3.1211 \(\int \frac{A+B x}{(d+e x) (b x+c x^2)^{5/2}} \, dx\)

Optimal. Leaf size=287 \[ \frac{2 \left (c x \left (2 b^2 c d e (A e+7 B d)-3 b^3 e^2 (B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]

[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)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d
- b*e)*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e)) + c*(16*A*c^3*d^3 - 3*b^3*e^2*(B*d - A*e) + 2
*b^2*c*d*e*(7*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*Sqrt[b*x + c*x^2]) - (e^3*(
B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(d^(5/2)*(c*d - b*e
)^(5/2))

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Rubi [A]  time = 0.354228, antiderivative size = 287, 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, 12, 724, 206} \[ \frac{2 \left (c x \left (2 b^2 c d e (A e+7 B d)-3 b^3 e^2 (B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )+b (c d-b e) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )\right )}{3 b^4 d^2 \sqrt{b x+c x^2} (c d-b e)^2}-\frac{2 (c x (2 A c d-b (A e+B d))+A b (c d-b e))}{3 b^2 d \left (b x+c x^2\right )^{3/2} (c d-b e)}-\frac{e^3 (B d-A e) \tanh ^{-1}\left (\frac{x (2 c d-b e)+b d}{2 \sqrt{d} \sqrt{b x+c x^2} \sqrt{c d-b e}}\right )}{d^{5/2} (c d-b e)^{5/2}} \]

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)*(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)*(b*x + c*x^2)^(3/2)) + (2*(b*(c*d
- b*e)*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e) - 4*b*c*d*(B*d + A*e)) + c*(16*A*c^3*d^3 - 3*b^3*e^2*(B*d - A*e) + 2
*b^2*c*d*e*(7*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*Sqrt[b*x + c*x^2]) - (e^3*(
B*d - A*e)*ArcTanh[(b*d + (2*c*d - b*e)*x)/(2*Sqrt[d]*Sqrt[c*d - b*e]*Sqrt[b*x + c*x^2])])/(d^(5/2)*(c*d - b*e
)^(5/2))

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 12

Int[(a_)*(u_), x_Symbol] :> Dist[a, Int[u, x], x] /; FreeQ[a, x] &&  !MatchQ[u, (b_)*(v_) /; FreeQ[b, x]]

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) \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) \left (b x+c x^2\right )^{3/2}}-\frac{2 \int \frac{\frac{1}{2} \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )-2 c e (b B d-2 A c d+A b e) x}{(d+e x) \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) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 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 \sqrt{b x+c x^2}}+\frac{4 \int -\frac{3 b^4 e^3 (B d-A e)}{4 (d+e x) \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) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 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 \sqrt{b x+c x^2}}-\frac{\left (e^3 (B d-A e)\right ) \int \frac{1}{(d+e x) \sqrt{b x+c x^2}} \, dx}{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) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 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 \sqrt{b x+c x^2}}+\frac{\left (2 e^3 (B d-A 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^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) \left (b x+c x^2\right )^{3/2}}+\frac{2 \left (b (c d-b e) \left (8 A c^2 d^2+3 b^2 e (B d-A e)-4 b c d (B d+A e)\right )+c \left (16 A c^3 d^3-3 b^3 e^2 (B d-A e)+2 b^2 c d e (7 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 \sqrt{b x+c x^2}}-\frac{e^3 (B d-A 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 )}{d^{5/2} (c d-b e)^{5/2}}\\ \end{align*}

