\(\int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx\) [171]

Optimal result

Integrand size = 24, antiderivative size = 206 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=-\frac {2 b^4 (b B-A c) \left (b x+c x^2\right )^{3/2}}{3 c^6 x^{3/2}}+\frac {2 b^3 (5 b B-4 A c) \left (b x+c x^2\right )^{5/2}}{5 c^6 x^{5/2}}-\frac {4 b^2 (5 b B-3 A c) \left (b x+c x^2\right )^{7/2}}{7 c^6 x^{7/2}}+\frac {4 b (5 b B-2 A c) \left (b x+c x^2\right )^{9/2}}{9 c^6 x^{9/2}}-\frac {2 (5 b B-A c) \left (b x+c x^2\right )^{11/2}}{11 c^6 x^{11/2}}+\frac {2 B \left (b x+c x^2\right )^{13/2}}{13 c^6 x^{13/2}} \] Output:

-2/3*b^4*(-A*c+B*b)*(c*x^2+b*x)^(3/2)/c^6/x^(3/2)+2/5*b^3*(-4*A*c+5*B*b)*( 
c*x^2+b*x)^(5/2)/c^6/x^(5/2)-4/7*b^2*(-3*A*c+5*B*b)*(c*x^2+b*x)^(7/2)/c^6/ 
x^(7/2)+4/9*b*(-2*A*c+5*B*b)*(c*x^2+b*x)^(9/2)/c^6/x^(9/2)-2/11*(-A*c+5*B* 
b)*(c*x^2+b*x)^(11/2)/c^6/x^(11/2)+2/13*B*(c*x^2+b*x)^(13/2)/c^6/x^(13/2)
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 113, normalized size of antiderivative = 0.55 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\frac {2 (x (b+c x))^{3/2} \left (-1280 b^5 B+315 c^5 x^4 (13 A+11 B x)+128 b^4 c (13 A+15 B x)-96 b^3 c^2 x (26 A+25 B x)+80 b^2 c^3 x^2 (39 A+35 B x)-70 b c^4 x^3 (52 A+45 B x)\right )}{45045 c^6 x^{3/2}} \] Input:

Integrate[x^(7/2)*(A + B*x)*Sqrt[b*x + c*x^2],x]
 

Output:

(2*(x*(b + c*x))^(3/2)*(-1280*b^5*B + 315*c^5*x^4*(13*A + 11*B*x) + 128*b^ 
4*c*(13*A + 15*B*x) - 96*b^3*c^2*x*(26*A + 25*B*x) + 80*b^2*c^3*x^2*(39*A 
+ 35*B*x) - 70*b*c^4*x^3*(52*A + 45*B*x)))/(45045*c^6*x^(3/2))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 197, normalized size of antiderivative = 0.96, number of steps used = 6, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {1221, 1128, 1128, 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.

\(\displaystyle \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx\)

\(\Big \downarrow \) 1221

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {(10 b B-13 A c) \int x^{7/2} \sqrt {c x^2+b x}dx}{13 c}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {(10 b B-13 A c) \left (\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {8 b \int x^{5/2} \sqrt {c x^2+b x}dx}{11 c}\right )}{13 c}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {(10 b B-13 A c) \left (\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {8 b \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {2 b \int x^{3/2} \sqrt {c x^2+b x}dx}{3 c}\right )}{11 c}\right )}{13 c}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {(10 b B-13 A c) \left (\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {8 b \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {2 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {4 b \int \sqrt {x} \sqrt {c x^2+b x}dx}{7 c}\right )}{3 c}\right )}{11 c}\right )}{13 c}\)

\(\Big \downarrow \) 1128

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {(10 b B-13 A c) \left (\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {8 b \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {2 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {2 b \int \frac {\sqrt {c x^2+b x}}{\sqrt {x}}dx}{5 c}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right )}{13 c}\)

