3.1244 \(\int \frac {A+B x}{(d+e x)^{3/2} (b x+c x^2)^2} \, dx\)

Optimal. Leaf size=254 \[ -\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-3 A b e-4 A c d+2 b B d)}{b^3 d^{5/2}}-\frac {e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)}+\frac {c^{3/2} \left (7 A b c e-4 A c^2 d-5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}} \]

[Out]

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

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Rubi [A]  time = 0.56, antiderivative size = 254, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 5, integrand size = 26, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.192, Rules used = {822, 828, 826, 1166, 208} \[ -\frac {e \left (b^2 (-e) (2 B d-3 A e)-b c d (2 A e+B d)+2 A c^2 d^2\right )}{b^2 d^2 \sqrt {d+e x} (c d-b e)^2}+\frac {c^{3/2} \left (7 A b c e-4 A c^2 d-5 b^2 B e+2 b B c d\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 (c d-b e)^{5/2}}-\frac {\tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-3 A b e-4 A c d+2 b B d)}{b^3 d^{5/2}}-\frac {c x (2 A c d-b (A e+B d))+A b (c d-b e)}{b^2 d \left (b x+c x^2\right ) \sqrt {d+e x} (c d-b e)} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 208

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

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 826

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

Rule 828

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

Rule 1166

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

Rubi steps

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

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Mathematica [C]  time = 0.18, size = 191, normalized size = 0.75 \[ \frac {-x (b+c x) \left (c d^2 \left (b c (7 A e+2 B d)-4 A c^2 d-5 b^2 B e\right ) \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {c (d+e x)}{c d-b e}\right )+(c d-b e)^2 \, _2F_1\left (-\frac {1}{2},1;\frac {1}{2};\frac {e x}{d}+1\right ) (3 A b e+4 A c d-2 b B d)\right )-A b^2 d (c d-b e)^2-b c d x (b e-c d) (A b e-2 A c d+b B d)}{b^3 d^2 x (b+c x) \sqrt {d+e x} (c d-b e)^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [B]  time = 36.32, size = 3240, normalized size = 12.76 \[ \text {result too large to display} \]

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)^2,x, algorithm="fricas")

[Out]

