\(\int \frac {(A+B x) (a+c x^2)^3}{(d+e x)^{7/2}} \, dx\) [120]

Optimal result

Integrand size = 24, antiderivative size = 346 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 (B d-A e) \left (c d^2+a e^2\right )^3}{5 e^8 (d+e x)^{5/2}}-\frac {2 \left (c d^2+a e^2\right )^2 \left (7 B c d^2-6 A c d e+a B e^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac {6 c \left (c d^2+a e^2\right ) \left (7 B c d^3-5 A c d^2 e+3 a B d e^2-a A e^3\right )}{e^8 \sqrt {d+e x}}-\frac {2 c \left (4 A c d e \left (5 c d^2+3 a e^2\right )-B \left (35 c^2 d^4+30 a c d^2 e^2+3 a^2 e^4\right )\right ) \sqrt {d+e x}}{e^8}-\frac {2 c^2 \left (35 B c d^3-15 A c d^2 e+15 a B d e^2-3 a A e^3\right ) (d+e x)^{3/2}}{3 e^8}+\frac {6 c^2 \left (7 B c d^2-2 A c d e+a B e^2\right ) (d+e x)^{5/2}}{5 e^8}-\frac {2 c^3 (7 B d-A e) (d+e x)^{7/2}}{7 e^8}+\frac {2 B c^3 (d+e x)^{9/2}}{9 e^8} \] Output:

2/5*(-A*e+B*d)*(a*e^2+c*d^2)^3/e^8/(e*x+d)^(5/2)-2/3*(a*e^2+c*d^2)^2*(-6*A 
*c*d*e+B*a*e^2+7*B*c*d^2)/e^8/(e*x+d)^(3/2)+6*c*(a*e^2+c*d^2)*(-A*a*e^3-5* 
A*c*d^2*e+3*B*a*d*e^2+7*B*c*d^3)/e^8/(e*x+d)^(1/2)-2*c*(4*A*c*d*e*(3*a*e^2 
+5*c*d^2)-B*(3*a^2*e^4+30*a*c*d^2*e^2+35*c^2*d^4))*(e*x+d)^(1/2)/e^8-2/3*c 
^2*(-3*A*a*e^3-15*A*c*d^2*e+15*B*a*d*e^2+35*B*c*d^3)*(e*x+d)^(3/2)/e^8+6/5 
*c^2*(-2*A*c*d*e+B*a*e^2+7*B*c*d^2)*(e*x+d)^(5/2)/e^8-2/7*c^3*(-A*e+7*B*d) 
*(e*x+d)^(7/2)/e^8+2/9*B*c^3*(e*x+d)^(9/2)/e^8
 

Mathematica [A] (verified)

Time = 0.59 (sec) , antiderivative size = 373, normalized size of antiderivative = 1.08 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \left (-9 A e \left (7 a^3 e^6+7 a^2 c e^4 \left (8 d^2+20 d e x+15 e^2 x^2\right )+7 a c^2 e^2 \left (128 d^4+320 d^3 e x+240 d^2 e^2 x^2+40 d e^3 x^3-5 e^4 x^4\right )+c^3 \left (1024 d^6+2560 d^5 e x+1920 d^4 e^2 x^2+320 d^3 e^3 x^3-40 d^2 e^4 x^4+12 d e^5 x^5-5 e^6 x^6\right )\right )+7 B \left (-3 a^3 e^6 (2 d+5 e x)+27 a^2 c e^4 \left (16 d^3+40 d^2 e x+30 d e^2 x^2+5 e^3 x^3\right )+9 a c^2 e^2 \left (256 d^5+640 d^4 e x+480 d^3 e^2 x^2+80 d^2 e^3 x^3-10 d e^4 x^4+3 e^5 x^5\right )+c^3 \left (2048 d^7+5120 d^6 e x+3840 d^5 e^2 x^2+640 d^4 e^3 x^3-80 d^3 e^4 x^4+24 d^2 e^5 x^5-10 d e^6 x^6+5 e^7 x^7\right )\right )\right )}{315 e^8 (d+e x)^{5/2}} \] Input:

Integrate[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(7/2),x]
 

Output:

(2*(-9*A*e*(7*a^3*e^6 + 7*a^2*c*e^4*(8*d^2 + 20*d*e*x + 15*e^2*x^2) + 7*a* 
c^2*e^2*(128*d^4 + 320*d^3*e*x + 240*d^2*e^2*x^2 + 40*d*e^3*x^3 - 5*e^4*x^ 
4) + c^3*(1024*d^6 + 2560*d^5*e*x + 1920*d^4*e^2*x^2 + 320*d^3*e^3*x^3 - 4 
0*d^2*e^4*x^4 + 12*d*e^5*x^5 - 5*e^6*x^6)) + 7*B*(-3*a^3*e^6*(2*d + 5*e*x) 
 + 27*a^2*c*e^4*(16*d^3 + 40*d^2*e*x + 30*d*e^2*x^2 + 5*e^3*x^3) + 9*a*c^2 
*e^2*(256*d^5 + 640*d^4*e*x + 480*d^3*e^2*x^2 + 80*d^2*e^3*x^3 - 10*d*e^4* 
x^4 + 3*e^5*x^5) + c^3*(2048*d^7 + 5120*d^6*e*x + 3840*d^5*e^2*x^2 + 640*d 
^4*e^3*x^3 - 80*d^3*e^4*x^4 + 24*d^2*e^5*x^5 - 10*d*e^6*x^6 + 5*e^7*x^7))) 
)/(315*e^8*(d + e*x)^(5/2))
 

Rubi [A] (verified)

Time = 0.47 (sec) , antiderivative size = 346, normalized size of antiderivative = 1.00, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.083, Rules used = {652, 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.

