3.1212 \(\int \frac{A+B x}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=449 \[ \frac{e \sqrt{b x+c x^2} \left (4 b^2 c^2 d^2 e (3 A e+10 B d)-2 b^3 c d e^2 (9 B d-10 A e)+3 b^4 e^3 (3 B d-5 A e)-16 b c^3 d^3 (4 A e+B d)+32 A c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{2 \left (c x \left (2 b^2 c d e (8 B d-A e)+b^3 \left (-e^2\right ) (3 B d-5 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (3 B d-5 A e)-2 b c d (A e+2 B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}-\frac{e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c*x^2)^(3/2)) + (
2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 5*A*e) - 2*b*c*d*(2*B*d + A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*
B*d - 5*A*e) + 2*b^2*c*d*e*(8*B*d - A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*(d + e*x)*S
qrt[b*x + c*x^2]) + (e*(32*A*c^4*d^4 - 2*b^3*c*d*e^2*(9*B*d - 10*A*e) + 3*b^4*e^3*(3*B*d - 5*A*e) + 4*b^2*c^2*
d^2*e*(10*B*d + 3*A*e) - 16*b*c^3*d^3*(B*d + 4*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) -
(e^3*(B*d*(8*c*d - 3*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sq
rt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))

________________________________________________________________________________________

Rubi [A]  time = 0.720008, antiderivative size = 449, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {822, 806, 724, 206} \[ \frac{e \sqrt{b x+c x^2} \left (4 b^2 c^2 d^2 e (3 A e+10 B d)-2 b^3 c d e^2 (9 B d-10 A e)+3 b^4 e^3 (3 B d-5 A e)-16 b c^3 d^3 (4 A e+B d)+32 A c^4 d^4\right )}{3 b^4 d^3 (d+e x) (c d-b e)^3}+\frac{2 \left (c x \left (2 b^2 c d e (8 B d-A e)+b^3 \left (-e^2\right ) (3 B d-5 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (b^2 e (3 B d-5 A e)-2 b c d (A e+2 B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (d+e x) (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (d+e x) (c d-b e)}-\frac{e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{2 d^{7/2} (c d-b e)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(-2*(A*b*(c*d - b*e) + c*(2*A*c*d - b*(B*d + A*e))*x))/(3*b^2*d*(c*d - b*e)*(d + e*x)*(b*x + c*x^2)^(3/2)) + (
2*(b*(c*d - b*e)*(8*A*c^2*d^2 + b^2*e*(3*B*d - 5*A*e) - 2*b*c*d*(2*B*d + A*e)) + c*(16*A*c^3*d^3 - b^3*e^2*(3*
B*d - 5*A*e) + 2*b^2*c*d*e*(8*B*d - A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x))/(3*b^4*d^2*(c*d - b*e)^2*(d + e*x)*S
qrt[b*x + c*x^2]) + (e*(32*A*c^4*d^4 - 2*b^3*c*d*e^2*(9*B*d - 10*A*e) + 3*b^4*e^3*(3*B*d - 5*A*e) + 4*b^2*c^2*
d^2*e*(10*B*d + 3*A*e) - 16*b*c^3*d^3*(B*d + 4*A*e))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*(d + e*x)) -
(e^3*(B*d*(8*c*d - 3*b*e) - 5*A*e*(2*c*d - b*e))*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sq
rt[b*x + c*x^2])])/(2*d^(7/2)*(c*d - b*e)^(7/2))

