\(\int x^2 (c+d x) (a x^2+b x^3)^{5/2} \, dx\) [258]

Optimal result

Integrand size = 24, antiderivative size = 314 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=-\frac {2 a^7 (b c-a d) \left (a x^2+b x^3\right )^{7/2}}{7 b^9 x^7}+\frac {2 a^6 (7 b c-8 a d) \left (a x^2+b x^3\right )^{9/2}}{9 b^9 x^9}-\frac {14 a^5 (3 b c-4 a d) \left (a x^2+b x^3\right )^{11/2}}{11 b^9 x^{11}}+\frac {14 a^4 (5 b c-8 a d) \left (a x^2+b x^3\right )^{13/2}}{13 b^9 x^{13}}-\frac {14 a^3 (b c-2 a d) \left (a x^2+b x^3\right )^{15/2}}{3 b^9 x^{15}}+\frac {14 a^2 (3 b c-8 a d) \left (a x^2+b x^3\right )^{17/2}}{17 b^9 x^{17}}-\frac {14 a (b c-4 a d) \left (a x^2+b x^3\right )^{19/2}}{19 b^9 x^{19}}+\frac {2 (b c-8 a d) \left (a x^2+b x^3\right )^{21/2}}{21 b^9 x^{21}}+\frac {2 d \left (a x^2+b x^3\right )^{23/2}}{23 b^9 x^{23}} \] Output:

-2/7*a^7*(-a*d+b*c)*(b*x^3+a*x^2)^(7/2)/b^9/x^7+2/9*a^6*(-8*a*d+7*b*c)*(b* 
x^3+a*x^2)^(9/2)/b^9/x^9-14/11*a^5*(-4*a*d+3*b*c)*(b*x^3+a*x^2)^(11/2)/b^9 
/x^11+14/13*a^4*(-8*a*d+5*b*c)*(b*x^3+a*x^2)^(13/2)/b^9/x^13-14/3*a^3*(-2* 
a*d+b*c)*(b*x^3+a*x^2)^(15/2)/b^9/x^15+14/17*a^2*(-8*a*d+3*b*c)*(b*x^3+a*x 
^2)^(17/2)/b^9/x^17-14/19*a*(-4*a*d+b*c)*(b*x^3+a*x^2)^(19/2)/b^9/x^19+2/2 
1*(-8*a*d+b*c)*(b*x^3+a*x^2)^(21/2)/b^9/x^21+2/23*d*(b*x^3+a*x^2)^(23/2)/b 
^9/x^23
 

Mathematica [A] (verified)

Time = 0.11 (sec) , antiderivative size = 175, normalized size of antiderivative = 0.56 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 x (a+b x)^4 \left (32768 a^8 d+138567 b^8 x^7 (23 c+21 d x)-48048 a^3 b^5 x^4 (23 c+24 d x)+29568 a^4 b^4 x^3 (23 c+26 d x)+7168 a^6 b^2 x (23 c+36 d x)-2048 a^7 b (23 c+56 d x)+24024 a^2 b^6 x^5 (69 c+68 d x)-5376 a^5 b^3 x^2 (69 c+88 d x)-14586 a b^7 x^6 (161 c+152 d x)\right )}{66927861 b^9 \sqrt {x^2 (a+b x)}} \] Input:

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

Output:

(2*x*(a + b*x)^4*(32768*a^8*d + 138567*b^8*x^7*(23*c + 21*d*x) - 48048*a^3 
*b^5*x^4*(23*c + 24*d*x) + 29568*a^4*b^4*x^3*(23*c + 26*d*x) + 7168*a^6*b^ 
2*x*(23*c + 36*d*x) - 2048*a^7*b*(23*c + 56*d*x) + 24024*a^2*b^6*x^5*(69*c 
 + 68*d*x) - 5376*a^5*b^3*x^2*(69*c + 88*d*x) - 14586*a*b^7*x^6*(161*c + 1 
52*d*x)))/(66927861*b^9*Sqrt[x^2*(a + b*x)])
 

Rubi [A] (verified)

Time = 1.06 (sec) , antiderivative size = 294, normalized size of antiderivative = 0.94, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.375, Rules used = {1945, 1922, 1922, 1908, 1922, 1922, 1922, 1922, 1920}

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.

