\(\int x^3 (c+d x)^{5/2} (a+b x^2) \, dx\) [480]

Optimal result

Integrand size = 20, antiderivative size = 156 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=-\frac {2 c^3 \left (b c^2+a d^2\right ) (c+d x)^{7/2}}{7 d^6}+\frac {2 c^2 \left (5 b c^2+3 a d^2\right ) (c+d x)^{9/2}}{9 d^6}-\frac {2 c \left (10 b c^2+3 a d^2\right ) (c+d x)^{11/2}}{11 d^6}+\frac {2 \left (10 b c^2+a d^2\right ) (c+d x)^{13/2}}{13 d^6}-\frac {2 b c (c+d x)^{15/2}}{3 d^6}+\frac {2 b (c+d x)^{17/2}}{17 d^6} \] Output:

-2/7*c^3*(a*d^2+b*c^2)*(d*x+c)^(7/2)/d^6+2/9*c^2*(3*a*d^2+5*b*c^2)*(d*x+c) 
^(9/2)/d^6-2/11*c*(3*a*d^2+10*b*c^2)*(d*x+c)^(11/2)/d^6+2/13*(a*d^2+10*b*c 
^2)*(d*x+c)^(13/2)/d^6-2/3*b*c*(d*x+c)^(15/2)/d^6+2/17*b*(d*x+c)^(17/2)/d^ 
6
 

Mathematica [A] (verified)

Time = 0.08 (sec) , antiderivative size = 107, normalized size of antiderivative = 0.69 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 (c+d x)^{7/2} \left (51 a d^2 \left (-16 c^3+56 c^2 d x-126 c d^2 x^2+231 d^3 x^3\right )+b \left (-256 c^5+896 c^4 d x-2016 c^3 d^2 x^2+3696 c^2 d^3 x^3-6006 c d^4 x^4+9009 d^5 x^5\right )\right )}{153153 d^6} \] Input:

Integrate[x^3*(c + d*x)^(5/2)*(a + b*x^2),x]
 

Output:

(2*(c + d*x)^(7/2)*(51*a*d^2*(-16*c^3 + 56*c^2*d*x - 126*c*d^2*x^2 + 231*d 
^3*x^3) + b*(-256*c^5 + 896*c^4*d*x - 2016*c^3*d^2*x^2 + 3696*c^2*d^3*x^3 
- 6006*c*d^4*x^4 + 9009*d^5*x^5)))/(153153*d^6)
 

Rubi [A] (verified)

Time = 0.49 (sec) , antiderivative size = 156, 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.

\(\displaystyle \int x^3 \left (a+b x^2\right ) (c+d x)^{5/2} \, dx\)

\(\Big \downarrow \) 522

\(\displaystyle \int \left (\frac {(c+d x)^{9/2} \left (-3 a c d^2-10 b c^3\right )}{d^5}+\frac {(c+d x)^{11/2} \left (a d^2+10 b c^2\right )}{d^5}+\frac {(c+d x)^{7/2} \left (3 a c^2 d^2+5 b c^4\right )}{d^5}+\frac {c^3 (c+d x)^{5/2} \left (-a d^2-b c^2\right )}{d^5}+\frac {b (c+d x)^{15/2}}{d^5}-\frac {5 b c (c+d x)^{13/2}}{d^5}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {2 (c+d x)^{13/2} \left (a d^2+10 b c^2\right )}{13 d^6}-\frac {2 c (c+d x)^{11/2} \left (3 a d^2+10 b c^2\right )}{11 d^6}+\frac {2 c^2 (c+d x)^{9/2} \left (3 a d^2+5 b c^2\right )}{9 d^6}-\frac {2 c^3 (c+d x)^{7/2} \left (a d^2+b c^2\right )}{7 d^6}+\frac {2 b (c+d x)^{17/2}}{17 d^6}-\frac {2 b c (c+d x)^{15/2}}{3 d^6}\)

Input:

Int[x^3*(c + d*x)^(5/2)*(a + b*x^2),x]
 

Output:

(-2*c^3*(b*c^2 + a*d^2)*(c + d*x)^(7/2))/(7*d^6) + (2*c^2*(5*b*c^2 + 3*a*d 
^2)*(c + d*x)^(9/2))/(9*d^6) - (2*c*(10*b*c^2 + 3*a*d^2)*(c + d*x)^(11/2)) 
/(11*d^6) + (2*(10*b*c^2 + a*d^2)*(c + d*x)^(13/2))/(13*d^6) - (2*b*c*(c + 
 d*x)^(15/2))/(3*d^6) + (2*b*(c + d*x)^(17/2))/(17*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 = 0.35 (sec) , antiderivative size = 93, normalized size of antiderivative = 0.60

method	result	size
pseudoelliptic	\(-\frac {32 \left (d x +c \right )^{\frac {7}{2}} \left (-\frac {231 \left (\frac {13 b \,x^{2}}{17}+a \right ) x^{3} d^{5}}{16}+\frac {63 \left (\frac {143 b \,x^{2}}{153}+a \right ) x^{2} c \,d^{4}}{8}-\frac {7 x \left (\frac {22 b \,x^{2}}{17}+a \right ) c^{2} d^{3}}{2}+c^{3} \left (\frac {42 b \,x^{2}}{17}+a \right ) d^{2}-\frac {56 b \,c^{4} d x}{51}+\frac {16 b \,c^{5}}{51}\right )}{3003 d^{6}}\)	\(93\)
gosper	\(-\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (-9009 b \,d^{5} x^{5}+6006 b c \,d^{4} x^{4}-11781 x^{3} a \,d^{5}-3696 b \,c^{2} d^{3} x^{3}+6426 x^{2} a c \,d^{4}+2016 b \,c^{3} d^{2} x^{2}-2856 a \,c^{2} d^{3} x -896 b \,c^{4} d x +816 a \,c^{3} d^{2}+256 b \,c^{5}\right )}{153153 d^{6}}\)	\(109\)
orering	\(-\frac {2 \left (d x +c \right )^{\frac {7}{2}} \left (-9009 b \,d^{5} x^{5}+6006 b c \,d^{4} x^{4}-11781 x^{3} a \,d^{5}-3696 b \,c^{2} d^{3} x^{3}+6426 x^{2} a c \,d^{4}+2016 b \,c^{3} d^{2} x^{2}-2856 a \,c^{2} d^{3} x -896 b \,c^{4} d x +816 a \,c^{3} d^{2}+256 b \,c^{5}\right )}{153153 d^{6}}\)	\(109\)
derivativedivides	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {17}{2}}}{17}-\frac {2 b c \left (d x +c \right )^{\frac {15}{2}}}{3}+\frac {2 \left (a \,d^{2}+10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {13}{2}}}{13}+\frac {2 \left (-7 b \,c^{3}-3 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}+\frac {2 \left (2 b \,c^{4}+3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\)	\(134\)
default	\(\frac {\frac {2 b \left (d x +c \right )^{\frac {17}{2}}}{17}-\frac {2 b c \left (d x +c \right )^{\frac {15}{2}}}{3}-\frac {2 \left (-a \,d^{2}-10 b \,c^{2}\right ) \left (d x +c \right )^{\frac {13}{2}}}{13}-\frac {2 \left (7 b \,c^{3}+3 c \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {11}{2}}}{11}-\frac {2 \left (-2 b \,c^{4}-3 c^{2} \left (a \,d^{2}+b \,c^{2}\right )\right ) \left (d x +c \right )^{\frac {9}{2}}}{9}-\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (d x +c \right )^{\frac {7}{2}}}{7}}{d^{6}}\)	\(135\)
trager	\(-\frac {2 \left (-9009 b \,d^{8} x^{8}-21021 b c \,d^{7} x^{7}-11781 a \,d^{8} x^{6}-12705 b \,c^{2} d^{6} x^{6}-28917 a c \,d^{7} x^{5}-63 b \,c^{3} d^{5} x^{5}-18921 a \,c^{2} d^{6} x^{4}+70 b \,c^{4} d^{4} x^{4}-255 a \,c^{3} d^{5} x^{3}-80 b \,c^{5} d^{3} x^{3}+306 a \,c^{4} d^{4} x^{2}+96 b \,c^{6} d^{2} x^{2}-408 a \,c^{5} d^{3} x -128 b \,c^{7} d x +816 d^{2} c^{6} a +256 b \,c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\)	\(181\)
risch	\(-\frac {2 \left (-9009 b \,d^{8} x^{8}-21021 b c \,d^{7} x^{7}-11781 a \,d^{8} x^{6}-12705 b \,c^{2} d^{6} x^{6}-28917 a c \,d^{7} x^{5}-63 b \,c^{3} d^{5} x^{5}-18921 a \,c^{2} d^{6} x^{4}+70 b \,c^{4} d^{4} x^{4}-255 a \,c^{3} d^{5} x^{3}-80 b \,c^{5} d^{3} x^{3}+306 a \,c^{4} d^{4} x^{2}+96 b \,c^{6} d^{2} x^{2}-408 a \,c^{5} d^{3} x -128 b \,c^{7} d x +816 d^{2} c^{6} a +256 b \,c^{8}\right ) \sqrt {d x +c}}{153153 d^{6}}\)	\(181\)