\(\displaystyle \int \frac {\left (a+c x^2\right )^3 (A+B x)}{(d+e x)^{7/2}} \, dx\)

\(\Big \downarrow \) 652

\(\displaystyle \int \left (-\frac {c \left (-3 a^2 B e^4+12 a A c d e^3-30 a B c d^2 e^2+20 A c^2 d^3 e-35 B c^2 d^4\right )}{e^7 \sqrt {d+e x}}-\frac {3 c^2 (d+e x)^{3/2} \left (-a B e^2+2 A c d e-7 B c d^2\right )}{e^7}+\frac {c^2 \sqrt {d+e x} \left (3 a A e^3-15 a B d e^2+15 A c d^2 e-35 B c d^3\right )}{e^7}+\frac {\left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{e^7 (d+e x)^{5/2}}+\frac {\left (a e^2+c d^2\right )^3 (A e-B d)}{e^7 (d+e x)^{7/2}}+\frac {3 c \left (a e^2+c d^2\right ) \left (a A e^3-3 a B d e^2+5 A c d^2 e-7 B c d^3\right )}{e^7 (d+e x)^{3/2}}+\frac {c^3 (d+e x)^{5/2} (A e-7 B d)}{e^7}+\frac {B c^3 (d+e x)^{7/2}}{e^7}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle -\frac {2 c \sqrt {d+e x} \left (4 A c d e \left (3 a e^2+5 c d^2\right )-B \left (3 a^2 e^4+30 a c d^2 e^2+35 c^2 d^4\right )\right )}{e^8}+\frac {6 c^2 (d+e x)^{5/2} \left (a B e^2-2 A c d e+7 B c d^2\right )}{5 e^8}-\frac {2 c^2 (d+e x)^{3/2} \left (-3 a A e^3+15 a B d e^2-15 A c d^2 e+35 B c d^3\right )}{3 e^8}-\frac {2 \left (a e^2+c d^2\right )^2 \left (a B e^2-6 A c d e+7 B c d^2\right )}{3 e^8 (d+e x)^{3/2}}+\frac {2 \left (a e^2+c d^2\right )^3 (B d-A e)}{5 e^8 (d+e x)^{5/2}}+\frac {6 c \left (a e^2+c d^2\right ) \left (-a A e^3+3 a B d e^2-5 A c d^2 e+7 B c d^3\right )}{e^8 \sqrt {d+e x}}-\frac {2 c^3 (d+e x)^{7/2} (7 B d-A e)}{7 e^8}+\frac {2 B c^3 (d+e x)^{9/2}}{9 e^8}\)

Input:

Int[((A + B*x)*(a + c*x^2)^3)/(d + e*x)^(7/2),x]
 

Output:

(2*(B*d - A*e)*(c*d^2 + a*e^2)^3)/(5*e^8*(d + e*x)^(5/2)) - (2*(c*d^2 + a* 
e^2)^2*(7*B*c*d^2 - 6*A*c*d*e + a*B*e^2))/(3*e^8*(d + e*x)^(3/2)) + (6*c*( 
c*d^2 + a*e^2)*(7*B*c*d^3 - 5*A*c*d^2*e + 3*a*B*d*e^2 - a*A*e^3))/(e^8*Sqr 
t[d + e*x]) - (2*c*(4*A*c*d*e*(5*c*d^2 + 3*a*e^2) - B*(35*c^2*d^4 + 30*a*c 
*d^2*e^2 + 3*a^2*e^4))*Sqrt[d + e*x])/e^8 - (2*c^2*(35*B*c*d^3 - 15*A*c*d^ 
2*e + 15*a*B*d*e^2 - 3*a*A*e^3)*(d + e*x)^(3/2))/(3*e^8) + (6*c^2*(7*B*c*d 
^2 - 2*A*c*d*e + a*B*e^2)*(d + e*x)^(5/2))/(5*e^8) - (2*c^3*(7*B*d - A*e)* 
(d + e*x)^(7/2))/(7*e^8) + (2*B*c^3*(d + e*x)^(9/2))/(9*e^8)
 

Defintions of rubi rules used

rule 652

Int[((d_.) + (e_.)*(x_))^(m_.)*((f_.) + (g_.)*(x_))^(n_.)*((a_) + (c_.)*(x_ 
)^2)^(p_.), x_Symbol] :> Int[ExpandIntegrand[(d + e*x)^m*(f + g*x)^n*(a + c 
*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, n}, x] && IGtQ[p, 0]
 

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

Maple [A] (verified)

Time = 1.07 (sec) , antiderivative size = 304, normalized size of antiderivative = 0.88

