3.1242 \(\int \frac{(A+B x) \sqrt{d+e x}}{(b x+c x^2)^2} \, dx\)

Optimal. Leaf size=158 \[ \frac{\left (3 A b c e-4 A c^2 d+b^2 (-B) e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c} \sqrt{c d-b e}}-\frac{\sqrt{d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (A b e-4 A c d+2 b B d)}{b^3 \sqrt{d}} \]

[Out]

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

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Rubi [A]  time = 0.313863, antiderivative size = 158, normalized size of antiderivative = 1., number of steps used = 5, number of rules used = 4, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.154, Rules used = {820, 826, 1166, 208} \[ \frac{\left (3 A b c e-4 A c^2 d+b^2 (-B) e+2 b B c d\right ) \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )}{b^3 \sqrt{c} \sqrt{c d-b e}}-\frac{\sqrt{d+e x} (A b-x (b B-2 A c))}{b^2 \left (b x+c x^2\right )}-\frac{\tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right ) (A b e-4 A c d+2 b B d)}{b^3 \sqrt{d}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

Rule 820

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_Symbol] :> Simp
[((d + e*x)^m*(a + b*x + c*x^2)^(p + 1)*(f*b - 2*a*g + (2*c*f - b*g)*x))/((p + 1)*(b^2 - 4*a*c)), x] + Dist[1/
((p + 1)*(b^2 - 4*a*c)), Int[(d + e*x)^(m - 1)*(a + b*x + c*x^2)^(p + 1)*Simp[g*(2*a*e*m + b*d*(2*p + 3)) - f*
(b*e*m + 2*c*d*(2*p + 3)) - e*(2*c*f - b*g)*(m + 2*p + 3)*x, x], 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] && LtQ[p, -1] && GtQ[m, 0] && (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 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]

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]

Rubi steps

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

Mathematica [A]  time = 0.656278, size = 232, normalized size = 1.47 \[ \frac{\frac{\frac{d \left (-b c (3 A e+2 B d)+4 A c^2 d+b^2 B e\right ) \left (\sqrt{c} \sqrt{d+e x}-\sqrt{c d-b e} \tanh ^{-1}\left (\frac{\sqrt{c} \sqrt{d+e x}}{\sqrt{c d-b e}}\right )\right )}{\sqrt{c} (c d-b e)}+\left (\sqrt{d+e x}-\sqrt{d} \tanh ^{-1}\left (\frac{\sqrt{d+e x}}{\sqrt{d}}\right )\right ) (A b e-4 A c d+2 b B d)}{b^2}-\frac{c (d+e x)^{3/2} (A b e-2 A c d+b B d)}{b (b+c x) (b e-c d)}-\frac{A (d+e x)^{3/2}}{x (b+c x)}}{b d} \]

Antiderivative was successfully verified.

[In]

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

[Out]

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

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Maple [B]  time = 0.02, size = 299, normalized size = 1.9 \begin{align*} -{\frac{A}{{b}^{2}x}\sqrt{ex+d}}-{\frac{Ae}{{b}^{2}}{\it Artanh} \left ({\sqrt{ex+d}{\frac{1}{\sqrt{d}}}} \right ){\frac{1}{\sqrt{d}}}}+4\,{\frac{\sqrt{d}Ac}{{b}^{3}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-2\,{\frac{\sqrt{d}B}{{b}^{2}}{\it Artanh} \left ({\frac{\sqrt{ex+d}}{\sqrt{d}}} \right ) }-{\frac{Ace}{{b}^{2} \left ( cex+be \right ) }\sqrt{ex+d}}+{\frac{Be}{b \left ( cex+be \right ) }\sqrt{ex+d}}-3\,{\frac{Ace}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+4\,{\frac{A{c}^{2}d}{{b}^{3}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) }+{\frac{Be}{b}\arctan \left ({c\sqrt{ex+d}{\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}} \right ){\frac{1}{\sqrt{ \left ( be-cd \right ) c}}}}-2\,{\frac{Bdc}{{b}^{2}\sqrt{ \left ( be-cd \right ) c}}\arctan \left ({\frac{\sqrt{ex+d}c}{\sqrt{ \left ( be-cd \right ) c}}} \right ) } \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^2,x)

