\(\int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx\) [98]

Optimal result

Integrand size = 27, antiderivative size = 279 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=-\frac {2 (3 b c f h-d (b f g+b e h+a f h)) \sqrt {e+f x}}{d^4}-\frac {\left (a d (3 d f g+4 d e h-7 c f h)+b \left (4 d^2 e g+11 c^2 f h-c d (7 f g+8 e h)\right )\right ) \sqrt {e+f x}}{4 d^4 (c+d x)}+\frac {2 b h (e+f x)^{3/2}}{3 d^3}+\frac {(b c-a d) (d g-c h) (e+f x)^{3/2}}{2 d^3 (c+d x)^2}-\frac {\left (3 a d f (d f g+4 d e h-5 c f h)+b \left (35 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-5 c d f (3 f g+8 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{4 d^{9/2} \sqrt {d e-c f}} \] Output:

-2*(3*b*c*f*h-d*(a*f*h+b*e*h+b*f*g))*(f*x+e)^(1/2)/d^4-1/4*(a*d*(-7*c*f*h+ 
4*d*e*h+3*d*f*g)+b*(4*d^2*e*g+11*c^2*f*h-c*d*(8*e*h+7*f*g)))*(f*x+e)^(1/2) 
/d^4/(d*x+c)+2/3*b*h*(f*x+e)^(3/2)/d^3+1/2*(-a*d+b*c)*(-c*h+d*g)*(f*x+e)^( 
3/2)/d^3/(d*x+c)^2-1/4*(3*a*d*f*(-5*c*f*h+4*d*e*h+d*f*g)+b*(35*c^2*f^2*h+4 
*d^2*e*(2*e*h+3*f*g)-5*c*d*f*(8*e*h+3*f*g)))*arctanh(d^(1/2)*(f*x+e)^(1/2) 
/(-c*f+d*e)^(1/2))/d^(9/2)/(-c*f+d*e)^(1/2)
 

Mathematica [A] (verified)

Time = 0.89 (sec) , antiderivative size = 275, normalized size of antiderivative = 0.99 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {\sqrt {e+f x} \left (b \left (-105 c^3 f h+5 c^2 d (9 f g+10 e h-35 f h x)+c d^2 \left (-6 e g+75 f g x+88 e h x-56 f h x^2\right )+4 d^3 x \left (-3 e g+6 f g x+8 e h x+2 f h x^2\right )\right )-3 a d \left (-15 c^2 f h+c d (3 f g+2 e h-25 f h x)+d^2 (f x (5 g-8 h x)+2 e (g+2 h x))\right )\right )}{12 d^4 (c+d x)^2}+\frac {\left (3 a d f (d f g+4 d e h-5 c f h)+b \left (35 c^2 f^2 h+4 d^2 e (3 f g+2 e h)-5 c d f (3 f g+8 e h)\right )\right ) \arctan \left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {-d e+c f}}\right )}{4 d^{9/2} \sqrt {-d e+c f}} \] Input:

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

Output:

(Sqrt[e + f*x]*(b*(-105*c^3*f*h + 5*c^2*d*(9*f*g + 10*e*h - 35*f*h*x) + c* 
d^2*(-6*e*g + 75*f*g*x + 88*e*h*x - 56*f*h*x^2) + 4*d^3*x*(-3*e*g + 6*f*g* 
x + 8*e*h*x + 2*f*h*x^2)) - 3*a*d*(-15*c^2*f*h + c*d*(3*f*g + 2*e*h - 25*f 
*h*x) + d^2*(f*x*(5*g - 8*h*x) + 2*e*(g + 2*h*x)))))/(12*d^4*(c + d*x)^2) 
+ ((3*a*d*f*(d*f*g + 4*d*e*h - 5*c*f*h) + b*(35*c^2*f^2*h + 4*d^2*e*(3*f*g 
 + 2*e*h) - 5*c*d*f*(3*f*g + 8*e*h)))*ArcTan[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[ 
-(d*e) + c*f]])/(4*d^(9/2)*Sqrt[-(d*e) + c*f])
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 317, normalized size of antiderivative = 1.14, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.185, Rules used = {162, 60, 60, 73, 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.

