3.1.12 \(\int \frac {\sqrt {c+d x} (e+f x)}{x (a+b x)} \, dx\)

Optimal. Leaf size=101 \[ \frac {2 \sqrt {b c-a d} (b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a b^{3/2}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {2 f \sqrt {c+d x}}{b} \]

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Rubi [A]  time = 0.11, antiderivative size = 101, normalized size of antiderivative = 1.00, number of steps used = 6, number of rules used = 4, integrand size = 25, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.160, Rules used = {154, 156, 63, 208} \begin {gather*} \frac {2 \sqrt {b c-a d} (b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a b^{3/2}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {2 f \sqrt {c+d x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Int[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)),x]

[Out]

(2*f*Sqrt[c + d*x])/b - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*(b*e - a*f)*ArcTan
h[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*b^(3/2))

Rule 63

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[{p = Denominator[m]}, Dist[p/b, Sub
st[Int[x^(p*(m + 1) - 1)*(c - (a*d)/b + (d*x^p)/b)^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] &
& NeQ[b*c - a*d, 0] && LtQ[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntLinearQ[a,
b, c, d, m, n, x]

Rule 154

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)), x_Symb
ol] :> Simp[(h*(a + b*x)^m*(c + d*x)^(n + 1)*(e + f*x)^(p + 1))/(d*f*(m + n + p + 2)), x] + Dist[1/(d*f*(m + n
+ p + 2)), Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2) - h*(b*c*e*m + a*(d*e*(
n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) + h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x]
, x], x] /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] && IntegersQ[2*m, 2
*n, 2*p]

Rule 156

Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))), x_Symbol] :>
Dist[(b*g - a*h)/(b*c - a*d), Int[(e + f*x)^p/(a + b*x), x], x] - Dist[(d*g - c*h)/(b*c - a*d), Int[(e + f*x)
^p/(c + d*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]

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

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Mathematica [A]  time = 0.12, size = 101, normalized size = 1.00 \begin {gather*} \frac {2 \sqrt {b c-a d} (b e-a f) \tanh ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x}}{\sqrt {b c-a d}}\right )}{a b^{3/2}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {2 f \sqrt {c+d x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

Integrate[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)),x]

[Out]

(2*f*Sqrt[c + d*x])/b - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a + (2*Sqrt[b*c - a*d]*(b*e - a*f)*ArcTan
h[(Sqrt[b]*Sqrt[c + d*x])/Sqrt[b*c - a*d]])/(a*b^(3/2))

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IntegrateAlgebraic [A]  time = 0.17, size = 111, normalized size = 1.10 \begin {gather*} -\frac {2 \sqrt {a d-b c} (b e-a f) \tan ^{-1}\left (\frac {\sqrt {b} \sqrt {c+d x} \sqrt {a d-b c}}{b c-a d}\right )}{a b^{3/2}}-\frac {2 \sqrt {c} e \tanh ^{-1}\left (\frac {\sqrt {c+d x}}{\sqrt {c}}\right )}{a}+\frac {2 f \sqrt {c+d x}}{b} \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(Sqrt[c + d*x]*(e + f*x))/(x*(a + b*x)),x]

[Out]

(2*f*Sqrt[c + d*x])/b - (2*Sqrt[-(b*c) + a*d]*(b*e - a*f)*ArcTan[(Sqrt[b]*Sqrt[-(b*c) + a*d]*Sqrt[c + d*x])/(b
*c - a*d)])/(a*b^(3/2)) - (2*Sqrt[c]*e*ArcTanh[Sqrt[c + d*x]/Sqrt[c]])/a

