\(\int \frac {\sqrt {c+d x} (a+b x^2)^2}{(e x)^{17/2}} \, dx\) [798]

Optimal result

Integrand size = 26, antiderivative size = 304 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}+\frac {8 a^2 d (c+d x)^{3/2}}{65 c^2 e^2 (e x)^{13/2}}-\frac {4 a \left (13 b c^2+4 a d^2\right ) (c+d x)^{3/2}}{143 c^3 e^3 (e x)^{11/2}}+\frac {32 a d \left (13 b c^2+4 a d^2\right ) (c+d x)^{3/2}}{1287 c^4 e^4 (e x)^{9/2}}-\frac {2 \left (429 b^2 c^4+416 a b c^2 d^2+128 a^2 d^4\right ) (c+d x)^{3/2}}{3003 c^5 e^5 (e x)^{7/2}}+\frac {8 d \left (429 b^2 c^4+416 a b c^2 d^2+128 a^2 d^4\right ) (c+d x)^{3/2}}{15015 c^6 e^6 (e x)^{5/2}}-\frac {16 d^2 \left (429 b^2 c^4+416 a b c^2 d^2+128 a^2 d^4\right ) (c+d x)^{3/2}}{45045 c^7 e^7 (e x)^{3/2}} \] Output:

-2/15*a^2*(d*x+c)^(3/2)/c/e/(e*x)^(15/2)+8/65*a^2*d*(d*x+c)^(3/2)/c^2/e^2/ 
(e*x)^(13/2)-4/143*a*(4*a*d^2+13*b*c^2)*(d*x+c)^(3/2)/c^3/e^3/(e*x)^(11/2) 
+32/1287*a*d*(4*a*d^2+13*b*c^2)*(d*x+c)^(3/2)/c^4/e^4/(e*x)^(9/2)-2/3003*( 
128*a^2*d^4+416*a*b*c^2*d^2+429*b^2*c^4)*(d*x+c)^(3/2)/c^5/e^5/(e*x)^(7/2) 
+8/15015*d*(128*a^2*d^4+416*a*b*c^2*d^2+429*b^2*c^4)*(d*x+c)^(3/2)/c^6/e^6 
/(e*x)^(5/2)-16/45045*d^2*(128*a^2*d^4+416*a*b*c^2*d^2+429*b^2*c^4)*(d*x+c 
)^(3/2)/c^7/e^7/(e*x)^(3/2)
 

Mathematica [A] (verified)

Time = 0.13 (sec) , antiderivative size = 178, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \sqrt {e x} (c+d x)^{3/2} \left (429 b^2 c^4 x^4 \left (15 c^2-12 c d x+8 d^2 x^2\right )+26 a b c^2 x^2 \left (315 c^4-280 c^3 d x+240 c^2 d^2 x^2-192 c d^3 x^3+128 d^4 x^4\right )+a^2 \left (3003 c^6-2772 c^5 d x+2520 c^4 d^2 x^2-2240 c^3 d^3 x^3+1920 c^2 d^4 x^4-1536 c d^5 x^5+1024 d^6 x^6\right )\right )}{45045 c^7 e^9 x^8} \] Input:

Integrate[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(17/2),x]
 

Output:

(-2*Sqrt[e*x]*(c + d*x)^(3/2)*(429*b^2*c^4*x^4*(15*c^2 - 12*c*d*x + 8*d^2* 
x^2) + 26*a*b*c^2*x^2*(315*c^4 - 280*c^3*d*x + 240*c^2*d^2*x^2 - 192*c*d^3 
*x^3 + 128*d^4*x^4) + a^2*(3003*c^6 - 2772*c^5*d*x + 2520*c^4*d^2*x^2 - 22 
40*c^3*d^3*x^3 + 1920*c^2*d^4*x^4 - 1536*c*d^5*x^5 + 1024*d^6*x^6)))/(4504 
5*c^7*e^9*x^8)
 

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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x}}{(e x)^{17/2}} \, dx\)

