3.1241 \(\int \frac {(A+B x) (d+e x)^{3/2}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=181 \[ -\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3}-\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}-\frac {\sqrt {c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}} \]

[Out]

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

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Rubi [A]  time = 0.34, antiderivative size = 181, 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 = {818, 826, 1166, 208} \[ -\frac {\sqrt {d+e x} \left (x \left (-b c (A e+B d)+2 A c^2 d+b^2 B e\right )+A b c d\right )}{b^2 c \left (b x+c x^2\right )}-\frac {\sqrt {c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{b^3 c^{3/2}}-\frac {\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (3 A b e-4 A c d+2 b B d)}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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 818

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

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

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Mathematica [A]  time = 0.33, size = 171, normalized size = 0.94 \[ \frac {-\frac {\sqrt {c d-b e} \left (-b c (A e+2 B d)+4 A c^2 d+b^2 (-B) e\right ) \tanh ^{-1}\left (\frac {\sqrt {c} \sqrt {d+e x}}{\sqrt {c d-b e}}\right )}{c^{3/2}}+\frac {b \sqrt {d+e x} (A c (-b d+b e x-2 c d x)+b B x (c d-b e))}{c x (b+c x)}+\sqrt {d} \tanh ^{-1}\left (\frac {\sqrt {d+e x}}{\sqrt {d}}\right ) (-3 A b e+4 A c d-2 b B d)}{b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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fricas [A]  time = 2.30, size = 1146, normalized size = 6.33 \[ \text {result too large to display} \]

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^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*
e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d - b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) + ((3*A
*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(d)*log((e*x - 2*sq
rt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*sqrt(e*x + d
))/(b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*
b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(c*d - b*e)/c)/(c*d - b*
e)) + ((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(d)*log(
(e*x - 2*sqrt(e*x + d)*sqrt(d) + 2*d)/x) - 2*(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*c)*e)*x)*s
qrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), 1/2*(2*((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*
(B*b^2*c - 2*A*b*c^2)*d)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) + ((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c +
A*b*c^2)*e)*x^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt((c*d - b*e)/c)*log((c*e*x + 2*c*d
- b*e + 2*sqrt(e*x + d)*c*sqrt((c*d - b*e)/c))/(c*x + b)) - 2*(A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3
- A*b^2*c)*e)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x), (((2*(B*b*c^2 - 2*A*c^3)*d + (B*b^2*c + A*b*c^2)*e)*x
^2 + (2*(B*b^2*c - 2*A*b*c^2)*d + (B*b^3 + A*b^2*c)*e)*x)*sqrt(-(c*d - b*e)/c)*arctan(-sqrt(e*x + d)*c*sqrt(-(
c*d - b*e)/c)/(c*d - b*e)) + ((3*A*b*c^2*e + 2*(B*b*c^2 - 2*A*c^3)*d)*x^2 + (3*A*b^2*c*e + 2*(B*b^2*c - 2*A*b*
c^2)*d)*x)*sqrt(-d)*arctan(sqrt(e*x + d)*sqrt(-d)/d) - (A*b^2*c*d - ((B*b^2*c - 2*A*b*c^2)*d - (B*b^3 - A*b^2*
c)*e)*x)*sqrt(e*x + d))/(b^3*c^2*x^2 + b^4*c*x)]

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giac [B]  time = 0.21, size = 334, normalized size = 1.85 \[ \frac {{\left (2 \, B b d^{2} - 4 \, A c d^{2} + 3 \, A b d e\right )} \arctan \left (\frac {\sqrt {x e + d}}{\sqrt {-d}}\right )}{b^{3} \sqrt {-d}} - \frac {{\left (2 \, B b c^{2} d^{2} - 4 \, A c^{3} d^{2} - B b^{2} c d e + 5 \, A b c^{2} d e - B b^{3} e^{2} - A b^{2} c e^{2}\right )} \arctan \left (\frac {\sqrt {x e + d} c}{\sqrt {-c^{2} d + b c e}}\right )}{\sqrt {-c^{2} d + b c e} b^{3} c} + \frac {{\left (x e + d\right )}^{\frac {3}{2}} B b c d e - 2 \, {\left (x e + d\right )}^{\frac {3}{2}} A c^{2} d e - \sqrt {x e + d} B b c d^{2} e + 2 \, \sqrt {x e + d} A c^{2} d^{2} e - {\left (x e + d\right )}^{\frac {3}{2}} B b^{2} e^{2} + {\left (x e + d\right )}^{\frac {3}{2}} A b c e^{2} + \sqrt {x e + d} B b^{2} d e^{2} - 2 \, \sqrt {x e + d} A b c d e^{2}}{{\left ({\left (x e + d\right )}^{2} c - 2 \, {\left (x e + d\right )} c d + c d^{2} + {\left (x e + d\right )} b e - b d e\right )} b^{2} c} \]

