\(\int \frac {\sqrt {c+d x} (a+b x^2)^2}{(e x)^{15/2}} \, dx\) [797]

Optimal result

Integrand size = 26, antiderivative size = 248 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}+\frac {20 a^2 d (c+d x)^{3/2}}{143 c^2 e^2 (e x)^{11/2}}-\frac {4 a \left (143 b c^2+40 a d^2\right ) (c+d x)^{3/2}}{1287 c^3 e^3 (e x)^{9/2}}+\frac {8 a d \left (143 b c^2+40 a d^2\right ) (c+d x)^{3/2}}{3003 c^4 e^4 (e x)^{7/2}}-\frac {2 \left (3003 b^2 c^4+2288 a b c^2 d^2+640 a^2 d^4\right ) (c+d x)^{3/2}}{15015 c^5 e^5 (e x)^{5/2}}+\frac {4 d \left (3003 b^2 c^4+2288 a b c^2 d^2+640 a^2 d^4\right ) (c+d x)^{3/2}}{45045 c^6 e^6 (e x)^{3/2}} \] Output:

-2/13*a^2*(d*x+c)^(3/2)/c/e/(e*x)^(13/2)+20/143*a^2*d*(d*x+c)^(3/2)/c^2/e^ 
2/(e*x)^(11/2)-4/1287*a*(40*a*d^2+143*b*c^2)*(d*x+c)^(3/2)/c^3/e^3/(e*x)^( 
9/2)+8/3003*a*d*(40*a*d^2+143*b*c^2)*(d*x+c)^(3/2)/c^4/e^4/(e*x)^(7/2)-2/1 
5015*(640*a^2*d^4+2288*a*b*c^2*d^2+3003*b^2*c^4)*(d*x+c)^(3/2)/c^5/e^5/(e* 
x)^(5/2)+4/45045*d*(640*a^2*d^4+2288*a*b*c^2*d^2+3003*b^2*c^4)*(d*x+c)^(3/ 
2)/c^6/e^6/(e*x)^(3/2)
 

Mathematica [A] (verified)

Time = 0.34 (sec) , antiderivative size = 146, normalized size of antiderivative = 0.59 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\frac {2 \sqrt {e x} (c+d x)^{3/2} \left (3003 b^2 c^4 x^4 (-3 c+2 d x)+286 a b c^2 x^2 \left (-35 c^3+30 c^2 d x-24 c d^2 x^2+16 d^3 x^3\right )-5 a^2 \left (693 c^5-630 c^4 d x+560 c^3 d^2 x^2-480 c^2 d^3 x^3+384 c d^4 x^4-256 d^5 x^5\right )\right )}{45045 c^6 e^8 x^7} \] Input:

Integrate[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(15/2),x]
 

Output:

(2*Sqrt[e*x]*(c + d*x)^(3/2)*(3003*b^2*c^4*x^4*(-3*c + 2*d*x) + 286*a*b*c^ 
2*x^2*(-35*c^3 + 30*c^2*d*x - 24*c*d^2*x^2 + 16*d^3*x^3) - 5*a^2*(693*c^5 
- 630*c^4*d*x + 560*c^3*d^2*x^2 - 480*c^2*d^3*x^3 + 384*c*d^4*x^4 - 256*d^ 
5*x^5)))/(45045*c^6*e^8*x^7)
 

Rubi [A] (verified)

Time = 0.77 (sec) , antiderivative size = 265, normalized size of antiderivative = 1.07, number of steps used = 9, number of rules used = 9, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.346, Rules used = {520, 27, 2124, 27, 520, 27, 87, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^2 \sqrt {c+d x}}{(e x)^{15/2}} \, dx\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {2 \int \frac {\sqrt {c+d x} \left (-13 b^2 c x^3-26 a b c x+10 a^2 d\right )}{2 (e x)^{13/2}}dx}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {\int \frac {\sqrt {c+d x} \left (-13 b^2 c x^3-26 a b c x+10 a^2 d\right )}{(e x)^{13/2}}dx}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 2124

\(\displaystyle -\frac {-\frac {2 \int \frac {\sqrt {c+d x} \left (143 b^2 c^2 x^2+2 a \left (143 b c^2+40 a d^2\right )\right )}{2 (e x)^{11/2}}dx}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\int \frac {\sqrt {c+d x} \left (143 b^2 c^2 x^2+2 a \left (143 b c^2+40 a d^2\right )\right )}{(e x)^{11/2}}dx}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 520

