3.13.83 \(\int \frac {A+B x}{(d+e x)^{3/2} (b x+c x^2)^{5/2}} \, dx\) [1283]

Optimal. Leaf size=706 \[ -\frac {2 (A b (c d-b e)+c (2 A c d-b (B d+A e)) x)}{3 b^2 d (c d-b e) \sqrt {d+e x} \left (b x+c x^2\right )^{3/2}}+\frac {2 \left (b (c d-b e) \left (8 A c^2 d^2+b^2 e (3 B d-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) x\right )}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)+b^4 \left (6 B d e^3-8 A e^4\right )\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}} \]

[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)/(c*x^2+b*x)^(3/2)/(e*x+d)^(1/2)+2/3*(b*(-b*e+
c*d)*(8*A*c^2*d^2+b^2*e*(-4*A*e+3*B*d)-b*c*d*(3*A*e+4*B*d))+c*(16*A*c^3*d^3+15*b^2*B*c*d^2*e-b^3*e^2*(-4*A*e+3
*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)^(1/2)/(c*x^2+b*x)^(1/2)-2/3*(16*A*c^4*d^4-b^3*c
*d*e^2*(-7*A*e+9*B*d)+2*b^4*e^3*(-4*A*e+3*B*d)-8*b*c^3*d^3*(4*A*e+B*d)+b^2*c^2*d^2*e*(9*A*e+19*B*d))*EllipticE
(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(e*x+d)^(1/2)/(-b)^(7/2)/d^3/(-b*
e+c*d)^3/(1+e*x/d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*(16*A*c^3*d^3+15*b^2*B*c*d^2*e-b^3*e^2*(-4*A*e+3*B*d)-8*b*c^2*d
^2*(3*A*e+B*d))*EllipticF(c^(1/2)*x^(1/2)/(-b)^(1/2),(b*e/c/d)^(1/2))*c^(1/2)*x^(1/2)*(c*x/b+1)^(1/2)*(1+e*x/d
)^(1/2)/(-b)^(7/2)/d^2/(-b*e+c*d)^2/(e*x+d)^(1/2)/(c*x^2+b*x)^(1/2)+2/3*e*(16*A*c^4*d^4-b^3*c*d*e^2*(-7*A*e+9*
B*d)-8*b*c^3*d^3*(4*A*e+B*d)+b^2*c^2*d^2*e*(9*A*e+19*B*d)+b^4*(-8*A*e^4+6*B*d*e^3))*(c*x^2+b*x)^(1/2)/b^4/d^3/
(-b*e+c*d)^3/(e*x+d)^(1/2)