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 806

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

Rule 724

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

Rule 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{A+B x}{(d+e x)^2 \left (b x+c x^2\right )^{5/2}} \, dx &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 A c^2 d^2+b^2 e (3 B d-5 A e)-2 b c d (2 B d+A e)\right )-3 c e (b B d-2 A c d+A b e) x}{(d+e x)^2 \left (b x+c x^2\right )^{3/2}} \, dx}{3 b^2 d (c d-b e)}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-5 A e)-2 b c d (2 B d+A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{4 \int \frac{\frac{1}{4} b e \left (16 A c^3 d^3-3 b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (6 B d-5 A e)-8 b c^2 d^2 (B d+2 A e)\right )+\frac{1}{2} c e \left (16 A c^3 d^3-b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^2 \sqrt{b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-5 A e)-2 b c d (2 B d+A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 A c^4 d^4-2 b^3 c d e^2 (9 B d-10 A e)+3 b^4 e^3 (3 B d-5 A e)+4 b^2 c^2 d^2 e (10 B d+3 A e)-16 b c^3 d^3 (B d+4 A e)\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}-\frac{\left (e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e))\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{2 d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-5 A e)-2 b c d (2 B d+A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 A c^4 d^4-2 b^3 c d e^2 (9 B d-10 A e)+3 b^4 e^3 (3 B d-5 A e)+4 b^2 c^2 d^2 e (10 B d+3 A e)-16 b c^3 d^3 (B d+4 A e)\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}+\frac{\left (e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e))\right ) \operatorname{Subst}\left (\int \frac{1}{4 c d^2-4 b d e-x^2} \, dx,x,\frac{-b d-(2 c d-b e) x}{\sqrt{b x+c x^2}}\right )}{d^3 (c d-b e)^3}\\ &=-\frac{2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) (d+e x) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-5 A e)-2 b c d (2 B d+A e)\right )+c \left (16 A c^3 d^3-b^3 e^2 (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 (d+e x) \sqrt{b x+c x^2}}+\frac{e \left (32 A c^4 d^4-2 b^3 c d e^2 (9 B d-10 A e)+3 b^4 e^3 (3 B d-5 A e)+4 b^2 c^2 d^2 e (10 B d+3 A e)-16 b c^3 d^3 (B d+4 A e)\right ) \sqrt{b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 (d+e x)}-\frac{e^3 (B d (8 c d-3 b e)-5 A e (2 c d-b e)) \tanh ^{-1}\left (\frac{b d+(2 c d-b e) x}{2 \sqrt{d} \sqrt{c d-b e} \sqrt{b x+c x^2}}\right )}{2 d^{7/2} (c d-b e)^{7/2}}\\ \end{align*}

Mathematica [A]  time = 0.908355, size = 557, normalized size = 1.24 \[ -\frac{\sqrt{d} \sqrt{b e-c d} \left (A \left (3 b^4 c^2 e \left (14 d^2 e^2 x^2+2 d^3 e x+2 d^4+10 d e^3 x^3-5 e^4 x^4\right )-2 b^3 c^3 d \left (3 d^2 e^2 x^2+13 d^3 e x+d^4-19 d e^3 x^3-10 e^4 x^4\right )+12 b^2 c^4 d^2 x \left (-7 d^2 e x+d^3-7 d e^2 x^2+e^3 x^3\right )-6 b^5 c e^2 \left (-3 d^2 e x+d^3+5 e^3 x^3\right )+b^6 e^3 \left (2 d^2-10 d e x-15 e^2 x^2\right )-16 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+32 c^6 d^4 x^3 (d+e x)\right )+b B d x \left (3 b^3 c^2 e \left (-6 d^2 e x+6 d^3-10 d e^2 x^2+3 e^3 x^3\right )-6 b^2 c^3 d \left (-9 d^2 e x+d^3-7 d e^2 x^2+3 e^3 x^3\right )-6 b^4 c e^2 \left (3 d^2+d e x-3 e^2 x^2\right )+3 b^5 e^3 (2 d+3 e x)+8 b c^4 d^2 x \left (-3 d^2+2 d e x+5 e^2 x^2\right )-16 c^5 d^3 x^2 (d+e x)\right )\right )+3 b^4 e^3 x^{3/2} (b+c x)^{3/2} (d+e x) (B d (3 b e-8 c d)-5 A e (b e-2 c d)) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{3 b^4 d^{7/2} (x (b+c x))^{3/2} (d+e x) (b e-c d)^{7/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(Sqrt[d]*Sqrt[-(c*d) + b*e]*(b*B*d*x*(-16*c^5*d^3*x^2*(d + e*x) + 3*b^5*e^3*(2*d + 3*e*x) - 6*b^4*c*e^2*(3*d^
2 + d*e*x - 3*e^2*x^2) + 8*b*c^4*d^2*x*(-3*d^2 + 2*d*e*x + 5*e^2*x^2) + 3*b^3*c^2*e*(6*d^3 - 6*d^2*e*x - 10*d*
e^2*x^2 + 3*e^3*x^3) - 6*b^2*c^3*d*(d^3 - 9*d^2*e*x - 7*d*e^2*x^2 + 3*e^3*x^3)) + A*(32*c^6*d^4*x^3*(d + e*x)
+ b^6*e^3*(2*d^2 - 10*d*e*x - 15*e^2*x^2) - 16*b*c^5*d^3*x^2*(-3*d^2 + d*e*x + 4*e^2*x^2) + 12*b^2*c^4*d^2*x*(
d^3 - 7*d^2*e*x - 7*d*e^2*x^2 + e^3*x^3) - 6*b^5*c*e^2*(d^3 - 3*d^2*e*x + 5*e^3*x^3) - 2*b^3*c^3*d*(d^4 + 13*d
^3*e*x + 3*d^2*e^2*x^2 - 19*d*e^3*x^3 - 10*e^4*x^4) + 3*b^4*c^2*e*(2*d^4 + 2*d^3*e*x + 14*d^2*e^2*x^2 + 10*d*e
^3*x^3 - 5*e^4*x^4))) + 3*b^4*e^3*(-5*A*e*(-2*c*d + b*e) + B*d*(-8*c*d + 3*b*e))*x^(3/2)*(b + c*x)^(3/2)*(d +
e*x)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(3*b^4*d^(7/2)*(-(c*d) + b*e)^(7/2)*(x*(b +
c*x))^(3/2)*(d + e*x))

