\(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{7/2}} \, dx\) [210]

Optimal result

Integrand size = 46, antiderivative size = 258 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {(2 c d-b e) (e f-d g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 (d+e x)^{3/2}}-\frac {2 (c e f-3 c d g+b e g) \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e^2 \sqrt {d+e x}}+\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e^2 (d+e x)^{3/2}}+\frac {\sqrt {2 c d-b e} (3 c e f-7 c d g+2 b e g) \text {arctanh}\left (\frac {\sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{\sqrt {2 c d-b e} \sqrt {d+e x}}\right )}{e^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.71 (sec) , antiderivative size = 192, normalized size of antiderivative = 0.74 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {\sqrt {(d+e x) (-b e+c (d-e x))} \left (\sqrt {c d-b e-c e x} \left (b e (-3 e f+11 d g+8 e g x)+2 c \left (-13 d^2 g+d e (6 f-9 g x)+e^2 x (3 f+g x)\right )\right )+3 \sqrt {-2 c d+b e} (-3 c e f+7 c d g-2 b e g) (d+e x) \arctan \left (\frac {\sqrt {c d-b e-c e x}}{\sqrt {-2 c d+b e}}\right )\right )}{3 e^2 (d+e x)^{3/2} \sqrt {-b e+c (d-e x)}} \] Input:

Integrate[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x 
)^(7/2),x]
 

Output:

-1/3*(Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(Sqrt[c*d - b*e - c*e*x]*(b*e 
*(-3*e*f + 11*d*g + 8*e*g*x) + 2*c*(-13*d^2*g + d*e*(6*f - 9*g*x) + e^2*x* 
(3*f + g*x))) + 3*Sqrt[-2*c*d + b*e]*(-3*c*e*f + 7*c*d*g - 2*b*e*g)*(d + e 
*x)*ArcTan[Sqrt[c*d - b*e - c*e*x]/Sqrt[-2*c*d + b*e]]))/(e^2*(d + e*x)^(3 
/2)*Sqrt[-(b*e) + c*(d - e*x)])
 

Rubi [A] (verified)

Time = 0.92 (sec) , antiderivative size = 276, normalized size of antiderivative = 1.07, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.109, Rules used = {1220, 1131, 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.

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

\(\Big \downarrow \) 1220

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

\(\Big \downarrow \) 1131

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

\(\Big \downarrow \) 1131

\(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \left ((2 c d-b e) \left ((2 c d-b e) \int \frac {1}{\sqrt {d+e x} \sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}dx+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\)

\(\Big \downarrow \) 1136

\(\displaystyle -\frac {(2 b e g-7 c d g+3 c e f) \left ((2 c d-b e) \left (2 e (2 c d-b e) \int \frac {1}{\frac {e^2 \left (-c x^2 e^2-b x e^2+d (c d-b e)\right )}{d+e x}-e^2 (2 c d-b e)}d\frac {\sqrt {-c x^2 e^2-b x e^2+d (c d-b e)}}{\sqrt {d+e x}}+\frac {2 \sqrt {d (c d-b e)-b e^2 x-c e^2 x^2}}{e \sqrt {d+e x}}\right )+\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{3/2}}{3 e (d+e x)^{3/2}}\right )}{2 e (2 c d-b e)}-\frac {(e f-d g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{e^2 (d+e x)^{7/2} (2 c d-b e)}\)

\(\Big \downarrow \) 221

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

Input:

Int[((f + g*x)*(c*d^2 - b*d*e - b*e^2*x - c*e^2*x^2)^(3/2))/(d + e*x)^(7/2 
),x]
 

Output:

-(((e*f - d*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5/2))/(e^2*(2*c*d - 
b*e)*(d + e*x)^(7/2))) - ((3*c*e*f - 7*c*d*g + 2*b*e*g)*((2*(d*(c*d - b*e) 
 - b*e^2*x - c*e^2*x^2)^(3/2))/(3*e*(d + e*x)^(3/2)) + (2*c*d - b*e)*((2*S 
qrt[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)))/(2*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 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 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. \(686\) vs. \(2(236)=472\).

