3.13.12 \(\int \frac {A+B x}{(d+e x)^2 (b x+c x^2)^{5/2}} \, dx\) [1212]

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

[Out]

-2/3*(A*b*(-b*e+c*d)+c*(2*A*c*d-b*(A*e+B*d))*x)/b^2/d/(-b*e+c*d)/(e*x+d)/(c*x^2+b*x)^(3/2)-1/2*e^3*(B*d*(-3*b*
e+8*c*d)-5*A*e*(-b*e+2*c*d))*arctanh(1/2*(b*d+(-b*e+2*c*d)*x)/d^(1/2)/(-b*e+c*d)^(1/2)/(c*x^2+b*x)^(1/2))/d^(7
/2)/(-b*e+c*d)^(7/2)+2/3*(b*(-b*e+c*d)*(8*A*c^2*d^2+b^2*e*(-5*A*e+3*B*d)-2*b*c*d*(A*e+2*B*d))+c*(16*A*c^3*d^3-
b^3*e^2*(-5*A*e+3*B*d)+2*b^2*c*d*e*(-A*e+8*B*d)-8*b*c^2*d^2*(3*A*e+B*d))*x)/b^4/d^2/(-b*e+c*d)^2/(e*x+d)/(c*x^
2+b*x)^(1/2)+1/3*e*(32*A*c^4*d^4-2*b^3*c*d*e^2*(-10*A*e+9*B*d)+3*b^4*e^3*(-5*A*e+3*B*d)+4*b^2*c^2*d^2*e*(3*A*e
+10*B*d)-16*b*c^3*d^3*(4*A*e+B*d))*(c*x^2+b*x)^(1/2)/b^4/d^3/(-b*e+c*d)^3/(e*x+d)

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Rubi [A]
time = 0.48, antiderivative size = 449, normalized size of antiderivative = 1.00, 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 = {836, 820, 738, 212} \begin {gather*} -\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 {2 \left (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 )+c x \left (b^3 \left (-e^2\right ) (3 B d-5 A e)+2 b^2 c d e (8 B d-A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} (d+e x) (c d-b e)^2}+\frac {e \sqrt {b x+c x^2} \left (3 b^4 e^3 (3 B d-5 A e)-2 b^3 c d e^2 (9 B d-10 A e)+4 b^2 c^2 d^2 e (3 A e+10 B d)-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 {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}} \end {gather*}

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 212

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

Rule 738

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 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Sim
p[(-(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[S
implify[m + 2*p + 3], 0]

Rule 836

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])

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 ) \text {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*}

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Mathematica [A]
time = 2.50, size = 568, normalized size = 1.27 \begin {gather*} \frac {x \left (\frac {\sqrt {d} (b+c x) \left (b B d x \left (16 c^5 d^3 x^2 (d+e x)-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 )+6 b^4 c e^2 \left (3 d^2+d e x-3 e^2 x^2\right )-3 b^3 c^2 e \left (6 d^3-6 d^2 e x-10 d e^2 x^2+3 e^3 x^3\right )+6 b^2 c^3 d \left (d^3-9 d^2 e x-7 d e^2 x^2+3 e^3 x^3\right )\right )+A \left (-32 c^6 d^4 x^3 (d+e x)+16 b c^5 d^3 x^2 \left (-3 d^2+d e x+4 e^2 x^2\right )+b^6 e^3 \left (-2 d^2+10 d e x+15 e^2 x^2\right )-12 b^2 c^4 d^2 x \left (d^3-7 d^2 e x-7 d e^2 x^2+e^3 x^3\right )+6 b^5 c e^2 \left (d^3-3 d^2 e x+5 e^3 x^3\right )+2 b^3 c^3 d \left (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\right )-3 b^4 c^2 e \left (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\right )\right )\right )}{b^4 (-c d+b e)^3 (d+e x)}-\frac {3 e^3 (B d (8 c d-3 b e)+5 A e (-2 c d+b e)) x^{3/2} (b+c x)^{5/2} \tan ^{-1}\left (\frac {-e \sqrt {x} \sqrt {b+c x}+\sqrt {c} (d+e x)}{\sqrt {d} \sqrt {-c d+b e}}\right )}{(-c d+b e)^{7/2}}\right )}{3 d^{7/2} (x (b+c x))^{5/2}} \end {gather*}