\(\Big \downarrow \) 1122

\(\displaystyle \frac {2 B x^{7/2} \left (b x+c x^2\right )^{3/2}}{13 c}-\frac {\left (\frac {2 x^{5/2} \left (b x+c x^2\right )^{3/2}}{11 c}-\frac {8 b \left (\frac {2 x^{3/2} \left (b x+c x^2\right )^{3/2}}{9 c}-\frac {2 b \left (\frac {2 \sqrt {x} \left (b x+c x^2\right )^{3/2}}{7 c}-\frac {4 b \left (\frac {2 \left (b x+c x^2\right )^{3/2}}{5 c \sqrt {x}}-\frac {4 b \left (b x+c x^2\right )^{3/2}}{15 c^2 x^{3/2}}\right )}{7 c}\right )}{3 c}\right )}{11 c}\right ) (10 b B-13 A c)}{13 c}\)

Input:

Int[x^(7/2)*(A + B*x)*Sqrt[b*x + c*x^2],x]
 

Output:

(2*B*x^(7/2)*(b*x + c*x^2)^(3/2))/(13*c) - ((10*b*B - 13*A*c)*((2*x^(5/2)* 
(b*x + c*x^2)^(3/2))/(11*c) - (8*b*((2*x^(3/2)*(b*x + c*x^2)^(3/2))/(9*c) 
- (2*b*((2*Sqrt[x]*(b*x + c*x^2)^(3/2))/(7*c) - (4*b*((-4*b*(b*x + c*x^2)^ 
(3/2))/(15*c^2*x^(3/2)) + (2*(b*x + c*x^2)^(3/2))/(5*c*Sqrt[x])))/(7*c)))/ 
(3*c)))/(11*c)))/(13*c)
 

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 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]
 

Maple [A] (verified)

Time = 0.91 (sec) , antiderivative size = 129, normalized size of antiderivative = 0.63

method	result	size
default	\(\frac {2 \left (c x +b \right ) \left (3465 B \,c^{5} x^{5}+4095 A \,c^{5} x^{4}-3150 B b \,c^{4} x^{4}-3640 A b \,c^{4} x^{3}+2800 B \,b^{2} c^{3} x^{3}+3120 A \,b^{2} c^{3} x^{2}-2400 B \,b^{3} c^{2} x^{2}-2496 A \,b^{3} c^{2} x +1920 B \,b^{4} c x +1664 A \,b^{4} c -1280 b^{5} B \right ) \sqrt {x \left (c x +b \right )}}{45045 c^{6} \sqrt {x}}\)	\(129\)
gosper	\(\frac {2 \left (c x +b \right ) \left (3465 B \,c^{5} x^{5}+4095 A \,c^{5} x^{4}-3150 B b \,c^{4} x^{4}-3640 A b \,c^{4} x^{3}+2800 B \,b^{2} c^{3} x^{3}+3120 A \,b^{2} c^{3} x^{2}-2400 B \,b^{3} c^{2} x^{2}-2496 A \,b^{3} c^{2} x +1920 B \,b^{4} c x +1664 A \,b^{4} c -1280 b^{5} B \right ) \sqrt {c \,x^{2}+b x}}{45045 c^{6} \sqrt {x}}\)	\(131\)
orering	\(\frac {2 \left (c x +b \right ) \left (3465 B \,c^{5} x^{5}+4095 A \,c^{5} x^{4}-3150 B b \,c^{4} x^{4}-3640 A b \,c^{4} x^{3}+2800 B \,b^{2} c^{3} x^{3}+3120 A \,b^{2} c^{3} x^{2}-2400 B \,b^{3} c^{2} x^{2}-2496 A \,b^{3} c^{2} x +1920 B \,b^{4} c x +1664 A \,b^{4} c -1280 b^{5} B \right ) \sqrt {c \,x^{2}+b x}}{45045 c^{6} \sqrt {x}}\)	\(131\)
risch	\(\frac {2 \left (c x +b \right ) \sqrt {x}\, \left (3465 B \,c^{6} x^{6}+4095 A \,c^{6} x^{5}+315 B b \,c^{5} x^{5}+455 A b \,c^{5} x^{4}-350 B \,b^{2} c^{4} x^{4}-520 A \,b^{2} c^{4} x^{3}+400 B \,b^{3} c^{3} x^{3}+624 A \,b^{3} c^{3} x^{2}-480 B \,b^{4} c^{2} x^{2}-832 A \,b^{4} c^{2} x +640 B \,b^{5} c x +1664 A \,b^{5} c -1280 B \,b^{6}\right )}{45045 \sqrt {x \left (c x +b \right )}\, c^{6}}\)	\(153\)