Time = 1.62 (sec) , antiderivative size = 687, normalized size of antiderivative = 2.66

method	result	size
default	\(\frac {\sqrt {-\left (e x +d \right ) \left (c e x +b e -c d \right )}\, \left (6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} e^{3} g x -33 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} g x +9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,e^{3} f x +42 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e g x -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d \,e^{2} f x -2 c \,e^{2} g \,x^{2} \sqrt {b e -2 c d}\, \sqrt {-c e x -b e +c d}+6 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b^{2} d \,e^{2} g -33 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c \,d^{2} e g +9 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b c d \,e^{2} f +42 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{3} g -18 \arctan \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c^{2} d^{2} e f -8 b \,e^{2} g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+18 c d e g x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-6 c \,e^{2} f x \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-11 b d e g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+3 b \,e^{2} f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}+26 c \,d^{2} g \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}-12 c d e f \sqrt {-c e x -b e +c d}\, \sqrt {b e -2 c d}\right )}{3 \left (e x +d \right )^{\frac {3}{2}} \sqrt {-c e x -b e +c d}\, e^{2} \sqrt {b e -2 c d}}\)	\(687\)

Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x,method= 
_RETURNVERBOSE)
 

Output:

1/3*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)*(6*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e 
-2*c*d)^(1/2))*b^2*e^3*g*x-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1 
/2))*b*c*d*e^2*g*x+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*b*c* 
e^3*f*x+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*g*x- 
18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d*e^2*f*x-2*c*e^2* 
g*x^2*(b*e-2*c*d)^(1/2)*(-c*e*x-b*e+c*d)^(1/2)+6*arctan((-c*e*x-b*e+c*d)^( 
1/2)/(b*e-2*c*d)^(1/2))*b^2*d*e^2*g-33*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e- 
2*c*d)^(1/2))*b*c*d^2*e*g+9*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2 
))*b*c*d*e^2*f+42*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^3 
*g-18*arctan((-c*e*x-b*e+c*d)^(1/2)/(b*e-2*c*d)^(1/2))*c^2*d^2*e*f-8*b*e^2 
*g*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+18*c*d*e*g*x*(-c*e*x-b*e+c*d 
)^(1/2)*(b*e-2*c*d)^(1/2)-6*c*e^2*f*x*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^( 
1/2)-11*b*d*e*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+3*b*e^2*f*(-c*e*x 
-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2)+26*c*d^2*g*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2 
*c*d)^(1/2)-12*c*d*e*f*(-c*e*x-b*e+c*d)^(1/2)*(b*e-2*c*d)^(1/2))/(e*x+d)^( 
3/2)/(-c*e*x-b*e+c*d)^(1/2)/e^2/(b*e-2*c*d)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.11 (sec) , antiderivative size = 641, normalized size of antiderivative = 2.48 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\left [\frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {2 \, c d - b e} \log \left (-\frac {c e^{2} x^{2} - 3 \, c d^{2} + 2 \, b d e - 2 \, {\left (c d e - b e^{2}\right )} x - 2 \, \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {2 \, c d - b e} \sqrt {e x + d}}{e^{2} x^{2} + 2 \, d e x + d^{2}}\right ) - 2 \, {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{6 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}, \frac {3 \, {\left (3 \, c d^{2} e f + {\left (3 \, c e^{3} f - {\left (7 \, c d e^{2} - 2 \, b e^{3}\right )} g\right )} x^{2} - {\left (7 \, c d^{3} - 2 \, b d^{2} e\right )} g + 2 \, {\left (3 \, c d e^{2} f - {\left (7 \, c d^{2} e - 2 \, b d e^{2}\right )} g\right )} x\right )} \sqrt {-2 \, c d + b e} \arctan \left (-\frac {\sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {-2 \, c d + b e} \sqrt {e x + d}}{2 \, c d^{2} - b d e + {\left (2 \, c d e - b e^{2}\right )} x}\right ) - {\left (2 \, c e^{2} g x^{2} + 3 \, {\left (4 \, c d e - b e^{2}\right )} f - {\left (26 \, c d^{2} - 11 \, b d e\right )} g + 2 \, {\left (3 \, c e^{2} f - {\left (9 \, c d e - 4 \, b e^{2}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{3 \, {\left (e^{4} x^{2} + 2 \, d e^{3} x + d^{2} e^{2}\right )}}\right ] \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, 
algorithm="fricas")
 