Input:

int(x^3*(d*x+c)^(5/2)*(b*x^2+a),x,method=_RETURNVERBOSE)
 

Output:

-32/3003*(d*x+c)^(7/2)*(-231/16*(13/17*b*x^2+a)*x^3*d^5+63/8*(143/153*b*x^ 
2+a)*x^2*c*d^4-7/2*x*(22/17*b*x^2+a)*c^2*d^3+c^3*(42/17*b*x^2+a)*d^2-56/51 
*b*c^4*d*x+16/51*b*c^5)/d^6
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 181, normalized size of antiderivative = 1.16 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (9009 \, b d^{8} x^{8} + 21021 \, b c d^{7} x^{7} - 256 \, b c^{8} - 816 \, a c^{6} d^{2} + 231 \, {\left (55 \, b c^{2} d^{6} + 51 \, a d^{8}\right )} x^{6} + 63 \, {\left (b c^{3} d^{5} + 459 \, a c d^{7}\right )} x^{5} - 7 \, {\left (10 \, b c^{4} d^{4} - 2703 \, a c^{2} d^{6}\right )} x^{4} + 5 \, {\left (16 \, b c^{5} d^{3} + 51 \, a c^{3} d^{5}\right )} x^{3} - 6 \, {\left (16 \, b c^{6} d^{2} + 51 \, a c^{4} d^{4}\right )} x^{2} + 8 \, {\left (16 \, b c^{7} d + 51 \, a c^{5} d^{3}\right )} x\right )} \sqrt {d x + c}}{153153 \, d^{6}} \] Input:

integrate(x^3*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="fricas")
 

Output:

2/153153*(9009*b*d^8*x^8 + 21021*b*c*d^7*x^7 - 256*b*c^8 - 816*a*c^6*d^2 + 
 231*(55*b*c^2*d^6 + 51*a*d^8)*x^6 + 63*(b*c^3*d^5 + 459*a*c*d^7)*x^5 - 7* 
(10*b*c^4*d^4 - 2703*a*c^2*d^6)*x^4 + 5*(16*b*c^5*d^3 + 51*a*c^3*d^5)*x^3 
- 6*(16*b*c^6*d^2 + 51*a*c^4*d^4)*x^2 + 8*(16*b*c^7*d + 51*a*c^5*d^3)*x)*s 
qrt(d*x + c)/d^6
 

Sympy [A] (verification not implemented)

Time = 0.84 (sec) , antiderivative size = 178, normalized size of antiderivative = 1.14 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\begin {cases} \frac {2 \left (- \frac {b c \left (c + d x\right )^{\frac {15}{2}}}{3 d^{2}} + \frac {b \left (c + d x\right )^{\frac {17}{2}}}{17 d^{2}} + \frac {\left (c + d x\right )^{\frac {13}{2}} \left (a d^{2} + 10 b c^{2}\right )}{13 d^{2}} + \frac {\left (c + d x\right )^{\frac {11}{2}} \left (- 3 a c d^{2} - 10 b c^{3}\right )}{11 d^{2}} + \frac {\left (c + d x\right )^{\frac {9}{2}} \cdot \left (3 a c^{2} d^{2} + 5 b c^{4}\right )}{9 d^{2}} + \frac {\left (c + d x\right )^{\frac {7}{2}} \left (- a c^{3} d^{2} - b c^{5}\right )}{7 d^{2}}\right )}{d^{4}} & \text {for}\: d \neq 0 \\c^{\frac {5}{2}} \left (\frac {a x^{4}}{4} + \frac {b x^{6}}{6}\right ) & \text {otherwise} \end {cases} \] Input:

integrate(x**3*(d*x+c)**(5/2)*(b*x**2+a),x)
 