method	result	size
pseudoelliptic	\(-\frac {2 \left (\left (-\frac {5 \left (\frac {7 B x}{9}+A \right ) x^{6} c^{3}}{7}-5 \left (\frac {3 B x}{5}+A \right ) x^{4} a \,c^{2}+15 a^{2} x^{2} \left (-B x +A \right ) c +a^{3} \left (\frac {5 B x}{3}+A \right )\right ) e^{7}+20 \left (\left (\frac {1}{18} B \,x^{6}+\frac {3}{35} A \,x^{5}\right ) c^{3}+2 \left (\frac {B x}{4}+A \right ) x^{3} a \,c^{2}+a^{2} x \left (-\frac {9 B x}{2}+A \right ) c +\frac {B \,a^{3}}{30}\right ) d \,e^{6}+8 c \,d^{2} \left (\left (-\frac {1}{3} B \,x^{5}-\frac {5}{7} A \,x^{4}\right ) c^{2}+30 x^{2} \left (-\frac {B x}{3}+A \right ) a c +a^{2} \left (-15 B x +A \right )\right ) e^{5}+320 c \,d^{3} \left (\frac {\left (\frac {7 B x}{36}+A \right ) x^{3} c^{2}}{7}+a x \left (-\frac {3 B x}{2}+A \right ) c -\frac {3 a^{2} B}{20}\right ) e^{4}+128 c^{2} \left (\left (-\frac {5}{9} B \,x^{3}+\frac {15}{7} A \,x^{2}\right ) c +a \left (-5 B x +A \right )\right ) d^{4} e^{3}+\frac {2560 c^{2} d^{5} \left (x \left (-\frac {7 B x}{6}+A \right ) c -\frac {7 B a}{10}\right ) e^{2}}{7}+\frac {1024 c^{3} d^{6} \left (-\frac {35 B x}{9}+A \right ) e}{7}-\frac {2048 B \,c^{3} d^{7}}{9}\right )}{5 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\)	\(304\)
risch	\(-\frac {2 c \left (-35 e^{4} B \,c^{2} x^{4}-45 e^{4} A \,c^{2} x^{3}+175 d \,e^{3} B \,c^{2} x^{3}+243 d \,e^{3} A \,c^{2} x^{2}-189 e^{4} B a c \,x^{2}-588 d^{2} e^{2} c^{2} B \,x^{2}-315 e^{4} A a c x -954 d^{2} e^{2} c^{2} A x +1197 d \,e^{3} B a c x +1834 c^{2} d^{3} e B x +3465 A a c d \,e^{3}+5058 A \,c^{2} d^{3} e -945 B \,e^{4} a^{2}-8064 B a c \,d^{2} e^{2}-8393 B \,c^{2} d^{4}\right ) \sqrt {e x +d}}{315 e^{8}}-\frac {2 \left (45 A \,x^{2} a c \,e^{5}+225 A \,x^{2} c^{2} d^{2} e^{3}-135 B \,x^{2} a c d \,e^{4}-315 B \,x^{2} c^{2} d^{3} e^{2}+60 A x a c d \,e^{4}+420 A x \,c^{2} d^{3} e^{2}+5 B x \,a^{2} e^{5}-230 B x a c \,d^{2} e^{3}-595 B x \,c^{2} d^{4} e +3 A \,a^{2} e^{5}+21 A a c \,d^{2} e^{3}+198 A \,c^{2} d^{4} e +2 B \,a^{2} d \,e^{4}-101 B a c \,d^{3} e^{2}-283 B \,c^{2} d^{5}\right ) \left (a \,e^{2}+c \,d^{2}\right )}{15 e^{8} \sqrt {e x +d}\, \left (e^{2} x^{2}+2 d e x +d^{2}\right )}\)	\(394\)
gosper	\(-\frac {2 \left (-35 B \,x^{7} c^{3} e^{7}-45 A \,x^{6} c^{3} e^{7}+70 B \,x^{6} c^{3} d \,e^{6}+108 A \,x^{5} c^{3} d \,e^{6}-189 B \,x^{5} a \,c^{2} e^{7}-168 B \,x^{5} c^{3} d^{2} e^{5}-315 A \,x^{4} a \,c^{2} e^{7}-360 A \,x^{4} c^{3} d^{2} e^{5}+630 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+2880 A \,x^{3} c^{3} d^{3} e^{4}-945 B \,x^{3} a^{2} c \,e^{7}-5040 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}+945 A \,x^{2} a^{2} c \,e^{7}+15120 A \,x^{2} a \,c^{2} d^{2} e^{5}+17280 A \,x^{2} c^{3} d^{4} e^{3}-5670 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-26880 B \,x^{2} c^{3} d^{5} e^{2}+1260 A x \,a^{2} c d \,e^{6}+20160 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+105 B x \,a^{3} e^{7}-7560 B x \,a^{2} c \,d^{2} e^{5}-40320 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +63 A \,a^{3} e^{7}+504 A \,a^{2} c \,d^{2} e^{5}+8064 A a \,c^{2} d^{4} e^{3}+9216 A \,c^{3} d^{6} e +42 B \,a^{3} d \,e^{6}-3024 B \,a^{2} c \,d^{3} e^{4}-16128 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\)	\(489\)
trager	\(-\frac {2 \left (-35 B \,x^{7} c^{3} e^{7}-45 A \,x^{6} c^{3} e^{7}+70 B \,x^{6} c^{3} d \,e^{6}+108 A \,x^{5} c^{3} d \,e^{6}-189 B \,x^{5} a \,c^{2} e^{7}-168 B \,x^{5} c^{3} d^{2} e^{5}-315 A \,x^{4} a \,c^{2} e^{7}-360 A \,x^{4} c^{3} d^{2} e^{5}+630 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+2880 A \,x^{3} c^{3} d^{3} e^{4}-945 B \,x^{3} a^{2} c \,e^{7}-5040 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}+945 A \,x^{2} a^{2} c \,e^{7}+15120 A \,x^{2} a \,c^{2} d^{2} e^{5}+17280 A \,x^{2} c^{3} d^{4} e^{3}-5670 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-26880 B \,x^{2} c^{3} d^{5} e^{2}+1260 A x \,a^{2} c d \,e^{6}+20160 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+105 B x \,a^{3} e^{7}-7560 B x \,a^{2} c \,d^{2} e^{5}-40320 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +63 A \,a^{3} e^{7}+504 A \,a^{2} c \,d^{2} e^{5}+8064 A a \,c^{2} d^{4} e^{3}+9216 A \,c^{3} d^{6} e +42 B \,a^{3} d \,e^{6}-3024 B \,a^{2} c \,d^{3} e^{4}-16128 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\)	\(489\)
orering	\(-\frac {2 \left (-35 B \,x^{7} c^{3} e^{7}-45 A \,x^{6} c^{3} e^{7}+70 B \,x^{6} c^{3} d \,e^{6}+108 A \,x^{5} c^{3} d \,e^{6}-189 B \,x^{5} a \,c^{2} e^{7}-168 B \,x^{5} c^{3} d^{2} e^{5}-315 A \,x^{4} a \,c^{2} e^{7}-360 A \,x^{4} c^{3} d^{2} e^{5}+630 B \,x^{4} a \,c^{2} d \,e^{6}+560 B \,x^{4} c^{3} d^{3} e^{4}+2520 A \,x^{3} a \,c^{2} d \,e^{6}+2880 A \,x^{3} c^{3} d^{3} e^{4}-945 B \,x^{3} a^{2} c \,e^{7}-5040 B \,x^{3} a \,c^{2} d^{2} e^{5}-4480 B \,x^{3} c^{3} d^{4} e^{3}+945 A \,x^{2} a^{2} c \,e^{7}+15120 A \,x^{2} a \,c^{2} d^{2} e^{5}+17280 A \,x^{2} c^{3} d^{4} e^{3}-5670 B \,x^{2} a^{2} c d \,e^{6}-30240 B \,x^{2} a \,c^{2} d^{3} e^{4}-26880 B \,x^{2} c^{3} d^{5} e^{2}+1260 A x \,a^{2} c d \,e^{6}+20160 A x a \,c^{2} d^{3} e^{4}+23040 A x \,c^{3} d^{5} e^{2}+105 B x \,a^{3} e^{7}-7560 B x \,a^{2} c \,d^{2} e^{5}-40320 B x a \,c^{2} d^{4} e^{3}-35840 B x \,c^{3} d^{6} e +63 A \,a^{3} e^{7}+504 A \,a^{2} c \,d^{2} e^{5}+8064 A a \,c^{2} d^{4} e^{3}+9216 A \,c^{3} d^{6} e +42 B \,a^{3} d \,e^{6}-3024 B \,a^{2} c \,d^{3} e^{4}-16128 B a \,c^{2} d^{5} e^{2}-14336 B \,c^{3} d^{7}\right )}{315 \left (e x +d \right )^{\frac {5}{2}} e^{8}}\)	\(489\)
derivativedivides	\(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}-2 B \,c^{3} d \left (e x +d \right )^{\frac {7}{2}}-\frac {12 A \,c^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {6 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {42 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+2 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+10 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-10 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {70 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-24 A a \,c^{2} d \,e^{3} \sqrt {e x +d}-40 A \,c^{3} d^{3} e \sqrt {e x +d}+6 B \,a^{2} c \,e^{4} \sqrt {e x +d}+60 B a \,c^{2} d^{2} e^{2} \sqrt {e x +d}+70 B \,c^{3} d^{4} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{\sqrt {e x +d}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\)	\(506\)
default	\(\frac {\frac {2 B \,c^{3} \left (e x +d \right )^{\frac {9}{2}}}{9}+\frac {2 A \,c^{3} e \left (e x +d \right )^{\frac {7}{2}}}{7}-2 B \,c^{3} d \left (e x +d \right )^{\frac {7}{2}}-\frac {12 A \,c^{3} d e \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {6 B a \,c^{2} e^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+\frac {42 B \,c^{3} d^{2} \left (e x +d \right )^{\frac {5}{2}}}{5}+2 A a \,c^{2} e^{3} \left (e x +d \right )^{\frac {3}{2}}+10 A \,c^{3} d^{2} e \left (e x +d \right )^{\frac {3}{2}}-10 B a \,c^{2} d \,e^{2} \left (e x +d \right )^{\frac {3}{2}}-\frac {70 B \,c^{3} d^{3} \left (e x +d \right )^{\frac {3}{2}}}{3}-24 A a \,c^{2} d \,e^{3} \sqrt {e x +d}-40 A \,c^{3} d^{3} e \sqrt {e x +d}+6 B \,a^{2} c \,e^{4} \sqrt {e x +d}+60 B a \,c^{2} d^{2} e^{2} \sqrt {e x +d}+70 B \,c^{3} d^{4} \sqrt {e x +d}-\frac {2 \left (A \,a^{3} e^{7}+3 A \,a^{2} c \,d^{2} e^{5}+3 A a \,c^{2} d^{4} e^{3}+A \,c^{3} d^{6} e -B \,a^{3} d \,e^{6}-3 B \,a^{2} c \,d^{3} e^{4}-3 B a \,c^{2} d^{5} e^{2}-B \,c^{3} d^{7}\right )}{5 \left (e x +d \right )^{\frac {5}{2}}}-\frac {6 c \left (A \,a^{2} e^{5}+6 A a c \,d^{2} e^{3}+5 A \,c^{2} d^{4} e -3 B \,a^{2} d \,e^{4}-10 B a c \,d^{3} e^{2}-7 B \,c^{2} d^{5}\right )}{\sqrt {e x +d}}-\frac {2 \left (-6 A \,a^{2} c d \,e^{5}-12 A a \,c^{2} d^{3} e^{3}-6 A \,c^{3} d^{5} e +B \,a^{3} e^{6}+9 B \,a^{2} c \,d^{2} e^{4}+15 B a \,c^{2} d^{4} e^{2}+7 B \,c^{3} d^{6}\right )}{3 \left (e x +d \right )^{\frac {3}{2}}}}{e^{8}}\)	\(506\)

Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x,method=_RETURNVERBOSE)
 

Output:

-2/5/(e*x+d)^(5/2)*((-5/7*(7/9*B*x+A)*x^6*c^3-5*(3/5*B*x+A)*x^4*a*c^2+15*a 
^2*x^2*(-B*x+A)*c+a^3*(5/3*B*x+A))*e^7+20*((1/18*B*x^6+3/35*A*x^5)*c^3+2*( 
1/4*B*x+A)*x^3*a*c^2+a^2*x*(-9/2*B*x+A)*c+1/30*B*a^3)*d*e^6+8*c*d^2*((-1/3 
*B*x^5-5/7*A*x^4)*c^2+30*x^2*(-1/3*B*x+A)*a*c+a^2*(-15*B*x+A))*e^5+320*c*d 
^3*(1/7*(7/36*B*x+A)*x^3*c^2+a*x*(-3/2*B*x+A)*c-3/20*a^2*B)*e^4+128*c^2*(( 
-5/9*B*x^3+15/7*A*x^2)*c+a*(-5*B*x+A))*d^4*e^3+2560/7*c^2*d^5*(x*(-7/6*B*x 
+A)*c-7/10*B*a)*e^2+1024/7*c^3*d^6*(-35/9*B*x+A)*e-2048/9*B*c^3*d^7)/e^8
 

Fricas [A] (verification not implemented)

Time = 0.09 (sec) , antiderivative size = 487, normalized size of antiderivative = 1.41 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (35 \, B c^{3} e^{7} x^{7} + 14336 \, B c^{3} d^{7} - 9216 \, A c^{3} d^{6} e + 16128 \, B a c^{2} d^{5} e^{2} - 8064 \, A a c^{2} d^{4} e^{3} + 3024 \, B a^{2} c d^{3} e^{4} - 504 \, A a^{2} c d^{2} e^{5} - 42 \, B a^{3} d e^{6} - 63 \, A a^{3} e^{7} - 5 \, {\left (14 \, B c^{3} d e^{6} - 9 \, A c^{3} e^{7}\right )} x^{6} + 3 \, {\left (56 \, B c^{3} d^{2} e^{5} - 36 \, A c^{3} d e^{6} + 63 \, B a c^{2} e^{7}\right )} x^{5} - 5 \, {\left (112 \, B c^{3} d^{3} e^{4} - 72 \, A c^{3} d^{2} e^{5} + 126 \, B a c^{2} d e^{6} - 63 \, A a c^{2} e^{7}\right )} x^{4} + 5 \, {\left (896 \, B c^{3} d^{4} e^{3} - 576 \, A c^{3} d^{3} e^{4} + 1008 \, B a c^{2} d^{2} e^{5} - 504 \, A a c^{2} d e^{6} + 189 \, B a^{2} c e^{7}\right )} x^{3} + 15 \, {\left (1792 \, B c^{3} d^{5} e^{2} - 1152 \, A c^{3} d^{4} e^{3} + 2016 \, B a c^{2} d^{3} e^{4} - 1008 \, A a c^{2} d^{2} e^{5} + 378 \, B a^{2} c d e^{6} - 63 \, A a^{2} c e^{7}\right )} x^{2} + 5 \, {\left (7168 \, B c^{3} d^{6} e - 4608 \, A c^{3} d^{5} e^{2} + 8064 \, B a c^{2} d^{4} e^{3} - 4032 \, A a c^{2} d^{3} e^{4} + 1512 \, B a^{2} c d^{2} e^{5} - 252 \, A a^{2} c d e^{6} - 21 \, B a^{3} e^{7}\right )} x\right )} \sqrt {e x + d}}{315 \, {\left (e^{11} x^{3} + 3 \, d e^{10} x^{2} + 3 \, d^{2} e^{9} x + d^{3} e^{8}\right )}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x, algorithm="fricas")
 

