\(\int \frac {(c+d x)^{5/2} (a+b x^2)^2}{(e x)^{21/2}} \, dx\) [823]

Optimal result

Integrand size = 26, antiderivative size = 304 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}+\frac {24 a^2 d (c+d x)^{7/2}}{323 c^2 e^2 (e x)^{17/2}}-\frac {4 a \left (323 b c^2+60 a d^2\right ) (c+d x)^{7/2}}{4845 c^3 e^3 (e x)^{15/2}}+\frac {32 a d \left (323 b c^2+60 a d^2\right ) (c+d x)^{7/2}}{62985 c^4 e^4 (e x)^{13/2}}-\frac {2 \left (20995 b^2 c^4+10336 a b c^2 d^2+1920 a^2 d^4\right ) (c+d x)^{7/2}}{230945 c^5 e^5 (e x)^{11/2}}+\frac {8 d \left (20995 b^2 c^4+10336 a b c^2 d^2+1920 a^2 d^4\right ) (c+d x)^{7/2}}{2078505 c^6 e^6 (e x)^{9/2}}-\frac {16 d^2 \left (20995 b^2 c^4+10336 a b c^2 d^2+1920 a^2 d^4\right ) (c+d x)^{7/2}}{14549535 c^7 e^7 (e x)^{7/2}} \] Output:

-2/19*a^2*(d*x+c)^(7/2)/c/e/(e*x)^(19/2)+24/323*a^2*d*(d*x+c)^(7/2)/c^2/e^ 
2/(e*x)^(17/2)-4/4845*a*(60*a*d^2+323*b*c^2)*(d*x+c)^(7/2)/c^3/e^3/(e*x)^( 
15/2)+32/62985*a*d*(60*a*d^2+323*b*c^2)*(d*x+c)^(7/2)/c^4/e^4/(e*x)^(13/2) 
-2/230945*(1920*a^2*d^4+10336*a*b*c^2*d^2+20995*b^2*c^4)*(d*x+c)^(7/2)/c^5 
/e^5/(e*x)^(11/2)+8/2078505*d*(1920*a^2*d^4+10336*a*b*c^2*d^2+20995*b^2*c^ 
4)*(d*x+c)^(7/2)/c^6/e^6/(e*x)^(9/2)-16/14549535*d^2*(1920*a^2*d^4+10336*a 
*b*c^2*d^2+20995*b^2*c^4)*(d*x+c)^(7/2)/c^7/e^7/(e*x)^(7/2)
 

Mathematica [A] (verified)

Time = 0.16 (sec) , antiderivative size = 179, normalized size of antiderivative = 0.59 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=-\frac {2 (c+d x)^{7/2} \left (20995 b^2 c^4 x^4 \left (63 c^2-28 c d x+8 d^2 x^2\right )+646 a b c^2 x^2 \left (3003 c^4-1848 c^3 d x+1008 c^2 d^2 x^2-448 c d^3 x^3+128 d^4 x^4\right )+15 a^2 \left (51051 c^6-36036 c^5 d x+24024 c^4 d^2 x^2-14784 c^3 d^3 x^3+8064 c^2 d^4 x^4-3584 c d^5 x^5+1024 d^6 x^6\right )\right )}{14549535 c^7 e^9 x^8 (e x)^{3/2}} \] Input:

Integrate[((c + d*x)^(5/2)*(a + b*x^2)^2)/(e*x)^(21/2),x]
 

Output:

(-2*(c + d*x)^(7/2)*(20995*b^2*c^4*x^4*(63*c^2 - 28*c*d*x + 8*d^2*x^2) + 6 
46*a*b*c^2*x^2*(3003*c^4 - 1848*c^3*d*x + 1008*c^2*d^2*x^2 - 448*c*d^3*x^3 
 + 128*d^4*x^4) + 15*a^2*(51051*c^6 - 36036*c^5*d*x + 24024*c^4*d^2*x^2 - 
14784*c^3*d^3*x^3 + 8064*c^2*d^4*x^4 - 3584*c*d^5*x^5 + 1024*d^6*x^6)))/(1 
4549535*c^7*e^9*x^8*(e*x)^(3/2))
 