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

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maple [B]  time = 0.07, size = 443, normalized size = 2.45 \[ \frac {A \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {5 A c d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {4 A \,c^{2} d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{3}}+\frac {B d e \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b}-\frac {2 B c \,d^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, b^{2}}+\frac {B \,e^{2} \arctan \left (\frac {\sqrt {e x +d}\, c}{\sqrt {\left (b e -c d \right ) c}}\right )}{\sqrt {\left (b e -c d \right ) c}\, c}+\frac {\sqrt {e x +d}\, A \,e^{2}}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, A c d e}{\left (c e x +b e \right ) b^{2}}+\frac {\sqrt {e x +d}\, B d e}{\left (c e x +b e \right ) b}-\frac {\sqrt {e x +d}\, B \,e^{2}}{\left (c e x +b e \right ) c}-\frac {3 A \sqrt {d}\, e \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}+\frac {4 A c \,d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{3}}-\frac {2 B \,d^{\frac {3}{2}} \arctanh \left (\frac {\sqrt {e x +d}}{\sqrt {d}}\right )}{b^{2}}-\frac {\sqrt {e x +d}\, A d}{b^{2} x} \]

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^2/b*(e*x+d)^(1/2)/(c*e*x+b*e)*A-e/b^2*(e*x+d)^(1/2)/(c*e*x+b*e)*A*c*d-e^2/c*(e*x+d)^(1/2)/(c*e*x+b*e)*B+e/b*
(e*x+d)^(1/2)/(c*e*x+b*e)*B*d+e^2/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A-5*e/b^2/
((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*c*d+4/b^3*c^2/((b*e-c*d)*c)^(1/2)*arctan((e*
x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*A*d^2+e^2/c/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*
B+e/b/((b*e-c*d)*c)^(1/2)*arctan((e*x+d)^(1/2)/((b*e-c*d)*c)^(1/2)*c)*B*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^2*c-d/b^2*A*(e*x+d)^(1/2)/x-3*e*d^(1/2)/b^2*arctanh((e*x+d)^(1/2)/d^(1/2
))*A+4*d^(3/2)/b^3*arctanh((e*x+d)^(1/2)/d^(1/2))*A*c-2*d^(3/2)/b^2*arctanh((e*x+d)^(1/2)/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 = 2.61, size = 4391, normalized size = 24.26 \[ \text {result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

