\(\int \frac {x (a+b x^2)^{5/2}}{(c+d x)^{10}} \, dx\) [1152]

Optimal result

Integrand size = 20, antiderivative size = 470 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=-\frac {5 a^3 b^3 d \left (10 b c^2-a d^2\right ) (a d-b c x) \sqrt {a+b x^2}}{128 \left (b c^2+a d^2\right )^6 (c+d x)^2}-\frac {5 a^2 b^2 d \left (10 b c^2-a d^2\right ) (a d-b c x) \left (a+b x^2\right )^{3/2}}{192 \left (b c^2+a d^2\right )^5 (c+d x)^4}+\frac {c \left (a+b x^2\right )^{5/2}}{9 d^2 (c+d x)^9}-\frac {\left (14 b c^2+9 a d^2\right ) \left (a+b x^2\right )^{5/2}}{72 d^2 \left (b c^2+a d^2\right ) (c+d x)^8}+\frac {5 b c \left (6 b c^2+17 a d^2\right ) \left (a+b x^2\right )^{5/2}}{504 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^7}+\frac {5 b \left (4 b^2 c^4+16 a b c^2 d^2-21 a^2 d^4\right ) \left (a+b x^2\right )^{5/2}}{1008 d^2 \left (b c^2+a d^2\right )^3 (c+d x)^6}+\frac {b^2 c \left (4 b^2 c^4+20 a b c^2 d^2-215 a^2 d^4\right ) \left (a+b x^2\right )^{5/2}}{1008 d^2 \left (b c^2+a d^2\right )^4 (c+d x)^5}-\frac {5 a^4 b^4 d \left (10 b c^2-a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{128 \left (b c^2+a d^2\right )^{13/2}} \] Output:

-5/128*a^3*b^3*d*(-a*d^2+10*b*c^2)*(-b*c*x+a*d)*(b*x^2+a)^(1/2)/(a*d^2+b*c 
^2)^6/(d*x+c)^2-5/192*a^2*b^2*d*(-a*d^2+10*b*c^2)*(-b*c*x+a*d)*(b*x^2+a)^( 
3/2)/(a*d^2+b*c^2)^5/(d*x+c)^4+1/9*c*(b*x^2+a)^(5/2)/d^2/(d*x+c)^9-1/72*(9 
*a*d^2+14*b*c^2)*(b*x^2+a)^(5/2)/d^2/(a*d^2+b*c^2)/(d*x+c)^8+5/504*b*c*(17 
*a*d^2+6*b*c^2)*(b*x^2+a)^(5/2)/d^2/(a*d^2+b*c^2)^2/(d*x+c)^7+5/1008*b*(-2 
1*a^2*d^4+16*a*b*c^2*d^2+4*b^2*c^4)*(b*x^2+a)^(5/2)/d^2/(a*d^2+b*c^2)^3/(d 
*x+c)^6+1/1008*b^2*c*(-215*a^2*d^4+20*a*b*c^2*d^2+4*b^2*c^4)*(b*x^2+a)^(5/ 
2)/d^2/(a*d^2+b*c^2)^4/(d*x+c)^5-5/128*a^4*b^4*d*(-a*d^2+10*b*c^2)*arctanh 
((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/(a*d^2+b*c^2)^(13/2)
 

Mathematica [A] (verified)

