3.23.44 \(\int \frac {(f+g x) (c d^2-b d e-b e^2 x-c e^2 x^2)^{3/2}}{(d+e x)^{3/2}} \, dx\) [2244]

3.23.44.1 Optimal result

Integrand size = 46, antiderivative size = 118 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (7 c e f-3 c d g-2 b e g) \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}} \]

output

-2/35*(-2*b*e*g-3*c*d*g+7*c*e*f)*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c^ 
2/e^2/(e*x+d)^(5/2)-2/7*g*(d*(-b*e+c*d)-b*e^2*x-c*e^2*x^2)^(5/2)/c/e^2/(e* 
x+d)^(3/2)
 

3.23.44.2 Mathematica [A] (verified)

Time = 0.07 (sec) , antiderivative size = 78, normalized size of antiderivative = 0.66 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 (-c d+b e+c e x)^2 \sqrt {(d+e x) (-b e+c (d-e x))} (-2 b e g+c (7 e f+2 d g+5 e g x))}{35 c^2 e^2 \sqrt {d+e x}} \]

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 
)^(3/2),x]
 

output

(-2*(-(c*d) + b*e + c*e*x)^2*Sqrt[(d + e*x)*(-(b*e) + c*(d - e*x))]*(-2*b* 
e*g + c*(7*e*f + 2*d*g + 5*e*g*x)))/(35*c^2*e^2*Sqrt[d + e*x])
 

3.23.44.3 Rubi [A] (verified)

Time = 0.33 (sec) , antiderivative size = 118, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.043, Rules used = {1221, 1122}

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.

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

\(\Big \downarrow \) 1221

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

\(\Big \downarrow \) 1122

\(\displaystyle -\frac {2 \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2} (-2 b e g-3 c d g+7 c e f)}{35 c^2 e^2 (d+e x)^{5/2}}-\frac {2 g \left (d (c d-b e)-b e^2 x-c e^2 x^2\right )^{5/2}}{7 c e^2 (d+e x)^{3/2}}\)

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)^(3/2 
),x]
 

output

(-2*(7*c*e*f - 3*c*d*g - 2*b*e*g)*(d*(c*d - b*e) - b*e^2*x - c*e^2*x^2)^(5 
/2))/(35*c^2*e^2*(d + e*x)^(5/2)) - (2*g*(d*(c*d - b*e) - b*e^2*x - c*e^2* 
x^2)^(5/2))/(7*c*e^2*(d + e*x)^(3/2))
 

3.23.44.3.1 Defintions of rubi rules used

rule 1122

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

3.23.44.4 Maple [A] (verified)

Time = 0.36 (sec) , antiderivative size = 73, normalized size of antiderivative = 0.62

method	result	size
default	\(\frac {2 \sqrt {-\left (e x +d \right ) \left (x c e +b e -c d \right )}\, \left (x c e +b e -c d \right )^{2} \left (-5 c e g x +2 b e g -2 c d g -7 c e f \right )}{35 \sqrt {e x +d}\, c^{2} e^{2}}\)	\(73\)
gosper	\(-\frac {2 \left (x c e +b e -c d \right ) \left (-5 c e g x +2 b e g -2 c d g -7 c e f \right ) \left (-c \,e^{2} x^{2}-b \,e^{2} x -b d e +c \,d^{2}\right )^{\frac {3}{2}}}{35 c^{2} e^{2} \left (e x +d \right )^{\frac {3}{2}}}\)	\(79\)

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)^(3/2),x,method= 
_RETURNVERBOSE)
 

output

2/35*(-(e*x+d)*(c*e*x+b*e-c*d))^(1/2)/(e*x+d)^(1/2)*(c*e*x+b*e-c*d)^2*(-5* 
c*e*g*x+2*b*e*g-2*c*d*g-7*c*e*f)/c^2/e^2
 

3.23.44.5 Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 229 vs. \(2 (106) = 212\).