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(1657\) vs. \(2(423)=846\).
time = 0.68, size = 1658, normalized size = 3.69

method	result	size
risch	\(\text {Expression too large to display}\)	\(1201\)
default	\(\text {Expression too large to display}\)	\(1658\)

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,method=_RETURNVERBOSE)

[Out]

(A*e-B*d)/e^3*(1/d/(b*e-c*d)*e^2/(x+d/e)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+5/2*e*(b*
e-2*c*d)/d/(b*e-c*d)*(-1/3/d/(b*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2*e*(
b*e-2*c*d)/d/(b*e-c*d)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e
)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x
+d/e)+1/e*(b*e-2*c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))-1/d/(b*e-c*d)*e^2*(-1/d/(b
*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+e*(b*e-2*c*d)/d/(b*e-c*d)*(2*c*(x+d/e)
+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/
e^2)^(1/2)+1/d/(b*e-c*d)*e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*
e-c*d)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2))/(x+d/e))))+4*c/d/(b*e-c*d)*e^2*
(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+
d/e)-d*(b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/
(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)))+B/e^2*(-1/3/d/(b*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*
e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+1/2*e*(b*e-2*c*d)/d/(b*e-c*d)*(2/3*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c
*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(3/2)+16/3*c/(-4*c
*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)^2*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(
b*e-c*d)/e^2)^(1/2))-1/d/(b*e-c*d)*e^2*(-1/d/(b*e-c*d)*e^2/(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^
2)^(1/2)+e*(b*e-2*c*d)/d/(b*e-c*d)*(2*c*(x+d/e)+1/e*(b*e-2*c*d))/(-4*c*d*(b*e-c*d)/e^2-1/e^2*(b*e-2*c*d)^2)/(c
*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c*d)/e^2)^(1/2)+1/d/(b*e-c*d)*e^2/(-d*(b*e-c*d)/e^2)^(1/2)*ln((-2*d*
(b*e-c*d)/e^2+1/e*(b*e-2*c*d)*(x+d/e)+2*(-d*(b*e-c*d)/e^2)^(1/2)*(c*(x+d/e)^2+1/e*(b*e-2*c*d)*(x+d/e)-d*(b*e-c
*d)/e^2)^(1/2))/(x+d/e))))

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Maxima [F(-2)]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Exception raised: ValueError} \end {gather*}

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 >> Computation failed since Maxima requested additional constraints; using the 'a
ssume' command before evaluation *may* help (example of legal syntax is 'assume(c*d-%e*b>0)', see `assume?` fo
r more detai

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Fricas [B] Leaf count of result is larger than twice the leaf count of optimal. 1304 vs. \(2 (447) = 894\).
time = 5.01, size = 2621, normalized size = 5.84 \begin {gather*} \text {Too large to display} \end {gather*}

