\(\int \frac {x^3 (a+b x^2)}{(c+d x)^{3/2}} \, dx\) [498]

Optimal result

Integrand size = 20, antiderivative size = 152 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2 c^3 \left (b c^2+a d^2\right )}{d^6 \sqrt {c+d x}}+\frac {2 c^2 \left (5 b c^2+3 a d^2\right ) \sqrt {c+d x}}{d^6}-\frac {2 c \left (10 b c^2+3 a d^2\right ) (c+d x)^{3/2}}{3 d^6}+\frac {2 \left (10 b c^2+a d^2\right ) (c+d x)^{5/2}}{5 d^6}-\frac {10 b c (c+d x)^{7/2}}{7 d^6}+\frac {2 b (c+d x)^{9/2}}{9 d^6} \] Output:

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

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.70 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \left (63 a d^2 \left (16 c^3+8 c^2 d x-2 c d^2 x^2+d^3 x^3\right )+5 b \left (256 c^5+128 c^4 d x-32 c^3 d^2 x^2+16 c^2 d^3 x^3-10 c d^4 x^4+7 d^5 x^5\right )\right )}{315 d^6 \sqrt {c+d x}} \] Input:

Integrate[(x^3*(a + b*x^2))/(c + d*x)^(3/2),x]
 

Output:

(2*(63*a*d^2*(16*c^3 + 8*c^2*d*x - 2*c*d^2*x^2 + d^3*x^3) + 5*b*(256*c^5 + 
 128*c^4*d*x - 32*c^3*d^2*x^2 + 16*c^2*d^3*x^3 - 10*c*d^4*x^4 + 7*d^5*x^5) 
))/(315*d^6*Sqrt[c + d*x])
 

Rubi [A] (verified)

Time = 0.46 (sec) , antiderivative size = 152, 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.

\(\displaystyle \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx\)

\(\Big \downarrow \) 522

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

\(\Big \downarrow \) 2009

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

Input:

Int[(x^3*(a + b*x^2))/(c + d*x)^(3/2),x]
 

Output:

(2*c^3*(b*c^2 + a*d^2))/(d^6*Sqrt[c + d*x]) + (2*c^2*(5*b*c^2 + 3*a*d^2)*S 
qrt[c + d*x])/d^6 - (2*c*(10*b*c^2 + 3*a*d^2)*(c + d*x)^(3/2))/(3*d^6) + ( 
2*(10*b*c^2 + a*d^2)*(c + d*x)^(5/2))/(5*d^6) - (10*b*c*(c + d*x)^(7/2))/( 
7*d^6) + (2*b*(c + d*x)^(9/2))/(9*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 = 0.26 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.61

method	result	size
pseudoelliptic	\(\frac {\frac {2 x^{3} \left (\frac {5 b \,x^{2}}{9}+a \right ) d^{5}}{5}-\frac {4 \left (\frac {25 b \,x^{2}}{63}+a \right ) x^{2} c \,d^{4}}{5}+\frac {16 \left (\frac {10 b \,x^{2}}{63}+a \right ) x \,c^{2} d^{3}}{5}+\frac {32 c^{3} \left (-\frac {10 b \,x^{2}}{63}+a \right ) d^{2}}{5}+\frac {256 b \,c^{4} d x}{63}+\frac {512 b \,c^{5}}{63}}{\sqrt {d x +c}\, d^{6}}\)	\(93\)
gosper	\(\frac {\frac {2}{9} b \,d^{5} x^{5}-\frac {20}{63} b c \,d^{4} x^{4}+\frac {2}{5} x^{3} a \,d^{5}+\frac {32}{63} b \,c^{2} d^{3} x^{3}-\frac {4}{5} x^{2} a c \,d^{4}-\frac {64}{63} b \,c^{3} d^{2} x^{2}+\frac {16}{5} a \,c^{2} d^{3} x +\frac {256}{63} b \,c^{4} d x +\frac {32}{5} a \,c^{3} d^{2}+\frac {512}{63} b \,c^{5}}{\sqrt {d x +c}\, d^{6}}\)	\(109\)
trager	\(\frac {\frac {2}{9} b \,d^{5} x^{5}-\frac {20}{63} b c \,d^{4} x^{4}+\frac {2}{5} x^{3} a \,d^{5}+\frac {32}{63} b \,c^{2} d^{3} x^{3}-\frac {4}{5} x^{2} a c \,d^{4}-\frac {64}{63} b \,c^{3} d^{2} x^{2}+\frac {16}{5} a \,c^{2} d^{3} x +\frac {256}{63} b \,c^{4} d x +\frac {32}{5} a \,c^{3} d^{2}+\frac {512}{63} b \,c^{5}}{\sqrt {d x +c}\, d^{6}}\)	\(109\)
orering	\(\frac {\frac {2}{9} b \,d^{5} x^{5}-\frac {20}{63} b c \,d^{4} x^{4}+\frac {2}{5} x^{3} a \,d^{5}+\frac {32}{63} b \,c^{2} d^{3} x^{3}-\frac {4}{5} x^{2} a c \,d^{4}-\frac {64}{63} b \,c^{3} d^{2} x^{2}+\frac {16}{5} a \,c^{2} d^{3} x +\frac {256}{63} b \,c^{4} d x +\frac {32}{5} a \,c^{3} d^{2}+\frac {512}{63} b \,c^{5}}{\sqrt {d x +c}\, d^{6}}\)	\(109\)
risch	\(\frac {2 \left (35 b \,x^{4} d^{4}-85 c b \,d^{3} x^{3}+63 a \,d^{4} x^{2}+165 b \,c^{2} d^{2} x^{2}-189 a c \,d^{3} x -325 b \,c^{3} d x +693 a \,c^{2} d^{2}+965 b \,c^{4}\right ) \sqrt {d x +c}}{315 d^{6}}+\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right )}{d^{6} \sqrt {d x +c}}\)	\(112\)
derivativedivides	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {10 b c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 a \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+4 b \,c^{2} \left (d x +c \right )^{\frac {5}{2}}-2 a c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {20 b \,c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} \sqrt {d x +c}+10 b \,c^{4} \sqrt {d x +c}+\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right )}{\sqrt {d x +c}}}{d^{6}}\)	\(132\)
default	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {10 b c \left (d x +c \right )^{\frac {7}{2}}}{7}+\frac {2 a \,d^{2} \left (d x +c \right )^{\frac {5}{2}}}{5}+4 b \,c^{2} \left (d x +c \right )^{\frac {5}{2}}-2 a c \,d^{2} \left (d x +c \right )^{\frac {3}{2}}-\frac {20 b \,c^{3} \left (d x +c \right )^{\frac {3}{2}}}{3}+6 a \,c^{2} d^{2} \sqrt {d x +c}+10 b \,c^{4} \sqrt {d x +c}+\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right )}{\sqrt {d x +c}}}{d^{6}}\)	\(132\)