Rubi [A] (verified)

Time = 0.82 (sec) , antiderivative size = 303, normalized size of antiderivative = 1.00, number of steps used = 10, number of rules used = 10, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.385, Rules used = {520, 27, 2124, 27, 520, 27, 87, 55, 55, 48}

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 (c+d x)^{5/2}}{(e x)^{21/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {(c+d x)^{5/2} \left (-19 b^2 c x^3-38 a b c x+12 a^2 d\right )}{2 (e x)^{19/2}}dx}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {(c+d x)^{5/2} \left (-19 b^2 c x^3-38 a b c x+12 a^2 d\right )}{(e x)^{19/2}}dx}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {(c+d x)^{5/2} \left (323 b^2 c^2 x^2+2 a \left (323 b c^2+60 a d^2\right )\right )}{2 (e x)^{17/2}}dx}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {(c+d x)^{5/2} \left (323 b^2 c^2 x^2+2 a \left (323 b c^2+60 a d^2\right )\right )}{(e x)^{17/2}}dx}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {\left (16 a d \left (323 b c^2+60 a d^2\right )-4845 b^2 c^3 x\right ) (c+d x)^{5/2}}{2 (e x)^{15/2}}dx}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (16 a d \left (323 b c^2+60 a d^2\right )-4845 b^2 c^3 x\right ) (c+d x)^{5/2}}{(e x)^{15/2}}dx}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (1920 a^2 d^4+10336 a b c^2 d^2+20995 b^2 c^4\right ) \int \frac {(c+d x)^{5/2}}{(e x)^{13/2}}dx}{13 c e}-\frac {32 a d (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{13 c e (e x)^{13/2}}}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (1920 a^2 d^4+10336 a b c^2 d^2+20995 b^2 c^4\right ) \left (-\frac {4 d \int \frac {(c+d x)^{5/2}}{(e x)^{11/2}}dx}{11 c e}-\frac {2 (c+d x)^{7/2}}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {32 a d (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{13 c e (e x)^{13/2}}}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (1920 a^2 d^4+10336 a b c^2 d^2+20995 b^2 c^4\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {(c+d x)^{5/2}}{(e x)^{9/2}}dx}{9 c e}-\frac {2 (c+d x)^{7/2}}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (c+d x)^{7/2}}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {32 a d (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{13 c e (e x)^{13/2}}}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {-\frac {-\frac {3 \left (1920 a^2 d^4+10336 a b c^2 d^2+20995 b^2 c^4\right ) \left (-\frac {4 d \left (\frac {4 d (c+d x)^{7/2}}{63 c^2 e^2 (e x)^{7/2}}-\frac {2 (c+d x)^{7/2}}{9 c e (e x)^{9/2}}\right )}{11 c e}-\frac {2 (c+d x)^{7/2}}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {32 a d (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{13 c e (e x)^{13/2}}}{15 c e}-\frac {4 a (c+d x)^{7/2} \left (60 a d^2+323 b c^2\right )}{15 c e (e x)^{15/2}}}{17 c e}-\frac {24 a^2 d (c+d x)^{7/2}}{17 c e (e x)^{17/2}}}{19 c e}-\frac {2 a^2 (c+d x)^{7/2}}{19 c e (e x)^{19/2}}\)

Input:

Int[((c + d*x)^(5/2)*(a + b*x^2)^2)/(e*x)^(21/2),x]
 

Output:

(-2*a^2*(c + d*x)^(7/2))/(19*c*e*(e*x)^(19/2)) - ((-24*a^2*d*(c + d*x)^(7/ 
2))/(17*c*e*(e*x)^(17/2)) - ((-4*a*(323*b*c^2 + 60*a*d^2)*(c + d*x)^(7/2)) 
/(15*c*e*(e*x)^(15/2)) - ((-32*a*d*(323*b*c^2 + 60*a*d^2)*(c + d*x)^(7/2)) 
/(13*c*e*(e*x)^(13/2)) - (3*(20995*b^2*c^4 + 10336*a*b*c^2*d^2 + 1920*a^2* 
d^4)*((-2*(c + d*x)^(7/2))/(11*c*e*(e*x)^(11/2)) - (4*d*((-2*(c + d*x)^(7/ 
2))/(9*c*e*(e*x)^(9/2)) + (4*d*(c + d*x)^(7/2))/(63*c^2*e^2*(e*x)^(7/2)))) 
/(11*c*e)))/(13*c*e))/(15*c*e))/(17*c*e))/(19*c*e)
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, x]]
 

rule 48

Int[((a_.) + (b_.)*(x_))^(m_.)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp 
[(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] /; FreeQ[{ 
a, b, c, d, m, n}, x] && EqQ[m + n + 2, 0] && NeQ[m, -1]
 

rule 55

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((b*c - a*d)*(m + 1))), x] - Simp[d*(S 
implify[m + n + 2]/((b*c - a*d)*(m + 1)))   Int[(a + b*x)^Simplify[m + 1]*( 
c + d*x)^n, x], x] /; FreeQ[{a, b, c, d, m, n}, x] && ILtQ[Simplify[m + n + 
 2], 0] && NeQ[m, -1] &&  !(LtQ[m, -1] && LtQ[n, -1] && (EqQ[a, 0] || (NeQ[ 
c, 0] && LtQ[m - n, 0] && IntegerQ[n]))) && (SumSimplerQ[m, 1] ||  !SumSimp 
lerQ[n, 1])
 

rule 87

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[(-(b*e - a*f))*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(f*(p 
+ 1)*(c*f - d*e))), x] - Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p 
+ 1)))/(f*(p + 1)*(c*f - d*e))   Int[(c + d*x)^n*(e + f*x)^(p + 1), x], x] 
/; FreeQ[{a, b, c, d, e, f, n}, x] && LtQ[p, -1] && ( !LtQ[n, -1] || Intege 
rQ[p] ||  !(IntegerQ[n] ||  !(EqQ[e, 0] ||  !(EqQ[c, 0] || LtQ[p, n]))))
 

rule 520

Int[((e_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_.), 
 x_Symbol] :> With[{Qx = PolynomialQuotient[(a + b*x^2)^p, e*x, x], R = Pol 
ynomialRemainder[(a + b*x^2)^p, e*x, x]}, Simp[R*(e*x)^(m + 1)*((c + d*x)^( 
n + 1)/((m + 1)*(e*c))), x] + Simp[1/((m + 1)*(e*c))   Int[(e*x)^(m + 1)*(c 
 + d*x)^n*ExpandToSum[(m + 1)*(e*c)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, e, n}, x] && IGtQ[p, 0] && LtQ[m, -1] &&  !IntegerQ[n]
 

rule 2124

Int[(Px_)*((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_.), x_Symbol] : 
> With[{Qx = PolynomialQuotient[Px, a + b*x, x], R = PolynomialRemainder[Px 
, a + b*x, x]}, Simp[R*(a + b*x)^(m + 1)*((c + d*x)^(n + 1)/((m + 1)*(b*c - 
 a*d))), x] + Simp[1/((m + 1)*(b*c - a*d))   Int[(a + b*x)^(m + 1)*(c + d*x 
)^n*ExpandToSum[(m + 1)*(b*c - a*d)*Qx - d*R*(m + n + 2), x], x], x]] /; Fr 
eeQ[{a, b, c, d, n}, x] && PolyQ[Px, x] && LtQ[m, -1] && (IntegerQ[m] ||  ! 
ILtQ[n, -1])
 

Maple [A] (verified)

