\(\int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx\) [10]

Optimal result

Integrand size = 36, antiderivative size = 307 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=-\frac {A \sqrt {a x+b x^2}}{5 a (c x)^{11/2}}+\frac {(9 A b-10 a B) \sqrt {a x+b x^2}}{40 a^2 c (c x)^{9/2}}-\frac {\left (63 A b^2-70 a b B+80 a^2 C\right ) \sqrt {a x+b x^2}}{240 a^3 c^2 (c x)^{7/2}}+\frac {\left (63 A b^3-70 a b^2 B+80 a^2 b C-96 a^3 D\right ) \sqrt {a x+b x^2}}{192 a^4 c^3 (c x)^{5/2}}-\frac {b \left (63 A b^3-70 a b^2 B+80 a^2 b C-96 a^3 D\right ) \sqrt {a x+b x^2}}{128 a^5 c^4 (c x)^{3/2}}+\frac {b^2 \left (63 A b^3-70 a b^2 B+80 a^2 b C-96 a^3 D\right ) \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {c x}}\right )}{128 a^{11/2} c^{11/2}} \] Output:

-1/5*A*(b*x^2+a*x)^(1/2)/a/(c*x)^(11/2)+1/40*(9*A*b-10*B*a)*(b*x^2+a*x)^(1 
/2)/a^2/c/(c*x)^(9/2)-1/240*(63*A*b^2-70*B*a*b+80*C*a^2)*(b*x^2+a*x)^(1/2) 
/a^3/c^2/(c*x)^(7/2)+1/192*(63*A*b^3-70*B*a*b^2+80*C*a^2*b-96*D*a^3)*(b*x^ 
2+a*x)^(1/2)/a^4/c^3/(c*x)^(5/2)-1/128*b*(63*A*b^3-70*B*a*b^2+80*C*a^2*b-9 
6*D*a^3)*(b*x^2+a*x)^(1/2)/a^5/c^4/(c*x)^(3/2)+1/128*b^2*(63*A*b^3-70*B*a* 
b^2+80*C*a^2*b-96*D*a^3)*arctanh(c^(1/2)*(b*x^2+a*x)^(1/2)/a^(1/2)/(c*x)^( 
1/2))/a^(11/2)/c^(11/2)
 

Mathematica [A] (verified)

Time = 0.64 (sec) , antiderivative size = 214, normalized size of antiderivative = 0.70 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\frac {\sqrt {c x} \left (-\sqrt {a} (a+b x) \left (945 A b^4 x^4-210 a b^3 x^3 (3 A+5 B x)+4 a^2 b^2 x^2 (126 A+25 x (7 B+12 C x))+32 a^4 \left (12 A+5 x \left (3 B+4 C x+6 D x^2\right )\right )-16 a^3 b x (27 A+5 x (7 B+2 x (5 C+9 D x)))\right )+15 b^2 \left (63 A b^3-70 a b^2 B+80 a^2 b C-96 a^3 D\right ) x^5 \sqrt {a+b x} \text {arctanh}\left (\frac {\sqrt {a+b x}}{\sqrt {a}}\right )\right )}{1920 a^{11/2} c^6 x^5 \sqrt {x (a+b x)}} \] Input:

Integrate[(A + B*x + C*x^2 + D*x^3)/((c*x)^(11/2)*Sqrt[a*x + b*x^2]),x]
 

Output:

(Sqrt[c*x]*(-(Sqrt[a]*(a + b*x)*(945*A*b^4*x^4 - 210*a*b^3*x^3*(3*A + 5*B* 
x) + 4*a^2*b^2*x^2*(126*A + 25*x*(7*B + 12*C*x)) + 32*a^4*(12*A + 5*x*(3*B 
 + 4*C*x + 6*D*x^2)) - 16*a^3*b*x*(27*A + 5*x*(7*B + 2*x*(5*C + 9*D*x))))) 
 + 15*b^2*(63*A*b^3 - 70*a*b^2*B + 80*a^2*b*C - 96*a^3*D)*x^5*Sqrt[a + b*x 
]*ArcTanh[Sqrt[a + b*x]/Sqrt[a]]))/(1920*a^(11/2)*c^6*x^5*Sqrt[x*(a + b*x) 
])
 

Rubi [A] (verified)

Time = 1.28 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.13, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.306, Rules used = {2169, 27, 2169, 27, 1220, 1135, 1135, 1135, 1135, 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.