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*sqrt(c*d^2 - b*d*e)*(5*(A*b^5*c^2*x^5 + 2*A*b^6*c*x^4 + A*b^7*x^3)*e^6 - (3*B*b^7*d*x^3 - 5*A*b^7*d*x
^2 + (3*B*b^5*c^2 + 10*A*b^4*c^3)*d*x^5 + 3*(2*B*b^6*c + 5*A*b^5*c^2)*d*x^4)*e^5 + (8*B*b^4*c^3*d^2*x^5 + (13*
B*b^5*c^2 - 10*A*b^4*c^3)*d^2*x^4 + 2*(B*b^6*c - 10*A*b^5*c^2)*d^2*x^3 - (3*B*b^7 + 10*A*b^6*c)*d^2*x^2)*e^4 +
8*(B*b^4*c^3*d^3*x^4 + 2*B*b^5*c^2*d^3*x^3 + B*b^6*c*d^3*x^2)*e^3)*log((2*c*d*x - b*x*e + b*d + 2*sqrt(c*d^2
- b*d*e)*sqrt(c*x^2 + b*x))/(x*e + d)) + 2*(2*A*b^3*c^4*d^7 + 16*(B*b*c^6 - 2*A*c^7)*d^7*x^3 + 24*(B*b^2*c^5 -
2*A*b*c^6)*d^7*x^2 + 6*(B*b^3*c^4 - 2*A*b^2*c^5)*d^7*x - 15*(A*b^5*c^2*d*x^4 + 2*A*b^6*c*d*x^3 + A*b^7*d*x^2)
*e^6 - (10*A*b^7*d^2*x - (9*B*b^5*c^2 + 35*A*b^4*c^3)*d^2*x^4 - 6*(3*B*b^6*c + 10*A*b^5*c^2)*d^2*x^3 - 3*(3*B*
b^7 + 5*A*b^6*c)*d^2*x^2)*e^5 + (2*A*b^7*d^3 - (27*B*b^4*c^3 + 8*A*b^3*c^4)*d^3*x^4 - 8*(6*B*b^5*c^2 - A*b^4*c
^3)*d^3*x^3 - 3*(5*B*b^6*c - 14*A*b^5*c^2)*d^3*x^2 + 2*(3*B*b^7 + 14*A*b^6*c)*d^3*x)*e^4 - 2*(4*A*b^6*c*d^4 -
(29*B*b^3*c^4 - 38*A*b^2*c^5)*d^4*x^4 - (36*B*b^4*c^3 - 61*A*b^3*c^4)*d^4*x^3 + 6*(B*b^5*c^2 + 4*A*b^4*c^3)*d^
4*x^2 + 6*(2*B*b^6*c + A*b^5*c^2)*d^4*x)*e^3 + 2*(6*A*b^5*c^2*d^5 - 4*(7*B*b^2*c^5 - 12*A*b*c^6)*d^5*x^4 - (13
*B*b^3*c^4 - 34*A*b^2*c^5)*d^5*x^3 + 3*(12*B*b^4*c^3 - 13*A*b^3*c^4)*d^5*x^2 + 2*(9*B*b^5*c^2 - 8*A*b^4*c^3)*d
^5*x)*e^2 - 2*(4*A*b^4*c^3*d^6 - 8*(B*b*c^6 - 2*A*c^7)*d^6*x^4 + 8*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*x^3 + 3*(13*B
*b^3*c^4 - 22*A*b^2*c^5)*d^6*x^2 + (12*B*b^4*c^3 - 19*A*b^3*c^4)*d^6*x)*e)*sqrt(c*x^2 + b*x))/(b^4*c^6*d^9*x^4
+ 2*b^5*c^5*d^9*x^3 + b^6*c^4*d^9*x^2 + (b^8*c^2*d^4*x^5 + 2*b^9*c*d^4*x^4 + b^10*d^4*x^3)*e^5 - (4*b^7*c^3*d
^5*x^5 + 7*b^8*c^2*d^5*x^4 + 2*b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^4 + 2*(3*b^6*c^4*d^6*x^5 + 4*b^7*c^3*d^6*x^4 -
b^8*c^2*d^6*x^3 - 2*b^9*c*d^6*x^2)*e^3 - 2*(2*b^5*c^5*d^7*x^5 + b^6*c^4*d^7*x^4 - 4*b^7*c^3*d^7*x^3 - 3*b^8*c^