Time = 0.28 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 x \left (d x +c \right )^{\frac {7}{2}} \left (15360 a^{2} d^{6} x^{6}+82688 a b \,c^{2} d^{4} x^{6}+167960 b^{2} c^{4} d^{2} x^{6}-53760 a^{2} c \,d^{5} x^{5}-289408 a b \,c^{3} d^{3} x^{5}-587860 b^{2} c^{5} d \,x^{5}+120960 a^{2} c^{2} d^{4} x^{4}+651168 a b \,c^{4} d^{2} x^{4}+1322685 b^{2} c^{6} x^{4}-221760 a^{2} c^{3} d^{3} x^{3}-1193808 a b \,c^{5} d \,x^{3}+360360 a^{2} c^{4} d^{2} x^{2}+1939938 a b \,c^{6} x^{2}-540540 a^{2} c^{5} d x +765765 a^{2} c^{6}\right )}{14549535 c^{7} \left (e x \right )^{\frac {21}{2}}}\)	\(200\)
orering	\(-\frac {2 x \left (d x +c \right )^{\frac {7}{2}} \left (15360 a^{2} d^{6} x^{6}+82688 a b \,c^{2} d^{4} x^{6}+167960 b^{2} c^{4} d^{2} x^{6}-53760 a^{2} c \,d^{5} x^{5}-289408 a b \,c^{3} d^{3} x^{5}-587860 b^{2} c^{5} d \,x^{5}+120960 a^{2} c^{2} d^{4} x^{4}+651168 a b \,c^{4} d^{2} x^{4}+1322685 b^{2} c^{6} x^{4}-221760 a^{2} c^{3} d^{3} x^{3}-1193808 a b \,c^{5} d \,x^{3}+360360 a^{2} c^{4} d^{2} x^{2}+1939938 a b \,c^{6} x^{2}-540540 a^{2} c^{5} d x +765765 a^{2} c^{6}\right )}{14549535 c^{7} \left (e x \right )^{\frac {21}{2}}}\)	\(200\)
default	\(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (15360 a^{2} d^{8} x^{8}+82688 a b \,c^{2} d^{6} x^{8}+167960 b^{2} c^{4} d^{4} x^{8}-23040 a^{2} c \,d^{7} x^{7}-124032 a b \,c^{3} d^{5} x^{7}-251940 b^{2} c^{5} d^{3} x^{7}+28800 a^{2} c^{2} d^{6} x^{6}+155040 a b \,c^{4} d^{4} x^{6}+314925 b^{2} c^{6} d^{2} x^{6}-33600 a^{2} c^{3} d^{5} x^{5}-180880 a b \,c^{5} d^{3} x^{5}+2057510 b^{2} c^{7} d \,x^{5}+37800 a^{2} c^{4} d^{4} x^{4}+203490 a b \,c^{6} d^{2} x^{4}+1322685 b^{2} c^{8} x^{4}-41580 a^{2} c^{5} d^{3} x^{3}+2686068 a b \,c^{7} d \,x^{3}+45045 a^{2} c^{6} d^{2} x^{2}+1939938 a b \,c^{8} x^{2}+990990 a^{2} c^{7} d x +765765 a^{2} c^{8}\right )}{14549535 x^{9} c^{7} e^{10} \sqrt {e x}}\)	\(287\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (15360 a^{2} d^{9} x^{9}+82688 a b \,c^{2} d^{7} x^{9}+167960 b^{2} c^{4} d^{5} x^{9}-7680 a^{2} c \,d^{8} x^{8}-41344 a b \,c^{3} d^{6} x^{8}-83980 b^{2} c^{5} d^{4} x^{8}+5760 a^{2} c^{2} d^{7} x^{7}+31008 a b \,c^{4} d^{5} x^{7}+62985 b^{2} c^{6} d^{3} x^{7}-4800 a^{2} c^{3} d^{6} x^{6}-25840 a b \,c^{5} d^{4} x^{6}+2372435 b^{2} c^{7} d^{2} x^{6}+4200 a^{2} c^{4} d^{5} x^{5}+22610 a b \,c^{6} d^{3} x^{5}+3380195 b^{2} c^{8} d \,x^{5}-3780 a^{2} c^{5} d^{4} x^{4}+2889558 a b \,c^{7} d^{2} x^{4}+1322685 b^{2} c^{9} x^{4}+3465 a^{2} c^{6} d^{3} x^{3}+4626006 a b \,c^{8} d \,x^{3}+1036035 a^{2} c^{7} d^{2} x^{2}+1939938 a b \,c^{9} x^{2}+1756755 a^{2} c^{8} d x +765765 a^{2} c^{9}\right )}{14549535 e^{10} \sqrt {e x}\, x^{9} c^{7}}\)	\(328\)