\(\Big \downarrow \) 520

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {5 \sqrt {c+d x} \left (13 b^2 c^2 x^2+2 a \left (13 b c^2+4 a d^2\right )\right )}{2 (e x)^{13/2}}dx}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {5 \int \frac {\sqrt {c+d x} \left (13 b^2 c^2 x^2+2 a \left (13 b c^2+4 a d^2\right )\right )}{(e x)^{13/2}}dx}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {5 \left (-\frac {2 \int \frac {\left (16 a d \left (13 b c^2+4 a d^2\right )-143 b^2 c^3 x\right ) \sqrt {c+d x}}{2 (e x)^{11/2}}dx}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {5 \left (-\frac {\int \frac {\left (16 a d \left (13 b c^2+4 a d^2\right )-143 b^2 c^3 x\right ) \sqrt {c+d x}}{(e x)^{11/2}}dx}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {5 \left (-\frac {-\frac {\left (128 a^2 d^4+416 a b c^2 d^2+429 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{(e x)^{9/2}}dx}{3 c e}-\frac {32 a d (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {5 \left (-\frac {-\frac {\left (128 a^2 d^4+416 a b c^2 d^2+429 b^2 c^4\right ) \left (-\frac {4 d \int \frac {\sqrt {c+d x}}{(e x)^{7/2}}dx}{7 c e}-\frac {2 (c+d x)^{3/2}}{7 c e (e x)^{7/2}}\right )}{3 c e}-\frac {32 a d (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {5 \left (-\frac {-\frac {\left (128 a^2 d^4+416 a b c^2 d^2+429 b^2 c^4\right ) \left (-\frac {4 d \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(e x)^{5/2}}dx}{5 c e}-\frac {2 (c+d x)^{3/2}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 (c+d x)^{3/2}}{7 c e (e x)^{7/2}}\right )}{3 c e}-\frac {32 a d (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {5 \left (-\frac {-\frac {\left (128 a^2 d^4+416 a b c^2 d^2+429 b^2 c^4\right ) \left (-\frac {4 d \left (\frac {4 d (c+d x)^{3/2}}{15 c^2 e^2 (e x)^{3/2}}-\frac {2 (c+d x)^{3/2}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {2 (c+d x)^{3/2}}{7 c e (e x)^{7/2}}\right )}{3 c e}-\frac {32 a d (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {4 a (c+d x)^{3/2} \left (4 a d^2+13 b c^2\right )}{11 c e (e x)^{11/2}}\right )}{13 c e}-\frac {8 a^2 d (c+d x)^{3/2}}{13 c e (e x)^{13/2}}}{5 c e}-\frac {2 a^2 (c+d x)^{3/2}}{15 c e (e x)^{15/2}}\)

Input:

Int[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(17/2),x]
 

Output:

(-2*a^2*(c + d*x)^(3/2))/(15*c*e*(e*x)^(15/2)) - ((-8*a^2*d*(c + d*x)^(3/2 
))/(13*c*e*(e*x)^(13/2)) - (5*((-4*a*(13*b*c^2 + 4*a*d^2)*(c + d*x)^(3/2)) 
/(11*c*e*(e*x)^(11/2)) - ((-32*a*d*(13*b*c^2 + 4*a*d^2)*(c + d*x)^(3/2))/( 
9*c*e*(e*x)^(9/2)) - ((429*b^2*c^4 + 416*a*b*c^2*d^2 + 128*a^2*d^4)*((-2*( 
c + d*x)^(3/2))/(7*c*e*(e*x)^(7/2)) - (4*d*((-2*(c + d*x)^(3/2))/(5*c*e*(e 
*x)^(5/2)) + (4*d*(c + d*x)^(3/2))/(15*c^2*e^2*(e*x)^(3/2))))/(7*c*e)))/(3 
*c*e))/(11*c*e)))/(13*c*e))/(5*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 = 0.26 (sec) , antiderivative size = 200, normalized size of antiderivative = 0.66