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Maple [B]  time = 0.016, size = 4295, normalized size = 9.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

-5/3*e^2/d^2/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c*A+5/3*e/d/(b*e-c*d)^2/(
(x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c*B-5*e^3/d^2/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/
e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c*B-80/3*B/e/(b*e-c*d)*c^2/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*
d)/e^2)^(1/2)-B*e^2/d^2/(b*e-c*d)^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-
d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))+8/3*B/d/(b*e-c*d)*c
/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+10/3*B/e/(b*e-c*d)/b/((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c+160/3/(b*e-c*d)^2*c^3/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2
)^(1/2)*x*B+160/3/(b*e-c*d)^2*c^4/b^4/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A-8/3*c/d/(b
*e-c*d)/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*A+64/3*c^2/d/(b*e-c*d)/b^3/((x+d/e)^2*c+(b
*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A-20/3/(b*e-c*d)^2/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*
d)/e^2)^(3/2)*x*c^3*A-20/3/(b*e-c*d)^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c^2*B-15*
e^3/d^2/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c*A+15*e^2/d/(b*e-c*d)^3/((x+d/e
)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c*B-5/3*e^2/d^2/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)-d*(b*e-c*d)/e^2)^(3/2)*b*A+5/3*e/d/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*b
*B+5*e/d/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*A+5*e^4/d^3/(b*e-c*d)^3/((x+d
/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*b*A-5*e^3/d^2/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d
/e)-d*(b*e-c*d)/e^2)^(1/2)*b*B-10*e/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^
2*B-5/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c*B-2/3*B/d/(b*e-c*d)/((x+d/e)^2*c
+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+2*B*e^2/d^2/(b*e-c*d)^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(
b*e-c*d)/e^2)^(1/2)*x*c-52/3*B*e/d/(b*e-c*d)^2/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x
*c^2+1/d/(b*e-c*d)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*A+80/3/(b*e-c*d)^2*c^2/b^
2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*B-1/e/(b*e-c*d)/(x+d/e)/((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*B+2*B*e^2/d^2/(b*e-c*d)^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^
(1/2)+80/3/(b*e-c*d)^2*c^3/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A-10/3/(b*e-c*d)^2/b/
((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*c^2*A-2/3*B/d/(b*e-c*d)/b/((x+d/e)^2*c+(b*e-2*c*d)/e
*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c+10/3/e/(b*e-c*d)^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(
3/2)*c^2*B*d-20*e/(b*e-c*d)^3/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3*B+128/3*c^3/
d/(b*e-c*d)/b^4/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A+20/3*B/e/(b*e-c*d)/b^2/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*c^2+16/3*B/d/(b*e-c*d)*c^2/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(
x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x-160/3*B/e/(b*e-c*d)*c^3/b^4/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2
)^(1/2)*x-26/3*B*e/d/(b*e-c*d)^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c-16/3*c^2/d/(b*e
-c*d)/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)*x*A-80/3*e/d/(b*e-c*d)^2*c^2/b^2/((x+d/e)^
2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A+5*e^4/d^3/(b*e-c*d)^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*
(b*e-c*d)/e^2)^(1/2)*x*c*A-80/3/e/(b*e-c*d)^2*c^3/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2
)*B*d+10*e^2/d/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*c^2*A+20/3*e^2/d^2/(b*e
-c*d)^2*c/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*A+20*e^2/d/(b*e-c*d)^3/b^2/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^3*A+20/3*e/d/(b*e-c*d)^2/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
-d*(b*e-c*d)/e^2)^(3/2)*x*c^2*A+20/3/e/(b*e-c*d)^2/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(3/
2)*x*c^3*B*d-160/3*e/d/(b*e-c*d)^2*c^3/b^3/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A-160/3
/e/(b*e-c*d)^2*c^4/b^4/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*B*d+40/3*e^2/d^2/(b*e-c*d)^
2*c^2/b^2/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*A-20*e^3/d^2/(b*e-c*d)^3/b/((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*A+20*e^2/d/(b*e-c*d)^3/b/((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)
-d*(b*e-c*d)/e^2)^(1/2)*x*c^2*B-5/2*e^4/d^3/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2
*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))
*b*A+5/2*e^3/d^2/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-
c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*b*B+5*e^3/d^2/(b*e-c*d)^3/
(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+
(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))*c*A-5*e^2/d/(b*e-c*d)^3/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2
*d*(b*e-c*d)/e^2+(b*e-2*c*d)/e*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*((x+d/e)^2*c+(b*e-2*c*d)/e*(x+d/e)-d*(b*e-c*
d)/e^2)^(1/2))/(x+d/e))*c*B