Input:

int(x^3*(b*x^2+a)/(d*x+c)^(3/2),x,method=_RETURNVERBOSE)
 

Output:

32/5/(d*x+c)^(1/2)*(1/16*x^3*(5/9*b*x^2+a)*d^5-1/8*(25/63*b*x^2+a)*x^2*c*d 
^4+1/2*(10/63*b*x^2+a)*x*c^2*d^3+c^3*(-10/63*b*x^2+a)*d^2+40/63*b*c^4*d*x+ 
80/63*b*c^5)/d^6
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.78 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (35 \, b d^{5} x^{5} - 50 \, b c d^{4} x^{4} + 1280 \, b c^{5} + 1008 \, a c^{3} d^{2} + {\left (80 \, b c^{2} d^{3} + 63 \, a d^{5}\right )} x^{3} - 2 \, {\left (80 \, b c^{3} d^{2} + 63 \, a c d^{4}\right )} x^{2} + 8 \, {\left (80 \, b c^{4} d + 63 \, a c^{2} d^{3}\right )} x\right )} \sqrt {d x + c}}{315 \, {\left (d^{7} x + c d^{6}\right )}} \] Input:

integrate(x^3*(b*x^2+a)/(d*x+c)^(3/2),x, algorithm="fricas")
 

Output:

2/315*(35*b*d^5*x^5 - 50*b*c*d^4*x^4 + 1280*b*c^5 + 1008*a*c^3*d^2 + (80*b 
*c^2*d^3 + 63*a*d^5)*x^3 - 2*(80*b*c^3*d^2 + 63*a*c*d^4)*x^2 + 8*(80*b*c^4 
*d + 63*a*c^2*d^3)*x)*sqrt(d*x + c)/(d^7*x + c*d^6)
 

Sympy [A] (verification not implemented)

Time = 1.61 (sec) , antiderivative size = 173, normalized size of antiderivative = 1.14 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\begin {cases} \frac {2 \left (- \frac {5 b c \left (c + d x\right )^{\frac {7}{2}}}{7 d^{2}} + \frac {b \left (c + d x\right )^{\frac {9}{2}}}{9 d^{2}} + \frac {c^{3} \left (a d^{2} + b c^{2}\right )}{d^{2} \sqrt {c + d x}} + \frac {\left (c + d x\right )^{\frac {5}{2}} \left (a d^{2} + 10 b c^{2}\right )}{5 d^{2}} + \frac {\left (c + d x\right )^{\frac {3}{2}} \left (- 3 a c d^{2} - 10 b c^{3}\right )}{3 d^{2}} + \frac {\sqrt {c + d x} \left (3 a c^{2} d^{2} + 5 b c^{4}\right )}{d^{2}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\\frac {\frac {a x^{4}}{4} + \frac {b x^{6}}{6}}{c^{\frac {3}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(b*x**2+a)/(d*x+c)**(3/2),x)
 

Output:

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

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 127, normalized size of antiderivative = 0.84 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (d x + c\right )}^{\frac {9}{2}} b - 225 \, {\left (d x + c\right )}^{\frac {7}{2}} b c + 63 \, {\left (10 \, b c^{2} + a d^{2}\right )} {\left (d x + c\right )}^{\frac {5}{2}} - 105 \, {\left (10 \, b c^{3} + 3 \, a c d^{2}\right )} {\left (d x + c\right )}^{\frac {3}{2}} + 315 \, {\left (5 \, b c^{4} + 3 \, a c^{2} d^{2}\right )} \sqrt {d x + c}}{d^{2}} + \frac {315 \, {\left (b c^{5} + a c^{3} d^{2}\right )}}{\sqrt {d x + c} d^{2}}\right )}}{315 \, d^{4}} \] Input:

integrate(x^3*(b*x^2+a)/(d*x+c)^(3/2),x, algorithm="maxima")
 

Output:

2/315*((35*(d*x + c)^(9/2)*b - 225*(d*x + c)^(7/2)*b*c + 63*(10*b*c^2 + a* 
d^2)*(d*x + c)^(5/2) - 105*(10*b*c^3 + 3*a*c*d^2)*(d*x + c)^(3/2) + 315*(5 
*b*c^4 + 3*a*c^2*d^2)*sqrt(d*x + c))/d^2 + 315*(b*c^5 + a*c^3*d^2)/(sqrt(d 
*x + c)*d^2))/d^4
 

Giac [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 151, normalized size of antiderivative = 0.99 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2 \, {\left (b c^{5} + a c^{3} d^{2}\right )}}{\sqrt {d x + c} d^{6}} + \frac {2 \, {\left (35 \, {\left (d x + c\right )}^{\frac {9}{2}} b d^{48} - 225 \, {\left (d x + c\right )}^{\frac {7}{2}} b c d^{48} + 630 \, {\left (d x + c\right )}^{\frac {5}{2}} b c^{2} d^{48} - 1050 \, {\left (d x + c\right )}^{\frac {3}{2}} b c^{3} d^{48} + 1575 \, \sqrt {d x + c} b c^{4} d^{48} + 63 \, {\left (d x + c\right )}^{\frac {5}{2}} a d^{50} - 315 \, {\left (d x + c\right )}^{\frac {3}{2}} a c d^{50} + 945 \, \sqrt {d x + c} a c^{2} d^{50}\right )}}{315 \, d^{54}} \] Input:

integrate(x^3*(b*x^2+a)/(d*x+c)^(3/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2*(b*c^5 + a*c^3*d^2)/(sqrt(d*x + c)*d^6) + 2/315*(35*(d*x + c)^(9/2)*b*d^ 
48 - 225*(d*x + c)^(7/2)*b*c*d^48 + 630*(d*x + c)^(5/2)*b*c^2*d^48 - 1050* 
(d*x + c)^(3/2)*b*c^3*d^48 + 1575*sqrt(d*x + c)*b*c^4*d^48 + 63*(d*x + c)^ 
(5/2)*a*d^50 - 315*(d*x + c)^(3/2)*a*c*d^50 + 945*sqrt(d*x + c)*a*c^2*d^50 
)/d^54
 

Mupad [B] (verification not implemented)

Time = 8.19 (sec) , antiderivative size = 133, normalized size of antiderivative = 0.88 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {2\,b\,c^5+2\,a\,c^3\,d^2}{d^6\,\sqrt {c+d\,x}}-\frac {\left (20\,b\,c^3+6\,a\,c\,d^2\right )\,{\left (c+d\,x\right )}^{3/2}}{3\,d^6}+\frac {\left (10\,b\,c^4+6\,a\,c^2\,d^2\right )\,\sqrt {c+d\,x}}{d^6}+\frac {2\,b\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}+\frac {\left (20\,b\,c^2+2\,a\,d^2\right )\,{\left (c+d\,x\right )}^{5/2}}{5\,d^6}-\frac {10\,b\,c\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6} \] Input:

int((x^3*(a + b*x^2))/(c + d*x)^(3/2),x)
 

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.72 \[ \int \frac {x^3 \left (a+b x^2\right )}{(c+d x)^{3/2}} \, dx=\frac {\frac {2}{9} b \,d^{5} x^{5}-\frac {20}{63} b c \,d^{4} x^{4}+\frac {2}{5} a \,d^{5} x^{3}+\frac {32}{63} b \,c^{2} d^{3} x^{3}-\frac {4}{5} a c \,d^{4} x^{2}-\frac {64}{63} b \,c^{3} d^{2} x^{2}+\frac {16}{5} a \,c^{2} d^{3} x +\frac {256}{63} b \,c^{4} d x +\frac {32}{5} a \,c^{3} d^{2}+\frac {512}{63} b \,c^{5}}{\sqrt {d x +c}\, d^{6}} \] Input:

int(x^3*(b*x^2+a)/(d*x+c)^(3/2),x)
 

Output:

(2*(1008*a*c**3*d**2 + 504*a*c**2*d**3*x - 126*a*c*d**4*x**2 + 63*a*d**5*x 
**3 + 1280*b*c**5 + 640*b*c**4*d*x - 160*b*c**3*d**2*x**2 + 80*b*c**2*d**3 
*x**3 - 50*b*c*d**4*x**4 + 35*b*d**5*x**5))/(315*sqrt(c + d*x)*d**6)