\(\int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx\) [91]

Optimal result

Integrand size = 25, antiderivative size = 180 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=-\frac {2 g \sqrt {d+e x}}{c}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {c} f-\sqrt {a} g\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}}+\frac {\sqrt {\sqrt {c} d+\sqrt {a} e} \left (\sqrt {c} f+\sqrt {a} g\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d+\sqrt {a} e}}\right )}{\sqrt {a} c^{5/4}} \] Output:

-2*g*(e*x+d)^(1/2)/c-(c^(1/2)*d-a^(1/2)*e)^(1/2)*(c^(1/2)*f-a^(1/2)*g)*arc 
tanh(c^(1/4)*(e*x+d)^(1/2)/(c^(1/2)*d-a^(1/2)*e)^(1/2))/a^(1/2)/c^(5/4)+(c 
^(1/2)*d+a^(1/2)*e)^(1/2)*(c^(1/2)*f+a^(1/2)*g)*arctanh(c^(1/4)*(e*x+d)^(1 
/2)/(c^(1/2)*d+a^(1/2)*e)^(1/2))/a^(1/2)/c^(5/4)
 

Mathematica [A] (verified)

Time = 0.54 (sec) , antiderivative size = 211, normalized size of antiderivative = 1.17 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=\frac {-2 \sqrt {a} \sqrt {c} g \sqrt {d+e x}-\sqrt {-c d-\sqrt {a} \sqrt {c} e} \left (\sqrt {c} f+\sqrt {a} g\right ) \arctan \left (\frac {\sqrt {-c d-\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d+\sqrt {a} e}\right )+\sqrt {-c d+\sqrt {a} \sqrt {c} e} \left (\sqrt {c} f-\sqrt {a} g\right ) \arctan \left (\frac {\sqrt {-c d+\sqrt {a} \sqrt {c} e} \sqrt {d+e x}}{\sqrt {c} d-\sqrt {a} e}\right )}{\sqrt {a} c^{3/2}} \] Input:

Integrate[(Sqrt[d + e*x]*(f + g*x))/(a - c*x^2),x]
 

Output:

(-2*Sqrt[a]*Sqrt[c]*g*Sqrt[d + e*x] - Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*(Sq 
rt[c]*f + Sqrt[a]*g)*ArcTan[(Sqrt[-(c*d) - Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x 
])/(Sqrt[c]*d + Sqrt[a]*e)] + Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*(Sqrt[c]*f 
- Sqrt[a]*g)*ArcTan[(Sqrt[-(c*d) + Sqrt[a]*Sqrt[c]*e]*Sqrt[d + e*x])/(Sqrt 
[c]*d - Sqrt[a]*e)])/(Sqrt[a]*c^(3/2))
 

Rubi [A] (verified)

Time = 0.43 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.06, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.200, Rules used = {653, 25, 654, 1480, 221}

Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.

\(\displaystyle \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx\)

\(\Big \downarrow \) 653

\(\displaystyle -\frac {\int -\frac {c d f+a e g+c (e f+d g) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 g \sqrt {d+e x}}{c}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {c d f+a e g+c (e f+d g) x}{\sqrt {d+e x} \left (a-c x^2\right )}dx}{c}-\frac {2 g \sqrt {d+e x}}{c}\)

\(\Big \downarrow \) 654

\(\displaystyle \frac {2 \int \frac {\left (c d^2-a e^2\right ) g-c (e f+d g) (d+e x)}{c d^2-2 c (d+e x) d-a e^2+c (d+e x)^2}d\sqrt {d+e x}}{c}-\frac {2 g \sqrt {d+e x}}{c}\)

\(\Big \downarrow \) 1480

\(\displaystyle \frac {2 \left (\frac {\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right ) \left (\sqrt {c} f-\sqrt {a} g\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d-\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}-\frac {\sqrt {c} \left (\sqrt {a} e+\sqrt {c} d\right ) \left (\sqrt {a} g+\sqrt {c} f\right ) \int \frac {1}{c (d+e x)-\sqrt {c} \left (\sqrt {c} d+\sqrt {a} e\right )}d\sqrt {d+e x}}{2 \sqrt {a}}\right )}{c}-\frac {2 g \sqrt {d+e x}}{c}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {2 \left (\frac {\sqrt {\sqrt {a} e+\sqrt {c} d} \left (\sqrt {a} g+\sqrt {c} f\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {a} e+\sqrt {c} d}}\right )}{2 \sqrt {a} \sqrt [4]{c}}-\frac {\sqrt {\sqrt {c} d-\sqrt {a} e} \left (\sqrt {c} f-\sqrt {a} g\right ) \text {arctanh}\left (\frac {\sqrt [4]{c} \sqrt {d+e x}}{\sqrt {\sqrt {c} d-\sqrt {a} e}}\right )}{2 \sqrt {a} \sqrt [4]{c}}\right )}{c}-\frac {2 g \sqrt {d+e x}}{c}\)