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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)^2/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.81262, size = 5225, normalized size = 11.64 \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)^2/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[-1/6*(3*((8*B*b^4*c^3*d^2*e^4 + 5*A*b^5*c^2*e^6 - (3*B*b^5*c^2 + 10*A*b^4*c^3)*d*e^5)*x^5 + (8*B*b^4*c^3*d^3*
e^3 + 10*A*b^6*c*e^6 + (13*B*b^5*c^2 - 10*A*b^4*c^3)*d^2*e^4 - 3*(2*B*b^6*c + 5*A*b^5*c^2)*d*e^5)*x^4 + (16*B*
b^5*c^2*d^3*e^3 - 3*B*b^7*d*e^5 + 5*A*b^7*e^6 + 2*(B*b^6*c - 10*A*b^5*c^2)*d^2*e^4)*x^3 + (8*B*b^6*c*d^3*e^3 +
5*A*b^7*d*e^5 - (3*B*b^7 + 10*A*b^6*c)*d^2*e^4)*x^2)*sqrt(c*d^2 - b*d*e)*log((b*d + (2*c*d - b*e)*x + 2*sqrt(
c*d^2 - b*d*e)*sqrt(c*x^2 + b*x))/(e*x + d)) + 2*(2*A*b^3*c^4*d^7 - 8*A*b^4*c^3*d^6*e + 12*A*b^5*c^2*d^5*e^2 -
8*A*b^6*c*d^4*e^3 + 2*A*b^7*d^3*e^4 - (15*A*b^5*c^2*d*e^6 - 16*(B*b*c^6 - 2*A*c^7)*d^6*e + 8*(7*B*b^2*c^5 - 1
2*A*b*c^6)*d^5*e^2 - 2*(29*B*b^3*c^4 - 38*A*b^2*c^5)*d^4*e^3 + (27*B*b^4*c^3 + 8*A*b^3*c^4)*d^3*e^4 - (9*B*b^5
*c^2 + 35*A*b^4*c^3)*d^2*e^5)*x^4 - 2*(15*A*b^6*c*d*e^6 - 8*(B*b*c^6 - 2*A*c^7)*d^7 + 8*(2*B*b^2*c^5 - 3*A*b*c
^6)*d^6*e + (13*B*b^3*c^4 - 34*A*b^2*c^5)*d^5*e^2 - (36*B*b^4*c^3 - 61*A*b^3*c^4)*d^4*e^3 + 4*(6*B*b^5*c^2 - A
*b^4*c^3)*d^3*e^4 - 3*(3*B*b^6*c + 10*A*b^5*c^2)*d^2*e^5)*x^3 - 3*(5*A*b^7*d*e^6 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d
^7 + 2*(13*B*b^3*c^4 - 22*A*b^2*c^5)*d^6*e - 2*(12*B*b^4*c^3 - 13*A*b^3*c^4)*d^5*e^2 + 4*(B*b^5*c^2 + 4*A*b^4*
c^3)*d^4*e^3 + (5*B*b^6*c - 14*A*b^5*c^2)*d^3*e^4 - (3*B*b^7 + 5*A*b^6*c)*d^2*e^5)*x^2 - 2*(5*A*b^7*d^2*e^5 -
3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^7 + (12*B*b^4*c^3 - 19*A*b^3*c^4)*d^6*e - 2*(9*B*b^5*c^2 - 8*A*b^4*c^3)*d^5*e^2
+ 6*(2*B*b^6*c + A*b^5*c^2)*d^4*e^3 - (3*B*b^7 + 14*A*b^6*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^4*c^6*d^8*e -
4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^4 + b^8*c^2*d^4*e^5)*x^5 + (b^4*c^6*d^9 - 2*b^5*c^5*d^
8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*d^5*e^4 + 2*b^9*c*d^4*e^5)*x^4 + (2*b^5*c^5*d^9 - 7*b^
6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*b^9*c*d^5*e^4 + b^10*d^4*e^5)*x^3 + (b^6*c^4*d^9 - 4*b