Output:

2/315*(35*B*c^3*e^7*x^7 + 14336*B*c^3*d^7 - 9216*A*c^3*d^6*e + 16128*B*a*c 
^2*d^5*e^2 - 8064*A*a*c^2*d^4*e^3 + 3024*B*a^2*c*d^3*e^4 - 504*A*a^2*c*d^2 
*e^5 - 42*B*a^3*d*e^6 - 63*A*a^3*e^7 - 5*(14*B*c^3*d*e^6 - 9*A*c^3*e^7)*x^ 
6 + 3*(56*B*c^3*d^2*e^5 - 36*A*c^3*d*e^6 + 63*B*a*c^2*e^7)*x^5 - 5*(112*B* 
c^3*d^3*e^4 - 72*A*c^3*d^2*e^5 + 126*B*a*c^2*d*e^6 - 63*A*a*c^2*e^7)*x^4 + 
 5*(896*B*c^3*d^4*e^3 - 576*A*c^3*d^3*e^4 + 1008*B*a*c^2*d^2*e^5 - 504*A*a 
*c^2*d*e^6 + 189*B*a^2*c*e^7)*x^3 + 15*(1792*B*c^3*d^5*e^2 - 1152*A*c^3*d^ 
4*e^3 + 2016*B*a*c^2*d^3*e^4 - 1008*A*a*c^2*d^2*e^5 + 378*B*a^2*c*d*e^6 - 
63*A*a^2*c*e^7)*x^2 + 5*(7168*B*c^3*d^6*e - 4608*A*c^3*d^5*e^2 + 8064*B*a* 
c^2*d^4*e^3 - 4032*A*a*c^2*d^3*e^4 + 1512*B*a^2*c*d^2*e^5 - 252*A*a^2*c*d* 
e^6 - 21*B*a^3*e^7)*x)*sqrt(e*x + d)/(e^11*x^3 + 3*d*e^10*x^2 + 3*d^2*e^9* 
x + d^3*e^8)
 

Sympy [A] (verification not implemented)

Time = 21.27 (sec) , antiderivative size = 476, normalized size of antiderivative = 1.38 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\begin {cases} \frac {2 \left (\frac {B c^{3} \left (d + e x\right )^{\frac {9}{2}}}{9 e^{7}} + \frac {3 c \left (a e^{2} + c d^{2}\right ) \left (- A a e^{3} - 5 A c d^{2} e + 3 B a d e^{2} + 7 B c d^{3}\right )}{e^{7} \sqrt {d + e x}} + \frac {\left (d + e x\right )^{\frac {7}{2}} \left (A c^{3} e - 7 B c^{3} d\right )}{7 e^{7}} + \frac {\left (d + e x\right )^{\frac {5}{2}} \left (- 6 A c^{3} d e + 3 B a c^{2} e^{2} + 21 B c^{3} d^{2}\right )}{5 e^{7}} + \frac {\left (d + e x\right )^{\frac {3}{2}} \cdot \left (3 A a c^{2} e^{3} + 15 A c^{3} d^{2} e - 15 B a c^{2} d e^{2} - 35 B c^{3} d^{3}\right )}{3 e^{7}} + \frac {\sqrt {d + e x} \left (- 12 A a c^{2} d e^{3} - 20 A c^{3} d^{3} e + 3 B a^{2} c e^{4} + 30 B a c^{2} d^{2} e^{2} + 35 B c^{3} d^{4}\right )}{e^{7}} - \frac {\left (a e^{2} + c d^{2}\right )^{2} \left (- 6 A c d e + B a e^{2} + 7 B c d^{2}\right )}{3 e^{7} \left (d + e x\right )^{\frac {3}{2}}} + \frac {\left (- A e + B d\right ) \left (a e^{2} + c d^{2}\right )^{3}}{5 e^{7} \left (d + e x\right )^{\frac {5}{2}}}\right )}{e} & \text {for}\: e \neq 0 \\\frac {A a^{3} x + A a^{2} c x^{3} + \frac {3 A a c^{2} x^{5}}{5} + \frac {A c^{3} x^{7}}{7} + \frac {B a^{3} x^{2}}{2} + \frac {3 B a^{2} c x^{4}}{4} + \frac {B a c^{2} x^{6}}{2} + \frac {B c^{3} x^{8}}{8}}{d^{\frac {7}{2}}} & \text {otherwise} \end {cases} \] Input:

integrate((B*x+A)*(c*x**2+a)**3/(e*x+d)**(7/2),x)
 

Output:

Piecewise((2*(B*c**3*(d + e*x)**(9/2)/(9*e**7) + 3*c*(a*e**2 + c*d**2)*(-A 
*a*e**3 - 5*A*c*d**2*e + 3*B*a*d*e**2 + 7*B*c*d**3)/(e**7*sqrt(d + e*x)) + 
 (d + e*x)**(7/2)*(A*c**3*e - 7*B*c**3*d)/(7*e**7) + (d + e*x)**(5/2)*(-6* 
A*c**3*d*e + 3*B*a*c**2*e**2 + 21*B*c**3*d**2)/(5*e**7) + (d + e*x)**(3/2) 
*(3*A*a*c**2*e**3 + 15*A*c**3*d**2*e - 15*B*a*c**2*d*e**2 - 35*B*c**3*d**3 
)/(3*e**7) + sqrt(d + e*x)*(-12*A*a*c**2*d*e**3 - 20*A*c**3*d**3*e + 3*B*a 
**2*c*e**4 + 30*B*a*c**2*d**2*e**2 + 35*B*c**3*d**4)/e**7 - (a*e**2 + c*d* 
*2)**2*(-6*A*c*d*e + B*a*e**2 + 7*B*c*d**2)/(3*e**7*(d + e*x)**(3/2)) + (- 
A*e + B*d)*(a*e**2 + c*d**2)**3/(5*e**7*(d + e*x)**(5/2)))/e, Ne(e, 0)), ( 
(A*a**3*x + A*a**2*c*x**3 + 3*A*a*c**2*x**5/5 + A*c**3*x**7/7 + B*a**3*x** 
2/2 + 3*B*a**2*c*x**4/4 + B*a*c**2*x**6/2 + B*c**3*x**8/8)/d**(7/2), True) 
)
 

Maxima [A] (verification not implemented)

