\(\int \frac {x (a+b x^2)^2}{\sqrt {c+d x}} \, dx\) [540]

Optimal result

Integrand size = 20, antiderivative size = 167 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=-\frac {2 c \left (b c^2+a d^2\right )^2 \sqrt {c+d x}}{d^6}+\frac {2 \left (b c^2+a d^2\right ) \left (5 b c^2+a d^2\right ) (c+d x)^{3/2}}{3 d^6}-\frac {4 b c \left (5 b c^2+3 a d^2\right ) (c+d x)^{5/2}}{5 d^6}+\frac {4 b \left (5 b c^2+a d^2\right ) (c+d x)^{7/2}}{7 d^6}-\frac {10 b^2 c (c+d x)^{9/2}}{9 d^6}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^6} \] Output:

-2*c*(a*d^2+b*c^2)^2*(d*x+c)^(1/2)/d^6+2/3*(a*d^2+b*c^2)*(a*d^2+5*b*c^2)*( 
d*x+c)^(3/2)/d^6-4/5*b*c*(3*a*d^2+5*b*c^2)*(d*x+c)^(5/2)/d^6+4/7*b*(a*d^2+ 
5*b*c^2)*(d*x+c)^(7/2)/d^6-10/9*b^2*c*(d*x+c)^(9/2)/d^6+2/11*b^2*(d*x+c)^( 
11/2)/d^6
 

Mathematica [A] (verified)

Time = 0.09 (sec) , antiderivative size = 126, normalized size of antiderivative = 0.75 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {c+d x} \left (1155 a^2 d^4 (-2 c+d x)+198 a b d^2 \left (-16 c^3+8 c^2 d x-6 c d^2 x^2+5 d^3 x^3\right )-5 b^2 \left (256 c^5-128 c^4 d x+96 c^3 d^2 x^2-80 c^2 d^3 x^3+70 c d^4 x^4-63 d^5 x^5\right )\right )}{3465 d^6} \] Input:

Integrate[(x*(a + b*x^2)^2)/Sqrt[c + d*x],x]
 

Output:

(2*Sqrt[c + d*x]*(1155*a^2*d^4*(-2*c + d*x) + 198*a*b*d^2*(-16*c^3 + 8*c^2 
*d*x - 6*c*d^2*x^2 + 5*d^3*x^3) - 5*b^2*(256*c^5 - 128*c^4*d*x + 96*c^3*d^ 
2*x^2 - 80*c^2*d^3*x^3 + 70*c*d^4*x^4 - 63*d^5*x^5)))/(3465*d^6)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 167, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.100, Rules used = {522, 2009}

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.

\(\displaystyle \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx\)

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {2 b (c+d x)^{5/2} \left (a d^2+5 b c^2\right )}{d^5}-\frac {2 b c (c+d x)^{3/2} \left (3 a d^2+5 b c^2\right )}{d^5}+\frac {\sqrt {c+d x} \left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right )}{d^5}-\frac {c \left (a d^2+b c^2\right )^2}{d^5 \sqrt {c+d x}}+\frac {b^2 (c+d x)^{9/2}}{d^5}-\frac {5 b^2 c (c+d x)^{7/2}}{d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {4 b (c+d x)^{7/2} \left (a d^2+5 b c^2\right )}{7 d^6}-\frac {4 b c (c+d x)^{5/2} \left (3 a d^2+5 b c^2\right )}{5 d^6}+\frac {2 (c+d x)^{3/2} \left (a d^2+b c^2\right ) \left (a d^2+5 b c^2\right )}{3 d^6}-\frac {2 c \sqrt {c+d x} \left (a d^2+b c^2\right )^2}{d^6}+\frac {2 b^2 (c+d x)^{11/2}}{11 d^6}-\frac {10 b^2 c (c+d x)^{9/2}}{9 d^6}\)

Input:

Int[(x*(a + b*x^2)^2)/Sqrt[c + d*x],x]
 

Output:

(-2*c*(b*c^2 + a*d^2)^2*Sqrt[c + d*x])/d^6 + (2*(b*c^2 + a*d^2)*(5*b*c^2 + 
 a*d^2)*(c + d*x)^(3/2))/(3*d^6) - (4*b*c*(5*b*c^2 + 3*a*d^2)*(c + d*x)^(5 
/2))/(5*d^6) + (4*b*(5*b*c^2 + a*d^2)*(c + d*x)^(7/2))/(7*d^6) - (10*b^2*c 
*(c + d*x)^(9/2))/(9*d^6) + (2*b^2*(c + d*x)^(11/2))/(11*d^6)
 

Defintions of rubi rules used

rule 522

Int[((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_.)*((a_) + (b_.)*(x_)^2)^(p_. 
), x_Symbol] :> Int[ExpandIntegrand[(e*x)^m*(c + d*x)^n*(a + b*x^2)^p, x], 
x] /; FreeQ[{a, b, c, d, e, m, n}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 0.34 (sec) , antiderivative size = 118, normalized size of antiderivative = 0.71

method	result	size
pseudoelliptic	\(-\frac {4 \left (\left (-\frac {3}{22} x^{5} d^{5}+\frac {5}{33} x^{4} c \,d^{4}-\frac {40}{231} c^{2} d^{3} x^{3}+\frac {16}{77} c^{3} d^{2} x^{2}-\frac {64}{231} c^{4} d x +\frac {128}{231} c^{5}\right ) b^{2}+\frac {48 d^{2} a \left (-\frac {5}{16} d^{3} x^{3}+\frac {3}{8} c \,d^{2} x^{2}-\frac {1}{2} c^{2} d x +c^{3}\right ) b}{35}+a^{2} d^{4} \left (-\frac {d x}{2}+c \right )\right ) \sqrt {d x +c}}{3 d^{6}}\)	\(118\)
gosper	\(-\frac {2 \sqrt {d x +c}\, \left (-315 b^{2} x^{5} d^{5}+350 b^{2} c \,x^{4} d^{4}-990 a b \,d^{5} x^{3}-400 c^{2} d^{3} x^{3} b^{2}+1188 a b c \,d^{4} x^{2}+480 b^{2} c^{3} d^{2} x^{2}-1155 a^{2} x \,d^{5}-1584 a b \,c^{2} d^{3} x -640 b^{2} c^{4} d x +2310 a^{2} c \,d^{4}+3168 a \,c^{3} d^{2} b +1280 c^{5} b^{2}\right )}{3465 d^{6}}\)	\(143\)
trager	\(-\frac {2 \sqrt {d x +c}\, \left (-315 b^{2} x^{5} d^{5}+350 b^{2} c \,x^{4} d^{4}-990 a b \,d^{5} x^{3}-400 c^{2} d^{3} x^{3} b^{2}+1188 a b c \,d^{4} x^{2}+480 b^{2} c^{3} d^{2} x^{2}-1155 a^{2} x \,d^{5}-1584 a b \,c^{2} d^{3} x -640 b^{2} c^{4} d x +2310 a^{2} c \,d^{4}+3168 a \,c^{3} d^{2} b +1280 c^{5} b^{2}\right )}{3465 d^{6}}\)	\(143\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-315 b^{2} x^{5} d^{5}+350 b^{2} c \,x^{4} d^{4}-990 a b \,d^{5} x^{3}-400 c^{2} d^{3} x^{3} b^{2}+1188 a b c \,d^{4} x^{2}+480 b^{2} c^{3} d^{2} x^{2}-1155 a^{2} x \,d^{5}-1584 a b \,c^{2} d^{3} x -640 b^{2} c^{4} d x +2310 a^{2} c \,d^{4}+3168 a \,c^{3} d^{2} b +1280 c^{5} b^{2}\right )}{3465 d^{6}}\)	\(143\)
orering	\(-\frac {2 \sqrt {d x +c}\, \left (-315 b^{2} x^{5} d^{5}+350 b^{2} c \,x^{4} d^{4}-990 a b \,d^{5} x^{3}-400 c^{2} d^{3} x^{3} b^{2}+1188 a b c \,d^{4} x^{2}+480 b^{2} c^{3} d^{2} x^{2}-1155 a^{2} x \,d^{5}-1584 a b \,c^{2} d^{3} x -640 b^{2} c^{4} d x +2310 a^{2} c \,d^{4}+3168 a \,c^{3} d^{2} b +1280 c^{5} b^{2}\right )}{3465 d^{6}}\)	\(143\)
derivativedivides	\(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {10 b^{2} c \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (8 b^{2} c^{2}+2 \left (a \,d^{2}+b \,c^{2}\right ) b \right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-c \left (2 \left (a \,d^{2}+b \,c^{2}\right ) b +4 b^{2} c^{2}\right )-4 \left (a \,d^{2}+b \,c^{2}\right ) b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (4 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) b +\left (a \,d^{2}+b \,c^{2}\right )^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d x +c}}{d^{6}}\)	\(178\)
default	\(\frac {\frac {2 b^{2} \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {10 b^{2} c \left (d x +c \right )^{\frac {9}{2}}}{9}+\frac {2 \left (8 b^{2} c^{2}+2 \left (a \,d^{2}+b \,c^{2}\right ) b \right ) \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 \left (-c \left (2 \left (a \,d^{2}+b \,c^{2}\right ) b +4 b^{2} c^{2}\right )-4 \left (a \,d^{2}+b \,c^{2}\right ) b c \right ) \left (d x +c \right )^{\frac {5}{2}}}{5}+\frac {2 \left (4 c^{2} \left (a \,d^{2}+b \,c^{2}\right ) b +\left (a \,d^{2}+b \,c^{2}\right )^{2}\right ) \left (d x +c \right )^{\frac {3}{2}}}{3}-2 c \left (a \,d^{2}+b \,c^{2}\right )^{2} \sqrt {d x +c}}{d^{6}}\)	\(178\)