Input:

int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(21/2),x,method=_RETURNVERBOSE)
 

Output:

-2/14549535*x*(d*x+c)^(7/2)*(15360*a^2*d^6*x^6+82688*a*b*c^2*d^4*x^6+16796 
0*b^2*c^4*d^2*x^6-53760*a^2*c*d^5*x^5-289408*a*b*c^3*d^3*x^5-587860*b^2*c^ 
5*d*x^5+120960*a^2*c^2*d^4*x^4+651168*a*b*c^4*d^2*x^4+1322685*b^2*c^6*x^4- 
221760*a^2*c^3*d^3*x^3-1193808*a*b*c^5*d*x^3+360360*a^2*c^4*d^2*x^2+193993 
8*a*b*c^6*x^2-540540*a^2*c^5*d*x+765765*a^2*c^6)/c^7/(e*x)^(21/2)
 

Fricas [A] (verification not implemented)

Time = 0.10 (sec) , antiderivative size = 309, normalized size of antiderivative = 1.02 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=-\frac {2 \, {\left (1756755 \, a^{2} c^{8} d x + 765765 \, a^{2} c^{9} + 8 \, {\left (20995 \, b^{2} c^{4} d^{5} + 10336 \, a b c^{2} d^{7} + 1920 \, a^{2} d^{9}\right )} x^{9} - 4 \, {\left (20995 \, b^{2} c^{5} d^{4} + 10336 \, a b c^{3} d^{6} + 1920 \, a^{2} c d^{8}\right )} x^{8} + 3 \, {\left (20995 \, b^{2} c^{6} d^{3} + 10336 \, a b c^{4} d^{5} + 1920 \, a^{2} c^{2} d^{7}\right )} x^{7} + 5 \, {\left (474487 \, b^{2} c^{7} d^{2} - 5168 \, a b c^{5} d^{4} - 960 \, a^{2} c^{3} d^{6}\right )} x^{6} + 35 \, {\left (96577 \, b^{2} c^{8} d + 646 \, a b c^{6} d^{3} + 120 \, a^{2} c^{4} d^{5}\right )} x^{5} + 63 \, {\left (20995 \, b^{2} c^{9} + 45866 \, a b c^{7} d^{2} - 60 \, a^{2} c^{5} d^{4}\right )} x^{4} + 231 \, {\left (20026 \, a b c^{8} d + 15 \, a^{2} c^{6} d^{3}\right )} x^{3} + 3003 \, {\left (646 \, a b c^{9} + 345 \, a^{2} c^{7} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{14549535 \, c^{7} e^{11} x^{10}} \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(21/2),x, algorithm="fricas")
 

Output:

-2/14549535*(1756755*a^2*c^8*d*x + 765765*a^2*c^9 + 8*(20995*b^2*c^4*d^5 + 
 10336*a*b*c^2*d^7 + 1920*a^2*d^9)*x^9 - 4*(20995*b^2*c^5*d^4 + 10336*a*b* 
c^3*d^6 + 1920*a^2*c*d^8)*x^8 + 3*(20995*b^2*c^6*d^3 + 10336*a*b*c^4*d^5 + 
 1920*a^2*c^2*d^7)*x^7 + 5*(474487*b^2*c^7*d^2 - 5168*a*b*c^5*d^4 - 960*a^ 
2*c^3*d^6)*x^6 + 35*(96577*b^2*c^8*d + 646*a*b*c^6*d^3 + 120*a^2*c^4*d^5)* 
x^5 + 63*(20995*b^2*c^9 + 45866*a*b*c^7*d^2 - 60*a^2*c^5*d^4)*x^4 + 231*(2 
0026*a*b*c^8*d + 15*a^2*c^6*d^3)*x^3 + 3003*(646*a*b*c^9 + 345*a^2*c^7*d^2 
)*x^2)*sqrt(d*x + c)*sqrt(e*x)/(c^7*e^11*x^10)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(5/2)*(b*x**2+a)**2/(e*x)**(21/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(21/2),x, algorithm="maxima")
 