2*d^7*x^2)*e^2 + (b^4*c^6*d^8*x^5 - 2*b^5*c^5*d^8*x^4 - 7*b^6*c^4*d^8*x^3 - 4*b^7*c^3*d^8*x^2)*e), -1/3*(3*sqr
t(-c*d^2 + b*d*e)*(5*(A*b^5*c^2*x^5 + 2*A*b^6*c*x^4 + A*b^7*x^3)*e^6 - (3*B*b^7*d*x^3 - 5*A*b^7*d*x^2 + (3*B*b
^5*c^2 + 10*A*b^4*c^3)*d*x^5 + 3*(2*B*b^6*c + 5*A*b^5*c^2)*d*x^4)*e^5 + (8*B*b^4*c^3*d^2*x^5 + (13*B*b^5*c^2 -
10*A*b^4*c^3)*d^2*x^4 + 2*(B*b^6*c - 10*A*b^5*c^2)*d^2*x^3 - (3*B*b^7 + 10*A*b^6*c)*d^2*x^2)*e^4 + 8*(B*b^4*c
^3*d^3*x^4 + 2*B*b^5*c^2*d^3*x^3 + B*b^6*c*d^3*x^2)*e^3)*arctan(-sqrt(-c*d^2 + b*d*e)*sqrt(c*x^2 + b*x)/(c*d*x
- b*x*e)) + (2*A*b^3*c^4*d^7 + 16*(B*b*c^6 - 2*A*c^7)*d^7*x^3 + 24*(B*b^2*c^5 - 2*A*b*c^6)*d^7*x^2 + 6*(B*b^3
*c^4 - 2*A*b^2*c^5)*d^7*x - 15*(A*b^5*c^2*d*x^4 + 2*A*b^6*c*d*x^3 + A*b^7*d*x^2)*e^6 - (10*A*b^7*d^2*x - (9*B*
b^5*c^2 + 35*A*b^4*c^3)*d^2*x^4 - 6*(3*B*b^6*c + 10*A*b^5*c^2)*d^2*x^3 - 3*(3*B*b^7 + 5*A*b^6*c)*d^2*x^2)*e^5
+ (2*A*b^7*d^3 - (27*B*b^4*c^3 + 8*A*b^3*c^4)*d^3*x^4 - 8*(6*B*b^5*c^2 - A*b^4*c^3)*d^3*x^3 - 3*(5*B*b^6*c - 1
4*A*b^5*c^2)*d^3*x^2 + 2*(3*B*b^7 + 14*A*b^6*c)*d^3*x)*e^4 - 2*(4*A*b^6*c*d^4 - (29*B*b^3*c^4 - 38*A*b^2*c^5)*
d^4*x^4 - (36*B*b^4*c^3 - 61*A*b^3*c^4)*d^4*x^3 + 6*(B*b^5*c^2 + 4*A*b^4*c^3)*d^4*x^2 + 6*(2*B*b^6*c + A*b^5*c
^2)*d^4*x)*e^3 + 2*(6*A*b^5*c^2*d^5 - 4*(7*B*b^2*c^5 - 12*A*b*c^6)*d^5*x^4 - (13*B*b^3*c^4 - 34*A*b^2*c^5)*d^5
*x^3 + 3*(12*B*b^4*c^3 - 13*A*b^3*c^4)*d^5*x^2 + 2*(9*B*b^5*c^2 - 8*A*b^4*c^3)*d^5*x)*e^2 - 2*(4*A*b^4*c^3*d^6
- 8*(B*b*c^6 - 2*A*c^7)*d^6*x^4 + 8*(2*B*b^2*c^5 - 3*A*b*c^6)*d^6*x^3 + 3*(13*B*b^3*c^4 - 22*A*b^2*c^5)*d^6*x
^2 + (12*B*b^4*c^3 - 19*A*b^3*c^4)*d^6*x)*e)*sqrt(c*x^2 + b*x))/(b^4*c^6*d^9*x^4 + 2*b^5*c^5*d^9*x^3 + b^6*c^4
*d^9*x^2 + (b^8*c^2*d^4*x^5 + 2*b^9*c*d^4*x^4 + b^10*d^4*x^3)*e^5 - (4*b^7*c^3*d^5*x^5 + 7*b^8*c^2*d^5*x^4 + 2
*b^9*c*d^5*x^3 - b^10*d^5*x^2)*e^4 + 2*(3*b^6*c^4*d^6*x^5 + 4*b^7*c^3*d^6*x^4 - b^8*c^2*d^6*x^3 - 2*b^9*c*d^6*
x^2)*e^3 - 2*(2*b^5*c^5*d^7*x^5 + b^6*c^4*d^7*x^4 - 4*b^7*c^3*d^7*x^3 - 3*b^8*c^2*d^7*x^2)*e^2 + (b^4*c^6*d^8*
x^5 - 2*b^5*c^5*d^8*x^4 - 7*b^6*c^4*d^8*x^3 - 4*b^7*c^3*d^8*x^2)*e)]

