\(\int \frac {\sqrt {c+d x} (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx\) [196]

Optimal result

Integrand size = 31, antiderivative size = 396 \[ \int \frac {\sqrt {c+d x} (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=-\frac {(12 a d f h-b (8 d f g+3 d e h+c f h)) \sqrt {c+d x} \sqrt {e+f x}}{4 b^3 d}+\frac {(2 b f g+b e h-3 a f h) \sqrt {c+d x} (e+f x)^{3/2}}{2 b^2 (b e-a f)}-\frac {(b g-a h) \sqrt {c+d x} (e+f x)^{5/2}}{b (b e-a f) (a+b x)}+\frac {\left (24 a^2 d^2 f^2 h-8 a b d f (2 d f g+3 d e h+c f h)-b^2 \left (c^2 f^2 h-3 d^2 e (4 f g+e h)-2 c d f (2 f g+3 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{4 b^4 d^{3/2} \sqrt {f}}-\frac {\sqrt {b e-a f} \left (6 a^2 d f h+b^2 (d e g+3 c f g+2 c e h)-a b (4 d f g+3 d e h+5 c f h)\right ) \text {arctanh}\left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {b c-a d} \sqrt {e+f x}}\right )}{b^4 \sqrt {b c-a d}} \] Output:

-1/4*(12*a*d*f*h-b*(c*f*h+3*d*e*h+8*d*f*g))*(d*x+c)^(1/2)*(f*x+e)^(1/2)/b^ 
3/d+1/2*(-3*a*f*h+b*e*h+2*b*f*g)*(d*x+c)^(1/2)*(f*x+e)^(3/2)/b^2/(-a*f+b*e 
)-(-a*h+b*g)*(d*x+c)^(1/2)*(f*x+e)^(5/2)/b/(-a*f+b*e)/(b*x+a)+1/4*(24*a^2* 
d^2*f^2*h-8*a*b*d*f*(c*f*h+3*d*e*h+2*d*f*g)-b^2*(c^2*f^2*h-3*d^2*e*(e*h+4* 
f*g)-2*c*d*f*(3*e*h+2*f*g)))*arctanh(f^(1/2)*(d*x+c)^(1/2)/d^(1/2)/(f*x+e) 
^(1/2))/b^4/d^(3/2)/f^(1/2)-(-a*f+b*e)^(1/2)*(6*a^2*d*f*h+b^2*(2*c*e*h+3*c 
*f*g+d*e*g)-a*b*(5*c*f*h+3*d*e*h+4*d*f*g))*arctanh((-a*f+b*e)^(1/2)*(d*x+c 
)^(1/2)/(-a*d+b*c)^(1/2)/(f*x+e)^(1/2))/b^4/(-a*d+b*c)^(1/2)
 

Mathematica [A] (verified)

Time = 1.76 (sec) , antiderivative size = 339, normalized size of antiderivative = 0.86 \[ \int \frac {\sqrt {c+d x} (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\frac {\frac {b \sqrt {c+d x} \sqrt {e+f x} \left (-12 a^2 d f h+b^2 (c f h x+2 d f x (2 g+h x)+d e (-4 g+5 h x))+a b (c f h+d (8 f g+9 e h-6 f h x))\right )}{d (a+b x)}+\frac {4 \sqrt {b e-a f} \left (6 a^2 d f h+b^2 (d e g+3 c f g+2 c e h)-a b (4 d f g+3 d e h+5 c f h)\right ) \arctan \left (\frac {\sqrt {b e-a f} \sqrt {c+d x}}{\sqrt {-b c+a d} \sqrt {e+f x}}\right )}{\sqrt {-b c+a d}}+\frac {\left (24 a^2 d^2 f^2 h-8 a b d f (2 d f g+3 d e h+c f h)+b^2 \left (-c^2 f^2 h+3 d^2 e (4 f g+e h)+2 c d f (2 f g+3 e h)\right )\right ) \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right )}{d^{3/2} \sqrt {f}}}{4 b^4} \] Input:

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

Output:

((b*Sqrt[c + d*x]*Sqrt[e + f*x]*(-12*a^2*d*f*h + b^2*(c*f*h*x + 2*d*f*x*(2 
*g + h*x) + d*e*(-4*g + 5*h*x)) + a*b*(c*f*h + d*(8*f*g + 9*e*h - 6*f*h*x) 
)))/(d*(a + b*x)) + (4*Sqrt[b*e - a*f]*(6*a^2*d*f*h + b^2*(d*e*g + 3*c*f*g 
 + 2*c*e*h) - a*b*(4*d*f*g + 3*d*e*h + 5*c*f*h))*ArcTan[(Sqrt[b*e - a*f]*S 
qrt[c + d*x])/(Sqrt[-(b*c) + a*d]*Sqrt[e + f*x])])/Sqrt[-(b*c) + a*d] + (( 
24*a^2*d^2*f^2*h - 8*a*b*d*f*(2*d*f*g + 3*d*e*h + c*f*h) + b^2*(-(c^2*f^2* 
h) + 3*d^2*e*(4*f*g + e*h) + 2*c*d*f*(2*f*g + 3*e*h)))*ArcTanh[(Sqrt[f]*Sq 
rt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(d^(3/2)*Sqrt[f]))/(4*b^4)
 

Rubi [A] (verified)

Time = 0.80 (sec) , antiderivative size = 425, normalized size of antiderivative = 1.07, number of steps used = 11, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.323, Rules used = {166, 27, 171, 27, 171, 27, 175, 66, 104, 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.

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

\(\Big \downarrow \) 166

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 171

\(\displaystyle -\frac {\frac {\int \frac {d (b e-a f) \sqrt {e+f x} (3 a (d e+3 c f) h-2 b (d e g+3 c f g+2 c e h)+(12 a d f h-b (8 d f g+3 d e h+c f h)) x)}{(a+b x) \sqrt {c+d x}}dx}{2 b d}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {(b e-a f) \int \frac {\sqrt {e+f x} (3 a (d e+3 c f) h-2 b (d e g+3 c f g+2 c e h)+(12 a d f h-b (8 d f g+3 d e h+c f h)) x)}{(a+b x) \sqrt {c+d x}}dx}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 171

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {\int \frac {2 b d e (3 a (d e+3 c f) h-2 b (d e g+3 c f g+2 c e h))-a (d e+c f) (12 a d f h-b (8 d f g+3 d e h+c f h))-\left (-\left (\left (-3 e (4 f g+e h) d^2-2 c f (2 f g+3 e h) d+c^2 f^2 h\right ) b^2\right )-8 a d f (2 d f g+3 d e h+c f h) b+24 a^2 d^2 f^2 h\right ) x}{2 (a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {\int \frac {2 b d e (3 a (d e+3 c f) h-2 b (d e g+3 c f g+2 c e h))-a (d e+c f) (12 a d f h-b (8 d f g+3 d e h+c f h))-\left (-\left (\left (-3 e (4 f g+e h) d^2-2 c f (2 f g+3 e h) d+c^2 f^2 h\right ) b^2\right )-8 a d f (2 d f g+3 d e h+c f h) b+24 a^2 d^2 f^2 h\right ) x}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 175

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {-\frac {\left (24 a^2 d^2 f^2 h-8 a b d f (c f h+3 d e h+2 d f g)-\left (b^2 \left (c^2 f^2 h-2 c d f (3 e h+2 f g)-3 d^2 e (e h+4 f g)\right )\right )\right ) \int \frac {1}{\sqrt {c+d x} \sqrt {e+f x}}dx}{b}-\frac {4 d (b e-a f) \left (6 a^2 d f h-a b (5 c f h+3 d e h+4 d f g)+b^2 (2 c e h+3 c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 66

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {-\frac {2 \left (24 a^2 d^2 f^2 h-8 a b d f (c f h+3 d e h+2 d f g)-\left (b^2 \left (c^2 f^2 h-2 c d f (3 e h+2 f g)-3 d^2 e (e h+4 f g)\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {4 d (b e-a f) \left (6 a^2 d f h-a b (5 c f h+3 d e h+4 d f g)+b^2 (2 c e h+3 c f g+d e g)\right ) \int \frac {1}{(a+b x) \sqrt {c+d x} \sqrt {e+f x}}dx}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 104