[Out]

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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^2,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [B]  time = 4.95055, size = 3279, normalized size = 20.75 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^2,x, algorithm="fricas")

[Out]

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

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Sympy [B]  time = 158.093, size = 1431, normalized size = 9.06 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)**(1/2)/(c*x**2+b*x)**2,x)

[Out]

2*A*c**2*d*e*sqrt(d + e*x)/(2*b**4*e**2 - 2*b**3*c*d*e + 2*b**3*c*e**2*x - 2*b**2*c**2*d*e*x) - 2*A*c*e**2*sqr
t(d + e*x)/(2*b**3*e**2 - 2*b**2*c*d*e + 2*b**2*c*e**2*x - 2*b*c**2*d*e*x) + A*c*e**2*sqrt(-1/(c*(b*e - c*d)**
3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*
(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - A*c*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e -
c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2
*b) - A*c**2*d*e*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c
*(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) + A*c**2*d*e*sqrt(-1/(c*(b
*e - c*d)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*s
qrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b**2) - A*d*e*sqrt(d**(-3))*log(-d**2*sqrt(d**(-3)) + sqrt(d +
e*x))/(2*b**2) + A*d*e*sqrt(d**(-3))*log(d**2*sqrt(d**(-3)) + sqrt(d + e*x))/(2*b**2) - 2*A*e*atan(sqrt(d + e*
x)/sqrt(b*e/c - d))/(b**2*sqrt(b*e/c - d)) + 2*A*e*atan(sqrt(d + e*x)/sqrt(-d))/(b**2*sqrt(-d)) - A*sqrt(d + e
*x)/(b**2*x) + 4*A*c*d*atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b**3*sqrt(b*e/c - d)) - 4*A*c*d*atan(sqrt(d + e*x)
/sqrt(-d))/(b**3*sqrt(-d)) - 2*B*c*d*e*sqrt(d + e*x)/(2*b**3*e**2 - 2*b**2*c*d*e + 2*b**2*c*e**2*x - 2*b*c**2*
d*e*x) - B*e**2*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*
(b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/2 + B*e**2*sqrt(-1/(c*(b*e - c*d)**3
))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(c*(b
*e - c*d)**3)) + sqrt(d + e*x))/2 + 2*B*e**2*sqrt(d + e*x)/(2*b**2*e**2 - 2*b*c*d*e + 2*b*c*e**2*x - 2*c**2*d*
e*x) + B*c*d*e*sqrt(-1/(c*(b*e - c*d)**3))*log(-b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) + 2*b*c*d*e*sqrt(-1/(c*(
b*e - c*d)**3)) - c**2*d**2*sqrt(-1/(c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - B*c*d*e*sqrt(-1/(c*(b*e - c*d
)**3))*log(b**2*e**2*sqrt(-1/(c*(b*e - c*d)**3)) - 2*b*c*d*e*sqrt(-1/(c*(b*e - c*d)**3)) + c**2*d**2*sqrt(-1/(
c*(b*e - c*d)**3)) + sqrt(d + e*x))/(2*b) - 2*B*d*atan(sqrt(d + e*x)/sqrt(b*e/c - d))/(b**2*sqrt(b*e/c - d)) +
2*B*d*atan(sqrt(d + e*x)/sqrt(-d))/(b**2*sqrt(-d))

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Giac [A]  time = 1.32656, size = 316, normalized size = 2. \begin{align*} -\frac{{\left (2 \, B b c d - 4 \, A c^{2} d - B b^{2} e + 3 \, A b c e\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}} + \frac{{\left (2 \, B b d - 4 \, A c d + A b e\right )} \arctan \left (\frac{\sqrt{x e + d}}{\sqrt{-d}}\right )}{b^{3} \sqrt{-d}} + \frac{{\left (x e + d\right )}^{\frac{3}{2}} B b e - 2 \,{\left (x e + d\right )}^{\frac{3}{2}} A c e - \sqrt{x e + d} B b d e + 2 \, \sqrt{x e + d} A c d e - \sqrt{x e + d} A b 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}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((B*x+A)*(e*x+d)^(1/2)/(c*x^2+b*x)^2,x, algorithm="giac")

[Out]

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