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

\(\Big \downarrow \) 1945

\(\displaystyle \frac {(23 b c-16 a d) \int x^2 \left (b x^3+a x^2\right )^{5/2}dx}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \int x \left (b x^3+a x^2\right )^{5/2}dx}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \int \left (b x^3+a x^2\right )^{5/2}dx}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1908

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x}dx}{17 b}\right )}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^2}dx}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^3}dx}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^4}dx}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1922

\(\displaystyle \frac {(23 b c-16 a d) \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {2 a \int \frac {\left (b x^3+a x^2\right )^{5/2}}{x^5}dx}{9 b}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}\right )}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

\(\Big \downarrow \) 1920

\(\displaystyle \frac {\left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{21 b}-\frac {2 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{19 b x}-\frac {12 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{17 b x^2}-\frac {10 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{15 b x^3}-\frac {8 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{13 b x^4}-\frac {6 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{11 b x^5}-\frac {4 a \left (\frac {2 \left (a x^2+b x^3\right )^{7/2}}{9 b x^6}-\frac {4 a \left (a x^2+b x^3\right )^{7/2}}{63 b^2 x^7}\right )}{11 b}\right )}{13 b}\right )}{15 b}\right )}{17 b}\right )}{19 b}\right )}{3 b}\right ) (23 b c-16 a d)}{23 b}+\frac {2 d x \left (a x^2+b x^3\right )^{7/2}}{23 b}\)

Input:

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

Output:

(2*d*x*(a*x^2 + b*x^3)^(7/2))/(23*b) + ((23*b*c - 16*a*d)*((2*(a*x^2 + b*x 
^3)^(7/2))/(21*b) - (2*a*((2*(a*x^2 + b*x^3)^(7/2))/(19*b*x) - (12*a*((2*( 
a*x^2 + b*x^3)^(7/2))/(17*b*x^2) - (10*a*((2*(a*x^2 + b*x^3)^(7/2))/(15*b* 
x^3) - (8*a*((2*(a*x^2 + b*x^3)^(7/2))/(13*b*x^4) - (6*a*((2*(a*x^2 + b*x^ 
3)^(7/2))/(11*b*x^5) - (4*a*((-4*a*(a*x^2 + b*x^3)^(7/2))/(63*b^2*x^7) + ( 
2*(a*x^2 + b*x^3)^(7/2))/(9*b*x^6)))/(11*b)))/(13*b)))/(15*b)))/(17*b)))/( 
19*b)))/(3*b)))/(23*b)
 

Defintions of rubi rules used

rule 1908

Int[((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol] :> Simp[(a*x^j + 
b*x^n)^(p + 1)/(a*(j*p + 1)*x^(j - 1)), x] - Simp[b*((n*p + n - j + 1)/(a*( 
j*p + 1)))   Int[x^(n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, j, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(n*p + n - j + 1)/(n 
- j)], 0] && NeQ[j*p + 1, 0]
 

rule 1920

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[(-c^(j - 1))*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(n - j 
)*(p + 1))), x] /; FreeQ[{a, b, c, j, m, n, p}, x] &&  !IntegerQ[p] && NeQ[ 
n, j] && EqQ[m + n*p + n - j + 1, 0] && (IntegerQ[j] || GtQ[c, 0])
 

rule 1922

Int[((c_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(n_.))^(p_), x_Symbol 
] :> Simp[c^(j - 1)*(c*x)^(m - j + 1)*((a*x^j + b*x^n)^(p + 1)/(a*(m + j*p 
+ 1))), x] - Simp[b*((m + n*p + n - j + 1)/(a*c^(n - j)*(m + j*p + 1)))   I 
nt[(c*x)^(m + n - j)*(a*x^j + b*x^n)^p, x], x] /; FreeQ[{a, b, c, j, m, n, 
p}, x] &&  !IntegerQ[p] && NeQ[n, j] && ILtQ[Simplify[(m + n*p + n - j + 1) 
/(n - j)], 0] && NeQ[m + j*p + 1, 0] && (IntegersQ[j, n] || GtQ[c, 0])
 