________________________________________________________________________________________

Rubi [A]
time = 1.25, antiderivative size = 706, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 8, integrand size = 28, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.286, Rules used = {836, 848, 857, 729, 113, 111, 118, 117} \begin {gather*} \frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {\frac {e x}{d}+1} \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right ) F\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}-\frac {2 \sqrt {c} \sqrt {x} \sqrt {\frac {c x}{b}+1} \sqrt {d+e x} \left (2 b^4 e^3 (3 B d-4 A e)-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right ) E\left (\text {ArcSin}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 \sqrt {b x+c x^2} \sqrt {\frac {e x}{d}+1} (c d-b e)^3}-\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} \sqrt {d+e x} (c d-b e)}+\frac {2 \left (b (c d-b e) \left (b^2 e (3 B d-4 A e)-b c d (3 A e+4 B d)+8 A c^2 d^2\right )+c x \left (b^3 \left (-e^2\right ) (3 B d-4 A e)-8 b c^2 d^2 (3 A e+B d)+16 A c^3 d^3+15 b^2 B c d^2 e\right )\right )}{3 b^4 d^2 \sqrt {b x+c x^2} \sqrt {d+e x} (c d-b e)^2}+\frac {2 e \sqrt {b x+c x^2} \left (b^4 \left (6 B d e^3-8 A e^4\right )-b^3 c d e^2 (9 B d-7 A e)+b^2 c^2 d^2 e (9 A e+19 B d)-8 b c^3 d^3 (4 A e+B d)+16 A c^4 d^4\right )}{3 b^4 d^3 \sqrt {d+e x} (c d-b e)^3} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(A + B*x)/((d + e*x)^(3/2)*(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)*Sqrt[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 - 4*A*e) - b*c*d*(4*B*d + 3*A*e)) + c*(16*A*c^3*d^3 + 15*b^2*
B*c*d^2*e - b^3*e^2*(3*B*d - 4*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[d + e*x]*Sq
rt[b*x + c*x^2]) + (2*e*(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) - 8*b*c^3*d^3*(B*d + 4*A*e) + b^2*c^2*d^2*
e*(19*B*d + 9*A*e) + b^4*(6*B*d*e^3 - 8*A*e^4))*Sqrt[b*x + c*x^2])/(3*b^4*d^3*(c*d - b*e)^3*Sqrt[d + e*x]) - (
2*Sqrt[c]*(16*A*c^4*d^4 - b^3*c*d*e^2*(9*B*d - 7*A*e) + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3*d^3*(B*d + 4*A*e)
+ b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*Sqrt[d + e*x]*EllipticE[ArcSin[(Sqrt[c]*Sqrt[x])/S
qrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^3*(c*d - b*e)^3*Sqrt[1 + (e*x)/d]*Sqrt[b*x + c*x^2]) + (2*Sqrt[c]*(16*
A*c^3*d^3 + 15*b^2*B*c*d^2*e - b^3*e^2*(3*B*d - 4*A*e) - 8*b*c^2*d^2*(B*d + 3*A*e))*Sqrt[x]*Sqrt[1 + (c*x)/b]*
Sqrt[1 + (e*x)/d]*EllipticF[ArcSin[(Sqrt[c]*Sqrt[x])/Sqrt[-b]], (b*e)/(c*d)])/(3*(-b)^(7/2)*d^2*(c*d - b*e)^2*
Sqrt[d + e*x]*Sqrt[b*x + c*x^2])

Rule 111

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[2*(Sqrt[e]/b)*Rt[-b/
d, 2]*EllipticE[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] && NeQ[
d*e - c*f, 0] && GtQ[c, 0] && GtQ[e, 0] &&  !LtQ[-b/d, 0]

Rule 113

Int[Sqrt[(e_) + (f_.)*(x_)]/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Dist[Sqrt[e + f*x]*(Sqrt[
1 + d*(x/c)]/(Sqrt[c + d*x]*Sqrt[1 + f*(x/e)])), Int[Sqrt[1 + f*(x/e)]/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]), x], x] /
; FreeQ[{b, c, d, e, f}, x] && NeQ[d*e - c*f, 0] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 117

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Simp[(2/(b*Sqrt[e]))*Rt
[-b/d, 2]*EllipticF[ArcSin[Sqrt[b*x]/(Sqrt[c]*Rt[-b/d, 2])], c*(f/(d*e))], x] /; FreeQ[{b, c, d, e, f}, x] &&
GtQ[c, 0] && GtQ[e, 0] && (PosQ[-b/d] || NegQ[-b/f])

Rule 118

Int[1/(Sqrt[(b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]*Sqrt[(e_) + (f_.)*(x_)]), x_Symbol] :> Dist[Sqrt[1 + d*(x/c)]*
(Sqrt[1 + f*(x/e)]/(Sqrt[c + d*x]*Sqrt[e + f*x])), Int[1/(Sqrt[b*x]*Sqrt[1 + d*(x/c)]*Sqrt[1 + f*(x/e)]), x],
x] /; FreeQ[{b, c, d, e, f}, x] &&  !(GtQ[c, 0] && GtQ[e, 0])

Rule 729

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

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])

Rule 848

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)/((m + 1)*(c*d^2 - b*d*e + a*e^2))), x] + Dist[1/((m
+ 1)*(c*d^2 - b*d*e + a*e^2)), Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p*Simp[(c*d*f - f*b*e + a*e*g)*(m + 1)
+ b*(d*g - e*f)*(p + 1) - c*(e*f - d*g)*(m + 2*p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, g, p}, x] &&
NeQ[b^2 - 4*a*c, 0] && NeQ[c*d^2 - b*d*e + a*e^2, 0] && LtQ[m, -1] && (IntegerQ[m] || IntegerQ[p] || IntegersQ
[2*m, 2*p])