^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^10*d^5*e^4)*x^2), -1/3*(3*((8*B*b^4*c^3*d^2*e^4 + 5*A*b
^5*c^2*e^6 - (3*B*b^5*c^2 + 10*A*b^4*c^3)*d*e^5)*x^5 + (8*B*b^4*c^3*d^3*e^3 + 10*A*b^6*c*e^6 + (13*B*b^5*c^2 -
10*A*b^4*c^3)*d^2*e^4 - 3*(2*B*b^6*c + 5*A*b^5*c^2)*d*e^5)*x^4 + (16*B*b^5*c^2*d^3*e^3 - 3*B*b^7*d*e^5 + 5*A*
b^7*e^6 + 2*(B*b^6*c - 10*A*b^5*c^2)*d^2*e^4)*x^3 + (8*B*b^6*c*d^3*e^3 + 5*A*b^7*d*e^5 - (3*B*b^7 + 10*A*b^6*c
)*d^2*e^4)*x^2)*sqrt(-c*d^2 + b*d*e)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/((c*d - b*e)*x)) + (2*A*b^
3*c^4*d^7 - 8*A*b^4*c^3*d^6*e + 12*A*b^5*c^2*d^5*e^2 - 8*A*b^6*c*d^4*e^3 + 2*A*b^7*d^3*e^4 - (15*A*b^5*c^2*d*e
^6 - 16*(B*b*c^6 - 2*A*c^7)*d^6*e + 8*(7*B*b^2*c^5 - 12*A*b*c^6)*d^5*e^2 - 2*(29*B*b^3*c^4 - 38*A*b^2*c^5)*d^4
*e^3 + (27*B*b^4*c^3 + 8*A*b^3*c^4)*d^3*e^4 - (9*B*b^5*c^2 + 35*A*b^4*c^3)*d^2*e^5)*x^4 - 2*(15*A*b^6*c*d*e^6
- 8*(B*b*c^6 - 2*A*c^7)*d^7 + 8*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*e + (13*B*b^3*c^4 - 34*A*b^2*c^5)*d^5*e^2 - (36*
B*b^4*c^3 - 61*A*b^3*c^4)*d^4*e^3 + 4*(6*B*b^5*c^2 - A*b^4*c^3)*d^3*e^4 - 3*(3*B*b^6*c + 10*A*b^5*c^2)*d^2*e^5
)*x^3 - 3*(5*A*b^7*d*e^6 - 8*(B*b^2*c^5 - 2*A*b*c^6)*d^7 + 2*(13*B*b^3*c^4 - 22*A*b^2*c^5)*d^6*e - 2*(12*B*b^4
*c^3 - 13*A*b^3*c^4)*d^5*e^2 + 4*(B*b^5*c^2 + 4*A*b^4*c^3)*d^4*e^3 + (5*B*b^6*c - 14*A*b^5*c^2)*d^3*e^4 - (3*B
*b^7 + 5*A*b^6*c)*d^2*e^5)*x^2 - 2*(5*A*b^7*d^2*e^5 - 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^7 + (12*B*b^4*c^3 - 19*A*b
^3*c^4)*d^6*e - 2*(9*B*b^5*c^2 - 8*A*b^4*c^3)*d^5*e^2 + 6*(2*B*b^6*c + A*b^5*c^2)*d^4*e^3 - (3*B*b^7 + 14*A*b^
6*c)*d^3*e^4)*x)*sqrt(c*x^2 + b*x))/((b^4*c^6*d^8*e - 4*b^5*c^5*d^7*e^2 + 6*b^6*c^4*d^6*e^3 - 4*b^7*c^3*d^5*e^
4 + b^8*c^2*d^4*e^5)*x^5 + (b^4*c^6*d^9 - 2*b^5*c^5*d^8*e - 2*b^6*c^4*d^7*e^2 + 8*b^7*c^3*d^6*e^3 - 7*b^8*c^2*
d^5*e^4 + 2*b^9*c*d^4*e^5)*x^4 + (2*b^5*c^5*d^9 - 7*b^6*c^4*d^8*e + 8*b^7*c^3*d^7*e^2 - 2*b^8*c^2*d^6*e^3 - 2*
b^9*c*d^5*e^4 + b^10*d^4*e^5)*x^3 + (b^6*c^4*d^9 - 4*b^7*c^3*d^8*e + 6*b^8*c^2*d^7*e^2 - 4*b^9*c*d^6*e^3 + b^1
0*d^5*e^4)*x^2)]

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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)**2/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Giac [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)^2/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

Timed out