\(\displaystyle \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx\)

\(\Big \downarrow \) 2169

\(\displaystyle -\frac {2 \int -\frac {(5 b C-6 a D) x^2 c^3+5 A b c^3+5 b B x c^3}{2 (c x)^{11/2} \sqrt {b x^2+a x}}dx}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(5 b C-6 a D) x^2 c^3+5 A b c^3+5 b B x c^3}{(c x)^{11/2} \sqrt {b x^2+a x}}dx}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {-\frac {2 \int -\frac {c^5 \left (35 A b^2+\left (48 D a^2-40 b C a+35 b^2 B\right ) x\right )}{2 (c x)^{11/2} \sqrt {b x^2+a x}}dx}{7 b c^2}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {c^3 \int \frac {35 A b^2+\left (48 D a^2-40 b C a+35 b^2 B\right ) x}{(c x)^{11/2} \sqrt {b x^2+a x}}dx}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1220

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \int \frac {1}{(c x)^{9/2} \sqrt {b x^2+a x}}dx}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \left (-\frac {7 b \int \frac {1}{(c x)^{7/2} \sqrt {b x^2+a x}}dx}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \left (-\frac {7 b \left (-\frac {5 b \int \frac {1}{(c x)^{5/2} \sqrt {b x^2+a x}}dx}{6 a c}-\frac {\sqrt {a x+b x^2}}{3 a (c x)^{7/2}}\right )}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \int \frac {1}{(c x)^{3/2} \sqrt {b x^2+a x}}dx}{4 a c}-\frac {\sqrt {a x+b x^2}}{2 a (c x)^{5/2}}\right )}{6 a c}-\frac {\sqrt {a x+b x^2}}{3 a (c x)^{7/2}}\right )}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1135

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{\sqrt {c x} \sqrt {b x^2+a x}}dx}{2 a c}-\frac {\sqrt {a x+b x^2}}{a (c x)^{3/2}}\right )}{4 a c}-\frac {\sqrt {a x+b x^2}}{2 a (c x)^{5/2}}\right )}{6 a c}-\frac {\sqrt {a x+b x^2}}{3 a (c x)^{7/2}}\right )}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 1136

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right ) \left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (-\frac {b \int \frac {1}{\frac {c \left (b x^2+a x\right )}{x}-a c}d\frac {\sqrt {b x^2+a x}}{\sqrt {c x}}}{a}-\frac {\sqrt {a x+b x^2}}{a (c x)^{3/2}}\right )}{4 a c}-\frac {\sqrt {a x+b x^2}}{2 a (c x)^{5/2}}\right )}{6 a c}-\frac {\sqrt {a x+b x^2}}{3 a (c x)^{7/2}}\right )}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

\(\Big \downarrow \) 221

\(\displaystyle \frac {\frac {c^3 \left (-\frac {\left (-\frac {7 b \left (-\frac {5 b \left (-\frac {3 b \left (\frac {b \text {arctanh}\left (\frac {\sqrt {c} \sqrt {a x+b x^2}}{\sqrt {a} \sqrt {c x}}\right )}{a^{3/2} c^{3/2}}-\frac {\sqrt {a x+b x^2}}{a (c x)^{3/2}}\right )}{4 a c}-\frac {\sqrt {a x+b x^2}}{2 a (c x)^{5/2}}\right )}{6 a c}-\frac {\sqrt {a x+b x^2}}{3 a (c x)^{7/2}}\right )}{8 a c}-\frac {\sqrt {a x+b x^2}}{4 a (c x)^{9/2}}\right ) \left (-96 a^3 D+80 a^2 b C-70 a b^2 B+63 A b^3\right )}{2 a c}-\frac {7 A b^2 \sqrt {a x+b x^2}}{a (c x)^{11/2}}\right )}{7 b}-\frac {2 c^2 \sqrt {a x+b x^2} (5 b C-6 a D)}{7 b (c x)^{9/2}}}{5 b c^3}-\frac {2 D \sqrt {a x+b x^2}}{5 b c^2 (c x)^{7/2}}\)