Rule 857

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_.), x_Symbol] :> Dis
t[g/e, Int[(d + e*x)^(m + 1)*(a + b*x + c*x^2)^p, x], x] + Dist[(e*f - d*g)/e, Int[(d + e*x)^m*(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]
&&  !IGtQ[m, 0]

Rubi steps

\begin {align*} \int \frac {A+B x}{(d+e x)^{3/2} \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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )-\frac {5}{2} c e (b B d-2 A c d+A b e) x}{(d+e x)^{3/2} \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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {4 \int \frac {\frac {1}{4} b e \left (8 A c^3 d^3+3 b^2 c d e (2 B d-A e)-b c^2 d^2 (4 B d+9 A e)-b^3 \left (6 B d e^2-8 A e^3\right )\right )+\frac {1}{4} c e \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) x}{(d+e x)^{3/2} \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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {8 \int \frac {\frac {1}{8} b c d e \left (8 A c^3 d^3+b^3 e^2 (3 B d-4 A e)+3 b^2 c d e (3 B d+A e)-b c^2 d^2 (4 B d+15 A e)\right )+\frac {1}{8} c e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) x}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right )\right ) \int \frac {1}{\sqrt {d+e x} \sqrt {b x+c x^2}} \, dx}{3 b^4 d^2 (c d-b e)^2}-\frac {\left (c \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right )\right ) \int \frac {\sqrt {d+e x}}{\sqrt {b x+c x^2}} \, dx}{3 b^4 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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}+\frac {\left (c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {1}{\sqrt {x} \sqrt {b+c x} \sqrt {d+e x}} \, dx}{3 b^4 d^2 (c d-b e)^2 \sqrt {b x+c x^2}}-\frac {\left (c \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {x} \sqrt {b+c x}\right ) \int \frac {\sqrt {d+e x}}{\sqrt {x} \sqrt {b+c x}} \, dx}{3 b^4 d^3 (c d-b e)^3 \sqrt {b x+c x^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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {\left (c \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x}\right ) \int \frac {\sqrt {1+\frac {e x}{d}}}{\sqrt {x} \sqrt {1+\frac {c x}{b}}} \, dx}{3 b^4 d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {\left (c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}\right ) \int \frac {1}{\sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}}} \, dx}{3 b^4 d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^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) \sqrt {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-4 A e)-b c d (4 B d+3 A e)\right )+c \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 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 {d+e x} \sqrt {b x+c x^2}}+\frac {2 e \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {b x+c x^2}}{3 b^4 d^3 (c d-b e)^3 \sqrt {d+e x}}-\frac {2 \sqrt {c} \left (16 A c^4 d^4-b^3 c d e^2 (9 B d-7 A e)+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {d+e x} E\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^3 (c d-b e)^3 \sqrt {1+\frac {e x}{d}} \sqrt {b x+c x^2}}+\frac {2 \sqrt {c} \left (16 A c^3 d^3+15 b^2 B c d^2 e-b^3 e^2 (3 B d-4 A e)-8 b c^2 d^2 (B d+3 A e)\right ) \sqrt {x} \sqrt {1+\frac {c x}{b}} \sqrt {1+\frac {e x}{d}} F\left (\sin ^{-1}\left (\frac {\sqrt {c} \sqrt {x}}{\sqrt {-b}}\right )|\frac {b e}{c d}\right )}{3 (-b)^{7/2} d^2 (c d-b e)^2 \sqrt {d+e x} \sqrt {b x+c x^2}}\\ \end {align*}