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {-\frac {2 \left (24 a^2 d^2 f^2 h-8 a b d f (c f h+3 d e h+2 d f g)-\left (b^2 \left (c^2 f^2 h-2 c d f (3 e h+2 f g)-3 d^2 e (e h+4 f g)\right )\right )\right ) \int \frac {1}{d-\frac {f (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}-\frac {8 d (b e-a f) \left (6 a^2 d f h-a b (5 c f h+3 d e h+4 d f g)+b^2 (2 c e h+3 c f g+d e g)\right ) \int \frac {1}{-b c+a d+\frac {(b e-a f) (c+d x)}{e+f x}}d\frac {\sqrt {c+d x}}{\sqrt {e+f x}}}{b}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

\(\Big \downarrow \) 221

\(\displaystyle -\frac {\frac {(b e-a f) \left (\frac {\frac {8 d \sqrt {b e-a f} \text {arctanh}\left (\frac {\sqrt {c+d x} \sqrt {b e-a f}}{\sqrt {e+f x} \sqrt {b c-a d}}\right ) \left (6 a^2 d f h-a b (5 c f h+3 d e h+4 d f g)+b^2 (2 c e h+3 c f g+d e g)\right )}{b \sqrt {b c-a d}}-\frac {2 \text {arctanh}\left (\frac {\sqrt {f} \sqrt {c+d x}}{\sqrt {d} \sqrt {e+f x}}\right ) \left (24 a^2 d^2 f^2 h-8 a b d f (c f h+3 d e h+2 d f g)-\left (b^2 \left (c^2 f^2 h-2 c d f (3 e h+2 f g)-3 d^2 e (e h+4 f g)\right )\right )\right )}{b \sqrt {d} \sqrt {f}}}{2 b d}+\frac {\sqrt {c+d x} \sqrt {e+f x} (12 a d f h-b (c f h+3 d e h+8 d f g))}{b d}\right )}{2 b}-\frac {\sqrt {c+d x} (e+f x)^{3/2} (-3 a f h+b e h+2 b f g)}{b}}{2 b (b e-a f)}-\frac {\sqrt {c+d x} (e+f x)^{5/2} (b g-a h)}{b (a+b x) (b e-a f)}\)

Input:

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

Output:

-(((b*g - a*h)*Sqrt[c + d*x]*(e + f*x)^(5/2))/(b*(b*e - a*f)*(a + b*x))) - 
 (-(((2*b*f*g + b*e*h - 3*a*f*h)*Sqrt[c + d*x]*(e + f*x)^(3/2))/b) + ((b*e 
 - a*f)*(((12*a*d*f*h - b*(8*d*f*g + 3*d*e*h + c*f*h))*Sqrt[c + d*x]*Sqrt[ 
e + f*x])/(b*d) + ((-2*(24*a^2*d^2*f^2*h - 8*a*b*d*f*(2*d*f*g + 3*d*e*h + 
c*f*h) - b^2*(c^2*f^2*h - 3*d^2*e*(4*f*g + e*h) - 2*c*d*f*(2*f*g + 3*e*h)) 
)*ArcTanh[(Sqrt[f]*Sqrt[c + d*x])/(Sqrt[d]*Sqrt[e + f*x])])/(b*Sqrt[d]*Sqr 
t[f]) + (8*d*Sqrt[b*e - a*f]*(6*a^2*d*f*h + b^2*(d*e*g + 3*c*f*g + 2*c*e*h 
) - a*b*(4*d*f*g + 3*d*e*h + 5*c*f*h))*ArcTanh[(Sqrt[b*e - a*f]*Sqrt[c + d 
*x])/(Sqrt[b*c - a*d]*Sqrt[e + f*x])])/(b*Sqrt[b*c - a*d]))/(2*b*d)))/(2*b 
))/(2*b*(b*e - a*f))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 66