[1/2*(((2*(B*b*c^3 - 2*A*c^4)*d^4*e - (5*B*b^2*c^2 - 7*A*b*c^3)*d^3*e^2)*x^3 + (2*(B*b*c^3 - 2*A*c^4)*d^5 - 3*
(B*b^2*c^2 - A*b*c^3)*d^4*e - (5*B*b^3*c - 7*A*b^2*c^2)*d^3*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^5 - (5*B*b
^3*c - 7*A*b^2*c^2)*d^4*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(
c/(c*d - b*e)))/(c*x + b)) + ((3*A*b^3*c*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^3*e + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e^2
- 2*(B*b^3*c + A*b^2*c^2)*d*e^3)*x^3 + (3*A*b^4*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^4 + (2*B*b^2*c^2 - A*b*c^3)*d^3
*e + (2*B*b^3*c - 7*A*b^2*c^2)*d^2*e^2 - (2*B*b^4 - A*b^3*c)*d*e^3)*x^2 + (3*A*b^4*d*e^3 - 2*(B*b^2*c^2 - 2*A*
b*c^3)*d^4 + (4*B*b^3*c - 5*A*b^2*c^2)*d^3*e - 2*(B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x +
d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c^2*d^4 - 2*A*b^3*c*d^3*e + A*b^4*d^2*e^2 + (3*A*b^3*c*d*e^3 - (B*b^2*c^2 - 2
*A*b*c^3)*d^3*e - 2*(B*b^3*c + A*b^2*c^2)*d^2*e^2)*x^2 - (A*b^2*c^2*d^3*e - 3*A*b^4*d*e^3 + (B*b^2*c^2 - 2*A*b
*c^3)*d^4 + (2*B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^3*c^3*d^5*e - 2*b^4*c^2*d^4*e^2 + b^5*c*d^3*e^3
)*x^3 + (b^3*c^3*d^6 - b^4*c^2*d^5*e - b^5*c*d^4*e^2 + b^6*d^3*e^3)*x^2 + (b^4*c^2*d^6 - 2*b^5*c*d^5*e + b^6*d
^4*e^2)*x), 1/2*(2*((2*(B*b*c^3 - 2*A*c^4)*d^4*e - (5*B*b^2*c^2 - 7*A*b*c^3)*d^3*e^2)*x^3 + (2*(B*b*c^3 - 2*A*
c^4)*d^5 - 3*(B*b^2*c^2 - A*b*c^3)*d^4*e - (5*B*b^3*c - 7*A*b^2*c^2)*d^3*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)
*d^5 - (5*B*b^3*c - 7*A*b^2*c^2)*d^4*e)*x)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d
- b*e))/(c*e*x + c*d)) + ((3*A*b^3*c*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^3*e + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e^2 -
2*(B*b^3*c + A*b^2*c^2)*d*e^3)*x^3 + (3*A*b^4*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^4 + (2*B*b^2*c^2 - A*b*c^3)*d^3*e
+ (2*B*b^3*c - 7*A*b^2*c^2)*d^2*e^2 - (2*B*b^4 - A*b^3*c)*d*e^3)*x^2 + (3*A*b^4*d*e^3 - 2*(B*b^2*c^2 - 2*A*b*c
^3)*d^4 + (4*B*b^3*c - 5*A*b^2*c^2)*d^3*e - 2*(B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(d)*log((e*x + 2*sqrt(e*x + d)
*sqrt(d) + 2*d)/x) - 2*(A*b^2*c^2*d^4 - 2*A*b^3*c*d^3*e + A*b^4*d^2*e^2 + (3*A*b^3*c*d*e^3 - (B*b^2*c^2 - 2*A*
b*c^3)*d^3*e - 2*(B*b^3*c + A*b^2*c^2)*d^2*e^2)*x^2 - (A*b^2*c^2*d^3*e - 3*A*b^4*d*e^3 + (B*b^2*c^2 - 2*A*b*c^
3)*d^4 + (2*B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^3*c^3*d^5*e - 2*b^4*c^2*d^4*e^2 + b^5*c*d^3*e^3)*x
^3 + (b^3*c^3*d^6 - b^4*c^2*d^5*e - b^5*c*d^4*e^2 + b^6*d^3*e^3)*x^2 + (b^4*c^2*d^6 - 2*b^5*c*d^5*e + b^6*d^4*
e^2)*x), -1/2*(2*((3*A*b^3*c*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^3*e + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e^2 - 2*(B*b^3*
c + A*b^2*c^2)*d*e^3)*x^3 + (3*A*b^4*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^4 + (2*B*b^2*c^2 - A*b*c^3)*d^3*e + (2*B*b^
3*c - 7*A*b^2*c^2)*d^2*e^2 - (2*B*b^4 - A*b^3*c)*d*e^3)*x^2 + (3*A*b^4*d*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^4 +
(4*B*b^3*c - 5*A*b^2*c^2)*d^3*e - 2*(B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) -
((2*(B*b*c^3 - 2*A*c^4)*d^4*e - (5*B*b^2*c^2 - 7*A*b*c^3)*d^3*e^2)*x^3 + (2*(B*b*c^3 - 2*A*c^4)*d^5 - 3*(B*b^
2*c^2 - A*b*c^3)*d^4*e - (5*B*b^3*c - 7*A*b^2*c^2)*d^3*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^5 - (5*B*b^3*c
- 7*A*b^2*c^2)*d^4*e)*x)*sqrt(c/(c*d - b*e))*log((c*e*x + 2*c*d - b*e + 2*(c*d - b*e)*sqrt(e*x + d)*sqrt(c/(c*
d - b*e)))/(c*x + b)) + 2*(A*b^2*c^2*d^4 - 2*A*b^3*c*d^3*e + A*b^4*d^2*e^2 + (3*A*b^3*c*d*e^3 - (B*b^2*c^2 - 2
*A*b*c^3)*d^3*e - 2*(B*b^3*c + A*b^2*c^2)*d^2*e^2)*x^2 - (A*b^2*c^2*d^3*e - 3*A*b^4*d*e^3 + (B*b^2*c^2 - 2*A*b
*c^3)*d^4 + (2*B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^3*c^3*d^5*e - 2*b^4*c^2*d^4*e^2 + b^5*c*d^3*e^3
)*x^3 + (b^3*c^3*d^6 - b^4*c^2*d^5*e - b^5*c*d^4*e^2 + b^6*d^3*e^3)*x^2 + (b^4*c^2*d^6 - 2*b^5*c*d^5*e + b^6*d
^4*e^2)*x), (((2*(B*b*c^3 - 2*A*c^4)*d^4*e - (5*B*b^2*c^2 - 7*A*b*c^3)*d^3*e^2)*x^3 + (2*(B*b*c^3 - 2*A*c^4)*d
^5 - 3*(B*b^2*c^2 - A*b*c^3)*d^4*e - (5*B*b^3*c - 7*A*b^2*c^2)*d^3*e^2)*x^2 + (2*(B*b^2*c^2 - 2*A*b*c^3)*d^5 -
(5*B*b^3*c - 7*A*b^2*c^2)*d^4*e)*x)*sqrt(-c/(c*d - b*e))*arctan(-(c*d - b*e)*sqrt(e*x + d)*sqrt(-c/(c*d - b*e
))/(c*e*x + c*d)) - ((3*A*b^3*c*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^3*e + (4*B*b^2*c^2 - 5*A*b*c^3)*d^2*e^2 - 2*(B*b
^3*c + A*b^2*c^2)*d*e^3)*x^3 + (3*A*b^4*e^4 - 2*(B*b*c^3 - 2*A*c^4)*d^4 + (2*B*b^2*c^2 - A*b*c^3)*d^3*e + (2*B
*b^3*c - 7*A*b^2*c^2)*d^2*e^2 - (2*B*b^4 - A*b^3*c)*d*e^3)*x^2 + (3*A*b^4*d*e^3 - 2*(B*b^2*c^2 - 2*A*b*c^3)*d^
4 + (4*B*b^3*c - 5*A*b^2*c^2)*d^3*e - 2*(B*b^4 + A*b^3*c)*d^2*e^2)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d
) - (A*b^2*c^2*d^4 - 2*A*b^3*c*d^3*e + A*b^4*d^2*e^2 + (3*A*b^3*c*d*e^3 - (B*b^2*c^2 - 2*A*b*c^3)*d^3*e - 2*(B
*b^3*c + A*b^2*c^2)*d^2*e^2)*x^2 - (A*b^2*c^2*d^3*e - 3*A*b^4*d*e^3 + (B*b^2*c^2 - 2*A*b*c^3)*d^4 + (2*B*b^4 +
A*b^3*c)*d^2*e^2)*x)*sqrt(e*x + d))/((b^3*c^3*d^5*e - 2*b^4*c^2*d^4*e^2 + b^5*c*d^3*e^3)*x^3 + (b^3*c^3*d^6 -
b^4*c^2*d^5*e - b^5*c*d^4*e^2 + b^6*d^3*e^3)*x^2 + (b^4*c^2*d^6 - 2*b^5*c*d^5*e + b^6*d^4*e^2)*x)]