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Mathematica [C] Result contains complex when optimal does not.
time = 23.77, size = 628, normalized size = 0.89 \begin {gather*} \frac {2 \left (b \left (3 b^4 e^4 (B d-A e) x^2 (b+c x)^2+b c^3 (b B-A c) d^3 (-c d+b e) x^2 (d+e x)+c^3 d^3 \left (8 A c^2 d+10 b^2 B e-b c (5 B d+13 A e)\right ) x^2 (b+c x) (d+e x)+A b d (-c d+b e)^3 (b+c x)^2 (d+e x)+(c d-b e)^3 (-3 b B d+8 A c d+5 A b e) x (b+c x)^2 (d+e x)\right )-\sqrt {\frac {b}{c}} c x (b+c x) \left (\sqrt {\frac {b}{c}} \left (16 A c^4 d^4+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^3 c d e^2 (-9 B d+7 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) (b+c x) (d+e x)+i b e \left (16 A c^4 d^4+2 b^4 e^3 (3 B d-4 A e)-8 b c^3 d^3 (B d+4 A e)+b^3 c d e^2 (-9 B d+7 A e)+b^2 c^2 d^2 e (19 B d+9 A e)\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} E\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )-i b e (c d-b e) \left (8 A c^3 d^3+3 b^2 c d e (2 B d-A e)-b c^2 d^2 (4 B d+9 A e)+b^3 \left (-6 B d e^2+8 A e^3\right )\right ) \sqrt {1+\frac {b}{c x}} \sqrt {1+\frac {d}{e x}} x^{3/2} F\left (i \sinh ^{-1}\left (\frac {\sqrt {\frac {b}{c}}}{\sqrt {x}}\right )|\frac {c d}{b e}\right )\right )\right )}{3 b^5 d^3 (c d-b e)^3 (x (b+c x))^{3/2} \sqrt {d+e x}} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(A + B*x)/((d + e*x)^(3/2)*(b*x + c*x^2)^(5/2)),x]

[Out]

(2*(b*(3*b^4*e^4*(B*d - A*e)*x^2*(b + c*x)^2 + b*c^3*(b*B - A*c)*d^3*(-(c*d) + b*e)*x^2*(d + e*x) + c^3*d^3*(8
*A*c^2*d + 10*b^2*B*e - b*c*(5*B*d + 13*A*e))*x^2*(b + c*x)*(d + e*x) + A*b*d*(-(c*d) + b*e)^3*(b + c*x)^2*(d
+ e*x) + (c*d - b*e)^3*(-3*b*B*d + 8*A*c*d + 5*A*b*e)*x*(b + c*x)^2*(d + e*x)) - Sqrt[b/c]*c*x*(b + c*x)*(Sqrt
[b/c]*(16*A*c^4*d^4 + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3*d^3*(B*d + 4*A*e) + b^3*c*d*e^2*(-9*B*d + 7*A*e) + b
^2*c^2*d^2*e*(19*B*d + 9*A*e))*(b + c*x)*(d + e*x) + I*b*e*(16*A*c^4*d^4 + 2*b^4*e^3*(3*B*d - 4*A*e) - 8*b*c^3
*d^3*(B*d + 4*A*e) + b^3*c*d*e^2*(-9*B*d + 7*A*e) + b^2*c^2*d^2*e*(19*B*d + 9*A*e))*Sqrt[1 + b/(c*x)]*Sqrt[1 +
d/(e*x)]*x^(3/2)*EllipticE[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)] - I*b*e*(c*d - b*e)*(8*A*c^3*d^3 + 3*b^
2*c*d*e*(2*B*d - A*e) - b*c^2*d^2*(4*B*d + 9*A*e) + b^3*(-6*B*d*e^2 + 8*A*e^3))*Sqrt[1 + b/(c*x)]*Sqrt[1 + d/(
e*x)]*x^(3/2)*EllipticF[I*ArcSinh[Sqrt[b/c]/Sqrt[x]], (c*d)/(b*e)])))/(3*b^5*d^3*(c*d - b*e)^3*(x*(b + c*x))^(
3/2)*Sqrt[d + e*x])