Int[1/(Sqrt[(a_) + (b_.)*(x_)]*Sqrt[(c_) + (d_.)*(x_)]), x_Symbol] :> Simp[ 
2   Subst[Int[1/(b - d*x^2), x], x, Sqrt[a + b*x]/Sqrt[c + d*x]], x] /; Fre 
eQ[{a, b, c, d}, x] &&  !GtQ[c - a*(d/b), 0]
 

rule 104

Int[(((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_))/((e_.) + (f_.)*(x 
_)), x_] :> With[{q = Denominator[m]}, Simp[q   Subst[Int[x^(q*(m + 1) - 1) 
/(b*e - a*f - (d*e - c*f)*x^q), x], x, (a + b*x)^(1/q)/(c + d*x)^(1/q)], x] 
] /; FreeQ[{a, b, c, d, e, f}, x] && EqQ[m + n + 1, 0] && RationalQ[n] && L 
tQ[-1, m, 0] && SimplerQ[a + b*x, c + d*x]
 

rule 166

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

rule 171

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) 
)^(p_)*((g_.) + (h_.)*(x_)), x_] :> Simp[h*(a + b*x)^m*(c + d*x)^(n + 1)*(( 
e + f*x)^(p + 1)/(d*f*(m + n + p + 2))), x] + Simp[1/(d*f*(m + n + p + 2)) 
  Int[(a + b*x)^(m - 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*g*(m + n + p + 2 
) - h*(b*c*e*m + a*(d*e*(n + 1) + c*f*(p + 1))) + (b*d*f*g*(m + n + p + 2) 
+ h*(a*d*f*m - b*(d*e*(m + n + 1) + c*f*(m + p + 1))))*x, x], x], x] /; Fre 
eQ[{a, b, c, d, e, f, g, h, n, p}, x] && GtQ[m, 0] && NeQ[m + n + p + 2, 0] 
 && IntegersQ[2*m, 2*n, 2*p]
 

rule 175

Int[(((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_ 
)))/((a_.) + (b_.)*(x_)), x_] :> Simp[h/b   Int[(c + d*x)^n*(e + f*x)^p, x] 
, x] + Simp[(b*g - a*h)/b   Int[(c + d*x)^n*((e + f*x)^p/(a + b*x)), x], x] 
 /; FreeQ[{a, b, c, d, e, f, g, h, n, p}, x]
 

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 [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(4510\) vs. \(2(354)=708\).

Time = 0.42 (sec) , antiderivative size = 4511, normalized size of antiderivative = 11.39

method	result	size
default	\(\text {Expression too large to display}\)	\(4511\)

Input:

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

Output:

1/8*(d*x+c)^(1/2)*(f*x+e)^(1/2)*(-ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2 
)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*b^4*c^2*f^2*h*x*((a^2*d*f-a*b*c*f-a*b* 
d*e+b^2*c*e)/b^2)^(1/2)+3*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^ 
(1/2)+c*f+d*e)/(d*f)^(1/2))*b^4*d^2*e^2*h*x*((a^2*d*f-a*b*c*f-a*b*d*e+b^2* 
c*e)/b^2)^(1/2)-16*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2 
)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b* 
x+a))*a^3*b*d^2*f^2*g*(d*f)^(1/2)+24*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^( 
1/2)*(d*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^3*b*d^2*f^2*h*((a^2*d*f-a*b*c*f-a 
*b*d*e+b^2*c*e)/b^2)^(1/2)-8*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d* 
x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2* 
b*c*e)/(b*x+a))*a*b^3*c*d*e^2*h*(d*f)^(1/2)+12*ln((-2*a*d*f*x+b*c*f*x+b*d* 
e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2 
)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b^3*d^2*e^2*h*x*(d*f)^(1/2)-8*ln((-2*a 
*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d*f-a*b*c*f-a*b*d*e 
+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*b^4*c*d*e^2*h*x*(d*f) 
^(1/2)+28*ln((-2*a*d*f*x+b*c*f*x+b*d*e*x+2*((f*x+e)*(d*x+c))^(1/2)*((a^2*d 
*f-a*b*c*f-a*b*d*e+b^2*c*e)/b^2)^(1/2)*b-a*c*f-a*d*e+2*b*c*e)/(b*x+a))*a*b 
^3*c*d*e*f*h*x*(d*f)^(1/2)-16*ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d 
*f)^(1/2)+c*f+d*e)/(d*f)^(1/2))*a^2*b^2*d^2*f^2*g*((a^2*d*f-a*b*c*f-a*b*d* 
e+b^2*c*e)/b^2)^(1/2)-ln(1/2*(2*d*f*x+2*((f*x+e)*(d*x+c))^(1/2)*(d*f)^(...
 