Input:

Int[(A + B*x + C*x^2 + D*x^3)/((c*x)^(11/2)*Sqrt[a*x + b*x^2]),x]
 

Output:

(-2*D*Sqrt[a*x + b*x^2])/(5*b*c^2*(c*x)^(7/2)) + ((-2*c^2*(5*b*C - 6*a*D)* 
Sqrt[a*x + b*x^2])/(7*b*(c*x)^(9/2)) + (c^3*((-7*A*b^2*Sqrt[a*x + b*x^2])/ 
(a*(c*x)^(11/2)) - ((63*A*b^3 - 70*a*b^2*B + 80*a^2*b*C - 96*a^3*D)*(-1/4* 
Sqrt[a*x + b*x^2]/(a*(c*x)^(9/2)) - (7*b*(-1/3*Sqrt[a*x + b*x^2]/(a*(c*x)^ 
(7/2)) - (5*b*(-1/2*Sqrt[a*x + b*x^2]/(a*(c*x)^(5/2)) - (3*b*(-(Sqrt[a*x + 
 b*x^2]/(a*(c*x)^(3/2))) + (b*ArcTanh[(Sqrt[c]*Sqrt[a*x + b*x^2])/(Sqrt[a] 
*Sqrt[c*x])])/(a^(3/2)*c^(3/2))))/(4*a*c)))/(6*a*c)))/(8*a*c)))/(2*a*c)))/ 
(7*b))/(5*b*c^3)
 

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 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 1135

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

rule 2169

Int[(Pq_)*((d_.) + (e_.)*(x_))^(m_.)*((a_.) + (b_.)*(x_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, S 
imp[f*(d + e*x)^(m + q - 1)*((a + b*x + c*x^2)^(p + 1)/(c*e^(q - 1)*(m + q 
+ 2*p + 1))), x] + Simp[1/(c*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + 
b*x + c*x^2)^p*ExpandToSum[c*e^q*(m + q + 2*p + 1)*Pq - c*f*(m + q + 2*p + 
1)*(d + e*x)^q + e*f*(m + p + q)*(d + e*x)^(q - 2)*(b*d - 2*a*e + (2*c*d - 
b*e)*x), x], x], x] /; NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, c, d, e, m, 
 p}, x] && PolyQ[Pq, x] && NeQ[b^2 - 4*a*c, 0] && EqQ[c*d^2 - b*d*e + a*e^2 
, 0]
 

Maple [A] (verified)

Time = 0.42 (sec) , antiderivative size = 482, normalized size of antiderivative = 1.57

method	result	size
default	\(\frac {\sqrt {x \left (b x +a \right )}\, \left (945 A \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (b x +a \right )}}{\sqrt {a c}}\right ) b^{5} c \,x^{5}-1050 B \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (b x +a \right )}}{\sqrt {a c}}\right ) a \,b^{4} c \,x^{5}+1200 C \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (b x +a \right )}}{\sqrt {a c}}\right ) a^{2} b^{3} c \,x^{5}-1440 D \,\operatorname {arctanh}\left (\frac {\sqrt {c \left (b x +a \right )}}{\sqrt {a c}}\right ) a^{3} b^{2} c \,x^{5}-945 A \,b^{4} x^{4} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+1050 B a \,b^{3} x^{4} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-1200 C \,a^{2} b^{2} x^{4} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+1440 D a^{3} b \,x^{4} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+630 A a \,b^{3} x^{3} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-700 B \,a^{2} b^{2} x^{3} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+800 C \,a^{3} b \,x^{3} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-960 D a^{4} x^{3} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-504 A \,a^{2} b^{2} x^{2} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+560 B \,a^{3} b \,x^{2} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-640 C \,a^{4} x^{2} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}+432 A \,a^{3} b x \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-480 B \,a^{4} x \sqrt {c \left (b x +a \right )}\, \sqrt {a c}-384 A \,a^{4} \sqrt {c \left (b x +a \right )}\, \sqrt {a c}\right )}{1920 c^{5} x^{5} \sqrt {c x}\, \sqrt {c \left (b x +a \right )}\, a^{5} \sqrt {a c}}\)	\(482\)