\(\displaystyle -\frac {-\frac {-\frac {2 \int \frac {3 \left (4 a d \left (143 b c^2+40 a d^2\right )-429 b^2 c^3 x\right ) \sqrt {c+d x}}{2 (e x)^{9/2}}dx}{9 c e}-\frac {4 a (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {-\frac {\int \frac {\left (4 a d \left (143 b c^2+40 a d^2\right )-429 b^2 c^3 x\right ) \sqrt {c+d x}}{(e x)^{9/2}}dx}{3 c e}-\frac {4 a (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 87

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (640 a^2 d^4+2288 a b c^2 d^2+3003 b^2 c^4\right ) \int \frac {\sqrt {c+d x}}{(e x)^{7/2}}dx}{7 c e}-\frac {8 a d (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 55

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (640 a^2 d^4+2288 a b c^2 d^2+3003 b^2 c^4\right ) \left (-\frac {2 d \int \frac {\sqrt {c+d x}}{(e x)^{5/2}}dx}{5 c e}-\frac {2 (c+d x)^{3/2}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {8 a d (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

\(\Big \downarrow \) 48

\(\displaystyle -\frac {-\frac {-\frac {-\frac {\left (640 a^2 d^4+2288 a b c^2 d^2+3003 b^2 c^4\right ) \left (\frac {4 d (c+d x)^{3/2}}{15 c^2 e^2 (e x)^{3/2}}-\frac {2 (c+d x)^{3/2}}{5 c e (e x)^{5/2}}\right )}{7 c e}-\frac {8 a d (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{7 c e (e x)^{7/2}}}{3 c e}-\frac {4 a (c+d x)^{3/2} \left (40 a d^2+143 b c^2\right )}{9 c e (e x)^{9/2}}}{11 c e}-\frac {20 a^2 d (c+d x)^{3/2}}{11 c e (e x)^{11/2}}}{13 c e}-\frac {2 a^2 (c+d x)^{3/2}}{13 c e (e x)^{13/2}}\)

Input:

Int[(Sqrt[c + d*x]*(a + b*x^2)^2)/(e*x)^(15/2),x]
 

Output:

(-2*a^2*(c + d*x)^(3/2))/(13*c*e*(e*x)^(13/2)) - ((-20*a^2*d*(c + d*x)^(3/ 
2))/(11*c*e*(e*x)^(11/2)) - ((-4*a*(143*b*c^2 + 40*a*d^2)*(c + d*x)^(3/2)) 
/(9*c*e*(e*x)^(9/2)) - ((-8*a*d*(143*b*c^2 + 40*a*d^2)*(c + d*x)^(3/2))/(7 
*c*e*(e*x)^(7/2)) - ((3003*b^2*c^4 + 2288*a*b*c^2*d^2 + 640*a^2*d^4)*((-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)
 

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 = 0.27 (sec) , antiderivative size = 159, normalized size of antiderivative = 0.64

method	result	size
gosper	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (-1280 a^{2} d^{5} x^{5}-4576 a b \,c^{2} d^{3} x^{5}-6006 b^{2} d \,x^{5} c^{4}+1920 a^{2} c \,d^{4} x^{4}+6864 a b \,c^{3} d^{2} x^{4}+9009 b^{2} c^{5} x^{4}-2400 a^{2} c^{2} d^{3} x^{3}-8580 a b \,c^{4} d \,x^{3}+2800 a^{2} c^{3} d^{2} x^{2}+10010 a b \,c^{5} x^{2}-3150 a^{2} c^{4} d x +3465 a^{2} c^{5}\right )}{45045 c^{6} \left (e x \right )^{\frac {15}{2}}}\)	\(159\)
orering	\(-\frac {2 x \left (d x +c \right )^{\frac {3}{2}} \left (-1280 a^{2} d^{5} x^{5}-4576 a b \,c^{2} d^{3} x^{5}-6006 b^{2} d \,x^{5} c^{4}+1920 a^{2} c \,d^{4} x^{4}+6864 a b \,c^{3} d^{2} x^{4}+9009 b^{2} c^{5} x^{4}-2400 a^{2} c^{2} d^{3} x^{3}-8580 a b \,c^{4} d \,x^{3}+2800 a^{2} c^{3} d^{2} x^{2}+10010 a b \,c^{5} x^{2}-3150 a^{2} c^{4} d x +3465 a^{2} c^{5}\right )}{45045 c^{6} \left (e x \right )^{\frac {15}{2}}}\)	\(159\)
default	\(-\frac {2 \left (d x +c \right )^{\frac {3}{2}} \left (-1280 a^{2} d^{5} x^{5}-4576 a b \,c^{2} d^{3} x^{5}-6006 b^{2} d \,x^{5} c^{4}+1920 a^{2} c \,d^{4} x^{4}+6864 a b \,c^{3} d^{2} x^{4}+9009 b^{2} c^{5} x^{4}-2400 a^{2} c^{2} d^{3} x^{3}-8580 a b \,c^{4} d \,x^{3}+2800 a^{2} c^{3} d^{2} x^{2}+10010 a b \,c^{5} x^{2}-3150 a^{2} c^{4} d x +3465 a^{2} c^{5}\right )}{45045 x^{6} c^{6} e^{7} \sqrt {e x}}\)	\(164\)
risch	\(-\frac {2 \sqrt {d x +c}\, \left (-1280 a^{2} d^{6} x^{6}-4576 a b \,c^{2} d^{4} x^{6}-6006 b^{2} c^{4} d^{2} x^{6}+640 a^{2} c \,d^{5} x^{5}+2288 a b \,c^{3} d^{3} x^{5}+3003 b^{2} c^{5} d \,x^{5}-480 a^{2} c^{2} d^{4} x^{4}-1716 a b \,c^{4} d^{2} x^{4}+9009 b^{2} c^{6} x^{4}+400 a^{2} c^{3} d^{3} x^{3}+1430 a b \,c^{5} d \,x^{3}-350 a^{2} c^{4} d^{2} x^{2}+10010 a b \,c^{6} x^{2}+315 a^{2} c^{5} d x +3465 a^{2} c^{6}\right )}{45045 e^{7} \sqrt {e x}\, x^{6} c^{6}}\)	\(205\)

Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(15/2),x,method=_RETURNVERBOSE)
 

Output:

-2/45045*x*(d*x+c)^(3/2)*(-1280*a^2*d^5*x^5-4576*a*b*c^2*d^3*x^5-6006*b^2* 
c^4*d*x^5+1920*a^2*c*d^4*x^4+6864*a*b*c^3*d^2*x^4+9009*b^2*c^5*x^4-2400*a^ 
2*c^2*d^3*x^3-8580*a*b*c^4*d*x^3+2800*a^2*c^3*d^2*x^2+10010*a*b*c^5*x^2-31 
50*a^2*c^4*d*x+3465*a^2*c^5)/c^6/(e*x)^(15/2)
 

Fricas [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 194, normalized size of antiderivative = 0.78 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {2 \, {\left (315 \, a^{2} c^{5} d x + 3465 \, a^{2} c^{6} - 2 \, {\left (3003 \, b^{2} c^{4} d^{2} + 2288 \, a b c^{2} d^{4} + 640 \, a^{2} d^{6}\right )} x^{6} + {\left (3003 \, b^{2} c^{5} d + 2288 \, a b c^{3} d^{3} + 640 \, a^{2} c d^{5}\right )} x^{5} + 3 \, {\left (3003 \, b^{2} c^{6} - 572 \, a b c^{4} d^{2} - 160 \, a^{2} c^{2} d^{4}\right )} x^{4} + 10 \, {\left (143 \, a b c^{5} d + 40 \, a^{2} c^{3} d^{3}\right )} x^{3} + 70 \, {\left (143 \, a b c^{6} - 5 \, a^{2} c^{4} d^{2}\right )} x^{2}\right )} \sqrt {d x + c} \sqrt {e x}}{45045 \, c^{6} e^{8} x^{7}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(15/2),x, algorithm="fricas")
 

Output:

-2/45045*(315*a^2*c^5*d*x + 3465*a^2*c^6 - 2*(3003*b^2*c^4*d^2 + 2288*a*b* 
c^2*d^4 + 640*a^2*d^6)*x^6 + (3003*b^2*c^5*d + 2288*a*b*c^3*d^3 + 640*a^2* 
c*d^5)*x^5 + 3*(3003*b^2*c^6 - 572*a*b*c^4*d^2 - 160*a^2*c^2*d^4)*x^4 + 10 
*(143*a*b*c^5*d + 40*a^2*c^3*d^3)*x^3 + 70*(143*a*b*c^6 - 5*a^2*c^4*d^2)*x 
^2)*sqrt(d*x + c)*sqrt(e*x)/(c^6*e^8*x^7)
 

Sympy [F(-1)]

Timed out. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\text {Timed out} \] Input:

integrate((d*x+c)**(1/2)*(b*x**2+a)**2/(e*x)**(15/2),x)
 

Output:

Timed out
 

Maxima [F(-2)]