Mathematica [A]  time = 0.944144, size = 288, normalized size = 1. \[ \frac{2 x \left (\frac{c x^2 (b+c x)^2 \left (2 b^2 c d e (A e+7 B d)+3 b^3 e^2 (A e-B d)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )}{b^3 d (c d-b e)^2}-\frac{c x^2 (b+c x) \left (3 b^2 e (B d-A e)-4 b c d (A e+B d)+8 A c^2 d^2\right )}{b^2 d (b e-c d)}+\frac{3 b e^3 x^{3/2} (b+c x)^{5/2} (A e-B d) \tan ^{-1}\left (\frac{\sqrt{x} \sqrt{b e-c d}}{\sqrt{d} \sqrt{b+c x}}\right )}{d^{3/2} (b e-c d)^{5/2}}-\frac{3 x (b+c x) (-A b e-2 A c d+b B d)}{b d}-A (b+c x)\right )}{3 b d (x (b+c x))^{5/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(2*x*(-(A*(b + c*x)) - (3*(b*B*d - 2*A*c*d - A*b*e)*x*(b + c*x))/(b*d) - (c*(8*A*c^2*d^2 + 3*b^2*e*(B*d - A*e)
- 4*b*c*d*(B*d + A*e))*x^2*(b + c*x))/(b^2*d*(-(c*d) + b*e)) + (c*(16*A*c^3*d^3 + 3*b^3*e^2*(-(B*d) + A*e) +
2*b^2*c*d*e*(7*B*d + A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*x^2*(b + c*x)^2)/(b^3*d*(c*d - b*e)^2) + (3*b*e^3*(-(B*
d) + A*e)*x^(3/2)*(b + c*x)^(5/2)*ArcTan[(Sqrt[-(c*d) + b*e]*Sqrt[x])/(Sqrt[d]*Sqrt[b + c*x])])/(d^(3/2)*(-(c*
d) + b*e)^(5/2))))/(3*b*d*(x*(b + c*x))^(5/2))

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Maple [B]  time = 0.013, size = 2000, normalized size = 7. \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)/(c*x^2+b*x)^(5/2),x)

[Out]

2/3/(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)*B-2/3/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*B*d+4*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)^(1/2)*x*c^2*B+16/3/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*d-2
*e^2/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*A+8/3*e/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)*A+e^2/d/(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))*B+4/3/(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*A+2/3/
(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*B+2*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)^(1/2)*c*B-e^3/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))*A-4/3*B/e/b^2/(c*x^2+b*x)^(3/2)*x*c-16/3/(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*B-32/3/(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*A+32/
3*B/e*c^2/b^4/(c*x^2+b*x)^(1/2)*x-2/3*B/e/b/(c*x^2+b*x)^(3/2)-8/3/(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)*B+2/3/(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*A-
16/3/(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)*A-2*e^2/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)^(1/2)*B+16/3*B/e*c/b^3/(c*x^2+b*x)^(1/2)-2/3*e/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)*A+2*e^3/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)*A-4*e^2/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*A+32/3/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*B*d+2*e^3/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*A-2/3*e/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*A-4/3/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*B*d+16/3*e/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*A-2*e^2/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)*x*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)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 2.17566, size = 2746, normalized size = 9.57 \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)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