Output:

Piecewise((2*(-b*c*(c + d*x)**(15/2)/(3*d**2) + b*(c + d*x)**(17/2)/(17*d* 
*2) + (c + d*x)**(13/2)*(a*d**2 + 10*b*c**2)/(13*d**2) + (c + d*x)**(11/2) 
*(-3*a*c*d**2 - 10*b*c**3)/(11*d**2) + (c + d*x)**(9/2)*(3*a*c**2*d**2 + 5 
*b*c**4)/(9*d**2) + (c + d*x)**(7/2)*(-a*c**3*d**2 - b*c**5)/(7*d**2))/d** 
4, Ne(d, 0)), (c**(5/2)*(a*x**4/4 + b*x**6/6), True))
 

Maxima [A] (verification not implemented)

Time = 0.03 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.76 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \, {\left (9009 \, {\left (d x + c\right )}^{\frac {17}{2}} b - 51051 \, {\left (d x + c\right )}^{\frac {15}{2}} b c + 11781 \, {\left (10 \, b c^{2} + a d^{2}\right )} {\left (d x + c\right )}^{\frac {13}{2}} - 13923 \, {\left (10 \, b c^{3} + 3 \, a c d^{2}\right )} {\left (d x + c\right )}^{\frac {11}{2}} + 17017 \, {\left (5 \, b c^{4} + 3 \, a c^{2} d^{2}\right )} {\left (d x + c\right )}^{\frac {9}{2}} - 21879 \, {\left (b c^{5} + a c^{3} d^{2}\right )} {\left (d x + c\right )}^{\frac {7}{2}}\right )}}{153153 \, d^{6}} \] Input:

integrate(x^3*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="maxima")
 

Output:

2/153153*(9009*(d*x + c)^(17/2)*b - 51051*(d*x + c)^(15/2)*b*c + 11781*(10 
*b*c^2 + a*d^2)*(d*x + c)^(13/2) - 13923*(10*b*c^3 + 3*a*c*d^2)*(d*x + c)^ 
(11/2) + 17017*(5*b*c^4 + 3*a*c^2*d^2)*(d*x + c)^(9/2) - 21879*(b*c^5 + a* 
c^3*d^2)*(d*x + c)^(7/2))/d^6
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 660 vs. \(2 (132) = 264\).

Time = 0.13 (sec) , antiderivative size = 660, normalized size of antiderivative = 4.23 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx =\text {Too large to display} \] Input:

integrate(x^3*(d*x+c)^(5/2)*(b*x^2+a),x, algorithm="giac")
 

Output:

2/765765*(21879*(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*c^3/d^3 + 7293*(35*(d*x + c)^(9/2) - 18 
0*(d*x + c)^(7/2)*c + 378*(d*x + c)^(5/2)*c^2 - 420*(d*x + c)^(3/2)*c^3 + 
315*sqrt(d*x + c)*c^4)*a*c^2/d^3 + 1105*(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*c^3/d^5 + 3315*(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)*a*c/d^3 + 765 
*(231*(d*x + c)^(13/2) - 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^ 
2 - 8580*(d*x + c)^(7/2)*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^( 
3/2)*c^5 + 3003*sqrt(d*x + c)*c^6)*b*c^2/d^5 + 255*(231*(d*x + c)^(13/2) - 
 1638*(d*x + c)^(11/2)*c + 5005*(d*x + c)^(9/2)*c^2 - 8580*(d*x + c)^(7/2) 
*c^3 + 9009*(d*x + c)^(5/2)*c^4 - 6006*(d*x + c)^(3/2)*c^5 + 3003*sqrt(d*x 
 + c)*c^6)*a/d^3 + 357*(429*(d*x + c)^(15/2) - 3465*(d*x + c)^(13/2)*c + 1 
2285*(d*x + c)^(11/2)*c^2 - 25025*(d*x + c)^(9/2)*c^3 + 32175*(d*x + c)^(7 
/2)*c^4 - 27027*(d*x + c)^(5/2)*c^5 + 15015*(d*x + c)^(3/2)*c^6 - 6435*sqr 
t(d*x + c)*c^7)*b*c/d^5 + 7*(6435*(d*x + c)^(17/2) - 58344*(d*x + c)^(15/2 
)*c + 235620*(d*x + c)^(13/2)*c^2 - 556920*(d*x + c)^(11/2)*c^3 + 850850*( 
d*x + c)^(9/2)*c^4 - 875160*(d*x + c)^(7/2)*c^5 + 612612*(d*x + c)^(5/2)*c 
^6 - 291720*(d*x + c)^(3/2)*c^7 + 109395*sqrt(d*x + c)*c^8)*b/d^5)/d
 