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

\(\Big \downarrow \) 162

\(\displaystyle \frac {\left (3 a d f (-5 c f h+4 d e h+d f g)+b \left (35 c^2 f^2 h-5 c d f (8 e h+3 f g)+4 d^2 e (2 e h+3 f g)\right )\right ) \int \frac {(e+f x)^{3/2}}{c+d x}dx}{8 d^2 (d e-c f)^2}-\frac {(e+f x)^{5/2} \left (d x \left (a d (-5 c f h+4 d e h+d f g)+b \left (9 c^2 f h-c d (8 e h+5 f g)+4 d^2 e g\right )\right )+a d \left (-3 c^2 f h-c d (f g-2 e h)+2 d^2 e g\right )+b c \left (7 c^2 f h-3 c d (2 e h+f g)+2 d^2 e g\right )\right )}{4 d^2 (c+d x)^2 (d e-c f)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\left (3 a d f (-5 c f h+4 d e h+d f g)+b \left (35 c^2 f^2 h-5 c d f (8 e h+3 f g)+4 d^2 e (2 e h+3 f g)\right )\right ) \left (\frac {(d e-c f) \int \frac {\sqrt {e+f x}}{c+d x}dx}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{8 d^2 (d e-c f)^2}-\frac {(e+f x)^{5/2} \left (d x \left (a d (-5 c f h+4 d e h+d f g)+b \left (9 c^2 f h-c d (8 e h+5 f g)+4 d^2 e g\right )\right )+a d \left (-3 c^2 f h-c d (f g-2 e h)+2 d^2 e g\right )+b c \left (7 c^2 f h-3 c d (2 e h+f g)+2 d^2 e g\right )\right )}{4 d^2 (c+d x)^2 (d e-c f)^2}\)

\(\Big \downarrow \) 60

\(\displaystyle \frac {\left (3 a d f (-5 c f h+4 d e h+d f g)+b \left (35 c^2 f^2 h-5 c d f (8 e h+3 f g)+4 d^2 e (2 e h+3 f g)\right )\right ) \left (\frac {(d e-c f) \left (\frac {(d e-c f) \int \frac {1}{(c+d x) \sqrt {e+f x}}dx}{d}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{8 d^2 (d e-c f)^2}-\frac {(e+f x)^{5/2} \left (d x \left (a d (-5 c f h+4 d e h+d f g)+b \left (9 c^2 f h-c d (8 e h+5 f g)+4 d^2 e g\right )\right )+a d \left (-3 c^2 f h-c d (f g-2 e h)+2 d^2 e g\right )+b c \left (7 c^2 f h-3 c d (2 e h+f g)+2 d^2 e g\right )\right )}{4 d^2 (c+d x)^2 (d e-c f)^2}\)

\(\Big \downarrow \) 73

\(\displaystyle \frac {\left (3 a d f (-5 c f h+4 d e h+d f g)+b \left (35 c^2 f^2 h-5 c d f (8 e h+3 f g)+4 d^2 e (2 e h+3 f g)\right )\right ) \left (\frac {(d e-c f) \left (\frac {2 (d e-c f) \int \frac {1}{c+\frac {d (e+f x)}{f}-\frac {d e}{f}}d\sqrt {e+f x}}{d f}+\frac {2 \sqrt {e+f x}}{d}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right )}{8 d^2 (d e-c f)^2}-\frac {(e+f x)^{5/2} \left (d x \left (a d (-5 c f h+4 d e h+d f g)+b \left (9 c^2 f h-c d (8 e h+5 f g)+4 d^2 e g\right )\right )+a d \left (-3 c^2 f h-c d (f g-2 e h)+2 d^2 e g\right )+b c \left (7 c^2 f h-3 c d (2 e h+f g)+2 d^2 e g\right )\right )}{4 d^2 (c+d x)^2 (d e-c f)^2}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\left (\frac {(d e-c f) \left (\frac {2 \sqrt {e+f x}}{d}-\frac {2 \sqrt {d e-c f} \text {arctanh}\left (\frac {\sqrt {d} \sqrt {e+f x}}{\sqrt {d e-c f}}\right )}{d^{3/2}}\right )}{d}+\frac {2 (e+f x)^{3/2}}{3 d}\right ) \left (3 a d f (-5 c f h+4 d e h+d f g)+b \left (35 c^2 f^2 h-5 c d f (8 e h+3 f g)+4 d^2 e (2 e h+3 f g)\right )\right )}{8 d^2 (d e-c f)^2}-\frac {(e+f x)^{5/2} \left (d x \left (a d (-5 c f h+4 d e h+d f g)+b \left (9 c^2 f h-c d (8 e h+5 f g)+4 d^2 e g\right )\right )+a d \left (-3 c^2 f h-c d (f g-2 e h)+2 d^2 e g\right )+b c \left (7 c^2 f h-3 c d (2 e h+f g)+2 d^2 e g\right )\right )}{4 d^2 (c+d x)^2 (d e-c f)^2}\)