Time = 0.38 (sec) , antiderivative size = 229, normalized size of antiderivative = 1.94 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (5 \, c^{3} e^{3} g x^{3} + {\left (7 \, c^{3} e^{3} f - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} g\right )} x^{2} + 7 \, {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} f + 2 \, {\left (c^{3} d^{3} - 3 \, b c^{2} d^{2} e + 3 \, b^{2} c d e^{2} - b^{3} e^{3}\right )} g - {\left (14 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} f - {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} g\right )} x\right )} \sqrt {-c e^{2} x^{2} - b e^{2} x + c d^{2} - b d e} \sqrt {e x + d}}{35 \, {\left (c^{2} e^{3} x + c^{2} d e^{2}\right )}} \]

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)^(3/2),x, 
algorithm="fricas")
 

output

-2/35*(5*c^3*e^3*g*x^3 + (7*c^3*e^3*f - 8*(c^3*d*e^2 - b*c^2*e^3)*g)*x^2 + 
 7*(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*f + 2*(c^3*d^3 - 3*b*c^2*d^2*e 
+ 3*b^2*c*d*e^2 - b^3*e^3)*g - (14*(c^3*d*e^2 - b*c^2*e^3)*f - (c^3*d^2*e 
- 2*b*c^2*d*e^2 + b^2*c*e^3)*g)*x)*sqrt(-c*e^2*x^2 - b*e^2*x + c*d^2 - b*d 
*e)*sqrt(e*x + d)/(c^2*e^3*x + c^2*d*e^2)
 

3.23.44.6 Sympy [F]

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

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)**(3/ 
2),x)
 

output

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

3.23.44.7 Maxima [A] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 197, normalized size of antiderivative = 1.67 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {2 \, {\left (c^{2} e^{2} x^{2} + c^{2} d^{2} - 2 \, b c d e + b^{2} e^{2} - 2 \, {\left (c^{2} d e - b c e^{2}\right )} x\right )} \sqrt {-c e x + c d - b e} f}{5 \, c e} - \frac {2 \, {\left (5 \, c^{3} e^{3} x^{3} + 2 \, c^{3} d^{3} - 6 \, b c^{2} d^{2} e + 6 \, b^{2} c d e^{2} - 2 \, b^{3} e^{3} - 8 \, {\left (c^{3} d e^{2} - b c^{2} e^{3}\right )} x^{2} + {\left (c^{3} d^{2} e - 2 \, b c^{2} d e^{2} + b^{2} c e^{3}\right )} x\right )} \sqrt {-c e x + c d - b e} g}{35 \, c^{2} e^{2}} \]

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)^(3/2),x, 
algorithm="maxima")
 

output

-2/5*(c^2*e^2*x^2 + c^2*d^2 - 2*b*c*d*e + b^2*e^2 - 2*(c^2*d*e - b*c*e^2)* 
x)*sqrt(-c*e*x + c*d - b*e)*f/(c*e) - 2/35*(5*c^3*e^3*x^3 + 2*c^3*d^3 - 6* 
b*c^2*d^2*e + 6*b^2*c*d*e^2 - 2*b^3*e^3 - 8*(c^3*d*e^2 - b*c^2*e^3)*x^2 + 
(c^3*d^2*e - 2*b*c^2*d*e^2 + b^2*c*e^3)*x)*sqrt(-c*e*x + c*d - b*e)*g/(c^2 
*e^2)
 

3.23.44.8 Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 901 vs. \(2 (106) = 212\).

Time = 0.33 (sec) , antiderivative size = 901, normalized size of antiderivative = 7.64 \[ \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)^{3/2}} \, dx=-\frac {2 \, {\left (35 \, c d f {\left (\frac {{\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}}{c} - \frac {2 \, \sqrt {2 \, c d - b e} c d - \sqrt {2 \, c d - b e} b e}{c}\right )} - 35 \, b e f {\left (\frac {{\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}}}{c} - \frac {2 \, \sqrt {2 \, c d - b e} c d - \sqrt {2 \, c d - b e} b e}{c}\right )} + c e g {\left (\frac {22 \, \sqrt {2 \, c d - b e} c^{3} d^{3} - 19 \, \sqrt {2 \, c d - b e} b c^{2} d^{2} e + 20 \, \sqrt {2 \, c d - b e} b^{2} c d e^{2} - 8 \, \sqrt {2 \, c d - b e} b^{3} e^{3}}{c^{3} e^{2}} - \frac {35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c^{2} d^{2} - 70 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b c d e + 35 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b^{2} e^{2} - 42 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} c d + 42 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e} b e - 15 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{3} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{c^{3} e^{2}}\right )} - 7 \, c f {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b e - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{c^{2}}\right )} - 7 \, b g {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b e - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{c^{2}}\right )} + \frac {7 \, c d g {\left (\frac {2 \, \sqrt {2 \, c d - b e} c^{2} d^{2} + 3 \, \sqrt {2 \, c d - b e} b c d e - 2 \, \sqrt {2 \, c d - b e} b^{2} e^{2}}{c^{2}} + \frac {5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} c d - 5 \, {\left (-{\left (e x + d\right )} c + 2 \, c d - b e\right )}^{\frac {3}{2}} b e - 3 \, {\left ({\left (e x + d\right )} c - 2 \, c d + b e\right )}^{2} \sqrt {-{\left (e x + d\right )} c + 2 \, c d - b e}}{c^{2}}\right )}}{e}\right )}}{105 \, e} \]