Time = 10.71 (sec) , antiderivative size = 641, normalized size of antiderivative = 1.36 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\frac {\sqrt {b c^2+a d^2} \sqrt {a+b x^2} \left (896 c \left (b c^2+a d^2\right )^8-112 \left (b c^2+a d^2\right )^6 \left (46 b^2 c^4+55 a b c^2 d^2+9 a^2 d^4\right ) (c+d x)+16 b c \left (b c^2+a d^2\right )^6 \left (758 b c^2+449 a d^2\right ) (c+d x)^2-8 b \left (b c^2+a d^2\right )^5 \left (1844 b^2 c^4+2196 a b c^2 d^2+357 a^2 d^4\right ) (c+d x)^3+8 b^2 c \left (b c^2+a d^2\right )^4 \left (1180 b^2 c^4+2352 a b c^2 d^2+1161 a^2 d^4\right ) (c+d x)^4-2 b^2 \left (b c^2+a d^2\right )^3 \left (1328 b^3 c^6+3992 a b^2 c^4 d^2+4002 a^2 b c^2 d^4+1239 a^3 d^6\right ) (c+d x)^5+2 b^3 c \left (b c^2+a d^2\right )^2 \left (16 b^3 c^6+72 a b^2 c^4 d^2+150 a^2 b c^2 d^4+325 a^3 d^6\right ) (c+d x)^6+b^3 \left (b c^2+a d^2\right ) \left (32 b^4 c^8+192 a b^3 c^6 d^2+540 a^2 b^2 c^4 d^4+1220 a^3 b c^2 d^6-315 a^4 d^8\right ) (c+d x)^7+b^4 c \left (32 b^4 c^8+224 a b^3 c^6 d^2+732 a^2 b^2 c^4 d^4+1760 a^3 b c^2 d^6-2245 a^4 d^8\right ) (c+d x)^8\right )-315 a^4 b^4 d^7 \left (-10 b c^2+a d^2\right ) (c+d x)^9 \log (c+d x)+315 a^4 b^4 d^7 \left (-10 b c^2+a d^2\right ) (c+d x)^9 \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{8064 d^6 \left (b c^2+a d^2\right )^{13/2} (c+d x)^9} \] Input:

Integrate[(x*(a + b*x^2)^(5/2))/(c + d*x)^10,x]
 

Output:

(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]*(896*c*(b*c^2 + a*d^2)^8 - 112*(b*c^2 
 + a*d^2)^6*(46*b^2*c^4 + 55*a*b*c^2*d^2 + 9*a^2*d^4)*(c + d*x) + 16*b*c*( 
b*c^2 + a*d^2)^6*(758*b*c^2 + 449*a*d^2)*(c + d*x)^2 - 8*b*(b*c^2 + a*d^2) 
^5*(1844*b^2*c^4 + 2196*a*b*c^2*d^2 + 357*a^2*d^4)*(c + d*x)^3 + 8*b^2*c*( 
b*c^2 + a*d^2)^4*(1180*b^2*c^4 + 2352*a*b*c^2*d^2 + 1161*a^2*d^4)*(c + d*x 
)^4 - 2*b^2*(b*c^2 + a*d^2)^3*(1328*b^3*c^6 + 3992*a*b^2*c^4*d^2 + 4002*a^ 
2*b*c^2*d^4 + 1239*a^3*d^6)*(c + d*x)^5 + 2*b^3*c*(b*c^2 + a*d^2)^2*(16*b^ 
3*c^6 + 72*a*b^2*c^4*d^2 + 150*a^2*b*c^2*d^4 + 325*a^3*d^6)*(c + d*x)^6 + 
b^3*(b*c^2 + a*d^2)*(32*b^4*c^8 + 192*a*b^3*c^6*d^2 + 540*a^2*b^2*c^4*d^4 
+ 1220*a^3*b*c^2*d^6 - 315*a^4*d^8)*(c + d*x)^7 + b^4*c*(32*b^4*c^8 + 224* 
a*b^3*c^6*d^2 + 732*a^2*b^2*c^4*d^4 + 1760*a^3*b*c^2*d^6 - 2245*a^4*d^8)*( 
c + d*x)^8) - 315*a^4*b^4*d^7*(-10*b*c^2 + a*d^2)*(c + d*x)^9*Log[c + d*x] 
 + 315*a^4*b^4*d^7*(-10*b*c^2 + a*d^2)*(c + d*x)^9*Log[a*d - b*c*x + Sqrt[ 
b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(8064*d^6*(b*c^2 + a*d^2)^(13/2)*(c + d*x 
)^9)
 

Rubi [A] (verified)

Time = 0.96 (sec) , antiderivative size = 432, normalized size of antiderivative = 0.92, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {594, 25, 688, 25, 27, 679, 486, 486, 486, 488, 219}

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.