Input:

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

Output:

-1/4*((e + f*x)^(5/2)*(a*d*(2*d^2*e*g - 3*c^2*f*h - c*d*(f*g - 2*e*h)) + b 
*c*(2*d^2*e*g + 7*c^2*f*h - 3*c*d*(f*g + 2*e*h)) + d*(a*d*(d*f*g + 4*d*e*h 
 - 5*c*f*h) + b*(4*d^2*e*g + 9*c^2*f*h - c*d*(5*f*g + 8*e*h)))*x))/(d^2*(d 
*e - c*f)^2*(c + d*x)^2) + ((3*a*d*f*(d*f*g + 4*d*e*h - 5*c*f*h) + b*(35*c 
^2*f^2*h + 4*d^2*e*(3*f*g + 2*e*h) - 5*c*d*f*(3*f*g + 8*e*h)))*((2*(e + f* 
x)^(3/2))/(3*d) + ((d*e - c*f)*((2*Sqrt[e + f*x])/d - (2*Sqrt[d*e - c*f]*A 
rcTanh[(Sqrt[d]*Sqrt[e + f*x])/Sqrt[d*e - c*f]])/d^(3/2)))/d))/(8*d^2*(d*e 
 - c*f)^2)
 

Defintions of rubi rules used

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 162

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.)*((e_) + (f_.)*(x_) 
)*((g_.) + (h_.)*(x_)), x_] :> Simp[((b^3*c*e*g*(m + 2) - a^3*d*f*h*(n + 2) 
 - a^2*b*(c*f*h*m - d*(f*g + e*h)*(m + n + 3)) - a*b^2*(c*(f*g + e*h) + d*e 
*g*(2*m + n + 4)) + b*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d*(f*g + 
e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2)))*x)/(b^2*(b 
*c - a*d)^2*(m + 1)*(m + 2)))*(a + b*x)^(m + 1)*(c + d*x)^(n + 1), x] + Sim 
p[(f*(h/b^2) - (d*(m + n + 3)*(a^2*d*f*h*(m - n) - a*b*(2*c*f*h*(m + 1) - d 
*(f*g + e*h)*(n + 1)) + b^2*(c*(f*g + e*h)*(m + 1) - d*e*g*(m + n + 2))))/( 
b^2*(b*c - a*d)^2*(m + 1)*(m + 2)))   Int[(a + b*x)^(m + 2)*(c + d*x)^n, x] 
, x] /; FreeQ[{a, b, c, d, e, f, g, h, m, n}, x] && (LtQ[m, -2] || (EqQ[m + 
 n + 3, 0] &&  !LtQ[n, -2]))
 

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]
 

Maple [A] (verified)

Time = 0.56 (sec) , antiderivative size = 276, normalized size of antiderivative = 0.99