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giac [B]  time = 0.26, size = 500, normalized size = 1.97 \[ -\frac {{\left (2 \, B b c^{3} d - 4 \, A c^{4} d - 5 \, B b^{2} c^{2} e + 7 \, A b c^{3} e\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{{\left (b^{3} c^{2} d^{2} - 2 \, b^{4} c d e + b^{5} e^{2}\right )} \sqrt {-c^{2} d + b c e}} + \frac {{\left (x e + d\right )}^{2} B b c^{2} d^{2} e - 2 \, {\left (x e + d\right )}^{2} A c^{3} d^{2} e - {\left (x e + d\right )} B b c^{2} d^{3} e + 2 \, {\left (x e + d\right )} A c^{3} d^{3} e + 2 \, {\left (x e + d\right )}^{2} B b^{2} c d e^{2} + 2 \, {\left (x e + d\right )}^{2} A b c^{2} d e^{2} - 4 \, {\left (x e + d\right )} B b^{2} c d^{2} e^{2} - 3 \, {\left (x e + d\right )} A b c^{2} d^{2} e^{2} + 2 \, B b^{2} c d^{3} e^{2} - 3 \, {\left (x e + d\right )}^{2} A b^{2} c e^{3} + 2 \, {\left (x e + d\right )} B b^{3} d e^{3} + 7 \, {\left (x e + d\right )} A b^{2} c d e^{3} - 2 \, B b^{3} d^{2} e^{3} - 2 \, A b^{2} c d^{2} e^{3} - 3 \, {\left (x e + d\right )} A b^{3} e^{4} + 2 \, A b^{3} d e^{4}}{{\left (b^{2} c^{2} d^{4} - 2 \, b^{3} c d^{3} e + b^{4} d^{2} e^{2}\right )} {\left ({\left (x e + d\right )}^{\frac {5}{2}} c - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} c d + \sqrt {x e + d} c d^{2} + {\left (x e + d\right )}^{\frac {3}{2}} b e - \sqrt {x e + d} b d e\right )}} + \frac {{\left (2 \, B b d - 4 \, A c d - 3 \, A b e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d} d^{2}} \]

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)^2,x, algorithm="giac")

[Out]

-(2*B*b*c^3*d - 4*A*c^4*d - 5*B*b^2*c^2*e + 7*A*b*c^3*e)*arctan(sqrt(x*e + d)*c/sqrt(-c^2*d + b*c*e))/((b^3*c^
2*d^2 - 2*b^4*c*d*e + b^5*e^2)*sqrt(-c^2*d + b*c*e)) + ((x*e + d)^2*B*b*c^2*d^2*e - 2*(x*e + d)^2*A*c^3*d^2*e
- (x*e + d)*B*b*c^2*d^3*e + 2*(x*e + d)*A*c^3*d^3*e + 2*(x*e + d)^2*B*b^2*c*d*e^2 + 2*(x*e + d)^2*A*b*c^2*d*e^
2 - 4*(x*e + d)*B*b^2*c*d^2*e^2 - 3*(x*e + d)*A*b*c^2*d^2*e^2 + 2*B*b^2*c*d^3*e^2 - 3*(x*e + d)^2*A*b^2*c*e^3
+ 2*(x*e + d)*B*b^3*d*e^3 + 7*(x*e + d)*A*b^2*c*d*e^3 - 2*B*b^3*d^2*e^3 - 2*A*b^2*c*d^2*e^3 - 3*(x*e + d)*A*b^
3*e^4 + 2*A*b^3*d*e^4)/((b^2*c^2*d^4 - 2*b^3*c*d^3*e + b^4*d^2*e^2)*((x*e + d)^(5/2)*c - 2*(x*e + d)^(3/2)*c*d
+ sqrt(x*e + d)*c*d^2 + (x*e + d)^(3/2)*b*e - sqrt(x*e + d)*b*d*e)) + (2*B*b*d - 4*A*c*d - 3*A*b*e)*arctan(sq
rt(x*e + d)/sqrt(-d))/(b^3*sqrt(-d)*d^2)

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maple [A]  time = 0.08, size = 427, normalized size = 1.68 \[ -\frac {7 A \,c^{3} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 A \,c^{4} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {5 B \,c^{2} e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b}-\frac {2 B \,c^{3} d \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\left (b e -c d \right )^{2} \sqrt {\left (b e -c d \right ) c}\, b^{2}}-\frac {\sqrt {e x +d}\, A \,c^{3} e}{\left (b e -c d \right )^{2} \left (c e x +b e \right ) b^{2}}+\frac {\sqrt {e x +d}\, B \,c^{2} e}{\left (b e -c d \right )^{2} \left (c e x +b e \right ) b}-\frac {2 A \,e^{3}}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d^{2}}+\frac {2 B \,e^{2}}{\left (b e -c d \right )^{2} \sqrt {e x +d}\, d}+\frac {3 A e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {5}{2}}}+\frac {4 A c \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3} d^{\frac {3}{2}}}-\frac {2 B \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2} d^{\frac {3}{2}}}-\frac {\sqrt {e x +d}\, A}{b^{2} d^{2} x} \]

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)^2,x)

[Out]

-e*c^3/(b*e-c*d)^2/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A+e*c^2/(b*e-c*d)^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*B-7*e*c^3/(b*
e-c*d)^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A+4*c^4/(b*e-c*d)^2/b^3/((b*e-c*d
)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d+5*e*c^2/(b*e-c*d)^2/b/((b*e-c*d)*c)^(1/2)*arctan((e
*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B-2*c^3/(b*e-c*d)^2/b^2/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)
*c)^(1/2)*c)*B*d-1/b^2/d^2*A*(e*x+d)^(1/2)/x+3*e/b^2/d^(5/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*A+4/b^3/d^(3/2)*ar
ctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2/b^2/d^(3/2)*arctanh((e*x+d)^(1/2)/d^(1/2))*B-2*e^3/(b*e-c*d)^2/d^2/(e*x+d)^
(1/2)*A+2*e^2/(b*e-c*d)^2/d/(e*x+d)^(1/2)*B

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maxima [F(-2)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Exception raised: ValueError} \]

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)^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(b*e-c*d>0)', see `assume?` for
more details)Is b*e-c*d positive or negative?