Output:

[1/6*(3*(3*c*d^2*e*f + (3*c*e^3*f - (7*c*d*e^2 - 2*b*e^3)*g)*x^2 - (7*c*d^ 
3 - 2*b*d^2*e)*g + 2*(3*c*d*e^2*f - (7*c*d^2*e - 2*b*d*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*sq 
rt(-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*(2*c*e^2*g*x^2 + 3*(4*c*d*e - b*e^2)*f - (2 
6*c*d^2 - 11*b*d*e)*g + 2*(3*c*e^2*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt(-c*e 
^2*x^2 - b*e^2*x + c*d^2 - b*d*e)*sqrt(e*x + d))/(e^4*x^2 + 2*d*e^3*x + d^ 
2*e^2), 1/3*(3*(3*c*d^2*e*f + (3*c*e^3*f - (7*c*d*e^2 - 2*b*e^3)*g)*x^2 - 
(7*c*d^3 - 2*b*d^2*e)*g + 2*(3*c*d*e^2*f - (7*c*d^2*e - 2*b*d*e^2)*g)*x)*s 
qrt(-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)) - (2* 
c*e^2*g*x^2 + 3*(4*c*d*e - b*e^2)*f - (26*c*d^2 - 11*b*d*e)*g + 2*(3*c*e^2 
*f - (9*c*d*e - 4*b*e^2)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)* 
sqrt(e*x + d))/(e^4*x^2 + 2*d*e^3*x + d^2*e^2)]
 

Sympy [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (- \left (d + e x\right ) \left (b e - c d + c e x\right )\right )^{\frac {3}{2}} \left (f + g x\right )}{\left (d + e x\right )^{\frac {7}{2}}}\, dx \] Input:

integrate((g*x+f)*(-c*e**2*x**2-b*e**2*x-b*d*e+c*d**2)**(3/2)/(e*x+d)**(7/ 
2),x)
 

Output:

Integral((-(d + e*x)*(b*e - c*d + c*e*x))**(3/2)*(f + g*x)/(d + e*x)**(7/2 
), x)
 

Maxima [F]

\[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int { \frac {{\left (-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e\right )}^{\frac {3}{2}} {\left (g x + f\right )}}{{\left (e x + d\right )}^{\frac {7}{2}}} \,d x } \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, 
algorithm="maxima")
 

Output:

integrate((-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d*e)^(3/2)*(g*x + f)/(e*x + d) 
^(7/2), x)
 

Giac [A] (verification not implemented)

Time = 0.37 (sec) , antiderivative size = 323, normalized size of antiderivative = 1.25 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=-\frac {6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} e f - 18 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{2} d g + 6 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c e g - 2 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c g + \frac {3 \, {\left (6 \, c^{3} d e f - 3 \, b c^{2} e^{2} f - 14 \, c^{3} d^{2} g + 11 \, b c^{2} d e g - 2 \, b^{2} c e^{2} g\right )} \arctan \left (\frac {\sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{\sqrt {-2 \, c d + b e}}\right )}{\sqrt {-2 \, c d + b e}} + \frac {3 \, {\left (2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d e f - \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} e^{2} f - 2 \, \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c^{3} d^{2} g + \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b c^{2} d e g\right )}}{{\left (e x + d\right )} c}}{3 \, c e^{2}} \] Input:

integrate((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x, 
algorithm="giac")
 