Input:

int((D*x^3+C*x^2+B*x+A)/(c*x)^(11/2)/(b*x^2+a*x)^(1/2),x,method=_RETURNVER 
BOSE)
 

Output:

1/1920*(x*(b*x+a))^(1/2)/c^5*(945*A*arctanh((c*(b*x+a))^(1/2)/(a*c)^(1/2)) 
*b^5*c*x^5-1050*B*arctanh((c*(b*x+a))^(1/2)/(a*c)^(1/2))*a*b^4*c*x^5+1200* 
C*arctanh((c*(b*x+a))^(1/2)/(a*c)^(1/2))*a^2*b^3*c*x^5-1440*D*arctanh((c*( 
b*x+a))^(1/2)/(a*c)^(1/2))*a^3*b^2*c*x^5-945*A*b^4*x^4*(c*(b*x+a))^(1/2)*( 
a*c)^(1/2)+1050*B*a*b^3*x^4*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-1200*C*a^2*b^2*x 
^4*(c*(b*x+a))^(1/2)*(a*c)^(1/2)+1440*D*a^3*b*x^4*(c*(b*x+a))^(1/2)*(a*c)^ 
(1/2)+630*A*a*b^3*x^3*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-700*B*a^2*b^2*x^3*(c*( 
b*x+a))^(1/2)*(a*c)^(1/2)+800*C*a^3*b*x^3*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-96 
0*D*a^4*x^3*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-504*A*a^2*b^2*x^2*(c*(b*x+a))^(1 
/2)*(a*c)^(1/2)+560*B*a^3*b*x^2*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-640*C*a^4*x^ 
2*(c*(b*x+a))^(1/2)*(a*c)^(1/2)+432*A*a^3*b*x*(c*(b*x+a))^(1/2)*(a*c)^(1/2 
)-480*B*a^4*x*(c*(b*x+a))^(1/2)*(a*c)^(1/2)-384*A*a^4*(c*(b*x+a))^(1/2)*(a 
*c)^(1/2))/x^5/(c*x)^(1/2)/(c*(b*x+a))^(1/2)/a^5/(a*c)^(1/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\left [-\frac {15 \, {\left (96 \, D a^{3} b^{2} - 80 \, C a^{2} b^{3} + 70 \, B a b^{4} - 63 \, A b^{5}\right )} \sqrt {a c} x^{6} \log \left (-\frac {b c x^{2} + 2 \, a c x + 2 \, \sqrt {b x^{2} + a x} \sqrt {a c} \sqrt {c x}}{x^{2}}\right ) + 2 \, {\left (384 \, A a^{5} - 15 \, {\left (96 \, D a^{4} b - 80 \, C a^{3} b^{2} + 70 \, B a^{2} b^{3} - 63 \, A a b^{4}\right )} x^{4} + 10 \, {\left (96 \, D a^{5} - 80 \, C a^{4} b + 70 \, B a^{3} b^{2} - 63 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (80 \, C a^{5} - 70 \, B a^{4} b + 63 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {c x}}{3840 \, a^{6} c^{6} x^{6}}, \frac {15 \, {\left (96 \, D a^{3} b^{2} - 80 \, C a^{2} b^{3} + 70 \, B a b^{4} - 63 \, A b^{5}\right )} \sqrt {-a c} x^{6} \arctan \left (\frac {\sqrt {b x^{2} + a x} \sqrt {-a c} \sqrt {c x}}{a c x}\right ) - {\left (384 \, A a^{5} - 15 \, {\left (96 \, D a^{4} b - 80 \, C a^{3} b^{2} + 70 \, B a^{2} b^{3} - 63 \, A a b^{4}\right )} x^{4} + 10 \, {\left (96 \, D a^{5} - 80 \, C a^{4} b + 70 \, B a^{3} b^{2} - 63 \, A a^{2} b^{3}\right )} x^{3} + 8 \, {\left (80 \, C a^{5} - 70 \, B a^{4} b + 63 \, A a^{3} b^{2}\right )} x^{2} + 48 \, {\left (10 \, B a^{5} - 9 \, A a^{4} b\right )} x\right )} \sqrt {b x^{2} + a x} \sqrt {c x}}{1920 \, a^{6} c^{6} x^{6}}\right ] \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(11/2)/(b*x^2+a*x)^(1/2),x, algorithm= 
"fricas")
                                                                                    