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mupad [B]  time = 6.08, size = 8946, normalized size = 35.22 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(atan(((((d + e*x)^(1/2)*(64*A^2*b^6*c^15*d^18*e^2 - 576*A^2*b^7*c^14*d^17*e^3 + 2228*A^2*b^8*c^13*d^16*e^4 -
4768*A^2*b^9*c^12*d^15*e^5 + 5960*A^2*b^10*c^11*d^14*e^6 - 3976*A^2*b^11*c^10*d^13*e^7 + 578*A^2*b^12*c^9*d^12
*e^8 + 1004*A^2*b^13*c^8*d^11*e^9 - 442*A^2*b^14*c^7*d^10*e^10 - 320*A^2*b^15*c^6*d^9*e^11 + 362*A^2*b^16*c^5*
d^8*e^12 - 132*A^2*b^17*c^4*d^7*e^13 + 18*A^2*b^18*c^3*d^6*e^14 + 16*B^2*b^8*c^13*d^18*e^2 - 168*B^2*b^9*c^12*
d^17*e^3 + 770*B^2*b^10*c^11*d^16*e^4 - 2020*B^2*b^11*c^10*d^15*e^5 + 3350*B^2*b^12*c^9*d^14*e^6 - 3664*B^2*b^
13*c^8*d^13*e^7 + 2678*B^2*b^14*c^7*d^12*e^8 - 1300*B^2*b^15*c^6*d^11*e^9 + 410*B^2*b^16*c^5*d^10*e^10 - 80*B^
2*b^17*c^4*d^9*e^11 + 8*B^2*b^18*c^3*d^8*e^12 - 64*A*B*b^7*c^14*d^18*e^2 + 624*A*B*b^8*c^13*d^17*e^3 - 2636*A*
B*b^9*c^12*d^16*e^4 + 6280*A*B*b^10*c^11*d^15*e^5 - 9140*A*B*b^11*c^10*d^14*e^6 + 8056*A*B*b^12*c^9*d^13*e^7 -
3620*A*B*b^13*c^8*d^12*e^8 - 224*A*B*b^14*c^7*d^11*e^9 + 1300*A*B*b^15*c^6*d^10*e^10 - 760*A*B*b^16*c^5*d^9*e
^11 + 208*A*B*b^17*c^4*d^8*e^12 - 24*A*B*b^18*c^3*d^7*e^13) - ((-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2
*e - 7*A*b*c*e - 2*B*b*c*d)*(((-c^3*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e -
2*B*b*c*d)*(16*b^12*c^13*d^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5
+ 4320*b^16*c^9*d^17*e^6 - 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^2
0*c^5*d^13*e^10 - 520*b^21*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^
5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) - 8*A*b^10*c^13*d^19*e^3 +
76*A*b^11*c^12*d^18*e^4 - 300*A*b^12*c^11*d^17*e^5 + 612*A*b^13*c^10*d^16*e^6 - 576*A*b^14*c^9*d^15*e^7 - 168
*A*b^15*c^8*d^14*e^8 + 1176*A*b^16*c^7*d^13*e^9 - 1560*A*b^17*c^6*d^12*e^10 + 1128*A*b^18*c^5*d^11*e^11 - 484*
A*b^19*c^4*d^10*e^12 + 116*A*b^20*c^3*d^9*e^13 - 12*A*b^21*c^2*d^8*e^14 + 4*B*b^11*c^12*d^19*e^3 - 56*B*b^12*c
^11*d^18*e^4 + 312*B*b^13*c^10*d^17*e^5 - 960*B*b^14*c^9*d^16*e^6 + 1848*B*b^15*c^8*d^15*e^7 - 2352*B*b^16*c^7
*d^14*e^8 + 2016*B*b^17*c^6*d^13*e^9 - 1152*B*b^18*c^5*d^12*e^10 + 420*B*b^19*c^4*d^11*e^11 - 88*B*b^20*c^3*d^
10*e^12 + 8*B*b^21*c^2*d^9*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^
2*d^2*e^3 - 5*b^7*c*d*e^4)))*(-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*1i)/(2
*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + (((d +
e*x)^(1/2)*(64*A^2*b^6*c^15*d^18*e^2 - 576*A^2*b^7*c^14*d^17*e^3 + 2228*A^2*b^8*c^13*d^16*e^4 - 4768*A^2*b^9*
c^12*d^15*e^5 + 5960*A^2*b^10*c^11*d^14*e^6 - 3976*A^2*b^11*c^10*d^13*e^7 + 578*A^2*b^12*c^9*d^12*e^8 + 1004*A
^2*b^13*c^8*d^11*e^9 - 442*A^2*b^14*c^7*d^10*e^10 - 320*A^2*b^15*c^6*d^9*e^11 + 362*A^2*b^16*c^5*d^8*e^12 - 13
2*A^2*b^17*c^4*d^7*e^13 + 18*A^2*b^18*c^3*d^6*e^14 + 16*B^2*b^8*c^13*d^18*e^2 - 168*B^2*b^9*c^12*d^17*e^3 + 77
0*B^2*b^10*c^11*d^16*e^4 - 2020*B^2*b^11*c^10*d^15*e^5 + 3350*B^2*b^12*c^9*d^14*e^6 - 3664*B^2*b^13*c^8*d^13*e