Input:

int(x*(b*x^2+a)^2/(d*x+c)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-4/3*((-3/22*x^5*d^5+5/33*x^4*c*d^4-40/231*c^2*d^3*x^3+16/77*c^3*d^2*x^2-6 
4/231*c^4*d*x+128/231*c^5)*b^2+48/35*d^2*a*(-5/16*d^3*x^3+3/8*c*d^2*x^2-1/ 
2*c^2*d*x+c^3)*b+a^2*d^4*(-1/2*d*x+c))*(d*x+c)^(1/2)/d^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 142, normalized size of antiderivative = 0.85 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (315 \, b^{2} d^{5} x^{5} - 350 \, b^{2} c d^{4} x^{4} - 1280 \, b^{2} c^{5} - 3168 \, a b c^{3} d^{2} - 2310 \, a^{2} c d^{4} + 10 \, {\left (40 \, b^{2} c^{2} d^{3} + 99 \, a b d^{5}\right )} x^{3} - 12 \, {\left (40 \, b^{2} c^{3} d^{2} + 99 \, a b c d^{4}\right )} x^{2} + {\left (640 \, b^{2} c^{4} d + 1584 \, a b c^{2} d^{3} + 1155 \, a^{2} d^{5}\right )} x\right )} \sqrt {d x + c}}{3465 \, d^{6}} \] Input:

integrate(x*(b*x^2+a)^2/(d*x+c)^(1/2),x, algorithm="fricas")
 

Output:

2/3465*(315*b^2*d^5*x^5 - 350*b^2*c*d^4*x^4 - 1280*b^2*c^5 - 3168*a*b*c^3* 
d^2 - 2310*a^2*c*d^4 + 10*(40*b^2*c^2*d^3 + 99*a*b*d^5)*x^3 - 12*(40*b^2*c 
^3*d^2 + 99*a*b*c*d^4)*x^2 + (640*b^2*c^4*d + 1584*a*b*c^2*d^3 + 1155*a^2* 
d^5)*x)*sqrt(d*x + c)/d^6
 

Sympy [A] (verification not implemented)

