\(\int \frac {(c x)^{5/2} (A+B x+C x^2+D x^3)}{\sqrt {a x+b x^2}} \, dx\) [2]

Optimal result

Integrand size = 36, antiderivative size = 289 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\frac {2 a^2 c^3 \left (A b^3-a \left (b^2 B-a b C+a^2 D\right )\right ) \sqrt {a x+b x^2}}{b^6 \sqrt {c x}}-\frac {2 a c^4 \left (2 A b^3-a \left (3 b^2 B-4 a b C+5 a^2 D\right )\right ) \left (a x+b x^2\right )^{3/2}}{3 b^6 (c x)^{3/2}}+\frac {2 c^5 \left (A b^3-a \left (3 b^2 B-6 a b C+10 a^2 D\right )\right ) \left (a x+b x^2\right )^{5/2}}{5 b^6 (c x)^{5/2}}+\frac {2 c^6 \left (b^2 B-4 a b C+10 a^2 D\right ) \left (a x+b x^2\right )^{7/2}}{7 b^6 (c x)^{7/2}}+\frac {2 c^7 (b C-5 a D) \left (a x+b x^2\right )^{9/2}}{9 b^6 (c x)^{9/2}}+\frac {2 c^8 D \left (a x+b x^2\right )^{11/2}}{11 b^6 (c x)^{11/2}} \] Output:

2*a^2*c^3*(A*b^3-a*(B*b^2-C*a*b+D*a^2))*(b*x^2+a*x)^(1/2)/b^6/(c*x)^(1/2)- 
2/3*a*c^4*(2*A*b^3-a*(3*B*b^2-4*C*a*b+5*D*a^2))*(b*x^2+a*x)^(3/2)/b^6/(c*x 
)^(3/2)+2/5*c^5*(A*b^3-a*(3*B*b^2-6*C*a*b+10*D*a^2))*(b*x^2+a*x)^(5/2)/b^6 
/(c*x)^(5/2)+2/7*c^6*(B*b^2-4*C*a*b+10*D*a^2)*(b*x^2+a*x)^(7/2)/b^6/(c*x)^ 
(7/2)+2/9*c^7*(C*b-5*D*a)*(b*x^2+a*x)^(9/2)/b^6/(c*x)^(9/2)+2/11*c^8*D*(b* 
x^2+a*x)^(11/2)/b^6/(c*x)^(11/2)
 

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.54 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\frac {2 c^3 \sqrt {x (a+b x)} \left (-1280 a^5 D+128 a^4 b (11 C+5 D x)-16 a^3 b^2 \left (99 B+44 C x+30 D x^2\right )+8 a^2 b^3 \left (231 A+x \left (99 B+66 C x+50 D x^2\right )\right )+b^5 x^2 (693 A+5 x (99 B+7 x (11 C+9 D x)))-2 a b^4 x (462 A+x (297 B+5 x (44 C+35 D x)))\right )}{3465 b^6 \sqrt {c x}} \] Input:

Integrate[((c*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a*x + b*x^2],x]
 

Output:

(2*c^3*Sqrt[x*(a + b*x)]*(-1280*a^5*D + 128*a^4*b*(11*C + 5*D*x) - 16*a^3* 
b^2*(99*B + 44*C*x + 30*D*x^2) + 8*a^2*b^3*(231*A + x*(99*B + 66*C*x + 50* 
D*x^2)) + b^5*x^2*(693*A + 5*x*(99*B + 7*x*(11*C + 9*D*x))) - 2*a*b^4*x*(4 
62*A + x*(297*B + 5*x*(44*C + 35*D*x)))))/(3465*b^6*Sqrt[c*x])
 

Rubi [A] (verified)

Time = 1.08 (sec) , antiderivative size = 270, normalized size of antiderivative = 0.93, number of steps used = 8, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.222, Rules used = {2169, 27, 2169, 27, 1221, 1128, 1128, 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.