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Sympy [F(-1)] Timed out
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

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 [B] Leaf count of result is larger than twice the leaf count of optimal. 2413 vs. \(2 (447) = 894\).
time = 2.10, size = 2413, normalized size = 5.37 \begin {gather*} \text {Too large to display} \end {gather*}

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]

1/6*((32*sqrt(c*d^2 - b*d*e)*B*b*c^4*d^4*e^2 - 64*sqrt(c*d^2 - b*d*e)*A*c^5*d^4*e^2 - 80*sqrt(c*d^2 - b*d*e)*B
*b^2*c^3*d^3*e^3 + 128*sqrt(c*d^2 - b*d*e)*A*b*c^4*d^3*e^3 - 24*B*b^4*c^(3/2)*d^2*e^5*log(abs(2*c*d - b*e - 2*
sqrt(c*d^2 - b*d*e)*sqrt(c))) + 36*sqrt(c*d^2 - b*d*e)*B*b^3*c^2*d^2*e^4 - 24*sqrt(c*d^2 - b*d*e)*A*b^2*c^3*d^
2*e^4 + 9*B*b^5*sqrt(c)*d*e^6*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 30*A*b^4*c^(3/2)*d*e^6*l
og(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) - 18*sqrt(c*d^2 - b*d*e)*B*b^4*c*d*e^5 - 40*sqrt(c*d^2 -
b*d*e)*A*b^3*c^2*d*e^5 - 15*A*b^5*sqrt(c)*e^7*log(abs(2*c*d - b*e - 2*sqrt(c*d^2 - b*d*e)*sqrt(c))) + 30*sqrt(
c*d^2 - b*d*e)*A*b^4*c*e^6)*sgn(1/(x*e + d))/(sqrt(c*d^2 - b*d*e)*b^4*c^(7/2)*d^6 - 3*sqrt(c*d^2 - b*d*e)*b^5*
c^(5/2)*d^5*e + 3*sqrt(c*d^2 - b*d*e)*b^6*c^(3/2)*d^4*e^2 - sqrt(c*d^2 - b*d*e)*b^7*sqrt(c)*d^3*e^3) + 2*((((4
*(4*B*b*c^5*d^7*e^16*sgn(1/(x*e + d)) - 8*A*c^6*d^7*e^16*sgn(1/(x*e + d)) - 16*B*b^2*c^4*d^6*e^17*sgn(1/(x*e +
d)) + 28*A*b*c^5*d^6*e^17*sgn(1/(x*e + d)) + 21*B*b^3*c^3*d^5*e^18*sgn(1/(x*e + d)) - 30*A*b^2*c^4*d^5*e^18*s
gn(1/(x*e + d)) - 18*B*b^4*c^2*d^4*e^19*sgn(1/(x*e + d)) + 5*A*b^3*c^3*d^4*e^19*sgn(1/(x*e + d)) + 12*B*b^5*c*
d^3*e^20*sgn(1/(x*e + d)) + 18*A*b^4*c^2*d^3*e^20*sgn(1/(x*e + d)) - 3*B*b^6*d^2*e^21*sgn(1/(x*e + d)) - 18*A*
b^5*c*d^2*e^21*sgn(1/(x*e + d)) + 5*A*b^6*d*e^22*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^
5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2) + 3
*(B*b^4*c^2*d^5*e^20*sgn(1/(x*e + d)) - 2*B*b^5*c*d^4*e^21*sgn(1/(x*e + d)) - A*b^4*c^2*d^4*e^21*sgn(1/(x*e +
d)) + B*b^6*d^3*e^22*sgn(1/(x*e + d)) + 2*A*b^5*c*d^3*e^22*sgn(1/(x*e + d)) - A*b^6*d^2*e^23*sgn(1/(x*e + d)))
*e^(-1)/((b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1
/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2)*(x*e + d)))*e^(-1)/(x*e + d) - 3*(16*B*b*c^5*d^6*e^15*sgn(1/(