[-1/3*(3*((B*b^4*c^2*d*e^3 - A*b^4*c^2*e^4)*x^4 + 2*(B*b^5*c*d*e^3 - A*b^5*c*e^4)*x^3 + (B*b^6*d*e^3 - A*b^6*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*(A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + (3*A*b^4*c^2*d*e^4 + 8*(B*b*c^5 -
2*A*c^6)*d^5 - 2*(11*B*b^2*c^4 - 20*A*b*c^5)*d^4*e + (17*B*b^3*c^3 - 26*A*b^2*c^4)*d^3*e^2 - (3*B*b^4*c^2 + A
*b^3*c^3)*d^2*e^3)*x^3 + 3*(2*A*b^5*c*d*e^4 + 4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 - 20*A*b^2*c^4)*d^
4*e + (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c + A*b^4*c^2)*d^2*e^3)*x^2 + 3*(A*b^6*d*e^4 + (B*b^3*c^
3 - 2*A*b^2*c^4)*d^5 - (3*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e + 3*(B*b^5*c - A*b^4*c^2)*d^3*e^2 - (B*b^6 + A*b^5*c)
*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*d^3*e^3)*x^4 + 2
*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^7*c^2*d^5*e + 3*
b^8*c*d^4*e^2 - b^9*d^3*e^3)*x^2), -2/3*(3*((B*b^4*c^2*d*e^3 - A*b^4*c^2*e^4)*x^4 + 2*(B*b^5*c*d*e^3 - A*b^5*c
*e^4)*x^3 + (B*b^6*d*e^3 - A*b^6*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)) + (A*b^3*c^3*d^5 - 3*A*b^4*c^2*d^4*e + 3*A*b^5*c*d^3*e^2 - A*b^6*d^2*e^3 + (3*A*b^4*c^2*d*e^
4 + 8*(B*b*c^5 - 2*A*c^6)*d^5 - 2*(11*B*b^2*c^4 - 20*A*b*c^5)*d^4*e + (17*B*b^3*c^3 - 26*A*b^2*c^4)*d^3*e^2 -
(3*B*b^4*c^2 + A*b^3*c^3)*d^2*e^3)*x^3 + 3*(2*A*b^5*c*d*e^4 + 4*(B*b^2*c^4 - 2*A*b*c^5)*d^5 - (11*B*b^3*c^3 -
20*A*b^2*c^4)*d^4*e + (9*B*b^4*c^2 - 13*A*b^3*c^3)*d^3*e^2 - (2*B*b^5*c + A*b^4*c^2)*d^2*e^3)*x^2 + 3*(A*b^6*d
*e^4 + (B*b^3*c^3 - 2*A*b^2*c^4)*d^5 - (3*B*b^4*c^2 - 5*A*b^3*c^3)*d^4*e + 3*(B*b^5*c - A*b^4*c^2)*d^3*e^2 - (
B*b^6 + A*b^5*c)*d^2*e^3)*x)*sqrt(c*x^2 + b*x))/((b^4*c^5*d^6 - 3*b^5*c^4*d^5*e + 3*b^6*c^3*d^4*e^2 - b^7*c^2*
d^3*e^3)*x^4 + 2*(b^5*c^4*d^6 - 3*b^6*c^3*d^5*e + 3*b^7*c^2*d^4*e^2 - b^8*c*d^3*e^3)*x^3 + (b^6*c^3*d^6 - 3*b^
7*c^2*d^5*e + 3*b^8*c*d^4*e^2 - b^9*d^3*e^3)*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)/(c*x**2+b*x)**(5/2),x)

[Out]

Timed out

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Giac [B]  time = 1.30004, size = 865, normalized size = 3.01 \begin{align*} \frac{2 \,{\left (B d e^{3} - A e^{4}\right )} \arctan \left (\frac{{\left (\sqrt{c} x - \sqrt{c x^{2} + b x}\right )} e + \sqrt{c} d}{\sqrt{-c d^{2} + b d e}}\right )}{{\left (c^{2} d^{4} - 2 \, b c d^{3} e + b^{2} d^{2} e^{2}\right )} \sqrt{-c d^{2} + b d e}} - \frac{{\left (x{\left (\frac{{\left (8 \, B b c^{6} d^{10} - 16 \, A c^{7} d^{10} - 30 \, B b^{2} c^{5} d^{9} e + 56 \, A b c^{6} d^{9} e + 39 \, B b^{3} c^{4} d^{8} e^{2} - 66 \, A b^{2} c^{5} d^{8} e^{2} - 20 \, B b^{4} c^{3} d^{7} e^{3} + 25 \, A b^{3} c^{4} d^{7} e^{3} + 3 \, B b^{5} c^{2} d^{6} e^{4} + 4 \, A b^{4} c^{3} d^{6} e^{4} - 3 \, A b^{5} c^{2} d^{5} e^{5}\right )} x}{b^{4} c^{2}} + \frac{3 \,{\left (4 \, B b^{2} c^{5} d^{10} - 8 \, A b c^{6} d^{10} - 15 \, B b^{3} c^{4} d^{9} e + 28 \, A b^{2} c^{5} d^{9} e + 20 \, B b^{4} c^{3} d^{8} e^{2} - 33 \, A b^{3} c^{4} d^{8} e^{2} - 11 \, B b^{5} c^{2} d^{7} e^{3} + 12 \, A b^{4} c^{3} d^{7} e^{3} + 2 \, B b^{6} c d^{6} e^{4} + 3 \, A b^{5} c^{2} d^{6} e^{4} - 2 \, A b^{6} c d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} + \frac{3 \,{\left (B b^{3} c^{4} d^{10} - 2 \, A b^{2} c^{5} d^{10} - 4 \, B b^{4} c^{3} d^{9} e + 7 \, A b^{3} c^{4} d^{9} e + 6 \, B b^{5} c^{2} d^{8} e^{2} - 8 \, A b^{4} c^{3} d^{8} e^{2} - 4 \, B b^{6} c d^{7} e^{3} + 2 \, A b^{5} c^{2} d^{7} e^{3} + B b^{7} d^{6} e^{4} + 2 \, A b^{6} c d^{6} e^{4} - A b^{7} d^{5} e^{5}\right )}}{b^{4} c^{2}}\right )} x + \frac{A b^{3} c^{4} d^{10} - 4 \, A b^{4} c^{3} d^{9} e + 6 \, A b^{5} c^{2} d^{8} e^{2} - 4 \, A b^{6} c d^{7} e^{3} + A b^{7} d^{6} e^{4}}{b^{4} c^{2}}}{3 \,{\left (c x^{2} + b x\right )}^{\frac{3}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