rule 1945

Int[((e_.)*(x_))^(m_.)*((a_.)*(x_)^(j_.) + (b_.)*(x_)^(jn_.))^(p_)*((c_) + 
(d_.)*(x_)^(n_.)), x_Symbol] :> Simp[d*e^(j - 1)*(e*x)^(m - j + 1)*((a*x^j 
+ b*x^(j + n))^(p + 1)/(b*(m + n + p*(j + n) + 1))), x] - Simp[(a*d*(m + j* 
p + 1) - b*c*(m + n + p*(j + n) + 1))/(b*(m + n + p*(j + n) + 1))   Int[(e* 
x)^m*(a*x^j + b*x^(j + n))^p, x], x] /; FreeQ[{a, b, c, d, e, j, m, n, p}, 
x] && EqQ[jn, j + n] &&  !IntegerQ[p] && NeQ[b*c - a*d, 0] && NeQ[m + n + p 
*(j + n) + 1, 0] && (GtQ[e, 0] || IntegerQ[j])
 

Maple [A] (verified)

Time = 0.64 (sec) , antiderivative size = 58, normalized size of antiderivative = 0.18

method	result	size
pseudoelliptic	\(-\frac {32 \left (b x +a \right )^{\frac {7}{2}} \left (-\frac {273 \left (\frac {11 d x}{13}+c \right ) x^{2} b^{3}}{16}+\frac {91 x \left (\frac {27 d x}{26}+c \right ) a \,b^{2}}{12}-\frac {13 \left (\frac {21 d x}{13}+c \right ) a^{2} b}{6}+a^{3} d \right )}{3003 b^{4}}\)	\(58\)
gosper	\(\frac {2 \left (b x +a \right ) \left (2909907 d \,x^{8} b^{8}-2217072 a \,b^{7} d \,x^{7}+3187041 b^{8} c \,x^{7}+1633632 a^{2} b^{6} d \,x^{6}-2348346 a \,b^{7} c \,x^{6}-1153152 a^{3} b^{5} d \,x^{5}+1657656 a^{2} b^{6} c \,x^{5}+768768 a^{4} b^{4} d \,x^{4}-1105104 a^{3} b^{5} c \,x^{4}-473088 a^{5} b^{3} d \,x^{3}+680064 a^{4} b^{4} c \,x^{3}+258048 a^{6} b^{2} d \,x^{2}-370944 a^{5} b^{3} c \,x^{2}-114688 a^{7} b d x +164864 a^{6} b^{2} c x +32768 a^{8} d -47104 a^{7} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{66927861 b^{9} x^{5}}\)	\(205\)
default	\(\frac {2 \left (b x +a \right ) \left (2909907 d \,x^{8} b^{8}-2217072 a \,b^{7} d \,x^{7}+3187041 b^{8} c \,x^{7}+1633632 a^{2} b^{6} d \,x^{6}-2348346 a \,b^{7} c \,x^{6}-1153152 a^{3} b^{5} d \,x^{5}+1657656 a^{2} b^{6} c \,x^{5}+768768 a^{4} b^{4} d \,x^{4}-1105104 a^{3} b^{5} c \,x^{4}-473088 a^{5} b^{3} d \,x^{3}+680064 a^{4} b^{4} c \,x^{3}+258048 a^{6} b^{2} d \,x^{2}-370944 a^{5} b^{3} c \,x^{2}-114688 a^{7} b d x +164864 a^{6} b^{2} c x +32768 a^{8} d -47104 a^{7} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{66927861 b^{9} x^{5}}\)	\(205\)