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fricas [A]  time = 1.22, size = 449, normalized size = 4.45 \begin {gather*} \left [\frac {b \sqrt {c} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {d x + c} a f - {\left (b e - a f\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right )}{a b}, \frac {b \sqrt {c} e \log \left (\frac {d x - 2 \, \sqrt {d x + c} \sqrt {c} + 2 \, c}{x}\right ) + 2 \, \sqrt {d x + c} a f + 2 \, {\left (b e - a f\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right )}{a b}, \frac {2 \, b \sqrt {-c} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + 2 \, \sqrt {d x + c} a f - {\left (b e - a f\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b d x + 2 \, b c - a d - 2 \, \sqrt {d x + c} b \sqrt {\frac {b c - a d}{b}}}{b x + a}\right )}{a b}, \frac {2 \, {\left (b \sqrt {-c} e \arctan \left (\frac {\sqrt {d x + c} \sqrt {-c}}{c}\right ) + \sqrt {d x + c} a f + {\left (b e - a f\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {\sqrt {d x + c} b \sqrt {-\frac {b c - a d}{b}}}{b c - a d}\right )\right )}}{a b}\right ] \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[(b*sqrt(c)*e*log((d*x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(d*x + c)*a*f - (b*e - a*f)*sqrt((b*c - a*d
)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/b))/(b*x + a)))/(a*b), (b*sqrt(c)*e*log((d*
x - 2*sqrt(d*x + c)*sqrt(c) + 2*c)/x) + 2*sqrt(d*x + c)*a*f + 2*(b*e - a*f)*sqrt(-(b*c - a*d)/b)*arctan(-sqrt(
d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)))/(a*b), (2*b*sqrt(-c)*e*arctan(sqrt(d*x + c)*sqrt(-c)/c) + 2*sqrt
(d*x + c)*a*f - (b*e - a*f)*sqrt((b*c - a*d)/b)*log((b*d*x + 2*b*c - a*d - 2*sqrt(d*x + c)*b*sqrt((b*c - a*d)/
b))/(b*x + a)))/(a*b), 2*(b*sqrt(-c)*e*arctan(sqrt(d*x + c)*sqrt(-c)/c) + sqrt(d*x + c)*a*f + (b*e - a*f)*sqrt
(-(b*c - a*d)/b)*arctan(-sqrt(d*x + c)*b*sqrt(-(b*c - a*d)/b)/(b*c - a*d)))/(a*b)]

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giac [A]  time = 1.35, size = 112, normalized size = 1.11 \begin {gather*} \frac {2 \, c \arctan \left (\frac {\sqrt {d x + c}}{\sqrt {-c}}\right ) e}{a \sqrt {-c}} + \frac {2 \, \sqrt {d x + c} f}{b} + \frac {2 \, {\left (a b c f - a^{2} d f - b^{2} c e + a b d e\right )} \arctan \left (\frac {\sqrt {d x + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} a b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

2*c*arctan(sqrt(d*x + c)/sqrt(-c))*e/(a*sqrt(-c)) + 2*sqrt(d*x + c)*f/b + 2*(a*b*c*f - a^2*d*f - b^2*c*e + a*b
*d*e)*arctan(sqrt(d*x + c)*b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*a*b)

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maple [B]  time = 0.02, size = 196, normalized size = 1.94 \begin {gather*} -\frac {2 a d f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, b}-\frac {2 b c e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, a}+\frac {2 c f \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}+\frac {2 d e \arctan \left (\frac {\sqrt {d x +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}}-\frac {2 \sqrt {c}\, e \arctanh \left (\frac {\sqrt {d x +c}}{\sqrt {c}}\right )}{a}+\frac {2 \sqrt {d x +c}\, f}{b} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a),x)

[Out]

2*f*(d*x+c)^(1/2)/b-2*a/b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*d*f+2/((a*d-b*c)*b)^
(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c*f+2/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)
^(1/2)*b)*d*e-2/a*b/((a*d-b*c)*b)^(1/2)*arctan((d*x+c)^(1/2)/((a*d-b*c)*b)^(1/2)*b)*c*e-2*e*arctanh((d*x+c)^(1
/2)/c^(1/2))*c^(1/2)/a

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(d*x+c)^(1/2)/x/(b*x+a),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(a*d-b*c>0)', see `assume?` for
more details)Is a*d-b*c positive or negative?