\(\displaystyle \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {2 \int \frac {(c x)^{5/2} \left ((11 b C-10 a D) x^2 c^3+11 A b c^3+11 b B x c^3\right )}{2 \sqrt {b x^2+a x}}dx}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\int \frac {(c x)^{5/2} \left ((11 b C-10 a D) x^2 c^3+11 A b c^3+11 b B x c^3\right )}{\sqrt {b x^2+a x}}dx}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 2169

\(\displaystyle \frac {\frac {2 \int \frac {c^5 (c x)^{5/2} \left (99 A b^2+\left (80 D a^2-88 b C a+99 b^2 B\right ) x\right )}{2 \sqrt {b x^2+a x}}dx}{9 b c^2}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {c^3 \int \frac {(c x)^{5/2} \left (99 A b^2+\left (80 D a^2-88 b C a+99 b^2 B\right ) x\right )}{\sqrt {b x^2+a x}}dx}{9 b}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {\frac {c^3 \left (\frac {3 \left (-160 a^3 D+176 a^2 b C-198 a b^2 B+231 A b^3\right ) \int \frac {(c x)^{5/2}}{\sqrt {b x^2+a x}}dx}{7 b}+\frac {2 (c x)^{5/2} \sqrt {a x+b x^2} \left (80 a^2 D-88 a b C+99 b^2 B\right )}{7 b}\right )}{9 b}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {\frac {c^3 \left (\frac {3 \left (-160 a^3 D+176 a^2 b C-198 a b^2 B+231 A b^3\right ) \left (\frac {2 c (c x)^{3/2} \sqrt {a x+b x^2}}{5 b}-\frac {4 a c \int \frac {(c x)^{3/2}}{\sqrt {b x^2+a x}}dx}{5 b}\right )}{7 b}+\frac {2 (c x)^{5/2} \sqrt {a x+b x^2} \left (80 a^2 D-88 a b C+99 b^2 B\right )}{7 b}\right )}{9 b}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {\frac {c^3 \left (\frac {3 \left (-160 a^3 D+176 a^2 b C-198 a b^2 B+231 A b^3\right ) \left (\frac {2 c (c x)^{3/2} \sqrt {a x+b x^2}}{5 b}-\frac {4 a c \left (\frac {2 c \sqrt {c x} \sqrt {a x+b x^2}}{3 b}-\frac {2 a c \int \frac {\sqrt {c x}}{\sqrt {b x^2+a x}}dx}{3 b}\right )}{5 b}\right )}{7 b}+\frac {2 (c x)^{5/2} \sqrt {a x+b x^2} \left (80 a^2 D-88 a b C+99 b^2 B\right )}{7 b}\right )}{9 b}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {\frac {c^3 \left (\frac {2 (c x)^{5/2} \sqrt {a x+b x^2} \left (80 a^2 D-88 a b C+99 b^2 B\right )}{7 b}+\frac {3 \left (\frac {2 c (c x)^{3/2} \sqrt {a x+b x^2}}{5 b}-\frac {4 a c \left (\frac {2 c \sqrt {c x} \sqrt {a x+b x^2}}{3 b}-\frac {4 a c^2 \sqrt {a x+b x^2}}{3 b^2 \sqrt {c x}}\right )}{5 b}\right ) \left (-160 a^3 D+176 a^2 b C-198 a b^2 B+231 A b^3\right )}{7 b}\right )}{9 b}+\frac {2 c^2 (c x)^{7/2} \sqrt {a x+b x^2} (11 b C-10 a D)}{9 b}}{11 b c^3}+\frac {2 D (c x)^{9/2} \sqrt {a x+b x^2}}{11 b c^2}\)

Input:

Int[((c*x)^(5/2)*(A + B*x + C*x^2 + D*x^3))/Sqrt[a*x + b*x^2],x]
 

Output:

(2*D*(c*x)^(9/2)*Sqrt[a*x + b*x^2])/(11*b*c^2) + ((2*c^2*(11*b*C - 10*a*D) 
*(c*x)^(7/2)*Sqrt[a*x + b*x^2])/(9*b) + (c^3*((2*(99*b^2*B - 88*a*b*C + 80 
*a^2*D)*(c*x)^(5/2)*Sqrt[a*x + b*x^2])/(7*b) + (3*(231*A*b^3 - 198*a*b^2*B 
 + 176*a^2*b*C - 160*a^3*D)*((2*c*(c*x)^(3/2)*Sqrt[a*x + b*x^2])/(5*b) - ( 
4*a*c*((-4*a*c^2*Sqrt[a*x + b*x^2])/(3*b^2*Sqrt[c*x]) + (2*c*Sqrt[c*x]*Sqr 
t[a*x + b*x^2])/(3*b)))/(5*b)))/(7*b)))/(9*b))/(11*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 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 1128