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Maple [B] Leaf count of result is larger than twice the leaf count of optimal. \(4000\) vs. \(2(646)=1292\).
time = 0.67, size = 4001, normalized size = 5.67

method	result	size
elliptic	\(\frac {\sqrt {x \left (e x +d \right ) \left (c x +b \right )}\, \left (\frac {2 c \left (A c -B b \right ) \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 b^{3} \left (b e -c d \right )^{2} \left (\frac {b}{c}+x \right )^{2}}+\frac {2 \left (c e \,x^{2}+c d x \right ) c^{2} \left (13 A b c e -8 A \,c^{2} d -10 b^{2} B e +5 B d c b \right )}{3 b^{4} \left (b e -c d \right )^{3} \sqrt {\left (\frac {b}{c}+x \right ) \left (c e \,x^{2}+c d x \right )}}-\frac {2 A \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}{3 d^{2} b^{3} x^{2}}+\frac {2 \left (c e \,x^{2}+b e x +c d x +b d \right ) \left (5 A b e +8 A c d -3 B b d \right )}{3 b^{4} d^{3} \sqrt {x \left (c e \,x^{2}+b e x +c d x +b d \right )}}+\frac {2 \left (c e \,x^{2}+b e x \right ) e^{3} \left (A e -B d \right )}{\left (b e -c d \right )^{3} d^{3} \sqrt {\left (x +\frac {d}{e}\right ) \left (c e \,x^{2}+b e x \right )}}+\frac {2 \left (\frac {\left (A c -B b \right ) c^{2} e}{3 b^{3} \left (b e -c d \right )^{2}}-\frac {c^{2} \left (13 A b c e -8 A \,c^{2} d -10 b^{2} B e +5 B d c b \right )}{3 \left (b e -c d \right )^{2} b^{4}}-\frac {c^{3} d \left (13 A b c e -8 A \,c^{2} d -10 b^{2} B e +5 B d c b \right )}{3 b^{4} \left (b e -c d \right )^{3}}-\frac {A c e}{3 b^{3} d^{2}}+\frac {e^{3} \left (A e -B d \right )}{\left (b e -c d \right )^{2} d^{3}}-\frac {b \,e^{4} \left (A e -B d \right )}{\left (b e -c d \right )^{3} d^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}+\frac {2 \left (-\frac {c^{3} e \left (13 A b c e -8 A \,c^{2} d -10 b^{2} B e +5 B d c b \right )}{3 b^{4} \left (b e -c d \right )^{3}}-\frac {c e \left (5 A b e +8 A c d -3 B b d \right )}{3 b^{4} d^{3}}-\frac {\left (A e -B d \right ) e^{4} c}{\left (b e -c d \right )^{3} d^{3}}\right ) b \sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}\, \sqrt {\frac {x +\frac {d}{e}}{-\frac {b}{c}+\frac {d}{e}}}\, \sqrt {-\frac {c x}{b}}\, \left (\left (-\frac {b}{c}+\frac {d}{e}\right ) \EllipticE \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )-\frac {d \EllipticF \left (\sqrt {\frac {\left (\frac {b}{c}+x \right ) c}{b}}, \sqrt {-\frac {b}{c \left (-\frac {b}{c}+\frac {d}{e}\right )}}\right )}{e}\right )}{c \sqrt {c e \,x^{3}+b e \,x^{2}+c d \,x^{2}+x b d}}\right )}{\sqrt {x \left (c x +b \right )}\, \sqrt {e x +d}}\)	\(896\)
default	\(\text {Expression too large to display}\)	\(4001\)

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x,method=_RETURNVERBOSE)

[Out]