\(\displaystyle \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx\)

\(\Big \downarrow \) 594

\(\displaystyle \frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}-\frac {\int -\frac {(9 a d+2 b c x) \left (b x^2+a\right )^{5/2}}{(c+d x)^9}dx}{9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\int \frac {(9 a d+2 b c x) \left (b x^2+a\right )^{5/2}}{(c+d x)^9}dx}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 688

\(\displaystyle \frac {\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}-\frac {\int -\frac {b \left (88 a c d+\left (2 b c^2-9 a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}}{(c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 25

\(\displaystyle \frac {\frac {\int \frac {b \left (88 a c d+\left (2 b c^2-9 a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}}{(c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {\frac {b \int \frac {\left (88 a c d+\left (2 b c^2-9 a d^2\right ) x\right ) \left (b x^2+a\right )^{5/2}}{(c+d x)^8}dx}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \int \frac {\left (b x^2+a\right )^{5/2}}{(c+d x)^7}dx}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5 a b \int \frac {\left (b x^2+a\right )^{3/2}}{(c+d x)^5}dx}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5 a b \left (\frac {3 a b \int \frac {\sqrt {b x^2+a}}{(c+d x)^3}dx}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 486

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5 a b \left (\frac {3 a b \left (\frac {a b \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5 a b \left (\frac {3 a b \left (-\frac {a b \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {b \left (\frac {9 a d \left (10 b c^2-a d^2\right ) \left (\frac {5 a b \left (\frac {3 a b \left (-\frac {a b \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{2 \left (a d^2+b c^2\right )^{3/2}}-\frac {\sqrt {a+b x^2} (a d-b c x)}{2 (c+d x)^2 \left (a d^2+b c^2\right )}\right )}{4 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{3/2} (a d-b c x)}{4 (c+d x)^4 \left (a d^2+b c^2\right )}\right )}{6 \left (a d^2+b c^2\right )}-\frac {\left (a+b x^2\right )^{5/2} (a d-b c x)}{6 (c+d x)^6 \left (a d^2+b c^2\right )}\right )}{a d^2+b c^2}+\frac {c \left (a+b x^2\right )^{7/2} \left (2 b c^2-97 a d^2\right )}{7 (c+d x)^7 \left (a d^2+b c^2\right )}\right )}{8 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{7/2} \left (2 b c^2-9 a d^2\right )}{8 (c+d x)^8 \left (a d^2+b c^2\right )}}{9 \left (a d^2+b c^2\right )}+\frac {c \left (a+b x^2\right )^{7/2}}{9 (c+d x)^9 \left (a d^2+b c^2\right )}\)

Input:

Int[(x*(a + b*x^2)^(5/2))/(c + d*x)^10,x]
 

Output:

(c*(a + b*x^2)^(7/2))/(9*(b*c^2 + a*d^2)*(c + d*x)^9) + (((2*b*c^2 - 9*a*d 
^2)*(a + b*x^2)^(7/2))/(8*(b*c^2 + a*d^2)*(c + d*x)^8) + (b*((c*(2*b*c^2 - 
 97*a*d^2)*(a + b*x^2)^(7/2))/(7*(b*c^2 + a*d^2)*(c + d*x)^7) + (9*a*d*(10 
*b*c^2 - a*d^2)*(-1/6*((a*d - b*c*x)*(a + b*x^2)^(5/2))/((b*c^2 + a*d^2)*( 
c + d*x)^6) + (5*a*b*(-1/4*((a*d - b*c*x)*(a + b*x^2)^(3/2))/((b*c^2 + a*d 
^2)*(c + d*x)^4) + (3*a*b*(-1/2*((a*d - b*c*x)*Sqrt[a + b*x^2])/((b*c^2 + 
a*d^2)*(c + d*x)^2) - (a*b*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt 
[a + b*x^2])])/(2*(b*c^2 + a*d^2)^(3/2))))/(4*(b*c^2 + a*d^2))))/(6*(b*c^2 
 + a*d^2))))/(b*c^2 + a*d^2)))/(8*(b*c^2 + a*d^2)))/(9*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], x]
 