Time = 0.04 (sec) , antiderivative size = 461, normalized size of antiderivative = 1.33 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (\frac {35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{3} - 45 \, {\left (7 \, B c^{3} d - A c^{3} e\right )} {\left (e x + d\right )}^{\frac {7}{2}} + 189 \, {\left (7 \, B c^{3} d^{2} - 2 \, A c^{3} d e + B a c^{2} e^{2}\right )} {\left (e x + d\right )}^{\frac {5}{2}} - 105 \, {\left (35 \, B c^{3} d^{3} - 15 \, A c^{3} d^{2} e + 15 \, B a c^{2} d e^{2} - 3 \, A a c^{2} e^{3}\right )} {\left (e x + d\right )}^{\frac {3}{2}} + 315 \, {\left (35 \, B c^{3} d^{4} - 20 \, A c^{3} d^{3} e + 30 \, B a c^{2} d^{2} e^{2} - 12 \, A a c^{2} d e^{3} + 3 \, B a^{2} c e^{4}\right )} \sqrt {e x + d}}{e^{7}} + \frac {21 \, {\left (3 \, B c^{3} d^{7} - 3 \, A c^{3} d^{6} e + 9 \, B a c^{2} d^{5} e^{2} - 9 \, A a c^{2} d^{4} e^{3} + 9 \, B a^{2} c d^{3} e^{4} - 9 \, A a^{2} c d^{2} e^{5} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7} + 45 \, {\left (7 \, B c^{3} d^{5} - 5 \, A c^{3} d^{4} e + 10 \, B a c^{2} d^{3} e^{2} - 6 \, A a c^{2} d^{2} e^{3} + 3 \, B a^{2} c d e^{4} - A a^{2} c e^{5}\right )} {\left (e x + d\right )}^{2} - 5 \, {\left (7 \, B c^{3} d^{6} - 6 \, A c^{3} d^{5} e + 15 \, B a c^{2} d^{4} e^{2} - 12 \, A a c^{2} d^{3} e^{3} + 9 \, B a^{2} c d^{2} e^{4} - 6 \, A a^{2} c d e^{5} + B a^{3} e^{6}\right )} {\left (e x + d\right )}\right )}}{{\left (e x + d\right )}^{\frac {5}{2}} e^{7}}\right )}}{315 \, e} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x, algorithm="maxima")
 

Output:

2/315*((35*(e*x + d)^(9/2)*B*c^3 - 45*(7*B*c^3*d - A*c^3*e)*(e*x + d)^(7/2 
) + 189*(7*B*c^3*d^2 - 2*A*c^3*d*e + B*a*c^2*e^2)*(e*x + d)^(5/2) - 105*(3 
5*B*c^3*d^3 - 15*A*c^3*d^2*e + 15*B*a*c^2*d*e^2 - 3*A*a*c^2*e^3)*(e*x + d) 
^(3/2) + 315*(35*B*c^3*d^4 - 20*A*c^3*d^3*e + 30*B*a*c^2*d^2*e^2 - 12*A*a* 
c^2*d*e^3 + 3*B*a^2*c*e^4)*sqrt(e*x + d))/e^7 + 21*(3*B*c^3*d^7 - 3*A*c^3* 
d^6*e + 9*B*a*c^2*d^5*e^2 - 9*A*a*c^2*d^4*e^3 + 9*B*a^2*c*d^3*e^4 - 9*A*a^ 
2*c*d^2*e^5 + 3*B*a^3*d*e^6 - 3*A*a^3*e^7 + 45*(7*B*c^3*d^5 - 5*A*c^3*d^4* 
e + 10*B*a*c^2*d^3*e^2 - 6*A*a*c^2*d^2*e^3 + 3*B*a^2*c*d*e^4 - A*a^2*c*e^5 
)*(e*x + d)^2 - 5*(7*B*c^3*d^6 - 6*A*c^3*d^5*e + 15*B*a*c^2*d^4*e^2 - 12*A 
*a*c^2*d^3*e^3 + 9*B*a^2*c*d^2*e^4 - 6*A*a^2*c*d*e^5 + B*a^3*e^6)*(e*x + d 
))/((e*x + d)^(5/2)*e^7))/e
 

Giac [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 599, normalized size of antiderivative = 1.73 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {2 \, {\left (315 \, {\left (e x + d\right )}^{2} B c^{3} d^{5} - 35 \, {\left (e x + d\right )} B c^{3} d^{6} + 3 \, B c^{3} d^{7} - 225 \, {\left (e x + d\right )}^{2} A c^{3} d^{4} e + 30 \, {\left (e x + d\right )} A c^{3} d^{5} e - 3 \, A c^{3} d^{6} e + 450 \, {\left (e x + d\right )}^{2} B a c^{2} d^{3} e^{2} - 75 \, {\left (e x + d\right )} B a c^{2} d^{4} e^{2} + 9 \, B a c^{2} d^{5} e^{2} - 270 \, {\left (e x + d\right )}^{2} A a c^{2} d^{2} e^{3} + 60 \, {\left (e x + d\right )} A a c^{2} d^{3} e^{3} - 9 \, A a c^{2} d^{4} e^{3} + 135 \, {\left (e x + d\right )}^{2} B a^{2} c d e^{4} - 45 \, {\left (e x + d\right )} B a^{2} c d^{2} e^{4} + 9 \, B a^{2} c d^{3} e^{4} - 45 \, {\left (e x + d\right )}^{2} A a^{2} c e^{5} + 30 \, {\left (e x + d\right )} A a^{2} c d e^{5} - 9 \, A a^{2} c d^{2} e^{5} - 5 \, {\left (e x + d\right )} B a^{3} e^{6} + 3 \, B a^{3} d e^{6} - 3 \, A a^{3} e^{7}\right )}}{15 \, {\left (e x + d\right )}^{\frac {5}{2}} e^{8}} + \frac {2 \, {\left (35 \, {\left (e x + d\right )}^{\frac {9}{2}} B c^{3} e^{64} - 315 \, {\left (e x + d\right )}^{\frac {7}{2}} B c^{3} d e^{64} + 1323 \, {\left (e x + d\right )}^{\frac {5}{2}} B c^{3} d^{2} e^{64} - 3675 \, {\left (e x + d\right )}^{\frac {3}{2}} B c^{3} d^{3} e^{64} + 11025 \, \sqrt {e x + d} B c^{3} d^{4} e^{64} + 45 \, {\left (e x + d\right )}^{\frac {7}{2}} A c^{3} e^{65} - 378 \, {\left (e x + d\right )}^{\frac {5}{2}} A c^{3} d e^{65} + 1575 \, {\left (e x + d\right )}^{\frac {3}{2}} A c^{3} d^{2} e^{65} - 6300 \, \sqrt {e x + d} A c^{3} d^{3} e^{65} + 189 \, {\left (e x + d\right )}^{\frac {5}{2}} B a c^{2} e^{66} - 1575 \, {\left (e x + d\right )}^{\frac {3}{2}} B a c^{2} d e^{66} + 9450 \, \sqrt {e x + d} B a c^{2} d^{2} e^{66} + 315 \, {\left (e x + d\right )}^{\frac {3}{2}} A a c^{2} e^{67} - 3780 \, \sqrt {e x + d} A a c^{2} d e^{67} + 945 \, \sqrt {e x + d} B a^{2} c e^{68}\right )}}{315 \, e^{72}} \] Input:

integrate((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x, algorithm="giac")
                                                                                    