method	result	size
gosper	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (1024 a^{2} d^{6} x^{6}+3328 a b \,c^{2} d^{4} x^{6}+3432 b^{2} c^{4} d^{2} x^{6}-1536 a^{2} c \,d^{5} x^{5}-4992 a b \,c^{3} d^{3} x^{5}-5148 b^{2} c^{5} d \,x^{5}+1920 a^{2} c^{2} d^{4} x^{4}+6240 a b \,c^{4} d^{2} x^{4}+6435 b^{2} c^{6} x^{4}-2240 a^{2} c^{3} d^{3} x^{3}-7280 a b \,c^{5} d \,x^{3}+2520 a^{2} c^{4} d^{2} x^{2}+8190 a b \,c^{6} x^{2}-2772 a^{2} c^{5} d x +3003 a^{2} c^{6}\right )}{45045 c^{7} \left (e x \right )^{\frac {17}{2}}}\)	\(200\)
orering	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (1024 a^{2} d^{6} x^{6}+3328 a b \,c^{2} d^{4} x^{6}+3432 b^{2} c^{4} d^{2} x^{6}-1536 a^{2} c \,d^{5} x^{5}-4992 a b \,c^{3} d^{3} x^{5}-5148 b^{2} c^{5} d \,x^{5}+1920 a^{2} c^{2} d^{4} x^{4}+6240 a b \,c^{4} d^{2} x^{4}+6435 b^{2} c^{6} x^{4}-2240 a^{2} c^{3} d^{3} x^{3}-7280 a b \,c^{5} d \,x^{3}+2520 a^{2} c^{4} d^{2} x^{2}+8190 a b \,c^{6} x^{2}-2772 a^{2} c^{5} d x +3003 a^{2} c^{6}\right )}{45045 c^{7} \left (e x \right )^{\frac {17}{2}}}\)	\(200\)
default	\(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (1024 a^{2} d^{6} x^{6}+3328 a b \,c^{2} d^{4} x^{6}+3432 b^{2} c^{4} d^{2} x^{6}-1536 a^{2} c \,d^{5} x^{5}-4992 a b \,c^{3} d^{3} x^{5}-5148 b^{2} c^{5} d \,x^{5}+1920 a^{2} c^{2} d^{4} x^{4}+6240 a b \,c^{4} d^{2} x^{4}+6435 b^{2} c^{6} x^{4}-2240 a^{2} c^{3} d^{3} x^{3}-7280 a b \,c^{5} d \,x^{3}+2520 a^{2} c^{4} d^{2} x^{2}+8190 a b \,c^{6} x^{2}-2772 a^{2} c^{5} d x +3003 a^{2} c^{6}\right )}{45045 x^{7} c^{7} e^{8} \sqrt {e x}}\)	\(205\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (1024 a^{2} d^{7} x^{7}+3328 a b \,c^{2} d^{5} x^{7}+3432 b^{2} c^{4} d^{3} x^{7}-512 a^{2} c \,d^{6} x^{6}-1664 a b \,c^{3} d^{4} x^{6}-1716 b^{2} c^{5} d^{2} x^{6}+384 a^{2} c^{2} d^{5} x^{5}+1248 a b \,c^{4} d^{3} x^{5}+1287 b^{2} c^{6} d \,x^{5}-320 a^{2} c^{3} d^{4} x^{4}-1040 a b \,c^{5} d^{2} x^{4}+6435 b^{2} c^{7} x^{4}+280 a^{2} c^{4} d^{3} x^{3}+910 a b \,c^{6} d \,x^{3}-252 a^{2} c^{5} d^{2} x^{2}+8190 a b \,c^{7} x^{2}+231 a^{2} c^{6} d x +3003 a^{2} c^{7}\right )}{45045 e^{8} \sqrt {e x}\, x^{7} c^{7}}\)	\(246\)

Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(17/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*x*(d*x+c)^(3/2)*(1024*a^2*d^6*x^6+3328*a*b*c^2*d^4*x^6+3432*b^2*c 
^4*d^2*x^6-1536*a^2*c*d^5*x^5-4992*a*b*c^3*d^3*x^5-5148*b^2*c^5*d*x^5+1920 
*a^2*c^2*d^4*x^4+6240*a*b*c^4*d^2*x^4+6435*b^2*c^6*x^4-2240*a^2*c^3*d^3*x^ 
3-7280*a*b*c^5*d*x^3+2520*a^2*c^4*d^2*x^2+8190*a*b*c^6*x^2-2772*a^2*c^5*d* 
x+3003*a^2*c^6)/c^7/(e*x)^(17/2)
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 233, normalized size of antiderivative = 0.77 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left (231 \, a^{2} c^{6} d x + 3003 \, a^{2} c^{7} + 8 \, {\left (429 \, b^{2} c^{4} d^{3} + 416 \, a b c^{2} d^{5} + 128 \, a^{2} d^{7}\right )} x^{7} - 4 \, {\left (429 \, b^{2} c^{5} d^{2} + 416 \, a b c^{3} d^{4} + 128 \, a^{2} c d^{6}\right )} x^{6} + 3 \, {\left (429 \, b^{2} c^{6} d + 416 \, a b c^{4} d^{3} + 128 \, a^{2} c^{2} d^{5}\right )} x^{5} + 5 \, {\left (1287 \, b^{2} c^{7} - 208 \, a b c^{5} d^{2} - 64 \, a^{2} c^{3} d^{4}\right )} x^{4} + 70 \, {\left (13 \, a b c^{6} d + 4 \, a^{2} c^{4} d^{3}\right )} x^{3} + 126 \, {\left (65 \, a b c^{7} - 2 \, a^{2} c^{5} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{45045 \, c^{7} e^{9} x^{8}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(17/2),x, algorithm="fricas")
 

Output:

-2/45045*(231*a^2*c^6*d*x + 3003*a^2*c^7 + 8*(429*b^2*c^4*d^3 + 416*a*b*c^ 
2*d^5 + 128*a^2*d^7)*x^7 - 4*(429*b^2*c^5*d^2 + 416*a*b*c^3*d^4 + 128*a^2* 
c*d^6)*x^6 + 3*(429*b^2*c^6*d + 416*a*b*c^4*d^3 + 128*a^2*c^2*d^5)*x^5 + 5 
*(1287*b^2*c^7 - 208*a*b*c^5*d^2 - 64*a^2*c^3*d^4)*x^4 + 70*(13*a*b*c^6*d 
+ 4*a^2*c^4*d^3)*x^3 + 126*(65*a*b*c^7 - 2*a^2*c^5*d^2)*x^2)*sqrt(d*x + c) 
*sqrt(e*x)/(c^7*e^9*x^8)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(b*x**2+a)**2/(e*x)**(17/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(17/2),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 = 0.19 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.31 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {2 \, {\left ({\left ({\left ({\left ({\left (d x + c\right )} {\left (4 \, {\left (d x + c\right )} {\left (\frac {2 \, {\left (429 \, b^{2} c^{4} d^{3} e^{7} + 416 \, a b c^{2} d^{5} e^{7} + 128 \, a^{2} d^{7} e^{7}\right )} {\left (d x + c\right )}}{c^{7}} - \frac {15 \, {\left (429 \, b^{2} c^{5} d^{3} e^{7} + 416 \, a b c^{3} d^{5} e^{7} + 128 \, a^{2} c d^{7} e^{7}\right )}}{c^{7}}\right )} + \frac {195 \, {\left (429 \, b^{2} c^{6} d^{3} e^{7} + 416 \, a b c^{4} d^{5} e^{7} + 128 \, a^{2} c^{2} d^{7} e^{7}\right )}}{c^{7}}\right )} - \frac {2860 \, {\left (51 \, b^{2} c^{7} d^{3} e^{7} + 52 \, a b c^{5} d^{5} e^{7} + 16 \, a^{2} c^{3} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )} + \frac {12870 \, {\left (11 \, b^{2} c^{8} d^{3} e^{7} + 13 \, a b c^{6} d^{5} e^{7} + 4 \, a^{2} c^{4} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )} - \frac {36036 \, {\left (2 \, b^{2} c^{9} d^{3} e^{7} + 3 \, a b c^{7} d^{5} e^{7} + a^{2} c^{5} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )} + \frac {15015 \, {\left (b^{2} c^{10} d^{3} e^{7} + 2 \, a b c^{8} d^{5} e^{7} + a^{2} c^{6} d^{7} e^{7}\right )}}{c^{7}}\right )} {\left (d x + c\right )}^{\frac {3}{2}} d^{9}}{45045 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {15}{2}} e^{8} {\left | d \right |}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(17/2),x, algorithm="giac")
 