Mupad [B] (verification not implemented)

Time = 0.05 (sec) , antiderivative size = 135, normalized size of antiderivative = 0.87 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {\left (10\,b\,c^4+6\,a\,c^2\,d^2\right )\,{\left (c+d\,x\right )}^{9/2}}{9\,d^6}-\frac {\left (2\,b\,c^5+2\,a\,c^3\,d^2\right )\,{\left (c+d\,x\right )}^{7/2}}{7\,d^6}-\frac {\left (20\,b\,c^3+6\,a\,c\,d^2\right )\,{\left (c+d\,x\right )}^{11/2}}{11\,d^6}+\frac {2\,b\,{\left (c+d\,x\right )}^{17/2}}{17\,d^6}+\frac {\left (20\,b\,c^2+2\,a\,d^2\right )\,{\left (c+d\,x\right )}^{13/2}}{13\,d^6}-\frac {2\,b\,c\,{\left (c+d\,x\right )}^{15/2}}{3\,d^6} \] Input:

int(x^3*(a + b*x^2)*(c + d*x)^(5/2),x)
 

Output:

((10*b*c^4 + 6*a*c^2*d^2)*(c + d*x)^(9/2))/(9*d^6) - ((2*b*c^5 + 2*a*c^3*d 
^2)*(c + d*x)^(7/2))/(7*d^6) - ((20*b*c^3 + 6*a*c*d^2)*(c + d*x)^(11/2))/( 
11*d^6) + (2*b*(c + d*x)^(17/2))/(17*d^6) + ((2*a*d^2 + 20*b*c^2)*(c + d*x 
)^(13/2))/(13*d^6) - (2*b*c*(c + d*x)^(15/2))/(3*d^6)
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.15 \[ \int x^3 (c+d x)^{5/2} \left (a+b x^2\right ) \, dx=\frac {2 \sqrt {d x +c}\, \left (9009 b \,d^{8} x^{8}+21021 b c \,d^{7} x^{7}+11781 a \,d^{8} x^{6}+12705 b \,c^{2} d^{6} x^{6}+28917 a c \,d^{7} x^{5}+63 b \,c^{3} d^{5} x^{5}+18921 a \,c^{2} d^{6} x^{4}-70 b \,c^{4} d^{4} x^{4}+255 a \,c^{3} d^{5} x^{3}+80 b \,c^{5} d^{3} x^{3}-306 a \,c^{4} d^{4} x^{2}-96 b \,c^{6} d^{2} x^{2}+408 a \,c^{5} d^{3} x +128 b \,c^{7} d x -816 a \,c^{6} d^{2}-256 b \,c^{8}\right )}{153153 d^{6}} \] Input:

int(x^3*(d*x+c)^(5/2)*(b*x^2+a),x)
 

Output:

(2*sqrt(c + d*x)*( - 816*a*c**6*d**2 + 408*a*c**5*d**3*x - 306*a*c**4*d**4 
*x**2 + 255*a*c**3*d**5*x**3 + 18921*a*c**2*d**6*x**4 + 28917*a*c*d**7*x** 
5 + 11781*a*d**8*x**6 - 256*b*c**8 + 128*b*c**7*d*x - 96*b*c**6*d**2*x**2 
+ 80*b*c**5*d**3*x**3 - 70*b*c**4*d**4*x**4 + 63*b*c**3*d**5*x**5 + 12705* 
b*c**2*d**6*x**6 + 21021*b*c*d**7*x**7 + 9009*b*d**8*x**8))/(153153*d**6)