^7 + 2678*B^2*b^14*c^7*d^12*e^8 - 1300*B^2*b^15*c^6*d^11*e^9 + 410*B^2*b^16*c^5*d^10*e^10 - 80*B^2*b^17*c^4*d^
9*e^11 + 8*B^2*b^18*c^3*d^8*e^12 - 64*A*B*b^7*c^14*d^18*e^2 + 624*A*B*b^8*c^13*d^17*e^3 - 2636*A*B*b^9*c^12*d^
16*e^4 + 6280*A*B*b^10*c^11*d^15*e^5 - 9140*A*B*b^11*c^10*d^14*e^6 + 8056*A*B*b^12*c^9*d^13*e^7 - 3620*A*B*b^1
3*c^8*d^12*e^8 - 224*A*B*b^14*c^7*d^11*e^9 + 1300*A*B*b^15*c^6*d^10*e^10 - 760*A*B*b^16*c^5*d^9*e^11 + 208*A*B
*b^17*c^4*d^8*e^12 - 24*A*B*b^18*c^3*d^7*e^13) - ((-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*
e - 2*B*b*c*d)*(((-c^3*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*(1
6*b^12*c^13*d^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 4320*b^16*c
^9*d^17*e^6 - 5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^20*c^5*d^13*e^
10 - 520*b^21*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4
*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + 8*A*b^10*c^13*d^19*e^3 - 76*A*b^11*c^
12*d^18*e^4 + 300*A*b^12*c^11*d^17*e^5 - 612*A*b^13*c^10*d^16*e^6 + 576*A*b^14*c^9*d^15*e^7 + 168*A*b^15*c^8*d
^14*e^8 - 1176*A*b^16*c^7*d^13*e^9 + 1560*A*b^17*c^6*d^12*e^10 - 1128*A*b^18*c^5*d^11*e^11 + 484*A*b^19*c^4*d^
10*e^12 - 116*A*b^20*c^3*d^9*e^13 + 12*A*b^21*c^2*d^8*e^14 - 4*B*b^11*c^12*d^19*e^3 + 56*B*b^12*c^11*d^18*e^4
- 312*B*b^13*c^10*d^17*e^5 + 960*B*b^14*c^9*d^16*e^6 - 1848*B*b^15*c^8*d^15*e^7 + 2352*B*b^16*c^7*d^14*e^8 - 2
016*B*b^17*c^6*d^13*e^9 + 1152*B*b^18*c^5*d^12*e^10 - 420*B*b^19*c^4*d^11*e^11 + 88*B*b^20*c^3*d^10*e^12 - 8*B
*b^21*c^2*d^9*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5
*b^7*c*d*e^4)))*(-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*1i)/(2*(b^8*e^5 - b
^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)))/((((d + e*x)^(1/2)*(
64*A^2*b^6*c^15*d^18*e^2 - 576*A^2*b^7*c^14*d^17*e^3 + 2228*A^2*b^8*c^13*d^16*e^4 - 4768*A^2*b^9*c^12*d^15*e^5
+ 5960*A^2*b^10*c^11*d^14*e^6 - 3976*A^2*b^11*c^10*d^13*e^7 + 578*A^2*b^12*c^9*d^12*e^8 + 1004*A^2*b^13*c^8*d
^11*e^9 - 442*A^2*b^14*c^7*d^10*e^10 - 320*A^2*b^15*c^6*d^9*e^11 + 362*A^2*b^16*c^5*d^8*e^12 - 132*A^2*b^17*c^
4*d^7*e^13 + 18*A^2*b^18*c^3*d^6*e^14 + 16*B^2*b^8*c^13*d^18*e^2 - 168*B^2*b^9*c^12*d^17*e^3 + 770*B^2*b^10*c^
11*d^16*e^4 - 2020*B^2*b^11*c^10*d^15*e^5 + 3350*B^2*b^12*c^9*d^14*e^6 - 3664*B^2*b^13*c^8*d^13*e^7 + 2678*B^2
*b^14*c^7*d^12*e^8 - 1300*B^2*b^15*c^6*d^11*e^9 + 410*B^2*b^16*c^5*d^10*e^10 - 80*B^2*b^17*c^4*d^9*e^11 + 8*B^
2*b^18*c^3*d^8*e^12 - 64*A*B*b^7*c^14*d^18*e^2 + 624*A*B*b^8*c^13*d^17*e^3 - 2636*A*B*b^9*c^12*d^16*e^4 + 6280
*A*B*b^10*c^11*d^15*e^5 - 9140*A*B*b^11*c^10*d^14*e^6 + 8056*A*B*b^12*c^9*d^13*e^7 - 3620*A*B*b^13*c^8*d^12*e^
8 - 224*A*B*b^14*c^7*d^11*e^9 + 1300*A*B*b^15*c^6*d^10*e^10 - 760*A*B*b^16*c^5*d^9*e^11 + 208*A*B*b^17*c^4*d^8
*e^12 - 24*A*B*b^18*c^3*d^7*e^13) - ((-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d
)*(((-c^3*(b*e - c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*(16*b^12*c^13*d