Output:

-2/45045*(((((d*x + c)*(4*(d*x + c)*(2*(429*b^2*c^4*d^3*e^7 + 416*a*b*c^2* 
d^5*e^7 + 128*a^2*d^7*e^7)*(d*x + c)/c^7 - 15*(429*b^2*c^5*d^3*e^7 + 416*a 
*b*c^3*d^5*e^7 + 128*a^2*c*d^7*e^7)/c^7) + 195*(429*b^2*c^6*d^3*e^7 + 416* 
a*b*c^4*d^5*e^7 + 128*a^2*c^2*d^7*e^7)/c^7) - 2860*(51*b^2*c^7*d^3*e^7 + 5 
2*a*b*c^5*d^5*e^7 + 16*a^2*c^3*d^7*e^7)/c^7)*(d*x + c) + 12870*(11*b^2*c^8 
*d^3*e^7 + 13*a*b*c^6*d^5*e^7 + 4*a^2*c^4*d^7*e^7)/c^7)*(d*x + c) - 36036* 
(2*b^2*c^9*d^3*e^7 + 3*a*b*c^7*d^5*e^7 + a^2*c^5*d^7*e^7)/c^7)*(d*x + c) + 
 15015*(b^2*c^10*d^3*e^7 + 2*a*b*c^8*d^5*e^7 + a^2*c^6*d^7*e^7)/c^7)*(d*x 
+ c)^(3/2)*d^9/(((d*x + c)*d*e - c*d*e)^(15/2)*e^8*abs(d))
 

Mupad [B] (verification not implemented)

Time = 9.10 (sec) , antiderivative size = 256, normalized size of antiderivative = 0.84 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2}{15\,e^8}-\frac {x^4\,\left (640\,a^2\,c^3\,d^4+2080\,a\,b\,c^5\,d^2-12870\,b^2\,c^7\right )}{45045\,c^7\,e^8}+\frac {x^7\,\left (2048\,a^2\,d^7+6656\,a\,b\,c^2\,d^5+6864\,b^2\,c^4\,d^3\right )}{45045\,c^7\,e^8}+\frac {x^5\,\left (768\,a^2\,c^2\,d^5+2496\,a\,b\,c^4\,d^3+2574\,b^2\,c^6\,d\right )}{45045\,c^7\,e^8}-\frac {x^6\,\left (1024\,a^2\,c\,d^6+3328\,a\,b\,c^3\,d^4+3432\,b^2\,c^5\,d^2\right )}{45045\,c^7\,e^8}+\frac {2\,a^2\,d\,x}{195\,c\,e^8}-\frac {4\,a\,x^2\,\left (2\,a\,d^2-65\,b\,c^2\right )}{715\,c^2\,e^8}+\frac {4\,a\,d\,x^3\,\left (13\,b\,c^2+4\,a\,d^2\right )}{1287\,c^3\,e^8}\right )}{x^7\,\sqrt {e\,x}} \] Input:

int(((a + b*x^2)^2*(c + d*x)^(1/2))/(e*x)^(17/2),x)
 

Output:

-((c + d*x)^(1/2)*((2*a^2)/(15*e^8) - (x^4*(640*a^2*c^3*d^4 - 12870*b^2*c^ 
7 + 2080*a*b*c^5*d^2))/(45045*c^7*e^8) + (x^7*(2048*a^2*d^7 + 6864*b^2*c^4 
*d^3 + 6656*a*b*c^2*d^5))/(45045*c^7*e^8) + (x^5*(2574*b^2*c^6*d + 768*a^2 
*c^2*d^5 + 2496*a*b*c^4*d^3))/(45045*c^7*e^8) - (x^6*(1024*a^2*c*d^6 + 343 
2*b^2*c^5*d^2 + 3328*a*b*c^3*d^4))/(45045*c^7*e^8) + (2*a^2*d*x)/(195*c*e^ 
8) - (4*a*x^2*(2*a*d^2 - 65*b*c^2))/(715*c^2*e^8) + (4*a*d*x^3*(4*a*d^2 + 
13*b*c^2))/(1287*c^3*e^8)))/(x^7*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 423, normalized size of antiderivative = 1.39 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{17/2}} \, dx=\frac {2 \sqrt {e}\, \left (-3003 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{7}-231 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{6} d x +252 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5} d^{2} x^{2}-280 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{3} x^{3}+320 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{4} x^{4}-384 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{5} x^{5}+512 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{6} x^{6}-1024 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{7} x^{7}-8190 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{7} x^{2}-910 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} d \,x^{3}+1040 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d^{2} x^{4}-1248 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{3} x^{5}+1664 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{4} x^{6}-3328 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{5} x^{7}-6435 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{7} x^{4}-1287 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} d \,x^{5}+1716 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d^{2} x^{6}-3432 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{3} x^{7}+1024 \sqrt {d}\, a^{2} d^{7} x^{8}+3328 \sqrt {d}\, a b \,c^{2} d^{5} x^{8}+3432 \sqrt {d}\, b^{2} c^{4} d^{3} x^{8}\right )}{45045 c^{7} e^{9} x^{8}} \] Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(17/2),x)
 

Output:

(2*sqrt(e)*( - 3003*sqrt(x)*sqrt(c + d*x)*a**2*c**7 - 231*sqrt(x)*sqrt(c + 
 d*x)*a**2*c**6*d*x + 252*sqrt(x)*sqrt(c + d*x)*a**2*c**5*d**2*x**2 - 280* 
sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**3*x**3 + 320*sqrt(x)*sqrt(c + d*x)*a**2 
*c**3*d**4*x**4 - 384*sqrt(x)*sqrt(c + d*x)*a**2*c**2*d**5*x**5 + 512*sqrt 
(x)*sqrt(c + d*x)*a**2*c*d**6*x**6 - 1024*sqrt(x)*sqrt(c + d*x)*a**2*d**7* 
x**7 - 8190*sqrt(x)*sqrt(c + d*x)*a*b*c**7*x**2 - 910*sqrt(x)*sqrt(c + d*x 
)*a*b*c**6*d*x**3 + 1040*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d**2*x**4 - 1248*s 
qrt(x)*sqrt(c + d*x)*a*b*c**4*d**3*x**5 + 1664*sqrt(x)*sqrt(c + d*x)*a*b*c 
**3*d**4*x**6 - 3328*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**5*x**7 - 6435*sqrt( 
x)*sqrt(c + d*x)*b**2*c**7*x**4 - 1287*sqrt(x)*sqrt(c + d*x)*b**2*c**6*d*x 
**5 + 1716*sqrt(x)*sqrt(c + d*x)*b**2*c**5*d**2*x**6 - 3432*sqrt(x)*sqrt(c 
 + d*x)*b**2*c**4*d**3*x**7 + 1024*sqrt(d)*a**2*d**7*x**8 + 3328*sqrt(d)*a 
*b*c**2*d**5*x**8 + 3432*sqrt(d)*b**2*c**4*d**3*x**8))/(45045*c**7*e**9*x* 
*8)