                                                                                    
 

Output:

[-1/3840*(15*(96*D*a^3*b^2 - 80*C*a^2*b^3 + 70*B*a*b^4 - 63*A*b^5)*sqrt(a* 
c)*x^6*log(-(b*c*x^2 + 2*a*c*x + 2*sqrt(b*x^2 + a*x)*sqrt(a*c)*sqrt(c*x))/ 
x^2) + 2*(384*A*a^5 - 15*(96*D*a^4*b - 80*C*a^3*b^2 + 70*B*a^2*b^3 - 63*A* 
a*b^4)*x^4 + 10*(96*D*a^5 - 80*C*a^4*b + 70*B*a^3*b^2 - 63*A*a^2*b^3)*x^3 
+ 8*(80*C*a^5 - 70*B*a^4*b + 63*A*a^3*b^2)*x^2 + 48*(10*B*a^5 - 9*A*a^4*b) 
*x)*sqrt(b*x^2 + a*x)*sqrt(c*x))/(a^6*c^6*x^6), 1/1920*(15*(96*D*a^3*b^2 - 
 80*C*a^2*b^3 + 70*B*a*b^4 - 63*A*b^5)*sqrt(-a*c)*x^6*arctan(sqrt(b*x^2 + 
a*x)*sqrt(-a*c)*sqrt(c*x)/(a*c*x)) - (384*A*a^5 - 15*(96*D*a^4*b - 80*C*a^ 
3*b^2 + 70*B*a^2*b^3 - 63*A*a*b^4)*x^4 + 10*(96*D*a^5 - 80*C*a^4*b + 70*B* 
a^3*b^2 - 63*A*a^2*b^3)*x^3 + 8*(80*C*a^5 - 70*B*a^4*b + 63*A*a^3*b^2)*x^2 
 + 48*(10*B*a^5 - 9*A*a^4*b)*x)*sqrt(b*x^2 + a*x)*sqrt(c*x))/(a^6*c^6*x^6) 
]
 

Sympy [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\text {Timed out} \] Input:

integrate((D*x**3+C*x**2+B*x+A)/(c*x)**(11/2)/(b*x**2+a*x)**(1/2),x)
 

Output:

Timed out
 

Maxima [F]

\[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\int { \frac {D x^{3} + C x^{2} + B x + A}{\sqrt {b x^{2} + a x} \left (c x\right )^{\frac {11}{2}}} \,d x } \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(11/2)/(b*x^2+a*x)^(1/2),x, algorithm= 
"maxima")
 

Output:

integrate((D*x^3 + C*x^2 + B*x + A)/(sqrt(b*x^2 + a*x)*(c*x)^(11/2)), x)
 

Giac [A] (verification not implemented)