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(e>0)', see `assume?` for more de 
tails)Is e
 

Giac [A] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 404, normalized size of antiderivative = 1.33 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=-\frac {2 \, {\left ({\left ({\left ({\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {2 \, {\left (20995 \, b^{2} c^{6} d^{5} e^{9} + 10336 \, a b c^{4} d^{7} e^{9} + 1920 \, a^{2} c^{2} d^{9} e^{9}\right )} {\left (d x + c\right )}}{c^{9}} - \frac {19 \, {\left (20995 \, b^{2} c^{7} d^{5} e^{9} + 10336 \, a b c^{5} d^{7} e^{9} + 1920 \, a^{2} c^{3} d^{9} e^{9}\right )}}{c^{9}}\right )} + \frac {323 \, {\left (20995 \, b^{2} c^{8} d^{5} e^{9} + 10336 \, a b c^{6} d^{7} e^{9} + 1920 \, a^{2} c^{4} d^{9} e^{9}\right )}}{c^{9}}\right )} - \frac {6460 \, {\left (2249 \, b^{2} c^{9} d^{5} e^{9} + 1292 \, a b c^{7} d^{7} e^{9} + 240 \, a^{2} c^{5} d^{9} e^{9}\right )}}{c^{9}}\right )} {\left (d x + c\right )} + \frac {41990 \, {\left (389 \, b^{2} c^{10} d^{5} e^{9} + 323 \, a b c^{8} d^{7} e^{9} + 60 \, a^{2} c^{6} d^{9} e^{9}\right )}}{c^{9}}\right )} {\left (d x + c\right )} - \frac {923780 \, {\left (10 \, b^{2} c^{11} d^{5} e^{9} + 13 \, a b c^{9} d^{7} e^{9} + 3 \, a^{2} c^{7} d^{9} e^{9}\right )}}{c^{9}}\right )} {\left (d x + c\right )} + \frac {2078505 \, {\left (b^{2} c^{12} d^{5} e^{9} + 2 \, a b c^{10} d^{7} e^{9} + a^{2} c^{8} d^{9} e^{9}\right )}}{c^{9}}\right )} {\left (d x + c\right )}^{\frac {7}{2}} d^{11}}{14549535 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {19}{2}} e^{10} {\left | d \right |}} \] Input:

integrate((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(21/2),x, algorithm="giac")
 

Output:

-2/14549535*(((((d*x + c)*(4*(d*x + c)*(2*(20995*b^2*c^6*d^5*e^9 + 10336*a 
*b*c^4*d^7*e^9 + 1920*a^2*c^2*d^9*e^9)*(d*x + c)/c^9 - 19*(20995*b^2*c^7*d 
^5*e^9 + 10336*a*b*c^5*d^7*e^9 + 1920*a^2*c^3*d^9*e^9)/c^9) + 323*(20995*b 
^2*c^8*d^5*e^9 + 10336*a*b*c^6*d^7*e^9 + 1920*a^2*c^4*d^9*e^9)/c^9) - 6460 
*(2249*b^2*c^9*d^5*e^9 + 1292*a*b*c^7*d^7*e^9 + 240*a^2*c^5*d^9*e^9)/c^9)* 
(d*x + c) + 41990*(389*b^2*c^10*d^5*e^9 + 323*a*b*c^8*d^7*e^9 + 60*a^2*c^6 
*d^9*e^9)/c^9)*(d*x + c) - 923780*(10*b^2*c^11*d^5*e^9 + 13*a*b*c^9*d^7*e^ 
9 + 3*a^2*c^7*d^9*e^9)/c^9)*(d*x + c) + 2078505*(b^2*c^12*d^5*e^9 + 2*a*b* 
c^10*d^7*e^9 + a^2*c^8*d^9*e^9)/c^9)*(d*x + c)^(7/2)*d^11/(((d*x + c)*d*e 
- c*d*e)^(19/2)*e^10*abs(d))
 