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*(m + 2*p + 
 1))), x] + Simp[Simplify[m + p]*((2*c*d - b*e)/(c*(m + 2*p + 1)))   Int[(d 
 + e*x)^(m - 1)*(a + b*x + c*x^2)^p, x], x] /; FreeQ[{a, b, c, d, e, m, p}, 
 x] && EqQ[c*d^2 - b*d*e + a*e^2, 0] && IGtQ[Simplify[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]
 

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 = 0.43 (sec) , antiderivative size = 196, normalized size of antiderivative = 0.68

method	result	size
default	\(\frac {2 c^{2} \sqrt {c x}\, \sqrt {x \left (b x +a \right )}\, \left (315 D x^{5} b^{5}+385 C \,b^{5} x^{4}-350 D a \,b^{4} x^{4}+495 B \,b^{5} x^{3}-440 C a \,b^{4} x^{3}+400 D a^{2} b^{3} x^{3}+693 A \,b^{5} x^{2}-594 B a \,b^{4} x^{2}+528 C \,a^{2} b^{3} x^{2}-480 D a^{3} b^{2} x^{2}-924 A a \,b^{4} x +792 B \,a^{2} b^{3} x -704 C \,a^{3} b^{2} x +640 D a^{4} b x +1848 A \,a^{2} b^{3}-1584 B \,a^{3} b^{2}+1408 C \,a^{4} b -1280 D a^{5}\right )}{3465 x \,b^{6}}\)	\(196\)
gosper	\(\frac {2 \left (b x +a \right ) \left (315 D x^{5} b^{5}+385 C \,b^{5} x^{4}-350 D a \,b^{4} x^{4}+495 B \,b^{5} x^{3}-440 C a \,b^{4} x^{3}+400 D a^{2} b^{3} x^{3}+693 A \,b^{5} x^{2}-594 B a \,b^{4} x^{2}+528 C \,a^{2} b^{3} x^{2}-480 D a^{3} b^{2} x^{2}-924 A a \,b^{4} x +792 B \,a^{2} b^{3} x -704 C \,a^{3} b^{2} x +640 D a^{4} b x +1848 A \,a^{2} b^{3}-1584 B \,a^{3} b^{2}+1408 C \,a^{4} b -1280 D a^{5}\right ) \left (c x \right )^{\frac {5}{2}}}{3465 b^{6} x^{2} \sqrt {b \,x^{2}+a x}}\)	\(200\)
orering	\(\frac {2 \left (b x +a \right ) \left (315 D x^{5} b^{5}+385 C \,b^{5} x^{4}-350 D a \,b^{4} x^{4}+495 B \,b^{5} x^{3}-440 C a \,b^{4} x^{3}+400 D a^{2} b^{3} x^{3}+693 A \,b^{5} x^{2}-594 B a \,b^{4} x^{2}+528 C \,a^{2} b^{3} x^{2}-480 D a^{3} b^{2} x^{2}-924 A a \,b^{4} x +792 B \,a^{2} b^{3} x -704 C \,a^{3} b^{2} x +640 D a^{4} b x +1848 A \,a^{2} b^{3}-1584 B \,a^{3} b^{2}+1408 C \,a^{4} b -1280 D a^{5}\right ) \left (c x \right )^{\frac {5}{2}}}{3465 b^{6} x^{2} \sqrt {b \,x^{2}+a x}}\)	\(200\)

Input:

int((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a*x)^(1/2),x,method=_RETURNVERB 
OSE)
 

Output:

2/3465*c^2/x*(c*x)^(1/2)*(x*(b*x+a))^(1/2)*(315*D*b^5*x^5+385*C*b^5*x^4-35 
0*D*a*b^4*x^4+495*B*b^5*x^3-440*C*a*b^4*x^3+400*D*a^2*b^3*x^3+693*A*b^5*x^ 
2-594*B*a*b^4*x^2+528*C*a^2*b^3*x^2-480*D*a^3*b^2*x^2-924*A*a*b^4*x+792*B* 
a^2*b^3*x-704*C*a^3*b^2*x+640*D*a^4*b*x+1848*A*a^2*b^3-1584*B*a^3*b^2+1408 
*C*a^4*b-1280*D*a^5)/b^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 206, normalized size of antiderivative = 0.71 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\frac {2 \, {\left (315 \, D b^{5} c^{2} x^{5} - 35 \, {\left (10 \, D a b^{4} - 11 \, C b^{5}\right )} c^{2} x^{4} + 5 \, {\left (80 \, D a^{2} b^{3} - 88 \, C a b^{4} + 99 \, B b^{5}\right )} c^{2} x^{3} - 3 \, {\left (160 \, D a^{3} b^{2} - 176 \, C a^{2} b^{3} + 198 \, B a b^{4} - 231 \, A b^{5}\right )} c^{2} x^{2} + 4 \, {\left (160 \, D a^{4} b - 176 \, C a^{3} b^{2} + 198 \, B a^{2} b^{3} - 231 \, A a b^{4}\right )} c^{2} x - 8 \, {\left (160 \, D a^{5} - 176 \, C a^{4} b + 198 \, B a^{3} b^{2} - 231 \, A a^{2} b^{3}\right )} c^{2}\right )} \sqrt {b x^{2} + a x} \sqrt {c x}}{3465 \, b^{6} x} \] Input:

integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a*x)^(1/2),x, algorithm=" 
fricas")
 

Output:

2/3465*(315*D*b^5*c^2*x^5 - 35*(10*D*a*b^4 - 11*C*b^5)*c^2*x^4 + 5*(80*D*a 
^2*b^3 - 88*C*a*b^4 + 99*B*b^5)*c^2*x^3 - 3*(160*D*a^3*b^2 - 176*C*a^2*b^3 
 + 198*B*a*b^4 - 231*A*b^5)*c^2*x^2 + 4*(160*D*a^4*b - 176*C*a^3*b^2 + 198 
*B*a^2*b^3 - 231*A*a*b^4)*c^2*x - 8*(160*D*a^5 - 176*C*a^4*b + 198*B*a^3*b 
^2 - 231*A*a^2*b^3)*c^2)*sqrt(b*x^2 + a*x)*sqrt(c*x)/(b^6*x)
                                                                                    
                                                                                    
 

Sympy [F]

\[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\int \frac {\left (c x\right )^{\frac {5}{2}} \left (A + B x + C x^{2} + D x^{3}\right )}{\sqrt {x \left (a + b x\right )}}\, dx \] Input:

integrate((c*x)**(5/2)*(D*x**3+C*x**2+B*x+A)/(b*x**2+a*x)**(1/2),x)
 

Output:

Integral((c*x)**(5/2)*(A + B*x + C*x**2 + D*x**3)/sqrt(x*(a + b*x)), x)
 

Maxima [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.06 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\frac {2 \, {\left (3 \, b^{3} c^{\frac {5}{2}} x^{3} - a b^{2} c^{\frac {5}{2}} x^{2} + 4 \, a^{2} b c^{\frac {5}{2}} x + 8 \, a^{3} c^{\frac {5}{2}}\right )} A}{15 \, \sqrt {b x + a} b^{3}} + \frac {2 \, {\left (5 \, b^{4} c^{\frac {5}{2}} x^{4} - a b^{3} c^{\frac {5}{2}} x^{3} + 2 \, a^{2} b^{2} c^{\frac {5}{2}} x^{2} - 8 \, a^{3} b c^{\frac {5}{2}} x - 16 \, a^{4} c^{\frac {5}{2}}\right )} B}{35 \, \sqrt {b x + a} b^{4}} + \frac {2 \, {\left (35 \, b^{5} c^{\frac {5}{2}} x^{5} - 5 \, a b^{4} c^{\frac {5}{2}} x^{4} + 8 \, a^{2} b^{3} c^{\frac {5}{2}} x^{3} - 16 \, a^{3} b^{2} c^{\frac {5}{2}} x^{2} + 64 \, a^{4} b c^{\frac {5}{2}} x + 128 \, a^{5} c^{\frac {5}{2}}\right )} C}{315 \, \sqrt {b x + a} b^{5}} + \frac {2 \, {\left (63 \, b^{6} c^{\frac {5}{2}} x^{6} - 7 \, a b^{5} c^{\frac {5}{2}} x^{5} + 10 \, a^{2} b^{4} c^{\frac {5}{2}} x^{4} - 16 \, a^{3} b^{3} c^{\frac {5}{2}} x^{3} + 32 \, a^{4} b^{2} c^{\frac {5}{2}} x^{2} - 128 \, a^{5} b c^{\frac {5}{2}} x - 256 \, a^{6} c^{\frac {5}{2}}\right )} D}{693 \, \sqrt {b x + a} b^{6}} \] Input:

integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a*x)^(1/2),x, algorithm=" 
maxima")
 

Output:

2/15*(3*b^3*c^(5/2)*x^3 - a*b^2*c^(5/2)*x^2 + 4*a^2*b*c^(5/2)*x + 8*a^3*c^ 
(5/2))*A/(sqrt(b*x + a)*b^3) + 2/35*(5*b^4*c^(5/2)*x^4 - a*b^3*c^(5/2)*x^3 
 + 2*a^2*b^2*c^(5/2)*x^2 - 8*a^3*b*c^(5/2)*x - 16*a^4*c^(5/2))*B/(sqrt(b*x 
 + a)*b^4) + 2/315*(35*b^5*c^(5/2)*x^5 - 5*a*b^4*c^(5/2)*x^4 + 8*a^2*b^3*c 
^(5/2)*x^3 - 16*a^3*b^2*c^(5/2)*x^2 + 64*a^4*b*c^(5/2)*x + 128*a^5*c^(5/2) 
)*C/(sqrt(b*x + a)*b^5) + 2/693*(63*b^6*c^(5/2)*x^6 - 7*a*b^5*c^(5/2)*x^5 
+ 10*a^2*b^4*c^(5/2)*x^4 - 16*a^3*b^3*c^(5/2)*x^3 + 32*a^4*b^2*c^(5/2)*x^2 
 - 128*a^5*b*c^(5/2)*x - 256*a^6*c^(5/2))*D/(sqrt(b*x + a)*b^6)
 

Giac [A] (verification not implemented)

Time = 0.39 (sec) , antiderivative size = 387, normalized size of antiderivative = 1.34 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=-\frac {2 \, c^{2} {\left (\frac {3465 \, {\left (D a^{5} c - C a^{4} b c + B a^{3} b^{2} c - A a^{2} b^{3} c\right )} \sqrt {b c x + a c}}{b^{6}} - \frac {8 \, {\left (160 \, \sqrt {a c} D a^{5} c - 176 \, \sqrt {a c} C a^{4} b c + 198 \, \sqrt {a c} B a^{3} b^{2} c - 231 \, \sqrt {a c} A a^{2} b^{3} c\right )}}{b^{6}} - \frac {5775 \, {\left (b c x + a c\right )}^{\frac {3}{2}} D a^{4} c^{4} - 4620 \, {\left (b c x + a c\right )}^{\frac {3}{2}} C a^{3} b c^{4} + 3465 \, {\left (b c x + a c\right )}^{\frac {3}{2}} B a^{2} b^{2} c^{4} - 2310 \, {\left (b c x + a c\right )}^{\frac {3}{2}} A a b^{3} c^{4} - 6930 \, {\left (b c x + a c\right )}^{\frac {5}{2}} D a^{3} c^{3} + 4158 \, {\left (b c x + a c\right )}^{\frac {5}{2}} C a^{2} b c^{3} - 2079 \, {\left (b c x + a c\right )}^{\frac {5}{2}} B a b^{2} c^{3} + 693 \, {\left (b c x + a c\right )}^{\frac {5}{2}} A b^{3} c^{3} + 4950 \, {\left (b c x + a c\right )}^{\frac {7}{2}} D a^{2} c^{2} - 1980 \, {\left (b c x + a c\right )}^{\frac {7}{2}} C a b c^{2} + 495 \, {\left (b c x + a c\right )}^{\frac {7}{2}} B b^{2} c^{2} - 1925 \, {\left (b c x + a c\right )}^{\frac {9}{2}} D a c + 385 \, {\left (b c x + a c\right )}^{\frac {9}{2}} C b c + 315 \, {\left (b c x + a c\right )}^{\frac {11}{2}} D}{b^{6} c^{4}}\right )}}{3465 \, {\left | c \right |}} \] Input:

integrate((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a*x)^(1/2),x, algorithm=" 
giac")
 