Input:

Int[(Sqrt[d + e*x]*(f + g*x))/(a - c*x^2),x]
 

Output:

(-2*g*Sqrt[d + e*x])/c + (2*(-1/2*(Sqrt[Sqrt[c]*d - Sqrt[a]*e]*(Sqrt[c]*f 
- Sqrt[a]*g)*ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d - Sqrt[a]*e]]) 
/(Sqrt[a]*c^(1/4)) + (Sqrt[Sqrt[c]*d + Sqrt[a]*e]*(Sqrt[c]*f + Sqrt[a]*g)* 
ArcTanh[(c^(1/4)*Sqrt[d + e*x])/Sqrt[Sqrt[c]*d + Sqrt[a]*e]])/(2*Sqrt[a]*c 
^(1/4))))/c
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

rule 221

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

rule 653

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

rule 654

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

rule 1480

Int[((d_) + (e_.)*(x_)^2)/((a_) + (b_.)*(x_)^2 + (c_.)*(x_)^4), x_Symbol] : 
> With[{q = Rt[b^2 - 4*a*c, 2]}, Simp[(e/2 + (2*c*d - b*e)/(2*q))   Int[1/( 
b/2 - q/2 + c*x^2), x], x] + Simp[(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]
 

Maple [A] (verified)

Time = 1.10 (sec) , antiderivative size = 190, normalized size of antiderivative = 1.06

method	result	size
derivativedivides	\(-\frac {2 g \sqrt {e x +d}}{c}+\frac {\left (a \,e^{2} g +c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-a \,e^{2} g -c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(190\)
risch	\(-\frac {2 g \sqrt {e x +d}}{c}+\frac {\left (a \,e^{2} g +c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-a \,e^{2} g -c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(190\)
pseudoelliptic	\(-\frac {2 g \sqrt {e x +d}}{c}+\frac {\left (a \,e^{2} g +c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}-\frac {\left (-a \,e^{2} g -c d e f +\sqrt {a c \,e^{2}}\, d g +\sqrt {a c \,e^{2}}\, e f \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(190\)
default	\(-\frac {2 g \sqrt {e x +d}}{c}-\frac {\left (-a \,e^{2} g -c d e f -\sqrt {a c \,e^{2}}\, d g -\sqrt {a c \,e^{2}}\, e f \right ) \operatorname {arctanh}\left (\frac {c \sqrt {e x +d}}{\sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (c d +\sqrt {a c \,e^{2}}\right ) c}}+\frac {\left (a \,e^{2} g +c d e f -\sqrt {a c \,e^{2}}\, d g -\sqrt {a c \,e^{2}}\, e f \right ) \arctan \left (\frac {c \sqrt {e x +d}}{\sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\right )}{\sqrt {a c \,e^{2}}\, \sqrt {\left (-c d +\sqrt {a c \,e^{2}}\right ) c}}\)	\(194\)

Input:

int((e*x+d)^(1/2)*(g*x+f)/(-c*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-2*g*(e*x+d)^(1/2)/c+(a*e^2*g+c*d*e*f+(a*c*e^2)^(1/2)*d*g+(a*c*e^2)^(1/2)* 
e*f)/(a*c*e^2)^(1/2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arctanh(c*(e*x+d)^(1/ 
2)/((c*d+(a*c*e^2)^(1/2))*c)^(1/2))-(-a*e^2*g-c*d*e*f+(a*c*e^2)^(1/2)*d*g+ 
(a*c*e^2)^(1/2)*e*f)/(a*c*e^2)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2)*arct 
an(c*(e*x+d)^(1/2)/((-c*d+(a*c*e^2)^(1/2))*c)^(1/2))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1530 vs. \(2 (130) = 260\).