Mupad [B] (verification not implemented)

Time = 9.25 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.12 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2\,c^2}{19\,e^{10}}+\frac {x^7\,\left (11520\,a^2\,c^2\,d^7+62016\,a\,b\,c^4\,d^5+125970\,b^2\,c^6\,d^3\right )}{14549535\,c^7\,e^{10}}-\frac {x^6\,\left (9600\,a^2\,c^3\,d^6+51680\,a\,b\,c^5\,d^4-4744870\,b^2\,c^7\,d^2\right )}{14549535\,c^7\,e^{10}}+\frac {x^4\,\left (-7560\,a^2\,c^5\,d^4+5779116\,a\,b\,c^7\,d^2+2645370\,b^2\,c^9\right )}{14549535\,c^7\,e^{10}}+\frac {x^9\,\left (30720\,a^2\,d^9+165376\,a\,b\,c^2\,d^7+335920\,b^2\,c^4\,d^5\right )}{14549535\,c^7\,e^{10}}-\frac {x^8\,\left (15360\,a^2\,c\,d^8+82688\,a\,b\,c^3\,d^6+167960\,b^2\,c^5\,d^4\right )}{14549535\,c^7\,e^{10}}+\frac {x^5\,\left (8400\,a^2\,c^4\,d^5+45220\,a\,b\,c^6\,d^3+6760390\,b^2\,c^8\,d\right )}{14549535\,c^7\,e^{10}}+\frac {2\,a\,x^2\,\left (646\,b\,c^2+345\,a\,d^2\right )}{4845\,e^{10}}+\frac {78\,a^2\,c\,d\,x}{323\,e^{10}}+\frac {2\,a\,d\,x^3\,\left (20026\,b\,c^2+15\,a\,d^2\right )}{62985\,c\,e^{10}}\right )}{x^9\,\sqrt {e\,x}} \] Input:

int(((a + b*x^2)^2*(c + d*x)^(5/2))/(e*x)^(21/2),x)
 

Output:

-((c + d*x)^(1/2)*((2*a^2*c^2)/(19*e^10) + (x^7*(11520*a^2*c^2*d^7 + 12597 
0*b^2*c^6*d^3 + 62016*a*b*c^4*d^5))/(14549535*c^7*e^10) - (x^6*(9600*a^2*c 
^3*d^6 - 4744870*b^2*c^7*d^2 + 51680*a*b*c^5*d^4))/(14549535*c^7*e^10) + ( 
x^4*(2645370*b^2*c^9 - 7560*a^2*c^5*d^4 + 5779116*a*b*c^7*d^2))/(14549535* 
c^7*e^10) + (x^9*(30720*a^2*d^9 + 335920*b^2*c^4*d^5 + 165376*a*b*c^2*d^7) 
)/(14549535*c^7*e^10) - (x^8*(15360*a^2*c*d^8 + 167960*b^2*c^5*d^4 + 82688 
*a*b*c^3*d^6))/(14549535*c^7*e^10) + (x^5*(6760390*b^2*c^8*d + 8400*a^2*c^ 
4*d^5 + 45220*a*b*c^6*d^3))/(14549535*c^7*e^10) + (2*a*x^2*(345*a*d^2 + 64 
6*b*c^2))/(4845*e^10) + (78*a^2*c*d*x)/(323*e^10) + (2*a*d*x^3*(15*a*d^2 + 
 20026*b*c^2))/(62985*c*e^10)))/(x^9*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.48 (sec) , antiderivative size = 553, normalized size of antiderivative = 1.82 \[ \int \frac {(c+d x)^{5/2} \left (a+b x^2\right )^2}{(e x)^{21/2}} \, dx=\frac {2 \sqrt {e}\, \left (-4626006 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{8} d \,x^{3}-2889558 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{7} d^{2} x^{4}-22610 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} d^{3} x^{5}+25840 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d^{4} x^{6}-31008 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{5} x^{7}+41344 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{6} x^{8}-82688 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{7} x^{9}-15360 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{9} x^{9}-1322685 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{9} x^{4}+167960 \sqrt {d}\, b^{2} c^{4} d^{5} x^{10}-765765 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{9}+15360 \sqrt {d}\, a^{2} d^{9} x^{10}-1756755 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{8} d x -1036035 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{7} d^{2} x^{2}-3465 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{6} d^{3} x^{3}+3780 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5} d^{4} x^{4}-4200 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{5} x^{5}+4800 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{6} x^{6}-5760 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{7} x^{7}+7680 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{8} x^{8}-1939938 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{9} x^{2}-3380195 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{8} d \,x^{5}-2372435 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{7} d^{2} x^{6}-62985 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d^{3} x^{7}+83980 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{4} x^{8}-167960 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{5} x^{9}+82688 \sqrt {d}\, a b \,c^{2} d^{7} x^{10}\right )}{14549535 c^{7} e^{11} x^{10}} \] Input:

int((d*x+c)^(5/2)*(b*x^2+a)^2/(e*x)^(21/2),x)
 

Output:

(2*sqrt(e)*( - 765765*sqrt(x)*sqrt(c + d*x)*a**2*c**9 - 1756755*sqrt(x)*sq 
rt(c + d*x)*a**2*c**8*d*x - 1036035*sqrt(x)*sqrt(c + d*x)*a**2*c**7*d**2*x 
**2 - 3465*sqrt(x)*sqrt(c + d*x)*a**2*c**6*d**3*x**3 + 3780*sqrt(x)*sqrt(c 
 + d*x)*a**2*c**5*d**4*x**4 - 4200*sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**5*x* 
*5 + 4800*sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**6*x**6 - 5760*sqrt(x)*sqrt(c 
+ d*x)*a**2*c**2*d**7*x**7 + 7680*sqrt(x)*sqrt(c + d*x)*a**2*c*d**8*x**8 - 
 15360*sqrt(x)*sqrt(c + d*x)*a**2*d**9*x**9 - 1939938*sqrt(x)*sqrt(c + d*x 
)*a*b*c**9*x**2 - 4626006*sqrt(x)*sqrt(c + d*x)*a*b*c**8*d*x**3 - 2889558* 
sqrt(x)*sqrt(c + d*x)*a*b*c**7*d**2*x**4 - 22610*sqrt(x)*sqrt(c + d*x)*a*b 
*c**6*d**3*x**5 + 25840*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d**4*x**6 - 31008*s 
qrt(x)*sqrt(c + d*x)*a*b*c**4*d**5*x**7 + 41344*sqrt(x)*sqrt(c + d*x)*a*b* 
c**3*d**6*x**8 - 82688*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**7*x**9 - 1322685* 
sqrt(x)*sqrt(c + d*x)*b**2*c**9*x**4 - 3380195*sqrt(x)*sqrt(c + d*x)*b**2* 
c**8*d*x**5 - 2372435*sqrt(x)*sqrt(c + d*x)*b**2*c**7*d**2*x**6 - 62985*sq 
rt(x)*sqrt(c + d*x)*b**2*c**6*d**3*x**7 + 83980*sqrt(x)*sqrt(c + d*x)*b**2 
*c**5*d**4*x**8 - 167960*sqrt(x)*sqrt(c + d*x)*b**2*c**4*d**5*x**9 + 15360 
*sqrt(d)*a**2*d**9*x**10 + 82688*sqrt(d)*a*b*c**2*d**7*x**10 + 167960*sqrt 
(d)*b**2*c**4*d**5*x**10))/(14549535*c**7*e**11*x**10)