2/3*(8*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c
*d))^(1/2))*x*b^7*e^5+8*A*x^3*b*c^6*d^4*e-10*A*x^3*b^4*c^3*d*e^4-12*B*x^3*b^5*c^2*d*e^4-20*B*x^3*b^3*c^4*d^3*e
^2-7*B*x^3*b^2*c^5*d^4*e+9*B*x*b^5*c^2*d^3*e^2-9*B*x*b^4*c^3*d^4*e+3*B*x^2*b^5*c^2*d^2*e^3+32*A*x^4*b*c^6*d^3*
e^2-7*A*x^4*b^3*c^4*d*e^4-9*A*x^4*b^2*c^5*d^2*e^3+9*B*x^2*b^4*c^3*d^3*e^2+43*A*x^2*b^2*c^5*d^4*e+A*x^2*b^5*c^2
*d*e^4-18*A*x^2*b^4*c^3*d^2*e^3-16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF((
(c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^6*d^5+8*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x
/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^6*c*e^5+16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/
(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b*c^6*d^5+8*B*((c*x+b)/
b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^
2*c^5*d^5-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x^2*b^2*c^5*d^5-16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^5*d^5+16*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^2*c^5*d^5+8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c
/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^3*c^4*d^5-6*B*((c*x+b)
/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^7
*d*e^4-8*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e
-c*d))^(1/2))*x*b^3*c^4*d^5-A*b^6*c*d^2*e^3+3*A*b^5*c^2*d^3*e^2-3*A*b^4*c^3*d^4*e-2*A*x^2*b^3*c^4*d^3*e^2-3*B*
x*b^6*c*d^2*e^3+15*B*x^3*b^4*c^3*d^2*e^3+4*A*x*b^6*c*d*e^4-6*A*x*b^5*c^2*d^2*e^3-6*A*x*b^4*c^3*d^3*e^2+14*A*x*
b^3*c^4*d^4*e-19*B*x^4*b^2*c^5*d^3*e^2+8*B*x^4*b*c^6*d^4*e-6*B*x^4*b^4*c^3*d*e^4+9*B*x^4*b^3*c^4*d^2*e^3-26*B*
x^2*b^3*c^4*d^4*e-6*B*x^2*b^6*c*d*e^4-22*A*x^3*b^3*c^4*d^2*e^3+40*A*x^3*b^2*c^5*d^3*e^2+4*A*((c*x+b)/b)^(1/2)*
(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*c^2*d*e
^4-4*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d
))^(1/2))*x^2*b^4*c^3*d^2*e^3-24*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c
*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^4*d^3*e^2+40*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*
(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^5*d^4*e-15*A*((c*x+b)/b)^(1/2)*(-(
e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*c^2*d*e^4-
2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^
(1/2))*x^2*b^4*c^3*d^2*e^3+41*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+
b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^4*d^3*e^2-48*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c
*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^2*c^5*d^4*e-3*B*((c*x+b)/b)^(1/2)*(-(e*x+
d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*c^2*d^2*e^3+18
*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(
1/2))*x^2*b^4*c^3*d^3*e^2-23*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b
)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^4*d^4*e-6*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b
)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^6*c*d*e^4+15*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/
(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^5*c^2*d^2*e^3-28*B*((
c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))
*x^2*b^4*c^3*d^3*e^2+27*B*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(((c*x+b)/b)^
(1/2),(b*e/(b*e-c*d))^(1/2))*x^2*b^3*c^4*d^4*e+4*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/
2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^6*c*d*e^4-4*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d
))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^5*c^2*d^2*e^3-24*A*((c*x+b)/b)^
(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^4*c^3
*d^3*e^2+40*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticF(((c*x+b)/b)^(1/2),(b*e/(
b*e-c*d))^(1/2))*x*b^3*c^4*d^4*e-15*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*x/b)^(1/2)*EllipticE(
((c*x+b)/b)^(1/2),(b*e/(b*e-c*d))^(1/2))*x*b^6*c*d*e^4-2*A*((c*x+b)/b)^(1/2)*(-(e*x+d)*c/(b*e-c*d))^(1/2)*(-c*
x/b)^(1/2)*EllipticE(((c*x+b)/b)^(1/2),(b*e/(b*...

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Maxima [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Failed to integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="maxima")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*(x*e + d)^(3/2)), x)

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Fricas [C] Result contains higher order function than in optimal. Order 9 vs. order 4.
time = 1.52, size = 2222, normalized size = 3.15 \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)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="fricas")

[Out]