Time = 0.12 (sec) , antiderivative size = 1530, normalized size of antiderivative = 8.50 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/2*(c*sqrt((c*d*f^2 + 2*a*e*f*g + a*d*g^2 + a*c^2*sqrt((c^2*e^2*f^4 + 4*c 
^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2 
*g^2)/(a*c^5)))/(a*c^2))*log(-(c^2*e*f^4 + 2*c^2*d*f^3*g - 2*a*c*d*f*g^3 - 
 a^2*e*g^4)*sqrt(e*x + d) + (a*c^2*e*f^2*g + 2*a*c^2*d*f*g^2 + a^2*c*e*g^3 
 - a*c^4*f*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2 
*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a*c^5)))*sqrt((c*d*f^2 + 2*a*e*f* 
g + a*d*g^2 + a*c^2*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 
+ a^2*e^2*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a*c^5)))/(a*c^2))) - c*s 
qrt((c*d*f^2 + 2*a*e*f*g + a*d*g^2 + a*c^2*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f 
^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a 
*c^5)))/(a*c^2))*log(-(c^2*e*f^4 + 2*c^2*d*f^3*g - 2*a*c*d*f*g^3 - a^2*e*g 
^4)*sqrt(e*x + d) - (a*c^2*e*f^2*g + 2*a*c^2*d*f*g^2 + a^2*c*e*g^3 - a*c^4 
*f*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2 
*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a*c^5)))*sqrt((c*d*f^2 + 2*a*e*f*g + a*d* 
g^2 + a*c^2*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f^3*g + 4*a*c*d*e*f*g^3 + a^2*e^ 
2*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a*c^5)))/(a*c^2))) + c*sqrt((c*d 
*f^2 + 2*a*e*f*g + a*d*g^2 - a*c^2*sqrt((c^2*e^2*f^4 + 4*c^2*d*e*f^3*g + 4 
*a*c*d*e*f*g^3 + a^2*e^2*g^4 + 2*(2*c^2*d^2 + a*c*e^2)*f^2*g^2)/(a*c^5)))/ 
(a*c^2))*log(-(c^2*e*f^4 + 2*c^2*d*f^3*g - 2*a*c*d*f*g^3 - a^2*e*g^4)*sqrt 
(e*x + d) + (a*c^2*e*f^2*g + 2*a*c^2*d*f*g^2 + a^2*c*e*g^3 + a*c^4*f*sq...
 

Sympy [F]

\[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=- \int \frac {f \sqrt {d + e x}}{- a + c x^{2}}\, dx - \int \frac {g x \sqrt {d + e x}}{- a + c x^{2}}\, dx \] Input:

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

Output:

-Integral(f*sqrt(d + e*x)/(-a + c*x**2), x) - Integral(g*x*sqrt(d + e*x)/( 
-a + c*x**2), x)
 

Maxima [F]

\[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=\int { -\frac {\sqrt {e x + d} {\left (g x + f\right )}}{c x^{2} - a} \,d x } \] Input:

integrate((e*x+d)^(1/2)*(g*x+f)/(-c*x^2+a),x, algorithm="maxima")
 

Output:

-integrate(sqrt(e*x + d)*(g*x + f)/(c*x^2 - a), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 316 vs. \(2 (130) = 260\).

Time = 0.15 (sec) , antiderivative size = 316, normalized size of antiderivative = 1.76 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=-\frac {2 \, \sqrt {e x + d} g}{c} - \frac {{\left (\sqrt {a c} c^{3} d^{2} e f - \sqrt {a c} a c^{2} e^{3} f + {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} g {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d + \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d - \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d - \sqrt {a c} c e} {\left | e \right |}} + \frac {{\left (\sqrt {a c} c^{3} d^{2} e f - \sqrt {a c} a c^{2} e^{3} f - {\left (a c^{2} d^{2} - a^{2} c e^{2}\right )} g {\left | c \right |} {\left | e \right |}\right )} \arctan \left (\frac {\sqrt {e x + d}}{\sqrt {-\frac {c^{2} d - \sqrt {c^{4} d^{2} - {\left (c^{2} d^{2} - a c e^{2}\right )} c^{2}}}{c^{2}}}}\right )}{{\left (a c^{3} d + \sqrt {a c} a c^{2} e\right )} \sqrt {-c^{2} d + \sqrt {a c} c e} {\left | e \right |}} \] Input:

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

Output:

-2*sqrt(e*x + d)*g/c - (sqrt(a*c)*c^3*d^2*e*f - sqrt(a*c)*a*c^2*e^3*f + (a 
*c^2*d^2 - a^2*c*e^2)*g*abs(c)*abs(e))*arctan(sqrt(e*x + d)/sqrt(-(c^2*d + 
 sqrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c^3*d - sqrt(a*c)*a*c^ 
2*e)*sqrt(-c^2*d - sqrt(a*c)*c*e)*abs(e)) + (sqrt(a*c)*c^3*d^2*e*f - sqrt( 
a*c)*a*c^2*e^3*f - (a*c^2*d^2 - a^2*c*e^2)*g*abs(c)*abs(e))*arctan(sqrt(e* 
x + d)/sqrt(-(c^2*d - sqrt(c^4*d^2 - (c^2*d^2 - a*c*e^2)*c^2))/c^2))/((a*c 
^3*d + sqrt(a*c)*a*c^2*e)*sqrt(-c^2*d + sqrt(a*c)*c*e)*abs(e))
 

Mupad [B] (verification not implemented)

Time = 0.29 (sec) , antiderivative size = 4278, normalized size of antiderivative = 23.77 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=\text {Too large to display} \] Input:

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

Output:

2*atanh((32*d*e^3*g^2*(a^3*c^5)^(1/2)*(d + e*x)^(1/2)*((d*g^2)/(4*c^2) + ( 
d*f^2)/(4*a*c) + (e*f*g)/(2*c^2) - (e*f^2*(a^3*c^5)^(1/2))/(4*a^2*c^4) - ( 
e*g^2*(a^3*c^5)^(1/2))/(4*a*c^5) - (d*f*g*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1 
/2))/(16*a*c^2*e^5*f^3 - 16*c^3*d^2*e^3*f^3 - (16*a*e^5*g^3*(a^3*c^5)^(1/2 
))/c^2 - (16*e^5*f^2*g*(a^3*c^5)^(1/2))/c + 16*a^2*c*e^5*f*g^2 + (16*d^2*e 
^3*g^3*(a^3*c^5)^(1/2))/c - 32*c^3*d^3*e^2*f^2*g + (16*d^2*e^3*f^2*g*(a^3* 
c^5)^(1/2))/a + (32*d^3*e^2*f*g^2*(a^3*c^5)^(1/2))/a - 16*a*c^2*d^2*e^3*f* 
g^2 - (32*d*e^4*f*g^2*(a^3*c^5)^(1/2))/c + 32*a*c^2*d*e^4*f^2*g) + (32*a*c 
^2*e^4*f^2*(d + e*x)^(1/2)*((d*g^2)/(4*c^2) + (d*f^2)/(4*a*c) + (e*f*g)/(2 
*c^2) - (e*f^2*(a^3*c^5)^(1/2))/(4*a^2*c^4) - (e*g^2*(a^3*c^5)^(1/2))/(4*a 
*c^5) - (d*f*g*(a^3*c^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*a^2*e^5*f*g^2 - 16 
*c^2*d^2*e^3*f^3 + 16*a*c*e^5*f^3 - (16*a*e^5*g^3*(a^3*c^5)^(1/2))/c^3 - ( 
16*e^5*f^2*g*(a^3*c^5)^(1/2))/c^2 + (16*d^2*e^3*g^3*(a^3*c^5)^(1/2))/c^2 - 
 32*c^2*d^3*e^2*f^2*g + 32*a*c*d*e^4*f^2*g - (32*d*e^4*f*g^2*(a^3*c^5)^(1/ 
2))/c^2 - 16*a*c*d^2*e^3*f*g^2 + (16*d^2*e^3*f^2*g*(a^3*c^5)^(1/2))/(a*c) 
+ (32*d^3*e^2*f*g^2*(a^3*c^5)^(1/2))/(a*c)) + (32*a^2*c*e^4*g^2*(d + e*x)^ 
(1/2)*((d*g^2)/(4*c^2) + (d*f^2)/(4*a*c) + (e*f*g)/(2*c^2) - (e*f^2*(a^3*c 
^5)^(1/2))/(4*a^2*c^4) - (e*g^2*(a^3*c^5)^(1/2))/(4*a*c^5) - (d*f*g*(a^3*c 
^5)^(1/2))/(2*a^2*c^4))^(1/2))/(16*a^2*e^5*f*g^2 - 16*c^2*d^2*e^3*f^3 + 16 
*a*c*e^5*f^3 - (16*a*e^5*g^3*(a^3*c^5)^(1/2))/c^3 - (16*e^5*f^2*g*(a^3*...
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 268, normalized size of antiderivative = 1.49 \[ \int \frac {\sqrt {d+e x} (f+g x)}{a-c x^2} \, dx=\frac {-2 \sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) c f +2 \sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}\, \mathit {atan} \left (\frac {\sqrt {e x +d}\, c}{\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e -c d}}\right ) a g -\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c f +\sqrt {a}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) c f -\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (-\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a g +\sqrt {c}\, \sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}\, \mathrm {log}\left (\sqrt {\sqrt {c}\, \sqrt {a}\, e +c d}+\sqrt {c}\, \sqrt {e x +d}\right ) a g -4 \sqrt {e x +d}\, a c g}{2 a \,c^{2}} \] Input:

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

Output:

( - 2*sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e - c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c 
)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*c*f + 2*sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - 
 c*d)*atan((sqrt(d + e*x)*c)/(sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e - c*d)))*a*g 
- sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c)*sqrt(a)*e + c* 
d) + sqrt(c)*sqrt(d + e*x))*c*f + sqrt(a)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*lo 
g(sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*c*f - sqrt(c)*sqr 
t(sqrt(c)*sqrt(a)*e + c*d)*log( - sqrt(sqrt(c)*sqrt(a)*e + c*d) + sqrt(c)* 
sqrt(d + e*x))*a*g + sqrt(c)*sqrt(sqrt(c)*sqrt(a)*e + c*d)*log(sqrt(sqrt(c 
)*sqrt(a)*e + c*d) + sqrt(c)*sqrt(d + e*x))*a*g - 4*sqrt(d + e*x)*a*c*g)/( 
2*a*c**2)