Time = 0.47 (sec) , antiderivative size = 474, normalized size of antiderivative = 1.54 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\frac {b^{5} {\left (\frac {15 \, {\left (96 \, D a^{3} - 80 \, C a^{2} b + 70 \, B a b^{2} - 63 \, A b^{3}\right )} \arctan \left (\frac {\sqrt {b c x + a c}}{\sqrt {-a c}}\right )}{\sqrt {-a c} a^{5} b^{3}} + \frac {2400 \, \sqrt {b c x + a c} D a^{7} c^{4} - 2640 \, \sqrt {b c x + a c} C a^{6} b c^{4} + 2790 \, \sqrt {b c x + a c} B a^{5} b^{2} c^{4} - 2895 \, \sqrt {b c x + a c} A a^{4} b^{3} c^{4} - 8640 \, {\left (b c x + a c\right )}^{\frac {3}{2}} D a^{6} c^{3} + 8480 \, {\left (b c x + a c\right )}^{\frac {3}{2}} C a^{5} b c^{3} - 7900 \, {\left (b c x + a c\right )}^{\frac {3}{2}} B a^{4} b^{2} c^{3} + 7110 \, {\left (b c x + a c\right )}^{\frac {3}{2}} A a^{3} b^{3} c^{3} + 11520 \, {\left (b c x + a c\right )}^{\frac {5}{2}} D a^{5} c^{2} - 10240 \, {\left (b c x + a c\right )}^{\frac {5}{2}} C a^{4} b c^{2} + 8960 \, {\left (b c x + a c\right )}^{\frac {5}{2}} B a^{3} b^{2} c^{2} - 8064 \, {\left (b c x + a c\right )}^{\frac {5}{2}} A a^{2} b^{3} c^{2} - 6720 \, {\left (b c x + a c\right )}^{\frac {7}{2}} D a^{4} c + 5600 \, {\left (b c x + a c\right )}^{\frac {7}{2}} C a^{3} b c - 4900 \, {\left (b c x + a c\right )}^{\frac {7}{2}} B a^{2} b^{2} c + 4410 \, {\left (b c x + a c\right )}^{\frac {7}{2}} A a b^{3} c + 1440 \, {\left (b c x + a c\right )}^{\frac {9}{2}} D a^{3} - 1200 \, {\left (b c x + a c\right )}^{\frac {9}{2}} C a^{2} b + 1050 \, {\left (b c x + a c\right )}^{\frac {9}{2}} B a b^{2} - 945 \, {\left (b c x + a c\right )}^{\frac {9}{2}} A b^{3}}{a^{5} b^{8} c^{5} x^{5}}\right )}}{1920 \, c^{4} {\left | c \right |}} \] Input:

integrate((D*x^3+C*x^2+B*x+A)/(c*x)^(11/2)/(b*x^2+a*x)^(1/2),x, algorithm= 
"giac")
 

Output:

1/1920*b^5*(15*(96*D*a^3 - 80*C*a^2*b + 70*B*a*b^2 - 63*A*b^3)*arctan(sqrt 
(b*c*x + a*c)/sqrt(-a*c))/(sqrt(-a*c)*a^5*b^3) + (2400*sqrt(b*c*x + a*c)*D 
*a^7*c^4 - 2640*sqrt(b*c*x + a*c)*C*a^6*b*c^4 + 2790*sqrt(b*c*x + a*c)*B*a 
^5*b^2*c^4 - 2895*sqrt(b*c*x + a*c)*A*a^4*b^3*c^4 - 8640*(b*c*x + a*c)^(3/ 
2)*D*a^6*c^3 + 8480*(b*c*x + a*c)^(3/2)*C*a^5*b*c^3 - 7900*(b*c*x + a*c)^( 
3/2)*B*a^4*b^2*c^3 + 7110*(b*c*x + a*c)^(3/2)*A*a^3*b^3*c^3 + 11520*(b*c*x 
 + a*c)^(5/2)*D*a^5*c^2 - 10240*(b*c*x + a*c)^(5/2)*C*a^4*b*c^2 + 8960*(b* 
c*x + a*c)^(5/2)*B*a^3*b^2*c^2 - 8064*(b*c*x + a*c)^(5/2)*A*a^2*b^3*c^2 - 
6720*(b*c*x + a*c)^(7/2)*D*a^4*c + 5600*(b*c*x + a*c)^(7/2)*C*a^3*b*c - 49 
00*(b*c*x + a*c)^(7/2)*B*a^2*b^2*c + 4410*(b*c*x + a*c)^(7/2)*A*a*b^3*c + 
1440*(b*c*x + a*c)^(9/2)*D*a^3 - 1200*(b*c*x + a*c)^(9/2)*C*a^2*b + 1050*( 
b*c*x + a*c)^(9/2)*B*a*b^2 - 945*(b*c*x + a*c)^(9/2)*A*b^3)/(a^5*b^8*c^5*x 
^5))/(c^4*abs(c))
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\int \frac {A+B\,x+C\,x^2+x^3\,D}{\sqrt {b\,x^2+a\,x}\,{\left (c\,x\right )}^{11/2}} \,d x \] Input:

int((A + B*x + C*x^2 + x^3*D)/((a*x + b*x^2)^(1/2)*(c*x)^(11/2)),x)
 

Output:

int((A + B*x + C*x^2 + x^3*D)/((a*x + b*x^2)^(1/2)*(c*x)^(11/2)), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 305, normalized size of antiderivative = 0.99 \[ \int \frac {A+B x+C x^2+D x^3}{(c x)^{11/2} \sqrt {a x+b x^2}} \, dx=\frac {\sqrt {c}\, \left (-768 \sqrt {b x +a}\, a^{5}-96 \sqrt {b x +a}\, a^{4} b x -1280 \sqrt {b x +a}\, a^{4} c \,x^{2}-1920 \sqrt {b x +a}\, a^{4} d \,x^{3}+112 \sqrt {b x +a}\, a^{3} b^{2} x^{2}+1600 \sqrt {b x +a}\, a^{3} b c \,x^{3}+2880 \sqrt {b x +a}\, a^{3} b d \,x^{4}-140 \sqrt {b x +a}\, a^{2} b^{3} x^{3}-2400 \sqrt {b x +a}\, a^{2} b^{2} c \,x^{4}+210 \sqrt {b x +a}\, a \,b^{4} x^{4}+1440 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a^{2} b^{2} d \,x^{5}-1200 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) a \,b^{3} c \,x^{5}+105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}-\sqrt {a}\right ) b^{5} x^{5}-1440 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a^{2} b^{2} d \,x^{5}+1200 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) a \,b^{3} c \,x^{5}-105 \sqrt {a}\, \mathrm {log}\left (\sqrt {b x +a}+\sqrt {a}\right ) b^{5} x^{5}\right )}{3840 a^{5} c^{6} x^{5}} \] Input:

int((D*x^3+C*x^2+B*x+A)/(c*x)^(11/2)/(b*x^2+a*x)^(1/2),x)
 

Output:

(sqrt(c)*( - 768*sqrt(a + b*x)*a**5 - 96*sqrt(a + b*x)*a**4*b*x - 1280*sqr 
t(a + b*x)*a**4*c*x**2 - 1920*sqrt(a + b*x)*a**4*d*x**3 + 112*sqrt(a + b*x 
)*a**3*b**2*x**2 + 1600*sqrt(a + b*x)*a**3*b*c*x**3 + 2880*sqrt(a + b*x)*a 
**3*b*d*x**4 - 140*sqrt(a + b*x)*a**2*b**3*x**3 - 2400*sqrt(a + b*x)*a**2* 
b**2*c*x**4 + 210*sqrt(a + b*x)*a*b**4*x**4 + 1440*sqrt(a)*log(sqrt(a + b* 
x) - sqrt(a))*a**2*b**2*d*x**5 - 1200*sqrt(a)*log(sqrt(a + b*x) - sqrt(a)) 
*a*b**3*c*x**5 + 105*sqrt(a)*log(sqrt(a + b*x) - sqrt(a))*b**5*x**5 - 1440 
*sqrt(a)*log(sqrt(a + b*x) + sqrt(a))*a**2*b**2*d*x**5 + 1200*sqrt(a)*log( 
sqrt(a + b*x) + sqrt(a))*a*b**3*c*x**5 - 105*sqrt(a)*log(sqrt(a + b*x) + s 
qrt(a))*b**5*x**5))/(3840*a**5*c**6*x**5)