Exception generated. \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\text {Exception raised: ValueError} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(15/2),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 = 0.21 (sec) , antiderivative size = 342, normalized size of antiderivative = 1.38 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\frac {2 \, {\left ({\left ({\left ({\left (d x + c\right )} {\left ({\left (d x + c\right )} {\left (\frac {2 \, {\left (3003 \, b^{2} c^{4} d^{9} e^{6} + 2288 \, a b c^{2} d^{11} e^{6} + 640 \, a^{2} d^{13} e^{6}\right )} {\left (d x + c\right )}}{c^{6}} - \frac {13 \, {\left (3003 \, b^{2} c^{5} d^{9} e^{6} + 2288 \, a b c^{3} d^{11} e^{6} + 640 \, a^{2} c d^{13} e^{6}\right )}}{c^{6}}\right )} + \frac {572 \, {\left (168 \, b^{2} c^{6} d^{9} e^{6} + 143 \, a b c^{4} d^{11} e^{6} + 40 \, a^{2} c^{2} d^{13} e^{6}\right )}}{c^{6}}\right )} - \frac {858 \, {\left (133 \, b^{2} c^{7} d^{9} e^{6} + 143 \, a b c^{5} d^{11} e^{6} + 40 \, a^{2} c^{3} d^{13} e^{6}\right )}}{c^{6}}\right )} {\left (d x + c\right )} + \frac {6006 \, {\left (11 \, b^{2} c^{8} d^{9} e^{6} + 16 \, a b c^{6} d^{11} e^{6} + 5 \, a^{2} c^{4} d^{13} e^{6}\right )}}{c^{6}}\right )} {\left (d x + c\right )} - \frac {15015 \, {\left (b^{2} c^{9} d^{9} e^{6} + 2 \, a b c^{7} d^{11} e^{6} + a^{2} c^{5} d^{13} e^{6}\right )}}{c^{6}}\right )} {\left (d x + c\right )}^{\frac {3}{2}} d}{45045 \, {\left ({\left (d x + c\right )} d e - c d e\right )}^{\frac {13}{2}} e^{7} {\left | d \right |}} \] Input:

integrate((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(15/2),x, algorithm="giac")
 

Output:

2/45045*((((d*x + c)*((d*x + c)*(2*(3003*b^2*c^4*d^9*e^6 + 2288*a*b*c^2*d^ 
11*e^6 + 640*a^2*d^13*e^6)*(d*x + c)/c^6 - 13*(3003*b^2*c^5*d^9*e^6 + 2288 
*a*b*c^3*d^11*e^6 + 640*a^2*c*d^13*e^6)/c^6) + 572*(168*b^2*c^6*d^9*e^6 + 
143*a*b*c^4*d^11*e^6 + 40*a^2*c^2*d^13*e^6)/c^6) - 858*(133*b^2*c^7*d^9*e^ 
6 + 143*a*b*c^5*d^11*e^6 + 40*a^2*c^3*d^13*e^6)/c^6)*(d*x + c) + 6006*(11* 
b^2*c^8*d^9*e^6 + 16*a*b*c^6*d^11*e^6 + 5*a^2*c^4*d^13*e^6)/c^6)*(d*x + c) 
 - 15015*(b^2*c^9*d^9*e^6 + 2*a*b*c^7*d^11*e^6 + a^2*c^5*d^13*e^6)/c^6)*(d 
*x + c)^(3/2)*d/(((d*x + c)*d*e - c*d*e)^(13/2)*e^7*abs(d))
 

Mupad [B] (verification not implemented)

Time = 8.86 (sec) , antiderivative size = 212, normalized size of antiderivative = 0.85 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=-\frac {\sqrt {c+d\,x}\,\left (\frac {2\,a^2}{13\,e^7}-\frac {x^4\,\left (960\,a^2\,c^2\,d^4+3432\,a\,b\,c^4\,d^2-18018\,b^2\,c^6\right )}{45045\,c^6\,e^7}-\frac {x^6\,\left (2560\,a^2\,d^6+9152\,a\,b\,c^2\,d^4+12012\,b^2\,c^4\,d^2\right )}{45045\,c^6\,e^7}+\frac {x^5\,\left (1280\,a^2\,c\,d^5+4576\,a\,b\,c^3\,d^3+6006\,b^2\,c^5\,d\right )}{45045\,c^6\,e^7}+\frac {2\,a^2\,d\,x}{143\,c\,e^7}-\frac {4\,a\,x^2\,\left (5\,a\,d^2-143\,b\,c^2\right )}{1287\,c^2\,e^7}+\frac {4\,a\,d\,x^3\,\left (143\,b\,c^2+40\,a\,d^2\right )}{9009\,c^3\,e^7}\right )}{x^6\,\sqrt {e\,x}} \] Input:

int(((a + b*x^2)^2*(c + d*x)^(1/2))/(e*x)^(15/2),x)
 