^21*e^2 - 168*b^13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 4320*b^16*c^9*d^17*e^6 -
5712*b^17*c^8*d^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^20*c^5*d^13*e^10 - 520*b^21
*c^4*d^12*e^11 + 96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e -
10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + 8*A*b^10*c^13*d^19*e^3 - 76*A*b^11*c^12*d^18*e^4 +
300*A*b^12*c^11*d^17*e^5 - 612*A*b^13*c^10*d^16*e^6 + 576*A*b^14*c^9*d^15*e^7 + 168*A*b^15*c^8*d^14*e^8 - 117
6*A*b^16*c^7*d^13*e^9 + 1560*A*b^17*c^6*d^12*e^10 - 1128*A*b^18*c^5*d^11*e^11 + 484*A*b^19*c^4*d^10*e^12 - 116
*A*b^20*c^3*d^9*e^13 + 12*A*b^21*c^2*d^8*e^14 - 4*B*b^11*c^12*d^19*e^3 + 56*B*b^12*c^11*d^18*e^4 - 312*B*b^13*
c^10*d^17*e^5 + 960*B*b^14*c^9*d^16*e^6 - 1848*B*b^15*c^8*d^15*e^7 + 2352*B*b^16*c^7*d^14*e^8 - 2016*B*b^17*c^
6*d^13*e^9 + 1152*B*b^18*c^5*d^12*e^10 - 420*B*b^19*c^4*d^11*e^11 + 88*B*b^20*c^3*d^10*e^12 - 8*B*b^21*c^2*d^9
*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)
))*(-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b
^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) - (((d + e*x)^(1/2)*(64*A^2*b^6*c^15*
d^18*e^2 - 576*A^2*b^7*c^14*d^17*e^3 + 2228*A^2*b^8*c^13*d^16*e^4 - 4768*A^2*b^9*c^12*d^15*e^5 + 5960*A^2*b^10
*c^11*d^14*e^6 - 3976*A^2*b^11*c^10*d^13*e^7 + 578*A^2*b^12*c^9*d^12*e^8 + 1004*A^2*b^13*c^8*d^11*e^9 - 442*A^
2*b^14*c^7*d^10*e^10 - 320*A^2*b^15*c^6*d^9*e^11 + 362*A^2*b^16*c^5*d^8*e^12 - 132*A^2*b^17*c^4*d^7*e^13 + 18*
A^2*b^18*c^3*d^6*e^14 + 16*B^2*b^8*c^13*d^18*e^2 - 168*B^2*b^9*c^12*d^17*e^3 + 770*B^2*b^10*c^11*d^16*e^4 - 20
20*B^2*b^11*c^10*d^15*e^5 + 3350*B^2*b^12*c^9*d^14*e^6 - 3664*B^2*b^13*c^8*d^13*e^7 + 2678*B^2*b^14*c^7*d^12*e
^8 - 1300*B^2*b^15*c^6*d^11*e^9 + 410*B^2*b^16*c^5*d^10*e^10 - 80*B^2*b^17*c^4*d^9*e^11 + 8*B^2*b^18*c^3*d^8*e
^12 - 64*A*B*b^7*c^14*d^18*e^2 + 624*A*B*b^8*c^13*d^17*e^3 - 2636*A*B*b^9*c^12*d^16*e^4 + 6280*A*B*b^10*c^11*d
^15*e^5 - 9140*A*B*b^11*c^10*d^14*e^6 + 8056*A*B*b^12*c^9*d^13*e^7 - 3620*A*B*b^13*c^8*d^12*e^8 - 224*A*B*b^14
*c^7*d^11*e^9 + 1300*A*B*b^15*c^6*d^10*e^10 - 760*A*B*b^16*c^5*d^9*e^11 + 208*A*B*b^17*c^4*d^8*e^12 - 24*A*B*b
^18*c^3*d^7*e^13) - ((-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*(((-c^3*(b*e -
c*d)^5)^(1/2)*(d + e*x)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d)*(16*b^12*c^13*d^21*e^2 - 168*b^
13*c^12*d^20*e^3 + 800*b^14*c^11*d^19*e^4 - 2280*b^15*c^10*d^18*e^5 + 4320*b^16*c^9*d^17*e^6 - 5712*b^17*c^8*d
^16*e^7 + 5376*b^18*c^7*d^15*e^8 - 3600*b^19*c^6*d^14*e^9 + 1680*b^20*c^5*d^13*e^10 - 520*b^21*c^4*d^12*e^11 +
96*b^22*c^3*d^11*e^12 - 8*b^23*c^2*d^10*e^13))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e
^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) - 8*A*b^10*c^13*d^19*e^3 + 76*A*b^11*c^12*d^18*e^4 - 300*A*b^12*c^11
*d^17*e^5 + 612*A*b^13*c^10*d^16*e^6 - 576*A*b^14*c^9*d^15*e^7 - 168*A*b^15*c^8*d^14*e^8 + 1176*A*b^16*c^7*d^1
3*e^9 - 1560*A*b^17*c^6*d^12*e^10 + 1128*A*b^18*c^5*d^11*e^11 - 484*A*b^19*c^4*d^10*e^12 + 116*A*b^20*c^3*d^9*