Fricas [F(-1)]

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

integrate((d*x+c)^(1/2)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x, algorithm="fric 
as")
 

Output:

Timed out
 

Sympy [F]

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

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

Output:

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

Maxima [F(-2)]

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

integrate((d*x+c)^(1/2)*(f*x+e)^(3/2)*(h*x+g)/(b*x+a)^2,x, algorithm="maxi 
ma")
 

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(2*a*d*f-b*c*f>0)', see `assume?` 
 for more
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1657 vs. \(2 (354) = 708\).

Time = 1.34 (sec) , antiderivative size = 1657, normalized size of antiderivative = 4.18 \[ \int \frac {\sqrt {c+d x} (e+f x)^{3/2} (g+h x)}{(a+b x)^2} \, dx=\text {Too large to display} \] Input:

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

Output:

1/4*sqrt(d^2*e + (d*x + c)*d*f - c*d*f)*sqrt(d*x + c)*(2*(d*x + c)*f*h*abs 
(d)/(b^2*d^3) + (4*b^7*d^4*f^3*g*abs(d) + 5*b^7*d^4*e*f^2*h*abs(d) - b^7*c 
*d^3*f^3*h*abs(d) - 8*a*b^6*d^4*f^3*h*abs(d))/(b^9*d^6*f^2)) - (sqrt(d*f)* 
b^3*d*e^2*g*abs(d) + 3*sqrt(d*f)*b^3*c*e*f*g*abs(d) - 5*sqrt(d*f)*a*b^2*d* 
e*f*g*abs(d) - 3*sqrt(d*f)*a*b^2*c*f^2*g*abs(d) + 4*sqrt(d*f)*a^2*b*d*f^2* 
g*abs(d) + 2*sqrt(d*f)*b^3*c*e^2*h*abs(d) - 3*sqrt(d*f)*a*b^2*d*e^2*h*abs( 
d) - 7*sqrt(d*f)*a*b^2*c*e*f*h*abs(d) + 9*sqrt(d*f)*a^2*b*d*e*f*h*abs(d) + 
 5*sqrt(d*f)*a^2*b*c*f^2*h*abs(d) - 6*sqrt(d*f)*a^3*d*f^2*h*abs(d))*arctan 
(-1/2*(b*d^2*e + b*c*d*f - 2*a*d^2*f - (sqrt(d*f)*sqrt(d*x + c) - sqrt(d^2 
*e + (d*x + c)*d*f - c*d*f))^2*b)/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c 
*d*f^2 - a^2*d^2*f^2)*d))/(sqrt(-b^2*c*d*e*f + a*b*d^2*e*f + a*b*c*d*f^2 - 
 a^2*d^2*f^2)*b^4*d) - 2*(sqrt(d*f)*b^3*d^3*e^3*g*abs(d) - 2*sqrt(d*f)*b^3 
*c*d^2*e^2*f*g*abs(d) - sqrt(d*f)*a*b^2*d^3*e^2*f*g*abs(d) + sqrt(d*f)*b^3 
*c^2*d*e*f^2*g*abs(d) + 2*sqrt(d*f)*a*b^2*c*d^2*e*f^2*g*abs(d) - sqrt(d*f) 
*a*b^2*c^2*d*f^3*g*abs(d) - sqrt(d*f)*a*b^2*d^3*e^3*h*abs(d) + 2*sqrt(d*f) 
*a*b^2*c*d^2*e^2*f*h*abs(d) + sqrt(d*f)*a^2*b*d^3*e^2*f*h*abs(d) - sqrt(d* 
f)*a*b^2*c^2*d*e*f^2*h*abs(d) - 2*sqrt(d*f)*a^2*b*c*d^2*e*f^2*h*abs(d) + s 
qrt(d*f)*a^2*b*c^2*d*f^3*h*abs(d) - sqrt(d*f)*(sqrt(d*f)*sqrt(d*x + c) - s 
qrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*b^3*d*e^2*g*abs(d) - sqrt(d*f)*(sqrt 
(d*f)*sqrt(d*x + c) - sqrt(d^2*e + (d*x + c)*d*f - c*d*f))^2*b^3*c*e*f*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [F]

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

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

Output:

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