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mupad [B]  time = 2.87, size = 2368, normalized size = 23.45

result too large to display

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(((e + f*x)*(c + d*x)^(1/2))/(x*(a + b*x)),x)

[Out]

(2*f*(c + d*x)^(1/2))/b - (c^(1/2)*e*atan(((c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b
^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d
^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b + (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b + (8*c^(1/2)*e*(a^3
*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)))/a)*1i)/a + (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 +
a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^
3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b - (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/
b - (8*c^(1/2)*e*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)))/a)*1i)/a)/((16*(b^3*c^2*d^3*e^3 - a*
b^2*c*d^4*e^3 - a^3*c*d^4*e*f^2 + b^3*c^3*d^2*e^2*f - 3*a*b^2*c^2*d^3*e^2*f - a*b^2*c^3*d^2*e*f^2 + 2*a^2*b*c^
2*d^3*e*f^2 + 2*a^2*b*c*d^4*e^2*f))/b - (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b^4*
c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*
e*f + 4*a^2*b^2*c*d^3*e*f))/b + (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b + (8*c^(1/2)*e*(a^3*b^
3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)))/a))/a + (c^(1/2)*e*((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^
2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2
- 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b - (c^(1/2)*e*((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b - (8
*c^(1/2)*e*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(c + d*x)^(1/2))/(a*b)))/a))/a))*2i)/a - (atan(((((8*(c + d*x)^(1/2
)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e
^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b + (((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d
^2*f))/b + (8*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2))/(a*b^4))*(
a*f - b*e)*(-b^3*(a*d - b*c))^(1/2))/(a*b^3))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*1i)/(a*b^3) + (((8*(c + d*x
)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c
*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b - (((8*(a^3*b^2*c*d^3*f - a^2*b^3
*c^2*d^2*f))/b - (8*(a^3*b^3*d^3 - 2*a^2*b^4*c*d^2)*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2))/(a*b
^4))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2))/(a*b^3))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*1i)/(a*b^3))/((16*(b^
3*c^2*d^3*e^3 - a*b^2*c*d^4*e^3 - a^3*c*d^4*e*f^2 + b^3*c^3*d^2*e^2*f - 3*a*b^2*c^2*d^3*e^2*f - a*b^2*c^3*d^2*
e*f^2 + 2*a^2*b*c^2*d^3*e*f^2 + 2*a^2*b*c*d^4*e^2*f))/b - (((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e^2
+ 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b^3*
c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b + (((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b + (8*(a^3*b^3*d^3 - 2*a^
2*b^4*c*d^2)*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2))/(a*b^4))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/
2))/(a*b^3))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2))/(a*b^3) + (((8*(c + d*x)^(1/2)*(a^4*d^4*f^2 + a^2*b^2*d^4*e
^2 + 2*b^4*c^2*d^2*e^2 - 2*a^3*b*d^4*e*f + a^2*b^2*c^2*d^2*f^2 - 2*a*b^3*c*d^3*e^2 - 2*a^3*b*c*d^3*f^2 - 2*a*b
^3*c^2*d^2*e*f + 4*a^2*b^2*c*d^3*e*f))/b - (((8*(a^3*b^2*c*d^3*f - a^2*b^3*c^2*d^2*f))/b - (8*(a^3*b^3*d^3 - 2
*a^2*b^4*c*d^2)*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*(c + d*x)^(1/2))/(a*b^4))*(a*f - b*e)*(-b^3*(a*d - b*c))^
(1/2))/(a*b^3))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2))/(a*b^3)))*(a*f - b*e)*(-b^3*(a*d - b*c))^(1/2)*2i)/(a*b^
3)

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sympy [A]  time = 27.33, size = 97, normalized size = 0.96 \begin {gather*} \frac {2 f \sqrt {c + d x}}{b} + \frac {2 c e \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {- c}} \right )}}{a \sqrt {- c}} - \frac {2 \left (a d - b c\right ) \left (a f - b e\right ) \operatorname {atan}{\left (\frac {\sqrt {c + d x}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{a b^{2} \sqrt {\frac {a d - b c}{b}}} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((f*x+e)*(d*x+c)**(1/2)/x/(b*x+a),x)

[Out]

2*f*sqrt(c + d*x)/b + 2*c*e*atan(sqrt(c + d*x)/sqrt(-c))/(a*sqrt(-c)) - 2*(a*d - b*c)*(a*f - b*e)*atan(sqrt(c
+ d*x)/sqrt((a*d - b*c)/b))/(a*b**2*sqrt((a*d - b*c)/b))

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