2*(B*d*e^3 - A*e^4)*arctan(((sqrt(c)*x - sqrt(c*x^2 + b*x))*e + sqrt(c)*d)/sqrt(-c*d^2 + b*d*e))/((c^2*d^4 - 2
*b*c*d^3*e + b^2*d^2*e^2)*sqrt(-c*d^2 + b*d*e)) - 1/3*((x*((8*B*b*c^6*d^10 - 16*A*c^7*d^10 - 30*B*b^2*c^5*d^9*
e + 56*A*b*c^6*d^9*e + 39*B*b^3*c^4*d^8*e^2 - 66*A*b^2*c^5*d^8*e^2 - 20*B*b^4*c^3*d^7*e^3 + 25*A*b^3*c^4*d^7*e
^3 + 3*B*b^5*c^2*d^6*e^4 + 4*A*b^4*c^3*d^6*e^4 - 3*A*b^5*c^2*d^5*e^5)*x/(b^4*c^2) + 3*(4*B*b^2*c^5*d^10 - 8*A*
b*c^6*d^10 - 15*B*b^3*c^4*d^9*e + 28*A*b^2*c^5*d^9*e + 20*B*b^4*c^3*d^8*e^2 - 33*A*b^3*c^4*d^8*e^2 - 11*B*b^5*
c^2*d^7*e^3 + 12*A*b^4*c^3*d^7*e^3 + 2*B*b^6*c*d^6*e^4 + 3*A*b^5*c^2*d^6*e^4 - 2*A*b^6*c*d^5*e^5)/(b^4*c^2)) +
3*(B*b^3*c^4*d^10 - 2*A*b^2*c^5*d^10 - 4*B*b^4*c^3*d^9*e + 7*A*b^3*c^4*d^9*e + 6*B*b^5*c^2*d^8*e^2 - 8*A*b^4*
c^3*d^8*e^2 - 4*B*b^6*c*d^7*e^3 + 2*A*b^5*c^2*d^7*e^3 + B*b^7*d^6*e^4 + 2*A*b^6*c*d^6*e^4 - A*b^7*d^5*e^5)/(b^
4*c^2))*x + (A*b^3*c^4*d^10 - 4*A*b^4*c^3*d^9*e + 6*A*b^5*c^2*d^8*e^2 - 4*A*b^6*c*d^7*e^3 + A*b^7*d^6*e^4)/(b^
4*c^2))/(c*x^2 + b*x)^(3/2)