Input:

int(x^(7/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

2/45045*(c*x+b)*(3465*B*c^5*x^5+4095*A*c^5*x^4-3150*B*b*c^4*x^4-3640*A*b*c 
^4*x^3+2800*B*b^2*c^3*x^3+3120*A*b^2*c^3*x^2-2400*B*b^3*c^2*x^2-2496*A*b^3 
*c^2*x+1920*B*b^4*c*x+1664*A*b^4*c-1280*B*b^5)*(x*(c*x+b))^(1/2)/c^6/x^(1/ 
2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 150, normalized size of antiderivative = 0.73 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (3465 \, B c^{6} x^{6} - 1280 \, B b^{6} + 1664 \, A b^{5} c + 315 \, {\left (B b c^{5} + 13 \, A c^{6}\right )} x^{5} - 35 \, {\left (10 \, B b^{2} c^{4} - 13 \, A b c^{5}\right )} x^{4} + 40 \, {\left (10 \, B b^{3} c^{3} - 13 \, A b^{2} c^{4}\right )} x^{3} - 48 \, {\left (10 \, B b^{4} c^{2} - 13 \, A b^{3} c^{3}\right )} x^{2} + 64 \, {\left (10 \, B b^{5} c - 13 \, A b^{4} c^{2}\right )} x\right )} \sqrt {c x^{2} + b x}}{45045 \, c^{6} \sqrt {x}} \] Input:

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x, algorithm="fricas")
 

Output:

2/45045*(3465*B*c^6*x^6 - 1280*B*b^6 + 1664*A*b^5*c + 315*(B*b*c^5 + 13*A* 
c^6)*x^5 - 35*(10*B*b^2*c^4 - 13*A*b*c^5)*x^4 + 40*(10*B*b^3*c^3 - 13*A*b^ 
2*c^4)*x^3 - 48*(10*B*b^4*c^2 - 13*A*b^3*c^3)*x^2 + 64*(10*B*b^5*c - 13*A* 
b^4*c^2)*x)*sqrt(c*x^2 + b*x)/(c^6*sqrt(x))
 

Sympy [F]

\[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\int x^{\frac {7}{2}} \sqrt {x \left (b + c x\right )} \left (A + B x\right )\, dx \] Input:

integrate(x**(7/2)*(B*x+A)*(c*x**2+b*x)**(1/2),x)
 

Output:

Integral(x**(7/2)*sqrt(x*(b + c*x))*(A + B*x), x)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.69 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (315 \, c^{5} x^{5} + 35 \, b c^{4} x^{4} - 40 \, b^{2} c^{3} x^{3} + 48 \, b^{3} c^{2} x^{2} - 64 \, b^{4} c x + 128 \, b^{5}\right )} \sqrt {c x + b} A}{3465 \, c^{5}} + \frac {2 \, {\left (693 \, c^{6} x^{6} + 63 \, b c^{5} x^{5} - 70 \, b^{2} c^{4} x^{4} + 80 \, b^{3} c^{3} x^{3} - 96 \, b^{4} c^{2} x^{2} + 128 \, b^{5} c x - 256 \, b^{6}\right )} \sqrt {c x + b} B}{9009 \, c^{6}} \] Input:

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x, algorithm="maxima")
 

Output:

2/3465*(315*c^5*x^5 + 35*b*c^4*x^4 - 40*b^2*c^3*x^3 + 48*b^3*c^2*x^2 - 64* 
b^4*c*x + 128*b^5)*sqrt(c*x + b)*A/c^5 + 2/9009*(693*c^6*x^6 + 63*b*c^5*x^ 
5 - 70*b^2*c^4*x^4 + 80*b^3*c^3*x^3 - 96*b^4*c^2*x^2 + 128*b^5*c*x - 256*b 
^6)*sqrt(c*x + b)*B/c^6
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\frac {2 \, {\left (315 \, {\left (c x + b\right )}^{\frac {11}{2}} - 1540 \, {\left (c x + b\right )}^{\frac {9}{2}} b + 2970 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{2} - 2772 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{3} + 1155 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{4}\right )} A}{3465 \, c^{5}} + \frac {2 \, {\left (693 \, {\left (c x + b\right )}^{\frac {13}{2}} - 4095 \, {\left (c x + b\right )}^{\frac {11}{2}} b + 10010 \, {\left (c x + b\right )}^{\frac {9}{2}} b^{2} - 12870 \, {\left (c x + b\right )}^{\frac {7}{2}} b^{3} + 9009 \, {\left (c x + b\right )}^{\frac {5}{2}} b^{4} - 3003 \, {\left (c x + b\right )}^{\frac {3}{2}} b^{5}\right )} B}{9009 \, c^{6}} \] Input:

integrate(x^(7/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x, algorithm="giac")
 

Output:

2/3465*(315*(c*x + b)^(11/2) - 1540*(c*x + b)^(9/2)*b + 2970*(c*x + b)^(7/ 
2)*b^2 - 2772*(c*x + b)^(5/2)*b^3 + 1155*(c*x + b)^(3/2)*b^4)*A/c^5 + 2/90 
09*(693*(c*x + b)^(13/2) - 4095*(c*x + b)^(11/2)*b + 10010*(c*x + b)^(9/2) 
*b^2 - 12870*(c*x + b)^(7/2)*b^3 + 9009*(c*x + b)^(5/2)*b^4 - 3003*(c*x + 
b)^(3/2)*b^5)*B/c^6
 

Mupad [F(-1)]

Timed out. \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\int x^{7/2}\,\sqrt {c\,x^2+b\,x}\,\left (A+B\,x\right ) \,d x \] Input:

int(x^(7/2)*(b*x + c*x^2)^(1/2)*(A + B*x),x)
 

Output:

int(x^(7/2)*(b*x + c*x^2)^(1/2)*(A + B*x), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 137, normalized size of antiderivative = 0.67 \[ \int x^{7/2} (A+B x) \sqrt {b x+c x^2} \, dx=\frac {2 \sqrt {c x +b}\, \left (3465 b \,c^{6} x^{6}+4095 a \,c^{6} x^{5}+315 b^{2} c^{5} x^{5}+455 a b \,c^{5} x^{4}-350 b^{3} c^{4} x^{4}-520 a \,b^{2} c^{4} x^{3}+400 b^{4} c^{3} x^{3}+624 a \,b^{3} c^{3} x^{2}-480 b^{5} c^{2} x^{2}-832 a \,b^{4} c^{2} x +640 b^{6} c x +1664 a \,b^{5} c -1280 b^{7}\right )}{45045 c^{6}} \] Input:

int(x^(7/2)*(B*x+A)*(c*x^2+b*x)^(1/2),x)
 

Output:

(2*sqrt(b + c*x)*(1664*a*b**5*c - 832*a*b**4*c**2*x + 624*a*b**3*c**3*x**2 
 - 520*a*b**2*c**4*x**3 + 455*a*b*c**5*x**4 + 4095*a*c**6*x**5 - 1280*b**7 
 + 640*b**6*c*x - 480*b**5*c**2*x**2 + 400*b**4*c**3*x**3 - 350*b**3*c**4* 
x**4 + 315*b**2*c**5*x**5 + 3465*b*c**6*x**6))/(45045*c**6)