orering	\(\frac {2 \left (b x +a \right ) \left (2909907 d \,x^{8} b^{8}-2217072 a \,b^{7} d \,x^{7}+3187041 b^{8} c \,x^{7}+1633632 a^{2} b^{6} d \,x^{6}-2348346 a \,b^{7} c \,x^{6}-1153152 a^{3} b^{5} d \,x^{5}+1657656 a^{2} b^{6} c \,x^{5}+768768 a^{4} b^{4} d \,x^{4}-1105104 a^{3} b^{5} c \,x^{4}-473088 a^{5} b^{3} d \,x^{3}+680064 a^{4} b^{4} c \,x^{3}+258048 a^{6} b^{2} d \,x^{2}-370944 a^{5} b^{3} c \,x^{2}-114688 a^{7} b d x +164864 a^{6} b^{2} c x +32768 a^{8} d -47104 a^{7} b c \right ) \left (b \,x^{3}+a \,x^{2}\right )^{\frac {5}{2}}}{66927861 b^{9} x^{5}}\)	\(205\)
risch	\(\frac {2 \sqrt {x^{2} \left (b x +a \right )}\, \left (2909907 b^{11} d \,x^{11}+6512649 a \,b^{10} d \,x^{10}+3187041 b^{11} c \,x^{10}+3712137 a^{2} b^{9} d \,x^{9}+7212777 a \,b^{10} c \,x^{9}+6435 a^{3} b^{8} d \,x^{8}+4173741 a^{2} b^{9} c \,x^{8}-6864 a^{4} b^{7} d \,x^{7}+9867 a^{3} b^{8} c \,x^{7}+7392 a^{5} b^{6} d \,x^{6}-10626 a^{4} b^{7} c \,x^{6}-8064 a^{6} b^{5} d \,x^{5}+11592 a^{5} b^{6} c \,x^{5}+8960 a^{7} b^{4} d \,x^{4}-12880 a^{6} b^{5} c \,x^{4}-10240 a^{8} b^{3} d \,x^{3}+14720 a^{7} b^{4} c \,x^{3}+12288 a^{9} b^{2} d \,x^{2}-17664 a^{8} b^{3} c \,x^{2}-16384 a^{10} b d x +23552 a^{9} b^{2} c x +32768 a^{11} d -47104 a^{10} b c \right )}{66927861 x \,b^{9}}\)	\(270\)
trager	\(\frac {2 \left (2909907 b^{11} d \,x^{11}+6512649 a \,b^{10} d \,x^{10}+3187041 b^{11} c \,x^{10}+3712137 a^{2} b^{9} d \,x^{9}+7212777 a \,b^{10} c \,x^{9}+6435 a^{3} b^{8} d \,x^{8}+4173741 a^{2} b^{9} c \,x^{8}-6864 a^{4} b^{7} d \,x^{7}+9867 a^{3} b^{8} c \,x^{7}+7392 a^{5} b^{6} d \,x^{6}-10626 a^{4} b^{7} c \,x^{6}-8064 a^{6} b^{5} d \,x^{5}+11592 a^{5} b^{6} c \,x^{5}+8960 a^{7} b^{4} d \,x^{4}-12880 a^{6} b^{5} c \,x^{4}-10240 a^{8} b^{3} d \,x^{3}+14720 a^{7} b^{4} c \,x^{3}+12288 a^{9} b^{2} d \,x^{2}-17664 a^{8} b^{3} c \,x^{2}-16384 a^{10} b d x +23552 a^{9} b^{2} c x +32768 a^{11} d -47104 a^{10} b c \right ) \sqrt {b \,x^{3}+a \,x^{2}}}{66927861 b^{9} x}\)	\(272\)

Input:

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

Output:

-32/3003*(b*x+a)^(7/2)*(-273/16*(11/13*d*x+c)*x^2*b^3+91/12*x*(27/26*d*x+c 
)*a*b^2-13/6*(21/13*d*x+c)*a^2*b+a^3*d)/b^4
 

Fricas [A] (verification not implemented)

