2*x*
(b + c*x)*(d + e*x)^(5/2))

________________________________________________________________________________________

Maple [A]  time = 0.04, size = 707, normalized size = 1.6 \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)^(7/2)/(c*x^2+b*x)^2,x)

[Out]

-2/5*e^3/d^2/(b*e-c*d)^2/(e*x+d)^(5/2)*A+2/5*e^2/d/(b*e-c*d)^2/(e*x+d)^(5/2)*B-4/3*e^4/d^3/(b*e-c*d)^3/(e*x+d)
^(3/2)*A*b+8/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*A*c+2/3*e^3/d^2/(b*e-c*d)^3/(e*x+d)^(3/2)*B*b-2*e^2/d/(b*e-c*
d)^3/(e*x+d)^(3/2)*B*c-6*e^5/d^4/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b^2+20*e^4/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*A*b*c-20
*e^3/d^2/(b*e-c*d)^4/(e*x+d)^(1/2)*A*c^2+2*e^4/d^3/(b*e-c*d)^4/(e*x+d)^(1/2)*B*b^2-8*e^3/d^2/(b*e-c*d)^4/(e*x+
d)^(1/2)*B*b*c+12*e^2/d/(b*e-c*d)^4/(e*x+d)^(1/2)*B*c^2-1/b^2/d^4*A*(e*x+d)^(1/2)/x+7*e/b^2/d^(9/2)*arctanh((e
*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(7/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2/b^2/d^(7/2)*arctanh((e*x+d)^(1/2)/d^
(1/2))*B-e*c^5/(b*e-c*d)^4/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e*c^4/(b*e-c*d)^4/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B-11*
e*c^5/(b*e-c*d)^4/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*A+4*c^6/(b*e-c*d)^4/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+9*e*c^4/(b*e-c*d)^4/b/((b*e-c*d)*c)^(1/2)*
arctan((e*x+d)^(1/2)*c/((b*e-c*d)*c)^(1/2))*B-2*c^5/(b*e-c*d)^4/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

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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)^(7/2)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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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)^(7/2)/(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)**(7/2)/(c*x**2+b*x)**2,x)

[Out]

Timed out

________________________________________________________________________________________

Giac [B]  time = 1.43881, size = 1170, normalized size = 2.56 \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)^(7/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

-(2*B*b*c^5*d - 4*A*c^6*d - 9*B*b^2*c^4*e + 11*A*b*c^5*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c
^4*d^4 - 4*b^4*c^3*d^3*e + 6*b^5*c^2*d^2*e^2 - 4*b^6*c*d*e^3 + b^7*e^4)*sqrt(-c^2*d + b*c*e)) + ((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)*A*b*c^4*d^3*e^2 - 5*sqrt(x*e + d)*A*b*c^4*d^4*e^2 - 6*(x*e + d)^(3/2)*A*b^2*c^3*d^2*e^3 + 1
0*sqrt(x*e + d)*A*b^2*c^3*d^3*e^3 + 4*(x*e + d)^(3/2)*A*b^3*c^2*d*e^4 - 10*sqrt(x*e + d)*A*b^3*c^2*d^2*e^4 - (
x*e + d)^(3/2)*A*b^4*c*e^5 + 5*sqrt(x*e + d)*A*b^4*c*d*e^5 - sqrt(x*e + d)*A*b^5*e^6)/((b^2*c^4*d^8 - 4*b^3*c^
3*d^7*e + 6*b^4*c^2*d^6*e^2 - 4*b^5*c*d^5*e^3 + b^6*d^4*e^4)*((x*e + d)^2*c - 2*(x*e + d)*c*d + c*d^2 + (x*e +
d)*b*e - b*d*e)) + 2/15*(90*(x*e + d)^2*B*c^2*d^3*e^2 + 15*(x*e + d)*B*c^2*d^4*e^2 + 3*B*c^2*d^5*e^2 - 60*(x*
e + d)^2*B*b*c*d^2*e^3 - 150*(x*e + d)^2*A*c^2*d^2*e^3 - 20*(x*e + d)*B*b*c*d^3*e^3 - 20*(x*e + d)*A*c^2*d^3*e
^3 - 6*B*b*c*d^4*e^3 - 3*A*c^2*d^4*e^3 + 15*(x*e + d)^2*B*b^2*d*e^4 + 150*(x*e + d)^2*A*b*c*d*e^4 + 5*(x*e + d
)*B*b^2*d^2*e^4 + 30*(x*e + d)*A*b*c*d^2*e^4 + 3*B*b^2*d^3*e^4 + 6*A*b*c*d^3*e^4 - 45*(x*e + d)^2*A*b^2*e^5 -
10*(x*e + d)*A*b^2*d*e^5 - 3*A*b^2*d^2*e^5)/((c^4*d^8 - 4*b*c^3*d^7*e + 6*b^2*c^2*d^6*e^2 - 4*b^3*c*d^5*e^3 +
b^4*d^4*e^4)*(x*e + d)^(5/2)) + (2*B*b*d - 4*A*c*d - 7*A*b*e)*arctan(sqrt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^4
)