(d^(1/2)*atan(((d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c
^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^
2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B
*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^4*c) + (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^
6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) + (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(d + e*x)^(1/2)*(3*A
*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/
(2*b^3) + (d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^
2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c
^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*
c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^4*c) - (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4
*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) - (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(d + e*x)^(1/2)*(3*A*b*e
- 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/(2*b^
3))/((2*(32*A^3*c^6*d^5*e^3 + 2*B^3*b^6*d^2*e^6 + 70*A^3*b^2*c^4*d^3*e^5 - 25*A^3*b^3*c^3*d^2*e^6 - 4*B^3*b^3*
c^3*d^5*e^3 - 2*B^3*b^4*c^2*d^4*e^4 + 3*A*B^2*b^6*d*e^7 - 80*A^3*b*c^5*d^4*e^4 + 3*A^3*b^4*c^2*d*e^7 + 4*B^3*b
^5*c*d^3*e^5 + 24*A*B^2*b^2*c^4*d^5*e^3 - 12*A*B^2*b^3*c^3*d^4*e^4 - 21*A*B^2*b^4*c^2*d^3*e^5 + 72*A^2*B*b^2*c
^4*d^4*e^4 - 9*A^2*B*b^3*c^3*d^3*e^5 - 21*A^2*B*b^4*c^2*d^2*e^6 + 6*A^2*B*b^5*c*d*e^7 + 6*A*B^2*b^5*c*d^2*e^6
- 48*A^2*B*b*c^5*d^5*e^3))/(b^6*c) + (d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*
d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B
^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*
b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^4*c) + (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*
b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) + (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2
)*(d + e*x)^(1/2)*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4
*A*c*d + 2*B*b*d))/(2*b^3) - (d^(1/2)*((2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2
+ 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c
*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*
d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^4*c) - (d^(1/2)*((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*
d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) - (d^(1/2)*(4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(d + e
*x)^(1/2)*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(b^7*c))*(3*A*b*e - 4*A*c*d + 2*B*b*d))/(2*b^3))*(3*A*b*e - 4*A*c*d +
2*B*b*d))/(2*b^3)))*(3*A*b*e - 4*A*c*d + 2*B*b*d)*1i)/b^3 - (((d + e*x)^(3/2)*(B*b^2*e^2 - A*b*c*e^2 + 2*A*c^
2*d*e - B*b*c*d*e))/(b^2*c) - ((d + e*x)^(1/2)*(2*A*c^2*d^2*e + B*b^2*d*e^2 - 2*A*b*c*d*e^2 - B*b*c*d^2*e))/(b
^2*c))/((b*e - 2*c*d)*(d + e*x) + c*(d + e*x)^2 + c*d^2 - b*d*e) + (atan((((-c^3*(b*e - c*d))^(1/2)*((2*(d + e
*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2
- 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d
*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d
^2*e^4))/(b^4*c) + (((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*
c) + ((4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*
e + 2*B*b*c*d))/(b^7*c^4))*(-c^3*(b*e - c*d))^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^3*c^3))*
(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d)*1i)/(2*b^3*c^3) + ((-c^3*(b*e - c*d))^(1/2)*((2*(d + e*x)^(1/2)*(B
^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^2*b^3
*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 + 2*A*B
*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4))/(b^
4*c) - (((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) - ((4*b^7
*c^3*e^3 - 8*b^6*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*
d))/(b^7*c^4))*(-c^3*(b*e - c*d))^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^3*c^3))*(B*b^2*e - 4
*A*c^2*d + A*b*c*e + 2*B*b*c*d)*1i)/(2*b^3*c^3))/((2*(32*A^3*c^6*d^5*e^3 + 2*B^3*b^6*d^2*e^6 + 70*A^3*b^2*c^4*
d^3*e^5 - 25*A^3*b^3*c^3*d^2*e^6 - 4*B^3*b^3*c^3*d^5*e^3 - 2*B^3*b^4*c^2*d^4*e^4 + 3*A*B^2*b^6*d*e^7 - 80*A^3*
b*c^5*d^4*e^4 + 3*A^3*b^4*c^2*d*e^7 + 4*B^3*b^5*c*d^3*e^5 + 24*A*B^2*b^2*c^4*d^5*e^3 - 12*A*B^2*b^3*c^3*d^4*e^
4 - 21*A*B^2*b^4*c^2*d^3*e^5 + 72*A^2*B*b^2*c^4*d^4*e^4 - 9*A^2*B*b^3*c^3*d^3*e^5 - 21*A^2*B*b^4*c^2*d^2*e^6 +
6*A^2*B*b^5*c*d*e^7 + 6*A*B^2*b^5*c*d^2*e^6 - 48*A^2*B*b*c^5*d^5*e^3))/(b^6*c) + ((-c^3*(b*e - c*d))^(1/2)*((
2*(d + e*x)^(1/2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4
*d^4*e^2 - 4*B^2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b
^3*c^3*d*e^5 + 2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b
^3*c^3*d^2*e^4))/(b^4*c) + (((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^
3)/(b^6*c) + ((4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(B*b^2*e - 4*A*c^2*d
+ A*b*c*e + 2*B*b*c*d))/(b^7*c^4))*(-c^3*(b*e - c*d))^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^
3*c^3))*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^3*c^3) - ((-c^3*(b*e - c*d))^(1/2)*((2*(d + e*x)^(1/
2)*(B^2*b^6*e^6 + A^2*b^4*c^2*e^6 + 32*A^2*c^6*d^4*e^2 + 42*A^2*b^2*c^4*d^2*e^4 + 8*B^2*b^2*c^4*d^4*e^2 - 4*B^
2*b^3*c^3*d^3*e^3 - 3*B^2*b^4*c^2*d^2*e^4 + 2*B^2*b^5*c*d*e^5 - 64*A^2*b*c^5*d^3*e^3 - 10*A^2*b^3*c^3*d*e^5 +
2*A*B*b^5*c*e^6 - 32*A*B*b*c^5*d^4*e^2 - 8*A*B*b^4*c^2*d*e^5 + 40*A*B*b^2*c^4*d^3*e^3 - 6*A*B*b^3*c^3*d^2*e^4)
)/(b^4*c) - (((8*A*b^7*c^3*d*e^4 - 4*B*b^8*c^2*d*e^4 - 8*A*b^6*c^4*d^2*e^3 + 4*B*b^7*c^3*d^2*e^3)/(b^6*c) - ((
4*b^7*c^3*e^3 - 8*b^6*c^4*d*e^2)*(-c^3*(b*e - c*d))^(1/2)*(d + e*x)^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B
*b*c*d))/(b^7*c^4))*(-c^3*(b*e - c*d))^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^3*c^3))*(B*b^2*
e - 4*A*c^2*d + A*b*c*e + 2*B*b*c*d))/(2*b^3*c^3)))*(-c^3*(b*e - c*d))^(1/2)*(B*b^2*e - 4*A*c^2*d + A*b*c*e +
2*B*b*c*d)*1i)/(b^3*c^3)

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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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