Time = 0.07 (sec) , antiderivative size = 273, normalized size of antiderivative = 0.87 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (2909907 \, b^{11} d x^{11} - 47104 \, a^{10} b c + 32768 \, a^{11} d + 138567 \, {\left (23 \, b^{11} c + 47 \, a b^{10} d\right )} x^{10} + 7293 \, {\left (989 \, a b^{10} c + 509 \, a^{2} b^{9} d\right )} x^{9} + 1287 \, {\left (3243 \, a^{2} b^{9} c + 5 \, a^{3} b^{8} d\right )} x^{8} + 429 \, {\left (23 \, a^{3} b^{8} c - 16 \, a^{4} b^{7} d\right )} x^{7} - 462 \, {\left (23 \, a^{4} b^{7} c - 16 \, a^{5} b^{6} d\right )} x^{6} + 504 \, {\left (23 \, a^{5} b^{6} c - 16 \, a^{6} b^{5} d\right )} x^{5} - 560 \, {\left (23 \, a^{6} b^{5} c - 16 \, a^{7} b^{4} d\right )} x^{4} + 640 \, {\left (23 \, a^{7} b^{4} c - 16 \, a^{8} b^{3} d\right )} x^{3} - 768 \, {\left (23 \, a^{8} b^{3} c - 16 \, a^{9} b^{2} d\right )} x^{2} + 1024 \, {\left (23 \, a^{9} b^{2} c - 16 \, a^{10} b d\right )} x\right )} \sqrt {b x^{3} + a x^{2}}}{66927861 \, b^{9} x} \] Input:

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

Output:

2/66927861*(2909907*b^11*d*x^11 - 47104*a^10*b*c + 32768*a^11*d + 138567*( 
23*b^11*c + 47*a*b^10*d)*x^10 + 7293*(989*a*b^10*c + 509*a^2*b^9*d)*x^9 + 
1287*(3243*a^2*b^9*c + 5*a^3*b^8*d)*x^8 + 429*(23*a^3*b^8*c - 16*a^4*b^7*d 
)*x^7 - 462*(23*a^4*b^7*c - 16*a^5*b^6*d)*x^6 + 504*(23*a^5*b^6*c - 16*a^6 
*b^5*d)*x^5 - 560*(23*a^6*b^5*c - 16*a^7*b^4*d)*x^4 + 640*(23*a^7*b^4*c - 
16*a^8*b^3*d)*x^3 - 768*(23*a^8*b^3*c - 16*a^9*b^2*d)*x^2 + 1024*(23*a^9*b 
^2*c - 16*a^10*b*d)*x)*sqrt(b*x^3 + a*x^2)/(b^9*x)
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 252, normalized size of antiderivative = 0.80 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \, {\left (138567 \, b^{10} x^{10} + 313599 \, a b^{9} x^{9} + 181467 \, a^{2} b^{8} x^{8} + 429 \, a^{3} b^{7} x^{7} - 462 \, a^{4} b^{6} x^{6} + 504 \, a^{5} b^{5} x^{5} - 560 \, a^{6} b^{4} x^{4} + 640 \, a^{7} b^{3} x^{3} - 768 \, a^{8} b^{2} x^{2} + 1024 \, a^{9} b x - 2048 \, a^{10}\right )} \sqrt {b x + a} c}{2909907 \, b^{8}} + \frac {2 \, {\left (2909907 \, b^{11} x^{11} + 6512649 \, a b^{10} x^{10} + 3712137 \, a^{2} b^{9} x^{9} + 6435 \, a^{3} b^{8} x^{8} - 6864 \, a^{4} b^{7} x^{7} + 7392 \, a^{5} b^{6} x^{6} - 8064 \, a^{6} b^{5} x^{5} + 8960 \, a^{7} b^{4} x^{4} - 10240 \, a^{8} b^{3} x^{3} + 12288 \, a^{9} b^{2} x^{2} - 16384 \, a^{10} b x + 32768 \, a^{11}\right )} \sqrt {b x + a} d}{66927861 \, b^{9}} \] Input:

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

Output:

2/2909907*(138567*b^10*x^10 + 313599*a*b^9*x^9 + 181467*a^2*b^8*x^8 + 429* 
a^3*b^7*x^7 - 462*a^4*b^6*x^6 + 504*a^5*b^5*x^5 - 560*a^6*b^4*x^4 + 640*a^ 
7*b^3*x^3 - 768*a^8*b^2*x^2 + 1024*a^9*b*x - 2048*a^10)*sqrt(b*x + a)*c/b^ 
8 + 2/66927861*(2909907*b^11*x^11 + 6512649*a*b^10*x^10 + 3712137*a^2*b^9* 
x^9 + 6435*a^3*b^8*x^8 - 6864*a^4*b^7*x^7 + 7392*a^5*b^6*x^6 - 8064*a^6*b^ 
5*x^5 + 8960*a^7*b^4*x^4 - 10240*a^8*b^3*x^3 + 12288*a^9*b^2*x^2 - 16384*a 
^10*b*x + 32768*a^11)*sqrt(b*x + a)*d/b^9
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1034 vs. \(2 (278) = 556\).