Time = 0.68 (sec) , antiderivative size = 219, normalized size of antiderivative = 1.31 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\begin {cases} \frac {2 \left (- \frac {5 b^{2} c \left (c + d x\right )^{\frac {9}{2}}}{9 d^{4}} + \frac {b^{2} \left (c + d x\right )^{\frac {11}{2}}}{11 d^{4}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \cdot \left (2 a b d^{2} + 10 b^{2} c^{2}\right )}{7 d^{4}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (- 6 a b c d^{2} - 10 b^{2} c^{3}\right )}{5 d^{4}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (a^{2} d^{4} + 6 a b c^{2} d^{2} + 5 b^{2} c^{4}\right )}{3 d^{4}} + \frac {\sqrt {c + d x} \left (- a^{2} c d^{4} - 2 a b c^{3} d^{2} - b^{2} c^{5}\right )}{d^{4}}\right )}{d^{2}} & \text {for}\: d \neq 0 \\\frac {\begin {cases} \frac {a^{2} x^{2}}{2} & \text {for}\: b = 0 \\\frac {\left (a + b x^{2}\right )^{3}}{6 b} & \text {otherwise} \end {cases}}{\sqrt {c}} & \text {otherwise} \end {cases} \] Input:

integrate(x*(b*x**2+a)**2/(d*x+c)**(1/2),x)
 

Output:

Piecewise((2*(-5*b**2*c*(c + d*x)**(9/2)/(9*d**4) + b**2*(c + d*x)**(11/2) 
/(11*d**4) + (c + d*x)**(7/2)*(2*a*b*d**2 + 10*b**2*c**2)/(7*d**4) + (c + 
d*x)**(5/2)*(-6*a*b*c*d**2 - 10*b**2*c**3)/(5*d**4) + (c + d*x)**(3/2)*(a* 
*2*d**4 + 6*a*b*c**2*d**2 + 5*b**2*c**4)/(3*d**4) + sqrt(c + d*x)*(-a**2*c 
*d**4 - 2*a*b*c**3*d**2 - b**2*c**5)/d**4)/d**2, Ne(d, 0)), (Piecewise((a* 
*2*x**2/2, Eq(b, 0)), ((a + b*x**2)**3/(6*b), True))/sqrt(c), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.90 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (315 \, {\left (d x + c\right )}^{\frac {11}{2}} b^{2} - 1925 \, {\left (d x + c\right )}^{\frac {9}{2}} b^{2} c + 990 \, {\left (5 \, b^{2} c^{2} + a b d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}} - 1386 \, {\left (5 \, b^{2} c^{3} + 3 \, a b c d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} + 1155 \, {\left (5 \, b^{2} c^{4} + 6 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} {\left (d x + c\right )}^{\frac {3}{2}} - 3465 \, {\left (b^{2} c^{5} + 2 \, a b c^{3} d^{2} + a^{2} c d^{4}\right )} \sqrt {d x + c}\right )}}{3465 \, d^{6}} \] Input:

integrate(x*(b*x^2+a)^2/(d*x+c)^(1/2),x, algorithm="maxima")
 

Output:

2/3465*(315*(d*x + c)^(11/2)*b^2 - 1925*(d*x + c)^(9/2)*b^2*c + 990*(5*b^2 
*c^2 + a*b*d^2)*(d*x + c)^(7/2) - 1386*(5*b^2*c^3 + 3*a*b*c*d^2)*(d*x + c) 
^(5/2) + 1155*(5*b^2*c^4 + 6*a*b*c^2*d^2 + a^2*d^4)*(d*x + c)^(3/2) - 3465 
*(b^2*c^5 + 2*a*b*c^3*d^2 + a^2*c*d^4)*sqrt(d*x + c))/d^6
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.95 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2 \, {\left (\frac {1155 \, {\left ({\left (d x + c\right )}^{\frac {3}{2}} - 3 \, \sqrt {d x + c} c\right )} a^{2}}{d} + \frac {198 \, {\left (5 \, {\left (d x + c\right )}^{\frac {7}{2}} - 21 \, {\left (d x + c\right )}^{\frac {5}{2}} c + 35 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{2} - 35 \, \sqrt {d x + c} c^{3}\right )} a b}{d^{3}} + \frac {5 \, {\left (63 \, {\left (d x + c\right )}^{\frac {11}{2}} - 385 \, {\left (d x + c\right )}^{\frac {9}{2}} c + 990 \, {\left (d x + c\right )}^{\frac {7}{2}} c^{2} - 1386 \, {\left (d x + c\right )}^{\frac {5}{2}} c^{3} + 1155 \, {\left (d x + c\right )}^{\frac {3}{2}} c^{4} - 693 \, \sqrt {d x + c} c^{5}\right )} b^{2}}{d^{5}}\right )}}{3465 \, d} \] Input:

integrate(x*(b*x^2+a)^2/(d*x+c)^(1/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/3465*(1155*((d*x + c)^(3/2) - 3*sqrt(d*x + c)*c)*a^2/d + 198*(5*(d*x + c 
)^(7/2) - 21*(d*x + c)^(5/2)*c + 35*(d*x + c)^(3/2)*c^2 - 35*sqrt(d*x + c) 
*c^3)*a*b/d^3 + 5*(63*(d*x + c)^(11/2) - 385*(d*x + c)^(9/2)*c + 990*(d*x 
+ c)^(7/2)*c^2 - 1386*(d*x + c)^(5/2)*c^3 + 1155*(d*x + c)^(3/2)*c^4 - 693 
*sqrt(d*x + c)*c^5)*b^2/d^5)/d
 

Mupad [B] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 154, normalized size of antiderivative = 0.92 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2\,b^2\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}-\frac {\left (20\,b^2\,c^3+12\,a\,b\,c\,d^2\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}+\frac {{\left (c+d\,x\right )}^{3/2}\,\left (2\,a^2\,d^4+12\,a\,b\,c^2\,d^2+10\,b^2\,c^4\right )}{3\,d^6}+\frac {\left (20\,b^2\,c^2+4\,a\,b\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {10\,b^2\,c\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}-\frac {2\,c\,{\left (b\,c^2+a\,d^2\right )}^2\,\sqrt {c+d\,x}}{d^6} \] Input:

int((x*(a + b*x^2)^2)/(c + d*x)^(1/2),x)
 

Output:

(2*b^2*(c + d*x)^(11/2))/(11*d^6) - ((20*b^2*c^3 + 12*a*b*c*d^2)*(c + d*x) 
^(5/2))/(5*d^6) + ((c + d*x)^(3/2)*(2*a^2*d^4 + 10*b^2*c^4 + 12*a*b*c^2*d^ 
2))/(3*d^6) + ((20*b^2*c^2 + 4*a*b*d^2)*(c + d*x)^(7/2))/(7*d^6) - (10*b^2 
*c*(c + d*x)^(9/2))/(9*d^6) - (2*c*(a*d^2 + b*c^2)^2*(c + d*x)^(1/2))/d^6
 

Reduce [B] (verification not implemented)

Time = 0.20 (sec) , antiderivative size = 141, normalized size of antiderivative = 0.84 \[ \int \frac {x \left (a+b x^2\right )^2}{\sqrt {c+d x}} \, dx=\frac {2 \sqrt {d x +c}\, \left (315 b^{2} d^{5} x^{5}-350 b^{2} c \,d^{4} x^{4}+990 a b \,d^{5} x^{3}+400 b^{2} c^{2} d^{3} x^{3}-1188 a b c \,d^{4} x^{2}-480 b^{2} c^{3} d^{2} x^{2}+1155 a^{2} d^{5} x +1584 a b \,c^{2} d^{3} x +640 b^{2} c^{4} d x -2310 a^{2} c \,d^{4}-3168 a b \,c^{3} d^{2}-1280 b^{2} c^{5}\right )}{3465 d^{6}} \] Input:

int(x*(b*x^2+a)^2/(d*x+c)^(1/2),x)
 

Output:

(2*sqrt(c + d*x)*( - 2310*a**2*c*d**4 + 1155*a**2*d**5*x - 3168*a*b*c**3*d 
**2 + 1584*a*b*c**2*d**3*x - 1188*a*b*c*d**4*x**2 + 990*a*b*d**5*x**3 - 12 
80*b**2*c**5 + 640*b**2*c**4*d*x - 480*b**2*c**3*d**2*x**2 + 400*b**2*c**2 
*d**3*x**3 - 350*b**2*c*d**4*x**4 + 315*b**2*d**5*x**5))/(3465*d**6)