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)^(3/2),x, 
algorithm="giac")
 

output

-2/105*(35*c*d*f*((-(e*x + d)*c + 2*c*d - b*e)^(3/2)/c - (2*sqrt(2*c*d - b 
*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) - 35*b*e*f*((-(e*x + d)*c + 2*c*d - b* 
e)^(3/2)/c - (2*sqrt(2*c*d - b*e)*c*d - sqrt(2*c*d - b*e)*b*e)/c) + c*e*g* 
((22*sqrt(2*c*d - b*e)*c^3*d^3 - 19*sqrt(2*c*d - b*e)*b*c^2*d^2*e + 20*sqr 
t(2*c*d - b*e)*b^2*c*d*e^2 - 8*sqrt(2*c*d - b*e)*b^3*e^3)/(c^3*e^2) - (35* 
(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c^2*d^2 - 70*(-(e*x + d)*c + 2*c*d - b* 
e)^(3/2)*b*c*d*e + 35*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b^2*e^2 - 42*((e* 
x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*c*d + 42*((e*x 
+ d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e)*b*e - 15*((e*x + 
d)*c - 2*c*d + b*e)^3*sqrt(-(e*x + d)*c + 2*c*d - b*e))/(c^3*e^2)) - 7*c*f 
*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c* 
d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e 
*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt( 
-(e*x + d)*c + 2*c*d - b*e))/c^2) - 7*b*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 
3*sqrt(2*c*d - b*e)*b*c*d*e - 2*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x 
 + d)*c + 2*c*d - b*e)^(3/2)*c*d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b* 
e - 3*((e*x + d)*c - 2*c*d + b*e)^2*sqrt(-(e*x + d)*c + 2*c*d - b*e))/c^2) 
 + 7*c*d*g*((2*sqrt(2*c*d - b*e)*c^2*d^2 + 3*sqrt(2*c*d - b*e)*b*c*d*e - 2 
*sqrt(2*c*d - b*e)*b^2*e^2)/c^2 + (5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*c* 
d - 5*(-(e*x + d)*c + 2*c*d - b*e)^(3/2)*b*e - 3*((e*x + d)*c - 2*c*d +...
 

3.23.44.9 Mupad [B] (verification not implemented)

Time = 11.36 (sec) , antiderivative size = 133, normalized size of antiderivative = 1.13 \[ \int \frac {(f+g x) \left (c d^2-b d e-b e^2 x-c e^2 x^2\right )^{3/2}}{(d+e x)^{3/2}} \, dx=-\frac {\left (x^2\,\left (\frac {16\,b\,e\,g}{35}-\frac {16\,c\,d\,g}{35}+\frac {2\,c\,e\,f}{5}\right )+\frac {2\,c\,e\,g\,x^3}{7}+\frac {2\,{\left (b\,e-c\,d\right )}^2\,\left (2\,c\,d\,g-2\,b\,e\,g+7\,c\,e\,f\right )}{35\,c^2\,e^2}+\frac {2\,x\,\left (b\,e-c\,d\right )\,\left (b\,e\,g-c\,d\,g+14\,c\,e\,f\right )}{35\,c\,e}\right )\,\sqrt {c\,d^2-b\,d\,e-c\,e^2\,x^2-b\,e^2\,x}}{\sqrt {d+e\,x}} \]

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)^(3/2 
),x)
 

output

-((x^2*((16*b*e*g)/35 - (16*c*d*g)/35 + (2*c*e*f)/5) + (2*c*e*g*x^3)/7 + ( 
2*(b*e - c*d)^2*(2*c*d*g - 2*b*e*g + 7*c*e*f))/(35*c^2*e^2) + (2*x*(b*e - 
c*d)*(b*e*g - c*d*g + 14*c*e*f))/(35*c*e))*(c*d^2 - c*e^2*x^2 - b*d*e - b* 
e^2*x)^(1/2))/(d + e*x)^(1/2)