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 219

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1/(Rt[a, 2]*Rt[-b, 2]))* 
ArcTanh[Rt[-b, 2]*(x/Rt[a, 2])], x] /; FreeQ[{a, b}, x] && NegQ[a/b] && (Gt 
Q[a, 0] || LtQ[b, 0])
 

rule 486

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

rule 488

Int[1/(((c_) + (d_.)*(x_))*Sqrt[(a_) + (b_.)*(x_)^2]), x_Symbol] :> -Subst[ 
Int[1/(b*c^2 + a*d^2 - x^2), x], x, (a*d - b*c*x)/Sqrt[a + b*x^2]] /; FreeQ 
[{a, b, c, d}, x]
 

rule 594

Int[(x_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
Simp[(-c)*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 1)*(b*c^2 + a*d^2))) 
, x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2) 
^p*(a*d*(n + 1) + b*c*(n + 2*p + 3)*x), x], x] /; FreeQ[{a, b, c, d, p}, x] 
 && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 679

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(-(e*f - d*g))*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1 
)/(2*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[(c*d*f + a*e*g)/(c*d^2 + a*e^2) 
 Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, d, e, f, g, m, 
 p}, x] && EqQ[Simplify[m + 2*p + 3], 0]
 

rule 688

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(e*f - d*g)*(d + e*x)^(m + 1)*((a + c*x^2)^(p + 1)/( 
(m + 1)*(c*d^2 + a*e^2))), x] + Simp[1/((m + 1)*(c*d^2 + a*e^2))   Int[(d + 
 e*x)^(m + 1)*(a + c*x^2)^p*Simp[(c*d*f + a*e*g)*(m + 1) - c*(e*f - d*g)*(m 
 + 2*p + 3)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g, p}, x] && LtQ[m, -1] 
&& (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(79914\) vs. \(2(434)=868\).

Time = 2.00 (sec) , antiderivative size = 79915, normalized size of antiderivative = 170.03

method	result	size
default	\(\text {Expression too large to display}\)	\(79915\)

Input:

int(x*(b*x^2+a)^(5/2)/(d*x+c)^10,x,method=_RETURNVERBOSE)
 

Output:

result too large to display
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2306 vs. \(2 (435) = 870\).

Time = 66.91 (sec) , antiderivative size = 4638, normalized size of antiderivative = 9.87 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^10,x, algorithm="fricas")
 

Output:

Too large to include
 

Sympy [F(-1)]

Timed out. \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\text {Timed out} \] Input:

integrate(x*(b*x**2+a)**(5/2)/(d*x+c)**10,x)
 

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 14371 vs. \(2 (435) = 870\).

Time = 0.67 (sec) , antiderivative size = 14371, normalized size of antiderivative = 30.58 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^10,x, algorithm="maxima")
 

Output:

55/256*b^11*c^13*arcsinh(b*x/sqrt(a*b))/(b^(17/2)*c^16*d^7 + 8*a*b^(15/2)* 
c^14*d^9 + 28*a^2*b^(13/2)*c^12*d^11 + 56*a^3*b^(11/2)*c^10*d^13 + 70*a^4* 
b^(9/2)*c^8*d^15 + 56*a^5*b^(7/2)*c^6*d^17 + 28*a^6*b^(5/2)*c^4*d^19 + 8*a 
^7*b^(3/2)*c^2*d^21 + a^8*sqrt(b)*d^23) + 55/256*a*b^10*c^11*arcsinh(b*x/s 
qrt(a*b))/(b^(17/2)*c^16*d^5 + 8*a*b^(15/2)*c^14*d^7 + 28*a^2*b^(13/2)*c^1 
2*d^9 + 56*a^3*b^(11/2)*c^10*d^11 + 70*a^4*b^(9/2)*c^8*d^13 + 56*a^5*b^(7/ 
2)*c^6*d^15 + 28*a^6*b^(5/2)*c^4*d^17 + 8*a^7*b^(3/2)*c^2*d^19 + a^8*sqrt( 
b)*d^21) - 55/256*sqrt(b*x^2 + a)*b^10*c^11*x/(b^8*c^16*d^5 + 8*a*b^7*c^14 
*d^7 + 28*a^2*b^6*c^12*d^9 + 56*a^3*b^5*c^10*d^11 + 70*a^4*b^4*c^8*d^13 + 
56*a^5*b^3*c^6*d^15 + 28*a^6*b^2*c^4*d^17 + 8*a^7*b*c^2*d^19 + a^8*d^21) - 
 195/128*b^10*c^11*arcsinh(b*x/sqrt(a*b))/(b^(15/2)*c^14*d^7 + 7*a*b^(13/2 
)*c^12*d^9 + 21*a^2*b^(11/2)*c^10*d^11 + 35*a^3*b^(9/2)*c^8*d^13 + 35*a^4* 
b^(7/2)*c^6*d^15 + 21*a^5*b^(5/2)*c^4*d^17 + 7*a^6*b^(3/2)*c^2*d^19 + a^7* 
sqrt(b)*d^21) + 55/384*(b*x^2 + a)^(3/2)*b^9*c^10/(b^8*c^16*d^4 + 8*a*b^7* 
c^14*d^6 + 28*a^2*b^6*c^12*d^8 + 56*a^3*b^5*c^10*d^10 + 70*a^4*b^4*c^8*d^1 
2 + 56*a^5*b^3*c^6*d^14 + 28*a^6*b^2*c^4*d^16 + 8*a^7*b*c^2*d^18 + a^8*d^2 
0) - 55/384*(b*x^2 + a)^(3/2)*b^9*c^9*x/(b^8*c^16*d^3 + 8*a*b^7*c^14*d^5 + 
 28*a^2*b^6*c^12*d^7 + 56*a^3*b^5*c^10*d^9 + 70*a^4*b^4*c^8*d^11 + 56*a^5* 
b^3*c^6*d^13 + 28*a^6*b^2*c^4*d^15 + 8*a^7*b*c^2*d^17 + a^8*d^19) - 55/256 
*sqrt(b*x^2 + a)*a*b^9*c^9*x/(b^8*c^16*d^3 + 8*a*b^7*c^14*d^5 + 28*a^2*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 4209 vs. \(2 (435) = 870\).

Time = 0.41 (sec) , antiderivative size = 4209, normalized size of antiderivative = 8.96 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^10,x, algorithm="giac")
 

Output:

5/64*(10*a^4*b^5*c^2*d - a^5*b^4*d^3)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a 
))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^6*c^12 + 6*a*b^5*c^10*d^2 + 15 
*a^2*b^4*c^8*d^4 + 20*a^3*b^3*c^6*d^6 + 15*a^4*b^2*c^4*d^8 + 6*a^5*b*c^2*d 
^10 + a^6*d^12)*sqrt(-b*c^2 - a*d^2)) - 1/4032*(3150*(sqrt(b)*x - sqrt(b*x 
^2 + a))^17*a^4*b^5*c^2*d^16 - 315*(sqrt(b)*x - sqrt(b*x^2 + a))^17*a^5*b^ 
4*d^18 + 53550*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^4*b^(11/2)*c^3*d^15 - 53 
55*(sqrt(b)*x - sqrt(b*x^2 + a))^16*a^5*b^(9/2)*c*d^17 - 10752*(sqrt(b)*x 
- sqrt(b*x^2 + a))^15*b^10*c^12*d^6 - 64512*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
15*a*b^9*c^10*d^8 - 161280*(sqrt(b)*x - sqrt(b*x^2 + a))^15*a^2*b^8*c^8*d^ 
10 - 215040*(sqrt(b)*x - sqrt(b*x^2 + a))^15*a^3*b^7*c^6*d^12 + 239820*(sq 
rt(b)*x - sqrt(b*x^2 + a))^15*a^4*b^6*c^4*d^14 - 131922*(sqrt(b)*x - sqrt( 
b*x^2 + a))^15*a^5*b^5*c^2*d^16 - 8022*(sqrt(b)*x - sqrt(b*x^2 + a))^15*a^ 
6*b^4*d^18 - 32256*(sqrt(b)*x - sqrt(b*x^2 + a))^14*b^(21/2)*c^13*d^5 - 19 
3536*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a*b^(19/2)*c^11*d^7 - 483840*(sqrt(b 
)*x - sqrt(b*x^2 + a))^14*a^2*b^(17/2)*c^9*d^9 - 645120*(sqrt(b)*x - sqrt( 
b*x^2 + a))^14*a^3*b^(15/2)*c^7*d^11 + 1248660*(sqrt(b)*x - sqrt(b*x^2 + a 
))^14*a^4*b^(13/2)*c^5*d^13 - 776286*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^5* 
b^(11/2)*c^3*d^15 + 8694*(sqrt(b)*x - sqrt(b*x^2 + a))^14*a^6*b^(9/2)*c*d^ 
17 - 64512*(sqrt(b)*x - sqrt(b*x^2 + a))^13*b^11*c^14*d^4 - 387072*(sqrt(b 
)*x - sqrt(b*x^2 + a))^13*a*b^10*c^12*d^6 - 967680*(sqrt(b)*x - sqrt(b*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx=\int \frac {x\,{\left (b\,x^2+a\right )}^{5/2}}{{\left (c+d\,x\right )}^{10}} \,d x \] Input:

int((x*(a + b*x^2)^(5/2))/(c + d*x)^10,x)
 

Output:

int((x*(a + b*x^2)^(5/2))/(c + d*x)^10, x)
 

Reduce [B] (verification not implemented)

Time = 24.94 (sec) , antiderivative size = 4541, normalized size of antiderivative = 9.66 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^{10}} \, dx =\text {Too large to display} \] Input:

int(x*(b*x^2+a)^(5/2)/(d*x+c)^10,x)
 

Output:

(315*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - 
 a*d + b*c*x)*a**5*b**4*c**9*d**3 + 2835*sqrt(a*d**2 + b*c**2)*log( - sqrt 
(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b**4*c**8*d**4*x + 
11340*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) 
- a*d + b*c*x)*a**5*b**4*c**7*d**5*x**2 + 26460*sqrt(a*d**2 + b*c**2)*log( 
 - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b**4*c**6*d* 
*6*x**3 + 39690*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 
+ b*c**2) - a*d + b*c*x)*a**5*b**4*c**5*d**7*x**4 + 39690*sqrt(a*d**2 + b* 
c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b* 
*4*c**4*d**8*x**5 + 26460*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq 
rt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b**4*c**3*d**9*x**6 + 11340*sqrt(a 
*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c* 
x)*a**5*b**4*c**2*d**10*x**7 + 2835*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + 
b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**5*b**4*c*d**11*x**8 + 315* 
sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
+ b*c*x)*a**5*b**4*d**12*x**9 - 3150*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + 
 b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b**5*c**11*d - 28350*sq 
rt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + 
b*c*x)*a**4*b**5*c**10*d**2*x - 113400*sqrt(a*d**2 + b*c**2)*log( - sqrt(a 
 + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**4*b**5*c**9*d**3*x**...