                                                                                    
 

Output:

2/15*(315*(e*x + d)^2*B*c^3*d^5 - 35*(e*x + d)*B*c^3*d^6 + 3*B*c^3*d^7 - 2 
25*(e*x + d)^2*A*c^3*d^4*e + 30*(e*x + d)*A*c^3*d^5*e - 3*A*c^3*d^6*e + 45 
0*(e*x + d)^2*B*a*c^2*d^3*e^2 - 75*(e*x + d)*B*a*c^2*d^4*e^2 + 9*B*a*c^2*d 
^5*e^2 - 270*(e*x + d)^2*A*a*c^2*d^2*e^3 + 60*(e*x + d)*A*a*c^2*d^3*e^3 - 
9*A*a*c^2*d^4*e^3 + 135*(e*x + d)^2*B*a^2*c*d*e^4 - 45*(e*x + d)*B*a^2*c*d 
^2*e^4 + 9*B*a^2*c*d^3*e^4 - 45*(e*x + d)^2*A*a^2*c*e^5 + 30*(e*x + d)*A*a 
^2*c*d*e^5 - 9*A*a^2*c*d^2*e^5 - 5*(e*x + d)*B*a^3*e^6 + 3*B*a^3*d*e^6 - 3 
*A*a^3*e^7)/((e*x + d)^(5/2)*e^8) + 2/315*(35*(e*x + d)^(9/2)*B*c^3*e^64 - 
 315*(e*x + d)^(7/2)*B*c^3*d*e^64 + 1323*(e*x + d)^(5/2)*B*c^3*d^2*e^64 - 
3675*(e*x + d)^(3/2)*B*c^3*d^3*e^64 + 11025*sqrt(e*x + d)*B*c^3*d^4*e^64 + 
 45*(e*x + d)^(7/2)*A*c^3*e^65 - 378*(e*x + d)^(5/2)*A*c^3*d*e^65 + 1575*( 
e*x + d)^(3/2)*A*c^3*d^2*e^65 - 6300*sqrt(e*x + d)*A*c^3*d^3*e^65 + 189*(e 
*x + d)^(5/2)*B*a*c^2*e^66 - 1575*(e*x + d)^(3/2)*B*a*c^2*d*e^66 + 9450*sq 
rt(e*x + d)*B*a*c^2*d^2*e^66 + 315*(e*x + d)^(3/2)*A*a*c^2*e^67 - 3780*sqr 
t(e*x + d)*A*a*c^2*d*e^67 + 945*sqrt(e*x + d)*B*a^2*c*e^68)/e^72
 

Mupad [B] (verification not implemented)

Time = 6.53 (sec) , antiderivative size = 455, normalized size of antiderivative = 1.32 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {\sqrt {d+e\,x}\,\left (6\,B\,a^2\,c\,e^4+60\,B\,a\,c^2\,d^2\,e^2-24\,A\,a\,c^2\,d\,e^3+70\,B\,c^3\,d^4-40\,A\,c^3\,d^3\,e\right )}{e^8}-\frac {\left (d+e\,x\right )\,\left (\frac {2\,B\,a^3\,e^6}{3}+6\,B\,a^2\,c\,d^2\,e^4-4\,A\,a^2\,c\,d\,e^5+10\,B\,a\,c^2\,d^4\,e^2-8\,A\,a\,c^2\,d^3\,e^3+\frac {14\,B\,c^3\,d^6}{3}-4\,A\,c^3\,d^5\,e\right )-{\left (d+e\,x\right )}^2\,\left (18\,B\,a^2\,c\,d\,e^4-6\,A\,a^2\,c\,e^5+60\,B\,a\,c^2\,d^3\,e^2-36\,A\,a\,c^2\,d^2\,e^3+42\,B\,c^3\,d^5-30\,A\,c^3\,d^4\,e\right )+\frac {2\,A\,a^3\,e^7}{5}-\frac {2\,B\,c^3\,d^7}{5}-\frac {2\,B\,a^3\,d\,e^6}{5}+\frac {2\,A\,c^3\,d^6\,e}{5}+\frac {6\,A\,a\,c^2\,d^4\,e^3}{5}+\frac {6\,A\,a^2\,c\,d^2\,e^5}{5}-\frac {6\,B\,a\,c^2\,d^5\,e^2}{5}-\frac {6\,B\,a^2\,c\,d^3\,e^4}{5}}{e^8\,{\left (d+e\,x\right )}^{5/2}}+\frac {{\left (d+e\,x\right )}^{5/2}\,\left (42\,B\,c^3\,d^2-12\,A\,c^3\,d\,e+6\,B\,a\,c^2\,e^2\right )}{5\,e^8}+\frac {2\,B\,c^3\,{\left (d+e\,x\right )}^{9/2}}{9\,e^8}+\frac {2\,c^2\,{\left (d+e\,x\right )}^{3/2}\,\left (-35\,B\,c\,d^3+15\,A\,c\,d^2\,e-15\,B\,a\,d\,e^2+3\,A\,a\,e^3\right )}{3\,e^8}+\frac {2\,c^3\,\left (A\,e-7\,B\,d\right )\,{\left (d+e\,x\right )}^{7/2}}{7\,e^8} \] Input:

int(((a + c*x^2)^3*(A + B*x))/(d + e*x)^(7/2),x)
 

Output:

((d + e*x)^(1/2)*(70*B*c^3*d^4 + 6*B*a^2*c*e^4 - 40*A*c^3*d^3*e + 60*B*a*c 
^2*d^2*e^2 - 24*A*a*c^2*d*e^3))/e^8 - ((d + e*x)*((2*B*a^3*e^6)/3 + (14*B* 
c^3*d^6)/3 - 4*A*c^3*d^5*e - 8*A*a*c^2*d^3*e^3 + 10*B*a*c^2*d^4*e^2 + 6*B* 
a^2*c*d^2*e^4 - 4*A*a^2*c*d*e^5) - (d + e*x)^2*(42*B*c^3*d^5 - 6*A*a^2*c*e 
^5 - 30*A*c^3*d^4*e - 36*A*a*c^2*d^2*e^3 + 60*B*a*c^2*d^3*e^2 + 18*B*a^2*c 
*d*e^4) + (2*A*a^3*e^7)/5 - (2*B*c^3*d^7)/5 - (2*B*a^3*d*e^6)/5 + (2*A*c^3 
*d^6*e)/5 + (6*A*a*c^2*d^4*e^3)/5 + (6*A*a^2*c*d^2*e^5)/5 - (6*B*a*c^2*d^5 
*e^2)/5 - (6*B*a^2*c*d^3*e^4)/5)/(e^8*(d + e*x)^(5/2)) + ((d + e*x)^(5/2)* 
(42*B*c^3*d^2 - 12*A*c^3*d*e + 6*B*a*c^2*e^2))/(5*e^8) + (2*B*c^3*(d + e*x 
)^(9/2))/(9*e^8) + (2*c^2*(d + e*x)^(3/2)*(3*A*a*e^3 - 35*B*c*d^3 - 15*B*a 
*d*e^2 + 15*A*c*d^2*e))/(3*e^8) + (2*c^3*(A*e - 7*B*d)*(d + e*x)^(7/2))/(7 
*e^8)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 508, normalized size of antiderivative = 1.47 \[ \int \frac {(A+B x) \left (a+c x^2\right )^3}{(d+e x)^{7/2}} \, dx=\frac {\frac {4096}{45} b \,c^{3} d^{7}-8 a^{3} c d \,e^{6} x +6 a^{2} b c \,e^{7} x^{3}-\frac {4}{15} a^{3} b d \,e^{6}-\frac {16}{5} a^{3} c \,d^{2} e^{5}-\frac {256}{5} a^{2} c^{2} d^{4} e^{3}-\frac {2048}{35} a \,c^{3} d^{6} e -\frac {2}{5} a^{4} e^{7}+48 a^{2} b c \,d^{2} e^{5} x +36 a^{2} b c d \,e^{6} x^{2}+256 a b \,c^{2} d^{4} e^{3} x +192 a b \,c^{2} d^{3} e^{4} x^{2}+32 a b \,c^{2} d^{2} e^{5} x^{3}-4 a b \,c^{2} d \,e^{6} x^{4}-\frac {2}{3} a^{3} b \,e^{7} x -6 a^{3} c \,e^{7} x^{2}+2 a^{2} c^{2} e^{7} x^{4}+\frac {2}{7} a \,c^{3} e^{7} x^{6}+\frac {2}{9} b \,c^{3} e^{7} x^{7}-128 a^{2} c^{2} d^{3} e^{4} x -96 a^{2} c^{2} d^{2} e^{5} x^{2}-16 a^{2} c^{2} d \,e^{6} x^{3}+\frac {6}{5} a b \,c^{2} e^{7} x^{5}-\frac {1024}{7} a \,c^{3} d^{5} e^{2} x -\frac {768}{7} a \,c^{3} d^{4} e^{3} x^{2}-\frac {128}{7} a \,c^{3} d^{3} e^{4} x^{3}+\frac {16}{7} a \,c^{3} d^{2} e^{5} x^{4}-\frac {24}{35} a \,c^{3} d \,e^{6} x^{5}+\frac {2048}{9} b \,c^{3} d^{6} e x +\frac {512}{3} b \,c^{3} d^{5} e^{2} x^{2}+\frac {256}{9} b \,c^{3} d^{4} e^{3} x^{3}-\frac {32}{9} b \,c^{3} d^{3} e^{4} x^{4}+\frac {16}{15} b \,c^{3} d^{2} e^{5} x^{5}-\frac {4}{9} b \,c^{3} d \,e^{6} x^{6}+\frac {96}{5} a^{2} b c \,d^{3} e^{4}+\frac {512}{5} a b \,c^{2} d^{5} e^{2}}{\sqrt {e x +d}\, e^{8} \left (e^{2} x^{2}+2 d e x +d^{2}\right )} \] Input:

int((B*x+A)*(c*x^2+a)^3/(e*x+d)^(7/2),x)
 

Output:

(2*( - 63*a**4*e**7 - 42*a**3*b*d*e**6 - 105*a**3*b*e**7*x - 504*a**3*c*d* 
*2*e**5 - 1260*a**3*c*d*e**6*x - 945*a**3*c*e**7*x**2 + 3024*a**2*b*c*d**3 
*e**4 + 7560*a**2*b*c*d**2*e**5*x + 5670*a**2*b*c*d*e**6*x**2 + 945*a**2*b 
*c*e**7*x**3 - 8064*a**2*c**2*d**4*e**3 - 20160*a**2*c**2*d**3*e**4*x - 15 
120*a**2*c**2*d**2*e**5*x**2 - 2520*a**2*c**2*d*e**6*x**3 + 315*a**2*c**2* 
e**7*x**4 + 16128*a*b*c**2*d**5*e**2 + 40320*a*b*c**2*d**4*e**3*x + 30240* 
a*b*c**2*d**3*e**4*x**2 + 5040*a*b*c**2*d**2*e**5*x**3 - 630*a*b*c**2*d*e* 
*6*x**4 + 189*a*b*c**2*e**7*x**5 - 9216*a*c**3*d**6*e - 23040*a*c**3*d**5* 
e**2*x - 17280*a*c**3*d**4*e**3*x**2 - 2880*a*c**3*d**3*e**4*x**3 + 360*a* 
c**3*d**2*e**5*x**4 - 108*a*c**3*d*e**6*x**5 + 45*a*c**3*e**7*x**6 + 14336 
*b*c**3*d**7 + 35840*b*c**3*d**6*e*x + 26880*b*c**3*d**5*e**2*x**2 + 4480* 
b*c**3*d**4*e**3*x**3 - 560*b*c**3*d**3*e**4*x**4 + 168*b*c**3*d**2*e**5*x 
**5 - 70*b*c**3*d*e**6*x**6 + 35*b*c**3*e**7*x**7))/(315*sqrt(d + e*x)*e** 
8*(d**2 + 2*d*e*x + e**2*x**2))