x*e + d)) - 32*A*c^6*d^6*e^15*sgn(1/(x*e + d)) - 56*B*b^2*c^4*d^5*e^16*sgn(1/(x*e + d)) + 96*A*b*c^5*d^5*e^16*
sgn(1/(x*e + d)) + 60*B*b^3*c^3*d^4*e^17*sgn(1/(x*e + d)) - 80*A*b^2*c^4*d^4*e^17*sgn(1/(x*e + d)) - 42*B*b^4*
c^2*d^3*e^18*sgn(1/(x*e + d)) + 20*B*b^5*c*d^2*e^19*sgn(1/(x*e + d)) + 46*A*b^4*c^2*d^2*e^19*sgn(1/(x*e + d))
- 3*B*b^6*d*e^20*sgn(1/(x*e + d)) - 30*A*b^5*c*d*e^20*sgn(1/(x*e + d)) + 5*A*b^6*e^21*sgn(1/(x*e + d)))/(b^4*c
^3*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 -
b^7*d^3*e^14*sgn(1/(x*e + d))^2))*e^(-1)/(x*e + d) + 6*(8*B*b*c^5*d^5*e^14*sgn(1/(x*e + d)) - 16*A*c^6*d^5*e^
14*sgn(1/(x*e + d)) - 24*B*b^2*c^4*d^4*e^15*sgn(1/(x*e + d)) + 40*A*b*c^5*d^4*e^15*sgn(1/(x*e + d)) + 19*B*b^3
*c^3*d^3*e^16*sgn(1/(x*e + d)) - 22*A*b^2*c^4*d^3*e^16*sgn(1/(x*e + d)) - 11*B*b^4*c^2*d^2*e^17*sgn(1/(x*e + d
)) - 7*A*b^3*c^3*d^2*e^17*sgn(1/(x*e + d)) + 3*B*b^5*c*d*e^18*sgn(1/(x*e + d)) + 15*A*b^4*c^2*d*e^18*sgn(1/(x*
e + d)) - 5*A*b^5*c*e^19*sgn(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*
e + d))^2 + 3*b^6*c*d^4*e^13*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2))*e^(-1)/(x*e + d) - (16*B*b
*c^5*d^4*e^13*sgn(1/(x*e + d)) - 32*A*c^6*d^4*e^13*sgn(1/(x*e + d)) - 40*B*b^2*c^4*d^3*e^14*sgn(1/(x*e + d)) +
64*A*b*c^5*d^3*e^14*sgn(1/(x*e + d)) + 18*B*b^3*c^3*d^2*e^15*sgn(1/(x*e + d)) - 12*A*b^2*c^4*d^2*e^15*sgn(1/(
x*e + d)) - 9*B*b^4*c^2*d*e^16*sgn(1/(x*e + d)) - 20*A*b^3*c^3*d*e^16*sgn(1/(x*e + d)) + 15*A*b^4*c^2*e^17*sgn
(1/(x*e + d)))/(b^4*c^3*d^6*e^11*sgn(1/(x*e + d))^2 - 3*b^5*c^2*d^5*e^12*sgn(1/(x*e + d))^2 + 3*b^6*c*d^4*e^13
*sgn(1/(x*e + d))^2 - b^7*d^3*e^14*sgn(1/(x*e + d))^2))/(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e +
d) - b*d*e/(x*e + d)^2)^(3/2) + 3*(8*B*c*d^2*e^6 - 3*B*b*d*e^7 - 10*A*c*d*e^7 + 5*A*b*e^8)*log(abs(2*c*d - b*e
- 2*sqrt(c*d^2 - b*d*e)*(sqrt(c - 2*c*d/(x*e + d) + c*d^2/(x*e + d)^2 + b*e/(x*e + d) - b*d*e/(x*e + d)^2) +
sqrt(c*d^2*e^2 - b*d*e^3)*e^(-1)/(x*e + d))))/((c^3*d^6*e - 3*b*c^2*d^5*e^2 + 3*b^2*c*d^4*e^3 - b^3*d^3*e^4)*s
qrt(c*d^2 - b*d*e)*sgn(1/(x*e + d))))*e^(-2)

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Mupad [F]
time = 0.00, size = -1, normalized size = -0.00 \begin {gather*} \int \frac {A+B\,x}{{\left (c\,x^2+b\,x\right )}^{5/2}\,{\left (d+e\,x\right )}^2} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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