2/9*((8*(B*b*c^6 - 2*A*c^7)*d^6*x^4 + 16*(B*b^2*c^5 - 2*A*b*c^6)*d^6*x^3 + 8*(B*b^3*c^4 - 2*A*b^2*c^5)*d^6*x^2
+ 8*(A*b^5*c^2*x^5 + 2*A*b^6*c*x^4 + A*b^7*x^3)*e^6 + (8*A*b^7*d*x^2 - (6*B*b^5*c^2 + 11*A*b^4*c^3)*d*x^5 - 2
*(6*B*b^6*c + 7*A*b^5*c^2)*d*x^4 - (6*B*b^7 - 5*A*b^6*c)*d*x^3)*e^5 - (29*A*b^5*c^2*d^2*x^3 - (12*B*b^4*c^3 -
7*A*b^3*c^4)*d^2*x^5 - (18*B*b^5*c^2 - 25*A*b^4*c^3)*d^2*x^4 + (6*B*b^7 + 11*A*b^6*c)*d^2*x^2)*e^4 + ((17*B*b^
3*c^4 - 22*A*b^2*c^5)*d^3*x^5 + (46*B*b^4*c^3 - 51*A*b^3*c^4)*d^3*x^4 + (41*B*b^5*c^2 - 36*A*b^4*c^3)*d^3*x^3
+ (12*B*b^6*c - 7*A*b^5*c^2)*d^3*x^2)*e^3 - ((23*B*b^2*c^5 - 40*A*b*c^6)*d^4*x^5 + 29*(B*b^3*c^4 - 2*A*b^2*c^5
)*d^4*x^4 - (11*B*b^4*c^3 - 4*A*b^3*c^4)*d^4*x^3 - (17*B*b^5*c^2 - 22*A*b^4*c^3)*d^4*x^2)*e^2 + (8*(B*b*c^6 -
2*A*c^7)*d^5*x^5 - (7*B*b^2*c^5 - 8*A*b*c^6)*d^5*x^4 - 2*(19*B*b^3*c^4 - 32*A*b^2*c^5)*d^5*x^3 - (23*B*b^4*c^3
- 40*A*b^3*c^4)*d^5*x^2)*e)*sqrt(c)*e^(1/2)*weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2,
-4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)
+ 3*(8*(A*b^4*c^3*x^5 + 2*A*b^5*c^2*x^4 + A*b^6*c*x^3)*e^6 + (8*A*b^6*c*d*x^2 - (6*B*b^4*c^3 + 7*A*b^3*c^4)*d
*x^5 - 6*(2*B*b^5*c^2 + A*b^4*c^3)*d*x^4 - 3*(2*B*b^6*c - 3*A*b^5*c^2)*d*x^3)*e^5 + (9*(B*b^3*c^4 - A*b^2*c^5)
*d^2*x^5 + (12*B*b^4*c^3 - 25*A*b^3*c^4)*d^2*x^4 - (3*B*b^5*c^2 + 23*A*b^4*c^3)*d^2*x^3 - (6*B*b^6*c + 7*A*b^5
*c^2)*d^2*x^2)*e^4 - ((19*B*b^2*c^5 - 32*A*b*c^6)*d^3*x^5 + (29*B*b^3*c^4 - 55*A*b^2*c^5)*d^3*x^4 + (B*b^4*c^3
- 14*A*b^3*c^4)*d^3*x^3 - 9*(B*b^5*c^2 - A*b^4*c^3)*d^3*x^2)*e^3 - (3*B*b^2*c^5*d^4*x^4 - 8*(B*b*c^6 - 2*A*c^
7)*d^4*x^5 + 6*(5*B*b^3*c^4 - 8*A*b^2*c^5)*d^4*x^3 + (19*B*b^4*c^3 - 32*A*b^3*c^4)*d^4*x^2)*e^2 + 8*((B*b*c^6
- 2*A*c^7)*d^5*x^4 + 2*(B*b^2*c^5 - 2*A*b*c^6)*d^5*x^3 + (B*b^3*c^4 - 2*A*b^2*c^5)*d^5*x^2)*e)*sqrt(c)*e^(1/2)
*weierstrassZeta(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3 - 3*b*c^2*d^2*e - 3*b^2*c*d*e^
2 + 2*b^3*e^3)*e^(-3)/c^3, weierstrassPInverse(4/3*(c^2*d^2 - b*c*d*e + b^2*e^2)*e^(-2)/c^2, -4/27*(2*c^3*d^3
- 3*b*c^2*d^2*e - 3*b^2*c*d*e^2 + 2*b^3*e^3)*e^(-3)/c^3, 1/3*(c*d + (3*c*x + b)*e)*e^(-1)/c)) + 3*sqrt(c*x^2 +
