\(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\) [199]

Optimal result

Integrand size = 46, antiderivative size = 186 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}-\frac {2 \sqrt {2 c d-b e} (e f-d g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \] Output:

2*(-d*g+e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(e*x+d)^(1/2)-2/3* 
g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(3/2)/c/e^2/(e*x+d)^(3/2)-2*(-b*e+2*c*d 
)^(1/2)*(-d*g+e*f)*arctanh((d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/(-b*e+2* 
c*d)^(1/2)/(e*x+d)^(1/2))/e^2
 

Mathematica [A] (verified)

Time = 0.36 (sec) , antiderivative size = 152, normalized size of antiderivative = 0.82 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {d+e x} \sqrt {c d-b e-c e x} \left (\sqrt {-b e+c (d-e x)} (b e g+c (3 e f-4 d g+e g x))+3 c \sqrt {-2 c d+b e} (-e f+d g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )\right )}{3 c e^2 \sqrt {(d+e x) (-b e+c (d-e x))}} \] Input:

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 
(3/2),x]
 

Output:

(2*Sqrt[d + e*x]*Sqrt[c*d - b*e - c*e*x]*(Sqrt[-(b*e) + c*(d - e*x)]*(b*e* 
g + c*(3*e*f - 4*d*g + e*g*x)) + 3*c*Sqrt[-2*c*d + b*e]*(-(e*f) + d*g)*Arc 
Tan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]]))/(3*c*e^2*Sqrt[(d + e*x)* 
(-(b*e) + c*(d - e*x))])
 

Rubi [A] (verified)

Time = 0.72 (sec) , antiderivative size = 183, normalized size of antiderivative = 0.98, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1221, 1131, 1136, 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.

\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{3/2}} \, dx\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {(e f-d g) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{3/2}}dx}{e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1131

\(\displaystyle \frac {(e f-d g) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(e f-d g) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )}{e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {(e f-d g) \left (\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}-\frac {2 \sqrt {2 c d-b e} \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e}\right )}{e}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 c e^2 (d+e x)^{3/2}}\)

Input:

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(3/2), 
x]
 

Output:

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

Defintions of rubi rules used

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 1131

Int[((d_.) + (e_.)*(x_))^(m_)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p_), x_S 
ymbol] :> Simp[(d + e*x)^(m + 1)*((a + b*x + c*x^2)^p/(e*(m + 2*p + 1))), x 
] - Simp[p*((2*c*d - b*e)/(e^2*(m + 2*p + 1)))   Int[(d + e*x)^(m + 1)*(a + 
 b*x + c*x^2)^(p - 1), x], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 - b 
*d*e + a*e^2, 0] && GtQ[p, 0] && (LeQ[-2, m, 0] || EqQ[m + p + 1, 0]) && Ne 
Q[m + 2*p + 1, 0] && IntegerQ[2*p]
 

rule 1136

Int[1/(Sqrt[(d_.) + (e_.)*(x_)]*Sqrt[(a_.) + (b_.)*(x_) + (c_.)*(x_)^2]), x 
_Symbol] :> Simp[2*e   Subst[Int[1/(2*c*d - b*e + e^2*x^2), x], x, Sqrt[a + 
 b*x + c*x^2]/Sqrt[d + e*x]], x] /; FreeQ[{a, b, c, d, e}, x] && EqQ[c*d^2 
- b*d*e + a*e^2, 0]
 

rule 1221

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

Maple [A] (verified)

Time = 2.10 (sec) , antiderivative size = 322, normalized size of antiderivative = 1.73

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d e g -3 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{2} f -6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} g +6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d e f +c e g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+b e g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-4 c d g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+3 c e f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\right )}{3 \sqrt {e x +d}\, \sqrt {-c e x -b e +c d}\, c \,e^{2} \sqrt {b e -2 c d}}\)	\(322\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2),x,method= 
_RETURNVERBOSE)
 

Output:

2/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e 
-2*c*d)^(1/2))*b*c*d*e*g-3*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2) 
)*b*c*e^2*f-6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*g+6 
*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e*f+c*e*g*x*(-c*e* 
x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+b*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d 
)^(1/2)-4*c*d*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*c*e*f*(-c*e*x-b 
*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^(1/2)/(-c*e*x-b*e+c*d)^(1/2)/c/e^ 
2/(b*e-2*c*d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 421, normalized size of antiderivative = 2.26 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\left [-\frac {3 \, {\left (c d e f - c d^{2} g + {\left (c e^{2} f - c d e g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e g x + 3 \, c e f - {\left (4 \, c d - b e\right )} g\right )} \sqrt {e x + d}}{3 \, {\left (c e^{3} x + c d e^{2}\right )}}, -\frac {2 \, {\left (3 \, {\left (c d e f - c d^{2} g + {\left (c e^{2} f - c d e g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x}\right ) - \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (c e g x + 3 \, c e f - {\left (4 \, c d - b e\right )} g\right )} \sqrt {e x + d}\right )}}{3 \, {\left (c e^{3} x + c d e^{2}\right )}}\right ] \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2),x, 
algorithm="fricas")
 

Output:

[-1/3*(3*(c*d*e*f - c*d^2*g + (c*e^2*f - c*d*e*g)*x)*sqrt(2*c*d - b*e)*log 
(-(c*e^2*x^2 - 3*c*d^2 + 2*b*d*e - 2*(c*d*e - b*e^2)*x - 2*sqrt(-c*e^2*x^2 
 - b*e^2*x + c*d^2 - b*d*e)*sqrt(2*c*d - b*e)*sqrt(e*x + d))/(e^2*x^2 + 2* 
d*e*x + d^2)) - 2*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(c*e*g*x + 3* 
c*e*f - (4*c*d - b*e)*g)*sqrt(e*x + d))/(c*e^3*x + c*d*e^2), -2/3*(3*(c*d* 
e*f - c*d^2*g + (c*e^2*f - c*d*e*g)*x)*sqrt(-2*c*d + b*e)*arctan(-sqrt(-c* 
e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(-2*c*d + b*e)*sqrt(e*x + d)/(2*c*d 
^2 - b*d*e + (2*c*d*e - b*e^2)*x)) - sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b 
*d*e)*(c*e*g*x + 3*c*e*f - (4*c*d - b*e)*g)*sqrt(e*x + d))/(c*e^3*x + c*d* 
e^2)]
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {3}{2}}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(1/2)/(e*x+d)**(3/ 
2),x)
 

Output:

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(3/2), 
x)
 

Maxima [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {3}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2),x, 
algorithm="maxima")
 

Output:

integrate(sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*(g*x + f)/(e*x + d)^( 
3/2), x)
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 157, normalized size of antiderivative = 0.84 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {3 \, {\left (2 \, c d e f - b e^{2} f - 2 \, c d^{2} g + b d e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} e f - 3 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d g - {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} g}{c^{3}}\right )}}{3 \, e^{2}} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2),x, 
algorithm="giac")
 

Output:

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

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{3/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(3/2 
),x)
 

Output:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(1/2))/(d + e*x)^(3/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 166, normalized size of antiderivative = 0.89 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{3/2}} \, dx=\frac {2 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d g -2 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c e f +\frac {2 \sqrt {-c e x -b e +c d}\, b e g}{3}-\frac {8 \sqrt {-c e x -b e +c d}\, c d g}{3}+2 \sqrt {-c e x -b e +c d}\, c e f +\frac {2 \sqrt {-c e x -b e +c d}\, c e g x}{3}}{c \,e^{2}} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(1/2)/(e*x+d)^(3/2),x)
 

Output:

(2*(3*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d)) 
*c*d*g - 3*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2* 
c*d))*c*e*f + sqrt( - b*e + c*d - c*e*x)*b*e*g - 4*sqrt( - b*e + c*d - c*e 
*x)*c*d*g + 3*sqrt( - b*e + c*d - c*e*x)*c*e*f + sqrt( - b*e + c*d - c*e*x 
)*c*e*g*x))/(3*c*e**2)