method	result	size
pseudoelliptic	\(\frac {-\frac {15 \left (\frac {\left (-g \,f^{2} a -4 e \left (a h +b g \right ) f -\frac {8 b \,e^{2} h}{3}\right ) d^{2}}{5}+c \left (\left (a h +b g \right ) f +\frac {8 e h b}{3}\right ) f d -\frac {7 b \,c^{2} f^{2} h}{3}\right ) \left (x d +c \right )^{2} \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4}+\frac {15 \left (\frac {\left (-\left (-\frac {8 b h \,x^{2}}{15}+\frac {8 \left (-a h -b g \right ) x}{5}+g a \right ) x f -\frac {2 e \left (-\frac {16 b h \,x^{2}}{3}+2 \left (a h +b g \right ) x +g a \right )}{5}\right ) d^{3}}{3}-\frac {2 c \left (\left (\frac {28 b h \,x^{2}}{3}+\frac {25 \left (-a h -b g \right ) x}{2}+\frac {3 g a}{2}\right ) f +e \left (-\frac {44}{3} b h x +a h +b g \right )\right ) d^{2}}{15}+\left (\left (-\frac {35}{9} b h x +a h +b g \right ) f +\frac {10 e h b}{9}\right ) c^{2} d -\frac {7 c^{3} h b f}{3}\right ) \sqrt {f x +e}\, \sqrt {\left (c f -d e \right ) d}}{4}}{\sqrt {\left (c f -d e \right ) d}\, \left (x d +c \right )^{2} d^{4}}\)	\(276\)
risch	\(\frac {2 \left (h f d b x +3 a d f h -9 b c f h +4 b e h d +3 b d f g \right ) \sqrt {f x +e}}{3 d^{4}}-\frac {\frac {2 \left (-\frac {9}{8} a c \,d^{2} f^{2} h +\frac {1}{2} a \,d^{3} e f h +\frac {5}{8} a \,d^{3} f^{2} g +\frac {13}{8} b \,c^{2} d \,f^{2} h -b c \,d^{2} e f h -\frac {9}{8} b c \,d^{2} f^{2} g +\frac {1}{2} b \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (7 a \,c^{2} d \,f^{2} h -11 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +4 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -11 b \,c^{3} f^{2} h +19 b \,c^{2} d e f h +7 b \,c^{2} d \,f^{2} g -8 b c \,d^{2} e^{2} h -11 b c \,d^{2} e f g +4 b \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{4}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a c d \,f^{2} h -12 a \,d^{2} e f h -3 a \,d^{2} f^{2} g -35 b \,c^{2} f^{2} h +40 b c d e f h +15 b c d \,f^{2} g -8 b \,d^{2} e^{2} h -12 b \,d^{2} e f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{4 \sqrt {\left (c f -d e \right ) d}}}{d^{4}}\)	\(386\)
derivativedivides	\(\frac {\frac {2 d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+2 a d f h \sqrt {f x +e}-6 b c f h \sqrt {f x +e}+2 b d e h \sqrt {f x +e}+2 b d f g \sqrt {f x +e}}{d^{4}}-\frac {2 \left (\frac {\left (-\frac {9}{8} a c \,d^{2} f^{2} h +\frac {1}{2} a \,d^{3} e f h +\frac {5}{8} a \,d^{3} f^{2} g +\frac {13}{8} b \,c^{2} d \,f^{2} h -b c \,d^{2} e f h -\frac {9}{8} b c \,d^{2} f^{2} g +\frac {1}{2} b \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (7 a \,c^{2} d \,f^{2} h -11 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +4 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -11 b \,c^{3} f^{2} h +19 b \,c^{2} d e f h +7 b \,c^{2} d \,f^{2} g -8 b c \,d^{2} e^{2} h -11 b c \,d^{2} e f g +4 b \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a c d \,f^{2} h -12 a \,d^{2} e f h -3 a \,d^{2} f^{2} g -35 b \,c^{2} f^{2} h +40 b c d e f h +15 b c d \,f^{2} g -8 b \,d^{2} e^{2} h -12 b \,d^{2} e f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}\)	\(409\)
default	\(\frac {\frac {2 d h \left (f x +e \right )^{\frac {3}{2}} b}{3}+2 a d f h \sqrt {f x +e}-6 b c f h \sqrt {f x +e}+2 b d e h \sqrt {f x +e}+2 b d f g \sqrt {f x +e}}{d^{4}}-\frac {2 \left (\frac {\left (-\frac {9}{8} a c \,d^{2} f^{2} h +\frac {1}{2} a \,d^{3} e f h +\frac {5}{8} a \,d^{3} f^{2} g +\frac {13}{8} b \,c^{2} d \,f^{2} h -b c \,d^{2} e f h -\frac {9}{8} b c \,d^{2} f^{2} g +\frac {1}{2} b \,d^{3} e f g \right ) \left (f x +e \right )^{\frac {3}{2}}-\frac {f \left (7 a \,c^{2} d \,f^{2} h -11 a c \,d^{2} e f h -3 a c \,d^{2} f^{2} g +4 a \,d^{3} e^{2} h +3 a \,d^{3} e f g -11 b \,c^{3} f^{2} h +19 b \,c^{2} d e f h +7 b \,c^{2} d \,f^{2} g -8 b c \,d^{2} e^{2} h -11 b c \,d^{2} e f g +4 b \,d^{3} e^{2} g \right ) \sqrt {f x +e}}{8}}{\left (\left (f x +e \right ) d +c f -d e \right )^{2}}+\frac {\left (15 a c d \,f^{2} h -12 a \,d^{2} e f h -3 a \,d^{2} f^{2} g -35 b \,c^{2} f^{2} h +40 b c d e f h +15 b c d \,f^{2} g -8 b \,d^{2} e^{2} h -12 b \,d^{2} e f g \right ) \arctan \left (\frac {d \sqrt {f x +e}}{\sqrt {\left (c f -d e \right ) d}}\right )}{8 \sqrt {\left (c f -d e \right ) d}}\right )}{d^{4}}\)	\(409\)