Time = 0.22 (sec) , antiderivative size = 1034, normalized size of antiderivative = 3.29 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\text {Too large to display} \] Input:

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

Output:

2/334639305*(52003*(429*(b*x + a)^(15/2) - 3465*(b*x + a)^(13/2)*a + 12285 
*(b*x + a)^(11/2)*a^2 - 25025*(b*x + a)^(9/2)*a^3 + 32175*(b*x + a)^(7/2)* 
a^4 - 27027*(b*x + a)^(5/2)*a^5 + 15015*(b*x + a)^(3/2)*a^6 - 6435*sqrt(b* 
x + a)*a^7)*a^3*c*sgn(x)/b^7 + 9177*(6435*(b*x + a)^(17/2) - 58344*(b*x + 
a)^(15/2)*a + 235620*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 
850850*(b*x + a)^(9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a) 
^(5/2)*a^6 - 291720*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*a^2*c* 
sgn(x)/b^7 + 3059*(6435*(b*x + a)^(17/2) - 58344*(b*x + a)^(15/2)*a + 2356 
20*(b*x + a)^(13/2)*a^2 - 556920*(b*x + a)^(11/2)*a^3 + 850850*(b*x + a)^( 
9/2)*a^4 - 875160*(b*x + a)^(7/2)*a^5 + 612612*(b*x + a)^(5/2)*a^6 - 29172 
0*(b*x + a)^(3/2)*a^7 + 109395*sqrt(b*x + a)*a^8)*a^3*d*sgn(x)/b^8 + 4347* 
(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554268*(b*x + a)^(15 
/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a)^(11/2)*a^4 - 32 
33230*(b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7/2)*a^6 - 1662804*(b*x + a 
)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8 - 230945*sqrt(b*x + a)*a^9)*a*c*s 
gn(x)/b^7 + 4347*(12155*(b*x + a)^(19/2) - 122265*(b*x + a)^(17/2)*a + 554 
268*(b*x + a)^(15/2)*a^2 - 1492260*(b*x + a)^(13/2)*a^3 + 2645370*(b*x + a 
)^(11/2)*a^4 - 3233230*(b*x + a)^(9/2)*a^5 + 2771340*(b*x + a)^(7/2)*a^6 - 
 1662804*(b*x + a)^(5/2)*a^7 + 692835*(b*x + a)^(3/2)*a^8 - 230945*sqrt(b* 
x + a)*a^9)*a^2*d*sgn(x)/b^8 + 345*(46189*(b*x + a)^(21/2) - 510510*(b*...
 

Mupad [B] (verification not implemented)

Time = 9.69 (sec) , antiderivative size = 232, normalized size of antiderivative = 0.74 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {\sqrt {b\,x^3+a\,x^2}\,\left (\frac {2\,a\,x^9\,\left (509\,a\,d+989\,b\,c\right )}{9177}+\frac {2\,b\,x^{10}\,\left (47\,a\,d+23\,b\,c\right )}{483}+\frac {2\,b^2\,d\,x^{11}}{23}+\frac {4096\,a^{10}\,\left (16\,a\,d-23\,b\,c\right )}{66927861\,b^9}-\frac {2048\,a^9\,x\,\left (16\,a\,d-23\,b\,c\right )}{66927861\,b^8}-\frac {2\,a^3\,x^7\,\left (16\,a\,d-23\,b\,c\right )}{156009\,b^2}+\frac {4\,a^4\,x^6\,\left (16\,a\,d-23\,b\,c\right )}{289731\,b^3}-\frac {16\,a^5\,x^5\,\left (16\,a\,d-23\,b\,c\right )}{1062347\,b^4}+\frac {160\,a^6\,x^4\,\left (16\,a\,d-23\,b\,c\right )}{9561123\,b^5}-\frac {1280\,a^7\,x^3\,\left (16\,a\,d-23\,b\,c\right )}{66927861\,b^6}+\frac {512\,a^8\,x^2\,\left (16\,a\,d-23\,b\,c\right )}{22309287\,b^7}+\frac {2\,a^2\,x^8\,\left (5\,a\,d+3243\,b\,c\right )}{52003\,b}\right )}{x} \] Input:

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

Output:

((a*x^2 + b*x^3)^(1/2)*((2*a*x^9*(509*a*d + 989*b*c))/9177 + (2*b*x^10*(47 
*a*d + 23*b*c))/483 + (2*b^2*d*x^11)/23 + (4096*a^10*(16*a*d - 23*b*c))/(6 
6927861*b^9) - (2048*a^9*x*(16*a*d - 23*b*c))/(66927861*b^8) - (2*a^3*x^7* 
(16*a*d - 23*b*c))/(156009*b^2) + (4*a^4*x^6*(16*a*d - 23*b*c))/(289731*b^ 
3) - (16*a^5*x^5*(16*a*d - 23*b*c))/(1062347*b^4) + (160*a^6*x^4*(16*a*d - 
 23*b*c))/(9561123*b^5) - (1280*a^7*x^3*(16*a*d - 23*b*c))/(66927861*b^6) 
+ (512*a^8*x^2*(16*a*d - 23*b*c))/(22309287*b^7) + (2*a^2*x^8*(5*a*d + 324 
3*b*c))/(52003*b)))/x
 

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 261, normalized size of antiderivative = 0.83 \[ \int x^2 (c+d x) \left (a x^2+b x^3\right )^{5/2} \, dx=\frac {2 \sqrt {b x +a}\, \left (2909907 b^{11} d \,x^{11}+6512649 a \,b^{10} d \,x^{10}+3187041 b^{11} c \,x^{10}+3712137 a^{2} b^{9} d \,x^{9}+7212777 a \,b^{10} c \,x^{9}+6435 a^{3} b^{8} d \,x^{8}+4173741 a^{2} b^{9} c \,x^{8}-6864 a^{4} b^{7} d \,x^{7}+9867 a^{3} b^{8} c \,x^{7}+7392 a^{5} b^{6} d \,x^{6}-10626 a^{4} b^{7} c \,x^{6}-8064 a^{6} b^{5} d \,x^{5}+11592 a^{5} b^{6} c \,x^{5}+8960 a^{7} b^{4} d \,x^{4}-12880 a^{6} b^{5} c \,x^{4}-10240 a^{8} b^{3} d \,x^{3}+14720 a^{7} b^{4} c \,x^{3}+12288 a^{9} b^{2} d \,x^{2}-17664 a^{8} b^{3} c \,x^{2}-16384 a^{10} b d x +23552 a^{9} b^{2} c x +32768 a^{11} d -47104 a^{10} b c \right )}{66927861 b^{9}} \] Input:

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

Output:

(2*sqrt(a + b*x)*(32768*a**11*d - 47104*a**10*b*c - 16384*a**10*b*d*x + 23 
552*a**9*b**2*c*x + 12288*a**9*b**2*d*x**2 - 17664*a**8*b**3*c*x**2 - 1024 
0*a**8*b**3*d*x**3 + 14720*a**7*b**4*c*x**3 + 8960*a**7*b**4*d*x**4 - 1288 
0*a**6*b**5*c*x**4 - 8064*a**6*b**5*d*x**5 + 11592*a**5*b**6*c*x**5 + 7392 
*a**5*b**6*d*x**6 - 10626*a**4*b**7*c*x**6 - 6864*a**4*b**7*d*x**7 + 9867* 
a**3*b**8*c*x**7 + 6435*a**3*b**8*d*x**8 + 4173741*a**2*b**9*c*x**8 + 3712 
137*a**2*b**9*d*x**9 + 7212777*a*b**10*c*x**9 + 6512649*a*b**10*d*x**10 + 
3187041*b**11*c*x**10 + 2909907*b**11*d*x**11))/(66927861*b**9)