\(\int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\) [201]

Optimal result

Integrand size = 46, antiderivative size = 220 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=-\frac {(e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{2 e^2 (d+e x)^{5/2}}+\frac {(c e f-9 c d g+4 b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{4 e^2 (2 c d-b e) (d+e x)^{3/2}}+\frac {c (c e f+7 c d g-4 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{4 e^2 (2 c d-b e)^{3/2}} \] Output:

-1/2*(-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)^(5/2)+1 
/4*(4*b*e*g-9*c*d*g+c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(1/2)/e^2/(-b* 
e+2*c*d)/(e*x+d)^(3/2)+1/4*c*(-4*b*e*g+7*c*d*g+c*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/(-b*e+2*c 
*d)^(3/2)
 

Mathematica [A] (verified)

Time = 0.79 (sec) , antiderivative size = 181, normalized size of antiderivative = 0.82 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\frac {c \sqrt {(d+e x) (-b e+c (d-e x))} \left (\frac {2 b e (d g+e (f+2 g x))+c \left (-5 d^2 g+e^2 f x-3 d e (f+3 g x)\right )}{c (2 c d-b e) (d+e x)^2}+\frac {(c e f+7 c d g-4 b e g) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )}{(-2 c d+b e)^{3/2} \sqrt {-b e+c (d-e x)}}\right )}{4 e^2 \sqrt {d+e x}} \] Input:

Integrate[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^ 
(7/2),x]
 

Output:

(c*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*((2*b*e*(d*g + e*(f + 2*g*x)) + 
c*(-5*d^2*g + e^2*f*x - 3*d*e*(f + 3*g*x)))/(c*(2*c*d - b*e)*(d + e*x)^2) 
+ ((c*e*f + 7*c*d*g - 4*b*e*g)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d 
+ b*e]])/((-2*c*d + b*e)^(3/2)*Sqrt[-(b*e) + c*(d - e*x)])))/(4*e^2*Sqrt[d 
 + e*x])
 

Rubi [A] (verified)

Time = 0.78 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.087, Rules used = {1220, 1130, 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.

\(\displaystyle \int \frac {(f+g x) \sqrt {-b d e-b e^2 x+c d^2-c e^2 x^2}}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \int \frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{(d+e x)^{5/2}}dx}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1130

\(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \left (-\frac {1}{2} c \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {(-4 b e g+7 c d g+c e f) \left (-c e \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right )}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {c \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {d+e x} \sqrt {2 c d-b e}}\right )}{e \sqrt {2 c d-b e}}-\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e (d+e x)^{3/2}}\right ) (-4 b e g+7 c d g+c e f)}{4 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{2 e^2 (d+e x)^{7/2} (2 c d-b e)}\)

Input:

Int[((f + g*x)*Sqrt[c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2])/(d + e*x)^(7/2), 
x]
 

Output:

-1/2*((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(3/2))/(e^2*(2*c*d 
 - b*e)*(d + e*x)^(7/2)) + ((c*e*f + 7*c*d*g - 4*b*e*g)*(-(Sqrt[d*(c*d - b 
*e) - b*e^2*x - c*e^2*x^2]/(e*(d + e*x)^(3/2))) + (c*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*Sqrt[2 
*c*d - b*e])))/(4*e*(2*c*d - b*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 1130

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 + p + 1))), x] 
- Simp[c*(p/(e^2*(m + p + 1)))   Int[(d + e*x)^(m + 2)*(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] && (LtQ[m, -2] || EqQ[m + 2*p + 1, 0]) && NeQ[m + 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 1220

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(621\) vs. \(2(198)=396\).

Time = 2.21 (sec) , antiderivative size = 622, normalized size of antiderivative = 2.83

method	result	size
default	\(-\frac {\left (4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}-7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}-\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}+8 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -14 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -2 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +4 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -7 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -\arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +4 b \,e^{2} g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-9 c d e g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+c \,e^{2} f x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+2 b d e g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+2 b \,e^{2} f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-5 c \,d^{2} g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-3 c d e f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\right ) \sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}}{4 \left (b e -2 c d \right )^{\frac {3}{2}} e^{2} \sqrt {-c e x -b e +c d}\, \left (e x +d \right )^{\frac {5}{2}}}\)	\(622\)

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)^(7/2),x,method= 
_RETURNVERBOSE)
 

Output:

-1/4*(4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*e^3*g*x^2-7*a 
rctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*g*x^2-arctan((-c 
*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*e^3*f*x^2+8*arctan((-c*e*x-b*e+ 
c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c*d*e^2*g*x-14*arctan((-c*e*x-b*e+c*d)^(1/ 
2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x-2*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2 
*c*d)^(1/2))*c^2*d*e^2*f*x+4*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/ 
2))*b*c*d^2*e*g-7*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3 
*g-arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f+4*b*e^2*g* 
x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)-9*c*d*e*g*x*(-c*e*x-b*e+c*d)^(1 
/2)*(b*e-2*c*d)^(1/2)+c*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2 
*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+2*b*e^2*f*(-c*e*x-b*e+c* 
d)^(1/2)*(b*e-2*c*d)^(1/2)-5*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1 
/2)-3*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))*(-(e*x+d)*(c*e*x+b 
*e-c*d))^(1/2)/(b*e-2*c*d)^(3/2)/e^2/(-c*e*x-b*e+c*d)^(1/2)/(e*x+d)^(5/2)
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 508 vs. \(2 (198) = 396\).

Time = 0.11 (sec) , antiderivative size = 1045, normalized size of antiderivative = 4.75 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx =\text {Too large to display} \] 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)^(7/2),x, 
algorithm="fricas")
 

Output:

[1/8*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3*(c^ 
2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c*d^3* 
e)*g + 3*(c^2*d^2*e^2*f + (7*c^2*d^3*e - 4*b*c*d^2*e^2)*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)*((6* 
c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)*f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2 
*d*e^2)*g - ((2*c^2*d*e^2 - b*c*e^3)*f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4* 
b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2*d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 
 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + b^2*e^7)*x^3 + 3*(4*c^2*d^3*e^4 - 4*b*c* 
d^2*e^5 + b^2*d*e^6)*x^2 + 3*(4*c^2*d^4*e^3 - 4*b*c*d^3*e^4 + b^2*d^2*e^5) 
*x), 1/4*((c^2*d^3*e*f + (c^2*e^4*f + (7*c^2*d*e^3 - 4*b*c*e^4)*g)*x^3 + 3 
*(c^2*d*e^3*f + (7*c^2*d^2*e^2 - 4*b*c*d*e^3)*g)*x^2 + (7*c^2*d^4 - 4*b*c* 
d^3*e)*g + 3*(c^2*d^2*e^2*f + (7*c^2*d^3*e - 4*b*c*d^2*e^2)*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)*((6*c^2*d^2*e - 7*b*c*d*e^2 + 2*b^2*e^3)* 
f + (10*c^2*d^3 - 9*b*c*d^2*e + 2*b^2*d*e^2)*g - ((2*c^2*d*e^2 - b*c*e^3)* 
f - (18*c^2*d^2*e - 17*b*c*d*e^2 + 4*b^2*e^3)*g)*x)*sqrt(e*x + d))/(4*c^2* 
d^5*e^2 - 4*b*c*d^4*e^3 + b^2*d^3*e^4 + (4*c^2*d^2*e^5 - 4*b*c*d*e^6 + ...
 

Sympy [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\sqrt {- \left (d + e x\right ) \left (b e - c d + c e x\right )} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] 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)**(7/ 
2),x)
 

Output:

Integral(sqrt(-(d + e*x)*(b*e - c*d + c*e*x))*(f + g*x)/(d + e*x)**(7/2), 
x)
 

Maxima [F]

\[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int { \frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] 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)^(7/2),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)^( 
7/2), x)
 

Giac [A] (verification not implemented)

Time = 0.36 (sec) , antiderivative size = 328, normalized size of antiderivative = 1.49 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=-\frac {\frac {{\left (c^{3} e f + 7 \, c^{3} d g - 4 \, b c^{2} e g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{{\left (2 \, c d - b e\right )} \sqrt {-2 \, c d + b e}} + \frac {2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} e^{2} f + 14 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{4} d^{2} g - 15 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{3} d e g + 4 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b^{2} c^{2} e^{2} g + {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} e f - 9 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{3} d g + 4 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c^{2} e g}{{\left (2 \, c d - b e\right )} {\left (e x + d\right )}^{2} c^{2}}}{4 \, c e^{2}} \] 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)^(7/2),x, 
algorithm="giac")
 

Output:

-1/4*((c^3*e*f + 7*c^3*d*g - 4*b*c^2*e*g)*arctan(sqrt(-(e*x + d)*c + 2*c*d 
 - b*e)/sqrt(-2*c*d + b*e))/((2*c*d - b*e)*sqrt(-2*c*d + b*e)) + (2*sqrt(- 
(e*x + d)*c + 2*c*d - b*e)*c^4*d*e*f - sqrt(-(e*x + d)*c + 2*c*d - b*e)*b* 
c^3*e^2*f + 14*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^4*d^2*g - 15*sqrt(-(e*x 
+ d)*c + 2*c*d - b*e)*b*c^3*d*e*g + 4*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b^2 
*c^2*e^2*g + (-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^3*e*f - 9*(-(e*x + d)*c 
+ 2*c*d - b*e)^(3/2)*c^3*d*g + 4*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*c^2* 
e*g)/((2*c*d - b*e)*(e*x + d)^2*c^2))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] 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)^(7/2 
),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)^(7/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 780, normalized size of antiderivative = 3.55 \[ \int \frac {(f+g x) \sqrt {c d^2-b d e-b e^2 x-c e^2 x^2}}{(d+e x)^{7/2}} \, dx=\frac {-4 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g -8 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x -4 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} g \,x^{2}+7 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g +\sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f +14 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x +2 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x +7 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} g \,x^{2}+\sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} e^{3} f \,x^{2}-2 \sqrt {-c e x -b e +c d}\, b^{2} d \,e^{2} g -2 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} f -4 \sqrt {-c e x -b e +c d}\, b^{2} e^{3} g x +9 \sqrt {-c e x -b e +c d}\, b c \,d^{2} e g +7 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} f +17 \sqrt {-c e x -b e +c d}\, b c d \,e^{2} g x -\sqrt {-c e x -b e +c d}\, b c \,e^{3} f x -10 \sqrt {-c e x -b e +c d}\, c^{2} d^{3} g -6 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e f -18 \sqrt {-c e x -b e +c d}\, c^{2} d^{2} e g x +2 \sqrt {-c e x -b e +c d}\, c^{2} d \,e^{2} f x}{4 e^{2} \left (b^{2} e^{4} x^{2}-4 b c d \,e^{3} x^{2}+4 c^{2} d^{2} e^{2} x^{2}+2 b^{2} d \,e^{3} x -8 b c \,d^{2} e^{2} x +8 c^{2} d^{3} e x +b^{2} d^{2} e^{2}-4 b c \,d^{3} e +4 c^{2} d^{4}\right )} \] 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)^(7/2),x)
 

Output:

( - 4*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d)) 
*b*c*d**2*e*g - 8*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b 
*e - 2*c*d))*b*c*d*e**2*g*x - 4*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - 
 c*e*x)/sqrt(b*e - 2*c*d))*b*c*e**3*g*x**2 + 7*sqrt(b*e - 2*c*d)*atan(sqrt 
( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**3*g + sqrt(b*e - 2*c*d)* 
atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2*e*f + 14*sqrt 
(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*d**2 
*e*g*x + 2*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2* 
c*d))*c**2*d*e**2*f*x + 7*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x 
)/sqrt(b*e - 2*c*d))*c**2*d*e**2*g*x**2 + sqrt(b*e - 2*c*d)*atan(sqrt( - b 
*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c**2*e**3*f*x**2 - 2*sqrt( - b*e + c* 
d - c*e*x)*b**2*d*e**2*g - 2*sqrt( - b*e + c*d - c*e*x)*b**2*e**3*f - 4*sq 
rt( - b*e + c*d - c*e*x)*b**2*e**3*g*x + 9*sqrt( - b*e + c*d - c*e*x)*b*c* 
d**2*e*g + 7*sqrt( - b*e + c*d - c*e*x)*b*c*d*e**2*f + 17*sqrt( - b*e + c* 
d - c*e*x)*b*c*d*e**2*g*x - sqrt( - b*e + c*d - c*e*x)*b*c*e**3*f*x - 10*s 
qrt( - b*e + c*d - c*e*x)*c**2*d**3*g - 6*sqrt( - b*e + c*d - c*e*x)*c**2* 
d**2*e*f - 18*sqrt( - b*e + c*d - c*e*x)*c**2*d**2*e*g*x + 2*sqrt( - b*e + 
 c*d - c*e*x)*c**2*d*e**2*f*x)/(4*e**2*(b**2*d**2*e**2 + 2*b**2*d*e**3*x + 
 b**2*e**4*x**2 - 4*b*c*d**3*e - 8*b*c*d**2*e**2*x - 4*b*c*d*e**3*x**2 + 4 
*c**2*d**4 + 8*c**2*d**3*e*x + 4*c**2*d**2*e**2*x**2))