e^13 - 12*A*b^21*c^2*d^8*e^14 + 4*B*b^11*c^12*d^19*e^3 - 56*B*b^12*c^11*d^18*e^4 + 312*B*b^13*c^10*d^17*e^5 -
960*B*b^14*c^9*d^16*e^6 + 1848*B*b^15*c^8*d^15*e^7 - 2352*B*b^16*c^7*d^14*e^8 + 2016*B*b^17*c^6*d^13*e^9 - 115
2*B*b^18*c^5*d^12*e^10 + 420*B*b^19*c^4*d^11*e^11 - 88*B*b^20*c^3*d^10*e^12 + 8*B*b^21*c^2*d^9*e^13))/(2*(b^8*
e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)))*(-c^3*(b*e -
c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e - 2*B*b*c*d))/(2*(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 1
0*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d*e^4)) + 64*A^3*b^4*c^15*d^16*e^3 - 512*A^3*b^5*c^14*d^15*e^
4 + 1804*A^3*b^6*c^13*d^14*e^5 - 3668*A^3*b^7*c^12*d^13*e^6 + 4606*A^3*b^8*c^11*d^12*e^7 - 3248*A^3*b^9*c^10*d
^11*e^8 + 322*A^3*b^10*c^9*d^10*e^9 + 1756*A^3*b^11*c^8*d^9*e^10 - 1742*A^3*b^12*c^7*d^8*e^11 + 744*A^3*b^13*c
^6*d^7*e^12 - 126*A^3*b^14*c^5*d^6*e^13 - 8*B^3*b^7*c^12*d^16*e^3 + 52*B^3*b^8*c^11*d^15*e^4 - 104*B^3*b^9*c^1
0*d^14*e^5 - 20*B^3*b^10*c^9*d^13*e^6 + 400*B^3*b^11*c^8*d^12*e^7 - 692*B^3*b^12*c^7*d^11*e^8 + 568*B^3*b^13*c
^6*d^10*e^9 - 236*B^3*b^14*c^5*d^9*e^10 + 40*B^3*b^15*c^4*d^8*e^11 + 48*A*B^2*b^6*c^13*d^16*e^3 - 336*A*B^2*b^
7*c^12*d^15*e^4 + 930*A*B^2*b^8*c^11*d^14*e^5 - 1332*A*B^2*b^9*c^10*d^13*e^6 + 1230*A*B^2*b^10*c^9*d^12*e^7 -
1248*A*B^2*b^11*c^8*d^11*e^8 + 1566*A*B^2*b^12*c^7*d^10*e^9 - 1380*A*B^2*b^13*c^6*d^9*e^10 + 642*A*B^2*b^14*c^
5*d^8*e^11 - 120*A*B^2*b^15*c^4*d^7*e^12 - 96*A^2*B*b^5*c^14*d^16*e^3 + 720*A^2*B*b^6*c^13*d^15*e^4 - 2346*A^2
*B*b^7*c^12*d^14*e^5 + 4524*A^2*B*b^8*c^11*d^13*e^6 - 6012*A^2*B*b^9*c^10*d^12*e^7 + 5916*A^2*B*b^10*c^9*d^11*
e^8 - 4080*A^2*B*b^11*c^8*d^10*e^9 + 1476*A^2*B*b^12*c^7*d^9*e^10 + 156*A^2*B*b^13*c^6*d^8*e^11 - 348*A^2*B*b^
14*c^5*d^7*e^12 + 90*A^2*B*b^15*c^4*d^6*e^13))*(-c^3*(b*e - c*d)^5)^(1/2)*(4*A*c^2*d + 5*B*b^2*e - 7*A*b*c*e -
2*B*b*c*d)*1i)/(b^8*e^5 - b^3*c^5*d^5 + 5*b^4*c^4*d^4*e - 10*b^5*c^3*d^3*e^2 + 10*b^6*c^2*d^2*e^3 - 5*b^7*c*d
*e^4) - (atan((B^3*b^14*d^13*e^11*(d + e*x)^(1/2)*8i - A^3*b^14*d^10*e^14*(d + e*x)^(1/2)*27i + A^3*b^13*c*d^1
1*e^13*(d + e*x)^(1/2)*189i - B^3*b^13*c*d^14*e^10*(d + e*x)^(1/2)*88i + A^3*b^3*c^11*d^21*e^3*(d + e*x)^(1/2)
*140i - A^3*b^4*c^10*d^20*e^4*(d + e*x)^(1/2)*1015i + A^3*b^5*c^9*d^19*e^5*(d + e*x)^(1/2)*2996i - A^3*b^6*c^8
*d^18*e^6*(d + e*x)^(1/2)*4375i + A^3*b^7*c^7*d^17*e^7*(d + e*x)^(1/2)*2561i + A^3*b^8*c^6*d^16*e^8*(d + e*x)^
(1/2)*1316i - A^3*b^9*c^5*d^15*e^9*(d + e*x)^(1/2)*3073i + A^3*b^10*c^4*d^14*e^10*(d + e*x)^(1/2)*1694i + A^3*
b^11*c^3*d^13*e^11*(d + e*x)^(1/2)*35i - A^3*b^12*c^2*d^12*e^12*(d + e*x)^(1/2)*441i - B^3*b^5*c^9*d^22*e^2*(d
+ e*x)^(1/2)*30i + B^3*b^6*c^8*d^21*e^3*(d + e*x)^(1/2)*260i - B^3*b^7*c^7*d^20*e^4*(d + e*x)^(1/2)*970i + B^
3*b^8*c^6*d^19*e^5*(d + e*x)^(1/2)*2048i - B^3*b^9*c^5*d^18*e^6*(d + e*x)^(1/2)*2698i + B^3*b^10*c^4*d^17*e^7*
(d + e*x)^(1/2)*2300i - B^3*b^11*c^3*d^16*e^8*(d + e*x)^(1/2)*1270i + B^3*b^12*c^2*d^15*e^9*(d + e*x)^(1/2)*44