Output:

-1/3*(6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^2*e*f - 18*sqrt(-(e*x + d)*c + 
2*c*d - b*e)*c^2*d*g + 6*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c*e*g - 2*(-(e 
*x + d)*c + 2*c*d - b*e)^(3/2)*c*g + 3*(6*c^3*d*e*f - 3*b*c^2*e^2*f - 14*c 
^3*d^2*g + 11*b*c^2*d*e*g - 2*b^2*c*e^2*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*(2*sqrt(-(e*x + d)*c + 
 2*c*d - b*e)*c^3*d*e*f - sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*c^2*e^2*f - 2 
*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c^3*d^2*g + sqrt(-(e*x + d)*c + 2*c*d - 
b*e)*b*c^2*d*e*g)/((e*x + d)*c))/(c*e^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\int \frac {\left (f+g\,x\right )\,{\left (c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x\right )}^{3/2}}{{\left (d+e\,x\right )}^{7/2}} \,d x \] Input:

int(((f + g*x)*(c*d^2 - c*e^2*x^2 - b*d*e - b*e^2*x)^(3/2))/(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)^(3/2))/(d + e*x)^(7/2 
), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 442, normalized size of antiderivative = 1.71 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{7/2}} \, dx=\frac {6 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b d e g +6 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) b \,e^{2} g x -21 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,d^{2} g +9 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e f -21 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c d e g x +9 \sqrt {b e -2 c d}\, \mathit {atan} \left (\frac {\sqrt {-c e x -b e +c d}}{\sqrt {b e -2 c d}}\right ) c \,e^{2} f x -11 \sqrt {-c e x -b e +c d}\, b d e g +3 \sqrt {-c e x -b e +c d}\, b \,e^{2} f -8 \sqrt {-c e x -b e +c d}\, b \,e^{2} g x +26 \sqrt {-c e x -b e +c d}\, c \,d^{2} g -12 \sqrt {-c e x -b e +c d}\, c d e f +18 \sqrt {-c e x -b e +c d}\, c d e g x -6 \sqrt {-c e x -b e +c d}\, c \,e^{2} f x -2 \sqrt {-c e x -b e +c d}\, c \,e^{2} g \,x^{2}}{3 e^{2} \left (e x +d \right )} \] Input:

int((g*x+f)*(-c*e^2*x^2-b*e^2*x-b*d*e+c*d^2)^(3/2)/(e*x+d)^(7/2),x)
 

Output:

(6*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*b* 
d*e*g + 6*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqrt(b*e - 2*c 
*d))*b*e**2*g*x - 21*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c*e*x)/sqr 
t(b*e - 2*c*d))*c*d**2*g + 9*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + c*d - c* 
e*x)/sqrt(b*e - 2*c*d))*c*d*e*f - 21*sqrt(b*e - 2*c*d)*atan(sqrt( - b*e + 
c*d - c*e*x)/sqrt(b*e - 2*c*d))*c*d*e*g*x + 9*sqrt(b*e - 2*c*d)*atan(sqrt( 
 - b*e + c*d - c*e*x)/sqrt(b*e - 2*c*d))*c*e**2*f*x - 11*sqrt( - b*e + c*d 
 - c*e*x)*b*d*e*g + 3*sqrt( - b*e + c*d - c*e*x)*b*e**2*f - 8*sqrt( - b*e 
+ c*d - c*e*x)*b*e**2*g*x + 26*sqrt( - b*e + c*d - c*e*x)*c*d**2*g - 12*sq 
rt( - b*e + c*d - c*e*x)*c*d*e*f + 18*sqrt( - b*e + c*d - c*e*x)*c*d*e*g*x 
 - 6*sqrt( - b*e + c*d - c*e*x)*c*e**2*f*x - 2*sqrt( - b*e + c*d - c*e*x)* 
c*e**2*g*x**2)/(3*e**2*(d + e*x))