Input:

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

Output:

15/4/((c*f-d*e)*d)^(1/2)*(-(1/5*(-g*f^2*a-4*e*(a*h+b*g)*f-8/3*b*e^2*h)*d^2 
+c*((a*h+b*g)*f+8/3*e*h*b)*f*d-7/3*b*c^2*f^2*h)*(d*x+c)^2*arctan(d*(f*x+e) 
^(1/2)/((c*f-d*e)*d)^(1/2))+(1/3*(-(-8/15*b*h*x^2+8/5*(-a*h-b*g)*x+g*a)*x* 
f-2/5*e*(-16/3*b*h*x^2+2*(a*h+b*g)*x+g*a))*d^3-2/15*c*((28/3*b*h*x^2+25/2* 
(-a*h-b*g)*x+3/2*g*a)*f+e*(-44/3*b*h*x+a*h+b*g))*d^2+((-35/9*b*h*x+a*h+b*g 
)*f+10/9*e*h*b)*c^2*d-7/3*c^3*h*b*f)*(f*x+e)^(1/2)*((c*f-d*e)*d)^(1/2))/(d 
*x+c)^2/d^4
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 754 vs. \(2 (253) = 506\).

Time = 0.18 (sec) , antiderivative size = 1522, normalized size of antiderivative = 5.46 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/24*(3*sqrt(d^2*e - c*d*f)*((3*(4*b*d^4*e*f - (5*b*c*d^3 - a*d^4)*f^2)* 
g + (8*b*d^4*e^2 - 4*(10*b*c*d^3 - 3*a*d^4)*e*f + 5*(7*b*c^2*d^2 - 3*a*c*d 
^3)*f^2)*h)*x^2 + 3*(4*b*c^2*d^2*e*f - (5*b*c^3*d - a*c^2*d^2)*f^2)*g + (8 
*b*c^2*d^2*e^2 - 4*(10*b*c^3*d - 3*a*c^2*d^2)*e*f + 5*(7*b*c^4 - 3*a*c^3*d 
)*f^2)*h + 2*(3*(4*b*c*d^3*e*f - (5*b*c^2*d^2 - a*c*d^3)*f^2)*g + (8*b*c*d 
^3*e^2 - 4*(10*b*c^2*d^2 - 3*a*c*d^3)*e*f + 5*(7*b*c^3*d - 3*a*c^2*d^2)*f^ 
2)*h)*x)*log((d*f*x + 2*d*e - c*f + 2*sqrt(d^2*e - c*d*f)*sqrt(f*x + e))/( 
d*x + c)) - 2*(8*(b*d^5*e*f - b*c*d^4*f^2)*h*x^3 + 8*(3*(b*d^5*e*f - b*c*d 
^4*f^2)*g + (4*b*d^5*e^2 - (11*b*c*d^4 - 3*a*d^5)*e*f + (7*b*c^2*d^3 - 3*a 
*c*d^4)*f^2)*h)*x^2 - 3*(2*(b*c*d^4 + a*d^5)*e^2 - (17*b*c^2*d^3 - a*c*d^4 
)*e*f + 3*(5*b*c^3*d^2 - a*c^2*d^3)*f^2)*g + (2*(25*b*c^2*d^3 - 3*a*c*d^4) 
*e^2 - (155*b*c^3*d^2 - 51*a*c^2*d^3)*e*f + 15*(7*b*c^4*d - 3*a*c^3*d^2)*f 
^2)*h - (3*(4*b*d^5*e^2 - (29*b*c*d^4 - 5*a*d^5)*e*f + 5*(5*b*c^2*d^3 - a* 
c*d^4)*f^2)*g - (4*(22*b*c*d^4 - 3*a*d^5)*e^2 - (263*b*c^2*d^3 - 87*a*c*d^ 
4)*e*f + 25*(7*b*c^3*d^2 - 3*a*c^2*d^3)*f^2)*h)*x)*sqrt(f*x + e))/(c^2*d^6 
*e - c^3*d^5*f + (d^8*e - c*d^7*f)*x^2 + 2*(c*d^7*e - c^2*d^6*f)*x), 1/12* 
(3*sqrt(-d^2*e + c*d*f)*((3*(4*b*d^4*e*f - (5*b*c*d^3 - a*d^4)*f^2)*g + (8 
*b*d^4*e^2 - 4*(10*b*c*d^3 - 3*a*d^4)*e*f + 5*(7*b*c^2*d^2 - 3*a*c*d^3)*f^ 
2)*h)*x^2 + 3*(4*b*c^2*d^2*e*f - (5*b*c^3*d - a*c^2*d^2)*f^2)*g + (8*b*c^2 
*d^2*e^2 - 4*(10*b*c^3*d - 3*a*c^2*d^2)*e*f + 5*(7*b*c^4 - 3*a*c^3*d)*f...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\text {Exception raised: ValueError} \] Input:

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(c*f-d*e>0)', see `assume?` for m 
ore detail
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 541 vs. \(2 (253) = 506\).

Time = 0.13 (sec) , antiderivative size = 541, normalized size of antiderivative = 1.94 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\frac {{\left (12 \, b d^{2} e f g - 15 \, b c d f^{2} g + 3 \, a d^{2} f^{2} g + 8 \, b d^{2} e^{2} h - 40 \, b c d e f h + 12 \, a d^{2} e f h + 35 \, b c^{2} f^{2} h - 15 \, a c d f^{2} h\right )} \arctan \left (\frac {\sqrt {f x + e} d}{\sqrt {-d^{2} e + c d f}}\right )}{4 \, \sqrt {-d^{2} e + c d f} d^{4}} - \frac {4 \, {\left (f x + e\right )}^{\frac {3}{2}} b d^{3} e f g - 4 \, \sqrt {f x + e} b d^{3} e^{2} f g - 9 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} f^{2} g + 5 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} f^{2} g + 11 \, \sqrt {f x + e} b c d^{2} e f^{2} g - 3 \, \sqrt {f x + e} a d^{3} e f^{2} g - 7 \, \sqrt {f x + e} b c^{2} d f^{3} g + 3 \, \sqrt {f x + e} a c d^{2} f^{3} g - 8 \, {\left (f x + e\right )}^{\frac {3}{2}} b c d^{2} e f h + 4 \, {\left (f x + e\right )}^{\frac {3}{2}} a d^{3} e f h + 8 \, \sqrt {f x + e} b c d^{2} e^{2} f h - 4 \, \sqrt {f x + e} a d^{3} e^{2} f h + 13 \, {\left (f x + e\right )}^{\frac {3}{2}} b c^{2} d f^{2} h - 9 \, {\left (f x + e\right )}^{\frac {3}{2}} a c d^{2} f^{2} h - 19 \, \sqrt {f x + e} b c^{2} d e f^{2} h + 11 \, \sqrt {f x + e} a c d^{2} e f^{2} h + 11 \, \sqrt {f x + e} b c^{3} f^{3} h - 7 \, \sqrt {f x + e} a c^{2} d f^{3} h}{4 \, {\left ({\left (f x + e\right )} d - d e + c f\right )}^{2} d^{4}} + \frac {2 \, {\left (3 \, \sqrt {f x + e} b d^{6} f g + {\left (f x + e\right )}^{\frac {3}{2}} b d^{6} h + 3 \, \sqrt {f x + e} b d^{6} e h - 9 \, \sqrt {f x + e} b c d^{5} f h + 3 \, \sqrt {f x + e} a d^{6} f h\right )}}{3 \, d^{9}} \] Input:

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

Output:

1/4*(12*b*d^2*e*f*g - 15*b*c*d*f^2*g + 3*a*d^2*f^2*g + 8*b*d^2*e^2*h - 40* 
b*c*d*e*f*h + 12*a*d^2*e*f*h + 35*b*c^2*f^2*h - 15*a*c*d*f^2*h)*arctan(sqr 
t(f*x + e)*d/sqrt(-d^2*e + c*d*f))/(sqrt(-d^2*e + c*d*f)*d^4) - 1/4*(4*(f* 
x + e)^(3/2)*b*d^3*e*f*g - 4*sqrt(f*x + e)*b*d^3*e^2*f*g - 9*(f*x + e)^(3/ 
2)*b*c*d^2*f^2*g + 5*(f*x + e)^(3/2)*a*d^3*f^2*g + 11*sqrt(f*x + e)*b*c*d^ 
2*e*f^2*g - 3*sqrt(f*x + e)*a*d^3*e*f^2*g - 7*sqrt(f*x + e)*b*c^2*d*f^3*g 
+ 3*sqrt(f*x + e)*a*c*d^2*f^3*g - 8*(f*x + e)^(3/2)*b*c*d^2*e*f*h + 4*(f*x 
 + e)^(3/2)*a*d^3*e*f*h + 8*sqrt(f*x + e)*b*c*d^2*e^2*f*h - 4*sqrt(f*x + e 
)*a*d^3*e^2*f*h + 13*(f*x + e)^(3/2)*b*c^2*d*f^2*h - 9*(f*x + e)^(3/2)*a*c 
*d^2*f^2*h - 19*sqrt(f*x + e)*b*c^2*d*e*f^2*h + 11*sqrt(f*x + e)*a*c*d^2*e 
*f^2*h + 11*sqrt(f*x + e)*b*c^3*f^3*h - 7*sqrt(f*x + e)*a*c^2*d*f^3*h)/((( 
f*x + e)*d - d*e + c*f)^2*d^4) + 2/3*(3*sqrt(f*x + e)*b*d^6*f*g + (f*x + e 
)^(3/2)*b*d^6*h + 3*sqrt(f*x + e)*b*d^6*e*h - 9*sqrt(f*x + e)*b*c*d^5*f*h 
+ 3*sqrt(f*x + e)*a*d^6*f*h)/d^9
 

Mupad [B] (verification not implemented)

Time = 0.24 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.61 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx=\sqrt {e+f\,x}\,\left (\frac {2\,a\,f\,h-4\,b\,e\,h+2\,b\,f\,g}{d^3}+\frac {2\,b\,h\,\left (3\,d^3\,e-3\,c\,d^2\,f\right )}{d^6}\right )+\frac {\sqrt {e+f\,x}\,\left (\frac {7\,a\,c^2\,d\,f^3\,h}{4}-\frac {3\,a\,c\,d^2\,f^3\,g}{4}-\frac {11\,b\,c^3\,f^3\,h}{4}+\frac {7\,b\,c^2\,d\,f^3\,g}{4}+\frac {3\,a\,d^3\,e\,f^2\,g}{4}+a\,d^3\,e^2\,f\,h+b\,d^3\,e^2\,f\,g-\frac {11\,a\,c\,d^2\,e\,f^2\,h}{4}-\frac {11\,b\,c\,d^2\,e\,f^2\,g}{4}-2\,b\,c\,d^2\,e^2\,f\,h+\frac {19\,b\,c^2\,d\,e\,f^2\,h}{4}\right )-{\left (e+f\,x\right )}^{3/2}\,\left (\frac {5\,a\,d^3\,f^2\,g}{4}-\frac {9\,a\,c\,d^2\,f^2\,h}{4}-\frac {9\,b\,c\,d^2\,f^2\,g}{4}+\frac {13\,b\,c^2\,d\,f^2\,h}{4}+a\,d^3\,e\,f\,h+b\,d^3\,e\,f\,g-2\,b\,c\,d^2\,e\,f\,h\right )}{d^6\,{\left (e+f\,x\right )}^2-\left (e+f\,x\right )\,\left (2\,d^6\,e-2\,c\,d^5\,f\right )+d^6\,e^2+c^2\,d^4\,f^2-2\,c\,d^5\,e\,f}+\frac {\mathrm {atan}\left (\frac {\sqrt {d}\,\sqrt {e+f\,x}}{\sqrt {c\,f-d\,e}}\right )\,\left (3\,a\,d^2\,f^2\,g+35\,b\,c^2\,f^2\,h+8\,b\,d^2\,e^2\,h-15\,a\,c\,d\,f^2\,h-15\,b\,c\,d\,f^2\,g+12\,a\,d^2\,e\,f\,h+12\,b\,d^2\,e\,f\,g-40\,b\,c\,d\,e\,f\,h\right )}{4\,d^{9/2}\,\sqrt {c\,f-d\,e}}+\frac {2\,b\,h\,{\left (e+f\,x\right )}^{3/2}}{3\,d^3} \] Input:

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

Output:

(e + f*x)^(1/2)*((2*a*f*h - 4*b*e*h + 2*b*f*g)/d^3 + (2*b*h*(3*d^3*e - 3*c 
*d^2*f))/d^6) + ((e + f*x)^(1/2)*((7*a*c^2*d*f^3*h)/4 - (3*a*c*d^2*f^3*g)/ 
4 - (11*b*c^3*f^3*h)/4 + (7*b*c^2*d*f^3*g)/4 + (3*a*d^3*e*f^2*g)/4 + a*d^3 
*e^2*f*h + b*d^3*e^2*f*g - (11*a*c*d^2*e*f^2*h)/4 - (11*b*c*d^2*e*f^2*g)/4 
 - 2*b*c*d^2*e^2*f*h + (19*b*c^2*d*e*f^2*h)/4) - (e + f*x)^(3/2)*((5*a*d^3 
*f^2*g)/4 - (9*a*c*d^2*f^2*h)/4 - (9*b*c*d^2*f^2*g)/4 + (13*b*c^2*d*f^2*h) 
/4 + a*d^3*e*f*h + b*d^3*e*f*g - 2*b*c*d^2*e*f*h))/(d^6*(e + f*x)^2 - (e + 
 f*x)*(2*d^6*e - 2*c*d^5*f) + d^6*e^2 + c^2*d^4*f^2 - 2*c*d^5*e*f) + (atan 
((d^(1/2)*(e + f*x)^(1/2))/(c*f - d*e)^(1/2))*(3*a*d^2*f^2*g + 35*b*c^2*f^ 
2*h + 8*b*d^2*e^2*h - 15*a*c*d*f^2*h - 15*b*c*d*f^2*g + 12*a*d^2*e*f*h + 1 
2*b*d^2*e*f*g - 40*b*c*d*e*f*h))/(4*d^(9/2)*(c*f - d*e)^(1/2)) + (2*b*h*(e 
 + f*x)^(3/2))/(3*d^3)
 

Reduce [B] (verification not implemented)

Time = 0.17 (sec) , antiderivative size = 1779, normalized size of antiderivative = 6.38 \[ \int \frac {(a+b x) (e+f x)^{3/2} (g+h x)}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 45*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*a*c**3*d*f**2*h + 36*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d) 
/(sqrt(d)*sqrt(c*f - d*e)))*a*c**2*d**2*e*f*h + 9*sqrt(d)*sqrt(c*f - d*e)* 
atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c**2*d**2*f**2*g - 90* 
sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))* 
a*c**2*d**2*f**2*h*x + 72*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/( 
sqrt(d)*sqrt(c*f - d*e)))*a*c*d**3*e*f*h*x + 18*sqrt(d)*sqrt(c*f - d*e)*at 
an((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c*d**3*f**2*g*x - 45*sqr 
t(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*c 
*d**3*f**2*h*x**2 + 36*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqr 
t(d)*sqrt(c*f - d*e)))*a*d**4*e*f*h*x**2 + 9*sqrt(d)*sqrt(c*f - d*e)*atan( 
(sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*a*d**4*f**2*g*x**2 + 105*sqrt 
(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c* 
*4*f**2*h - 120*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sq 
rt(c*f - d*e)))*b*c**3*d*e*f*h - 45*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + 
 f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**3*d*f**2*g + 210*sqrt(d)*sqrt(c*f 
 - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**3*d*f**2*h* 
x + 24*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x)*d)/(sqrt(d)*sqrt(c*f - 
d*e)))*b*c**2*d**2*e**2*h + 36*sqrt(d)*sqrt(c*f - d*e)*atan((sqrt(e + f*x) 
*d)/(sqrt(d)*sqrt(c*f - d*e)))*b*c**2*d**2*e*f*g - 240*sqrt(d)*sqrt(c*f...