b*x)*(8*(A*b^4*c^3*x^4 + 2*A*b^5*c^2*x^3 + A*b^6*c*x^2)*e^6 + (4*A*b^6*c*d*x - (6*B*b^4*c^3 + 7*A*b^3*c^4)*d*
x^4 - 2*(6*B*b^5*c^2 + 5*A*b^4*c^3)*d*x^3 - (6*B*b^6*c - A*b^5*c^2)*d*x^2)*e^5 - (A*b^6*c*d^2 - 9*(B*b^3*c^4 -
A*b^2*c^5)*d^2*x^4 - (15*B*b^4*c^3 - 22*A*b^3*c^4)*d^2*x^3 - 3*(B*b^5*c^2 - 6*A*b^4*c^3)*d^2*x^2 + 3*(B*b^6*c
+ 2*A*b^5*c^2)*d^2*x)*e^4 + (3*A*b^5*c^2*d^3 - (19*B*b^2*c^5 - 32*A*b*c^6)*d^3*x^4 - 20*(B*b^3*c^4 - 2*A*b^2*
c^5)*d^3*x^3 + (9*B*b^4*c^3 - 2*A*b^3*c^4)*d^3*x^2 + 3*(3*B*b^5*c^2 - 2*A*b^4*c^3)*d^3*x)*e^3 - (3*A*b^4*c^3*d
^4 - 8*(B*b*c^6 - 2*A*c^7)*d^4*x^4 + (7*B*b^2*c^5 - 8*A*b*c^6)*d^4*x^3 + (26*B*b^3*c^4 - 43*A*b^2*c^5)*d^4*x^2
+ (9*B*b^4*c^3 - 14*A*b^3*c^4)*d^4*x)*e^2 + (A*b^3*c^4*d^5 + 8*(B*b*c^6 - 2*A*c^7)*d^5*x^3 + 12*(B*b^2*c^5 -
2*A*b*c^6)*d^5*x^2 + 3*(B*b^3*c^4 - 2*A*b^2*c^5)*d^5*x)*e)*sqrt(x*e + d))/((b^7*c^3*d^3*x^5 + 2*b^8*c^2*d^3*x^
4 + b^9*c*d^3*x^3)*e^5 - (3*b^6*c^4*d^4*x^5 + 5*b^7*c^3*d^4*x^4 + b^8*c^2*d^4*x^3 - b^9*c*d^4*x^2)*e^4 + 3*(b^
5*c^5*d^5*x^5 + b^6*c^4*d^5*x^4 - b^7*c^3*d^5*x^3 - b^8*c^2*d^5*x^2)*e^3 - (b^4*c^6*d^6*x^5 - b^5*c^5*d^6*x^4
- 5*b^6*c^4*d^6*x^3 - 3*b^7*c^3*d^6*x^2)*e^2 - (b^4*c^6*d^7*x^4 + 2*b^5*c^5*d^7*x^3 + b^6*c^4*d^7*x^2)*e)

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Sympy [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {A + B x}{\left (x \left (b + c x\right )\right )^{\frac {5}{2}} \left (d + e x\right )^{\frac {3}{2}}}\, dx \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)**(3/2)/(c*x**2+b*x)**(5/2),x)

[Out]

Integral((A + B*x)/((x*(b + c*x))**(5/2)*(d + e*x)**(3/2)), x)

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Giac [F]
time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {could not integrate} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)/(e*x+d)^(3/2)/(c*x^2+b*x)^(5/2),x, algorithm="giac")

[Out]

integrate((B*x + A)/((c*x^2 + b*x)^(5/2)*(x*e + d)^(3/2)), x)

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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 )}^{3/2}} \,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)^(3/2)),x)

[Out]

int((A + B*x)/((b*x + c*x^2)^(5/2)*(d + e*x)^(3/2)), x)

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