0i - A*B^2*b^14*d^12*e^12*(d + e*x)^(1/2)*36i + A^2*B*b^14*d^11*e^13*(d + e*x)^(1/2)*54i + A*B^2*b^13*c*d^13*e
^11*(d + e*x)^(1/2)*348i - A^2*B*b^13*c*d^12*e^12*(d + e*x)^(1/2)*450i + A*B^2*b^4*c^10*d^22*e^2*(d + e*x)^(1/
2)*120i - A*B^2*b^5*c^9*d^21*e^3*(d + e*x)^(1/2)*915i + A*B^2*b^6*c^8*d^20*e^4*(d + e*x)^(1/2)*2850i - A*B^2*b
^7*c^7*d^19*e^5*(d + e*x)^(1/2)*4473i + A*B^2*b^8*c^6*d^18*e^6*(d + e*x)^(1/2)*3072i + A*B^2*b^9*c^5*d^17*e^7*
(d + e*x)^(1/2)*951i - A*B^2*b^10*c^4*d^16*e^8*(d + e*x)^(1/2)*3690i + A*B^2*b^11*c^3*d^15*e^9*(d + e*x)^(1/2)
*3225i - A*B^2*b^12*c^2*d^14*e^10*(d + e*x)^(1/2)*1452i - A^2*B*b^3*c^11*d^22*e^2*(d + e*x)^(1/2)*120i + A^2*B
*b^4*c^10*d^21*e^3*(d + e*x)^(1/2)*720i - A^2*B*b^5*c^9*d^20*e^4*(d + e*x)^(1/2)*1380i - A^2*B*b^6*c^8*d^19*e^
5*(d + e*x)^(1/2)*204i + A^2*B*b^7*c^7*d^18*e^6*(d + e*x)^(1/2)*4878i - A^2*B*b^8*c^6*d^17*e^7*(d + e*x)^(1/2)
*8130i + A^2*B*b^9*c^5*d^16*e^8*(d + e*x)^(1/2)*5646i - A^2*B*b^10*c^4*d^15*e^9*(d + e*x)^(1/2)*450i - A^2*B*b
^11*c^3*d^14*e^10*(d + e*x)^(1/2)*2046i + A^2*B*b^12*c^2*d^13*e^11*(d + e*x)^(1/2)*1482i)/(d^5*(d^5)^(1/2)*(d^
5*(d^5*(2561*A^3*b^7*c^7*e^7 - d^5*(30*B^3*b^5*c^9*e^2 - 120*A*B^2*b^4*c^10*e^2 + 120*A^2*B*b^3*c^11*e^2) + 23
00*B^3*b^10*c^4*e^7 + 140*A^3*b^3*c^11*d^4*e^3 - 1015*A^3*b^4*c^10*d^3*e^4 + 2996*A^3*b^5*c^9*d^2*e^5 + 260*B^
3*b^6*c^8*d^4*e^3 - 970*B^3*b^7*c^7*d^3*e^4 + 2048*B^3*b^8*c^6*d^2*e^5 + 951*A*B^2*b^9*c^5*e^7 - 8130*A^2*B*b^
8*c^6*e^7 - 4375*A^3*b^6*c^8*d*e^6 - 2698*B^3*b^9*c^5*d*e^6 - 915*A*B^2*b^5*c^9*d^4*e^3 + 2850*A*B^2*b^6*c^8*d
^3*e^4 - 4473*A*B^2*b^7*c^7*d^2*e^5 + 720*A^2*B*b^4*c^10*d^4*e^3 - 1380*A^2*B*b^5*c^9*d^3*e^4 - 204*A^2*B*b^6*
c^8*d^2*e^5 + 3072*A*B^2*b^8*c^6*d*e^6 + 4878*A^2*B*b^7*c^7*d*e^6) - 441*A^3*b^12*c^2*e^12 - 36*A*B^2*b^14*e^1
2 + 8*B^3*b^14*d*e^11 + 1316*A^3*b^8*c^6*d^4*e^8 - 3073*A^3*b^9*c^5*d^3*e^9 + 1694*A^3*b^10*c^4*d^2*e^10 - 127
0*B^3*b^11*c^3*d^4*e^8 + 440*B^3*b^12*c^2*d^3*e^9 - 450*A^2*B*b^13*c*e^12 + 35*A^3*b^11*c^3*d*e^11 - 88*B^3*b^
13*c*d^2*e^10 - 3690*A*B^2*b^10*c^4*d^4*e^8 + 3225*A*B^2*b^11*c^3*d^3*e^9 - 1452*A*B^2*b^12*c^2*d^2*e^10 + 564
6*A^2*B*b^9*c^5*d^4*e^8 - 450*A^2*B*b^10*c^4*d^3*e^9 - 2046*A^2*B*b^11*c^3*d^2*e^10 + 348*A*B^2*b^13*c*d*e^11
+ 1482*A^2*B*b^12*c^2*d*e^11) - 27*A^3*b^14*d^3*e^14 + 54*A^2*B*b^14*d^4*e^13 + 189*A^3*b^13*c*d^4*e^13)))*(3*
A*b*e + 4*A*c*d - 2*B*b*d)*1i)/(b^3*(d^5)^(1/2)) - ((2*(A*e^3 - B*d*e^2))/(c*d^2 - b*d*e) + ((d + e*x)*(3*A*b^
3*e^4 - 2*A*c^3*d^3*e - 2*B*b^3*d*e^3 + 3*A*b*c^2*d^2*e^2 + 4*B*b^2*c*d^2*e^2 - 7*A*b^2*c*d*e^3 + B*b*c^2*d^3*
e))/(b^2*(c*d^2 - b*d*e)^2) - ((d + e*x)^2*(2*A*b*c^2*d*e^2 - 2*A*c^3*d^2*e - 3*A*b^2*c*e^3 + B*b*c^2*d^2*e +
2*B*b^2*c*d*e^2))/(b^2*(c*d^2 - b*d*e)^2))/(c*(d + e*x)^(5/2) + (c*d^2 - b*d*e)*(d + e*x)^(1/2) + (b*e - 2*c*d
)*(d + e*x)^(3/2))

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sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]

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)**2,x)

[Out]

Timed out

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