Output:

-((c + d*x)^(1/2)*((2*a^2)/(13*e^7) - (x^4*(960*a^2*c^2*d^4 - 18018*b^2*c^ 
6 + 3432*a*b*c^4*d^2))/(45045*c^6*e^7) - (x^6*(2560*a^2*d^6 + 12012*b^2*c^ 
4*d^2 + 9152*a*b*c^2*d^4))/(45045*c^6*e^7) + (x^5*(1280*a^2*c*d^5 + 6006*b 
^2*c^5*d + 4576*a*b*c^3*d^3))/(45045*c^6*e^7) + (2*a^2*d*x)/(143*c*e^7) - 
(4*a*x^2*(5*a*d^2 - 143*b*c^2))/(1287*c^2*e^7) + (4*a*d*x^3*(40*a*d^2 + 14 
3*b*c^2))/(9009*c^3*e^7)))/(x^6*(e*x)^(1/2))
 

Reduce [B] (verification not implemented)

Time = 0.28 (sec) , antiderivative size = 358, normalized size of antiderivative = 1.44 \[ \int \frac {\sqrt {c+d x} \left (a+b x^2\right )^2}{(e x)^{15/2}} \, dx=\frac {2 \sqrt {e}\, \left (-3465 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{6}-315 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{5} d x +350 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{4} d^{2} x^{2}-400 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{3} d^{3} x^{3}+480 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c^{2} d^{4} x^{4}-640 \sqrt {x}\, \sqrt {d x +c}\, a^{2} c \,d^{5} x^{5}+1280 \sqrt {x}\, \sqrt {d x +c}\, a^{2} d^{6} x^{6}-10010 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{6} x^{2}-1430 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{5} d \,x^{3}+1716 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{4} d^{2} x^{4}-2288 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{3} d^{3} x^{5}+4576 \sqrt {x}\, \sqrt {d x +c}\, a b \,c^{2} d^{4} x^{6}-9009 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{6} x^{4}-3003 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{5} d \,x^{5}+6006 \sqrt {x}\, \sqrt {d x +c}\, b^{2} c^{4} d^{2} x^{6}-1280 \sqrt {d}\, a^{2} d^{6} x^{7}-4576 \sqrt {d}\, a b \,c^{2} d^{4} x^{7}-6006 \sqrt {d}\, b^{2} c^{4} d^{2} x^{7}\right )}{45045 c^{6} e^{8} x^{7}} \] Input:

int((d*x+c)^(1/2)*(b*x^2+a)^2/(e*x)^(15/2),x)
 

Output:

(2*sqrt(e)*( - 3465*sqrt(x)*sqrt(c + d*x)*a**2*c**6 - 315*sqrt(x)*sqrt(c + 
 d*x)*a**2*c**5*d*x + 350*sqrt(x)*sqrt(c + d*x)*a**2*c**4*d**2*x**2 - 400* 
sqrt(x)*sqrt(c + d*x)*a**2*c**3*d**3*x**3 + 480*sqrt(x)*sqrt(c + d*x)*a**2 
*c**2*d**4*x**4 - 640*sqrt(x)*sqrt(c + d*x)*a**2*c*d**5*x**5 + 1280*sqrt(x 
)*sqrt(c + d*x)*a**2*d**6*x**6 - 10010*sqrt(x)*sqrt(c + d*x)*a*b*c**6*x**2 
 - 1430*sqrt(x)*sqrt(c + d*x)*a*b*c**5*d*x**3 + 1716*sqrt(x)*sqrt(c + d*x) 
*a*b*c**4*d**2*x**4 - 2288*sqrt(x)*sqrt(c + d*x)*a*b*c**3*d**3*x**5 + 4576 
*sqrt(x)*sqrt(c + d*x)*a*b*c**2*d**4*x**6 - 9009*sqrt(x)*sqrt(c + d*x)*b** 
2*c**6*x**4 - 3003*sqrt(x)*sqrt(c + d*x)*b**2*c**5*d*x**5 + 6006*sqrt(x)*s 
qrt(c + d*x)*b**2*c**4*d**2*x**6 - 1280*sqrt(d)*a**2*d**6*x**7 - 4576*sqrt 
(d)*a*b*c**2*d**4*x**7 - 6006*sqrt(d)*b**2*c**4*d**2*x**7))/(45045*c**6*e* 
*8*x**7)