Output:

-2/3465*c^2*(3465*(D*a^5*c - C*a^4*b*c + B*a^3*b^2*c - A*a^2*b^3*c)*sqrt(b 
*c*x + a*c)/b^6 - 8*(160*sqrt(a*c)*D*a^5*c - 176*sqrt(a*c)*C*a^4*b*c + 198 
*sqrt(a*c)*B*a^3*b^2*c - 231*sqrt(a*c)*A*a^2*b^3*c)/b^6 - (5775*(b*c*x + a 
*c)^(3/2)*D*a^4*c^4 - 4620*(b*c*x + a*c)^(3/2)*C*a^3*b*c^4 + 3465*(b*c*x + 
 a*c)^(3/2)*B*a^2*b^2*c^4 - 2310*(b*c*x + a*c)^(3/2)*A*a*b^3*c^4 - 6930*(b 
*c*x + a*c)^(5/2)*D*a^3*c^3 + 4158*(b*c*x + a*c)^(5/2)*C*a^2*b*c^3 - 2079* 
(b*c*x + a*c)^(5/2)*B*a*b^2*c^3 + 693*(b*c*x + a*c)^(5/2)*A*b^3*c^3 + 4950 
*(b*c*x + a*c)^(7/2)*D*a^2*c^2 - 1980*(b*c*x + a*c)^(7/2)*C*a*b*c^2 + 495* 
(b*c*x + a*c)^(7/2)*B*b^2*c^2 - 1925*(b*c*x + a*c)^(9/2)*D*a*c + 385*(b*c* 
x + a*c)^(9/2)*C*b*c + 315*(b*c*x + a*c)^(11/2)*D)/(b^6*c^4))/abs(c)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\int \frac {{\left (c\,x\right )}^{5/2}\,\left (A+B\,x+C\,x^2+x^3\,D\right )}{\sqrt {b\,x^2+a\,x}} \,d x \] Input:

int(((c*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a*x + b*x^2)^(1/2),x)
 

Output:

int(((c*x)^(5/2)*(A + B*x + C*x^2 + x^3*D))/(a*x + b*x^2)^(1/2), x)
 

Reduce [B] (verification not implemented)

Time = 0.25 (sec) , antiderivative size = 156, normalized size of antiderivative = 0.54 \[ \int \frac {(c x)^{5/2} \left (A+B x+C x^2+D x^3\right )}{\sqrt {a x+b x^2}} \, dx=\frac {2 \sqrt {c}\, \sqrt {b x +a}\, c^{2} \left (315 b^{5} d \,x^{5}-350 a \,b^{4} d \,x^{4}+385 b^{5} c \,x^{4}+400 a^{2} b^{3} d \,x^{3}-440 a \,b^{4} c \,x^{3}+495 b^{6} x^{3}-480 a^{3} b^{2} d \,x^{2}+528 a^{2} b^{3} c \,x^{2}+99 a \,b^{5} x^{2}+640 a^{4} b d x -704 a^{3} b^{2} c x -132 a^{2} b^{4} x -1280 a^{5} d +1408 a^{4} b c +264 a^{3} b^{3}\right )}{3465 b^{6}} \] Input:

int((c*x)^(5/2)*(D*x^3+C*x^2+B*x+A)/(b*x^2+a*x)^(1/2),x)
 

Output:

(2*sqrt(c)*sqrt(a + b*x)*c**2*( - 1280*a**5*d + 1408*a**4*b*c + 640*a**4*b 
*d*x + 264*a**3*b**3 - 704*a**3*b**2*c*x - 480*a**3*b**2*d*x**2 - 132*a**2 
*b**4*x + 528*a**2*b**3*c*x**2 + 400*a**2*b**3*d*x**3 + 99*a*b**5*x**2 - 4 
40*a*b**4*c*x**3 - 350*a*b**4*d*x**4 + 495*b**6*x**3 + 385*b**5*c*x**4 + 3 
15*b**5*d*x**5))/(3465*b**6)