\(\int \frac {x (a+b x^2)^{5/2}}{(c+d x)^8} \, dx\) [1150]

Optimal result

Integrand size = 20, antiderivative size = 294 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=-\frac {5 a^3 b^2 d (a d-b c x) \sqrt {a+b x^2}}{16 \left (b c^2+a d^2\right )^4 (c+d x)^2}-\frac {5 a^2 b d (a d-b c x) \left (a+b x^2\right )^{3/2}}{24 \left (b c^2+a d^2\right )^3 (c+d x)^4}+\frac {c \left (a+b x^2\right )^{5/2}}{7 d^2 (c+d x)^7}-\frac {\left (12 b c^2+7 a d^2\right ) \left (a+b x^2\right )^{5/2}}{42 d^2 \left (b c^2+a d^2\right ) (c+d x)^6}+\frac {b c \left (6 b c^2+13 a d^2\right ) \left (a+b x^2\right )^{5/2}}{42 d^2 \left (b c^2+a d^2\right )^2 (c+d x)^5}-\frac {5 a^4 b^3 d \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{16 \left (b c^2+a d^2\right )^{9/2}} \] Output:

-5/16*a^3*b^2*d*(-b*c*x+a*d)*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^4/(d*x+c)^2-5/2 
4*a^2*b*d*(-b*c*x+a*d)*(b*x^2+a)^(3/2)/(a*d^2+b*c^2)^3/(d*x+c)^4+1/7*c*(b* 
x^2+a)^(5/2)/d^2/(d*x+c)^7-1/42*(7*a*d^2+12*b*c^2)*(b*x^2+a)^(5/2)/d^2/(a* 
d^2+b*c^2)/(d*x+c)^6+1/42*b*c*(13*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)^5-5/16*a^4*b^3*d*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)^(9/2)
 

Mathematica [A] (verified)

Time = 10.47 (sec) , antiderivative size = 413, normalized size of antiderivative = 1.40 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\frac {1}{336} \left (\frac {\sqrt {a+b x^2} \left (48 b^6 c^7 x^6-8 a^6 d^6 (c+7 d x)+8 a b^5 c^5 x^4 \left (18 c^2+7 c d x+25 d^2 x^2\right )-2 a^5 b d^4 \left (19 c^3+133 c^2 d x+61 c d^2 x^2+91 d^3 x^3\right )+2 a^2 b^4 c^3 x^2 \left (72 c^4+91 c^3 d x+349 c^2 d^2 x^2+133 c d^3 x^3+163 d^4 x^4\right )-a^4 b^2 d^2 \left (87 c^5+609 c^4 d x+800 c^3 d^2 x^2+1232 c^2 d^3 x^3+465 c d^4 x^4+231 d^5 x^5\right )+a^3 b^3 c \left (48 c^6+231 c^5 d x+1041 c^4 d^2 x^2+1232 c^3 d^3 x^3+1664 c^2 d^4 x^4+609 c d^5 x^5+279 d^6 x^6\right )\right )}{\left (b c^2+a d^2\right )^4 (c+d x)^7}+\frac {105 a^4 b^3 d \log (c+d x)}{\left (b c^2+a d^2\right )^{9/2}}-\frac {105 a^4 b^3 d \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{9/2}}\right ) \] Input:

Integrate[(x*(a + b*x^2)^(5/2))/(c + d*x)^8,x]
 

Output:

((Sqrt[a + b*x^2]*(48*b^6*c^7*x^6 - 8*a^6*d^6*(c + 7*d*x) + 8*a*b^5*c^5*x^ 
4*(18*c^2 + 7*c*d*x + 25*d^2*x^2) - 2*a^5*b*d^4*(19*c^3 + 133*c^2*d*x + 61 
*c*d^2*x^2 + 91*d^3*x^3) + 2*a^2*b^4*c^3*x^2*(72*c^4 + 91*c^3*d*x + 349*c^ 
2*d^2*x^2 + 133*c*d^3*x^3 + 163*d^4*x^4) - a^4*b^2*d^2*(87*c^5 + 609*c^4*d 
*x + 800*c^3*d^2*x^2 + 1232*c^2*d^3*x^3 + 465*c*d^4*x^4 + 231*d^5*x^5) + a 
^3*b^3*c*(48*c^6 + 231*c^5*d*x + 1041*c^4*d^2*x^2 + 1232*c^3*d^3*x^3 + 166 
4*c^2*d^4*x^4 + 609*c*d^5*x^5 + 279*d^6*x^6)))/((b*c^2 + a*d^2)^4*(c + d*x 
)^7) + (105*a^4*b^3*d*Log[c + d*x])/(b*c^2 + a*d^2)^(9/2) - (105*a^4*b^3*d 
*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^2)^( 
9/2))/336
 

Rubi [A] (verified)

Time = 0.66 (sec) , antiderivative size = 284, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.300, Rules used = {588, 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.

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

\(\Big \downarrow \) 588

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

\(\Big \downarrow \) 486

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

\(\Big \downarrow \) 486

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

\(\Big \downarrow \) 486

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[(x*(a + b*x^2)^(5/2))/(c + d*x)^8,x]
 

Output:

(c*(a + b*x^2)^(7/2))/(7*(b*c^2 + a*d^2)*(c + d*x)^7) + (a*d*(-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*Ar 
cTanh[(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)
 

Defintions of rubi rules used

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 588

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)/(2*(p + 1)*(b*c^2 + a*d^2))) 
, x] + Simp[a*(d/(b*c^2 + a*d^2))   Int[(c + d*x)^(n + 1)*(a + b*x^2)^p, x] 
, x] /; FreeQ[{a, b, c, d, n, p}, x] && EqQ[Simplify[n + 2*p + 3], 0] && Ne 
Q[b*c^2 + a*d^2, 0]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(31825\) vs. \(2(266)=532\).

Time = 1.07 (sec) , antiderivative size = 31826, normalized size of antiderivative = 108.25

method	result	size
default	\(\text {Expression too large to display}\)	\(31826\)

Input:

int(x*(b*x^2+a)^(5/2)/(d*x+c)^8,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. 1281 vs. \(2 (267) = 534\).

Time = 18.04 (sec) , antiderivative size = 2588, normalized size of antiderivative = 8.80 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^8,x, algorithm="fricas")
 

Output:

[1/672*(105*(a^4*b^3*d^8*x^7 + 7*a^4*b^3*c*d^7*x^6 + 21*a^4*b^3*c^2*d^6*x^ 
5 + 35*a^4*b^3*c^3*d^5*x^4 + 35*a^4*b^3*c^4*d^4*x^3 + 21*a^4*b^3*c^5*d^3*x 
^2 + 7*a^4*b^3*c^6*d^2*x + a^4*b^3*c^7*d)*sqrt(b*c^2 + a*d^2)*log((2*a*b*c 
*d*x - a*b*c^2 - 2*a^2*d^2 - (2*b^2*c^2 + a*b*d^2)*x^2 - 2*sqrt(b*c^2 + a* 
d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(48*a^3 
*b^4*c^9 - 39*a^4*b^3*c^7*d^2 - 125*a^5*b^2*c^5*d^4 - 46*a^6*b*c^3*d^6 - 8 
*a^7*c*d^8 + (48*b^7*c^9 + 248*a*b^6*c^7*d^2 + 526*a^2*b^5*c^5*d^4 + 605*a 
^3*b^4*c^3*d^6 + 279*a^4*b^3*c*d^8)*x^6 + 7*(8*a*b^6*c^8*d + 46*a^2*b^5*c^ 
6*d^3 + 125*a^3*b^4*c^4*d^5 + 54*a^4*b^3*c^2*d^7 - 33*a^5*b^2*d^9)*x^5 + ( 
144*a*b^6*c^9 + 842*a^2*b^5*c^7*d^2 + 2362*a^3*b^4*c^5*d^4 + 1199*a^4*b^3* 
c^3*d^6 - 465*a^5*b^2*c*d^8)*x^4 + 14*(13*a^2*b^5*c^8*d + 101*a^3*b^4*c^6* 
d^3 - 101*a^5*b^2*c^2*d^7 - 13*a^6*b*d^9)*x^3 + (144*a^2*b^5*c^9 + 1185*a^ 
3*b^4*c^7*d^2 + 241*a^4*b^3*c^5*d^4 - 922*a^5*b^2*c^3*d^6 - 122*a^6*b*c*d^ 
8)*x^2 + 7*(33*a^3*b^4*c^8*d - 54*a^4*b^3*c^6*d^3 - 125*a^5*b^2*c^4*d^5 - 
46*a^6*b*c^2*d^7 - 8*a^7*d^9)*x)*sqrt(b*x^2 + a))/(b^5*c^17 + 5*a*b^4*c^15 
*d^2 + 10*a^2*b^3*c^13*d^4 + 10*a^3*b^2*c^11*d^6 + 5*a^4*b*c^9*d^8 + a^5*c 
^7*d^10 + (b^5*c^10*d^7 + 5*a*b^4*c^8*d^9 + 10*a^2*b^3*c^6*d^11 + 10*a^3*b 
^2*c^4*d^13 + 5*a^4*b*c^2*d^15 + a^5*d^17)*x^7 + 7*(b^5*c^11*d^6 + 5*a*b^4 
*c^9*d^8 + 10*a^2*b^3*c^7*d^10 + 10*a^3*b^2*c^5*d^12 + 5*a^4*b*c^3*d^14 + 
a^5*c*d^16)*x^6 + 21*(b^5*c^12*d^5 + 5*a*b^4*c^10*d^7 + 10*a^2*b^3*c^8*...
 

Sympy [F]

\[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\int \frac {x \left (a + b x^{2}\right )^{\frac {5}{2}}}{\left (c + d x\right )^{8}}\, dx \] Input:

integrate(x*(b*x**2+a)**(5/2)/(d*x+c)**8,x)
 

Output:

Integral(x*(a + b*x**2)**(5/2)/(c + d*x)**8, x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 7534 vs. \(2 (267) = 534\).

Time = 0.43 (sec) , antiderivative size = 7534, normalized size of antiderivative = 25.63 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^8,x, algorithm="maxima")
 

Output:

5/32*b^9*c^11*arcsinh(b*x/sqrt(a*b))/(b^(13/2)*c^12*d^7 + 6*a*b^(11/2)*c^1 
0*d^9 + 15*a^2*b^(9/2)*c^8*d^11 + 20*a^3*b^(7/2)*c^6*d^13 + 15*a^4*b^(5/2) 
*c^4*d^15 + 6*a^5*b^(3/2)*c^2*d^17 + a^6*sqrt(b)*d^19) + 5/32*a*b^8*c^9*ar 
csinh(b*x/sqrt(a*b))/(b^(13/2)*c^12*d^5 + 6*a*b^(11/2)*c^10*d^7 + 15*a^2*b 
^(9/2)*c^8*d^9 + 20*a^3*b^(7/2)*c^6*d^11 + 15*a^4*b^(5/2)*c^4*d^13 + 6*a^5 
*b^(3/2)*c^2*d^15 + a^6*sqrt(b)*d^17) - 5/32*sqrt(b*x^2 + a)*b^8*c^9*x/(b^ 
6*c^12*d^5 + 6*a*b^5*c^10*d^7 + 15*a^2*b^4*c^8*d^9 + 20*a^3*b^3*c^6*d^11 + 
 15*a^4*b^2*c^4*d^13 + 6*a^5*b*c^2*d^15 + a^6*d^17) - 35/32*b^8*c^9*arcsin 
h(b*x/sqrt(a*b))/(b^(11/2)*c^10*d^7 + 5*a*b^(9/2)*c^8*d^9 + 10*a^2*b^(7/2) 
*c^6*d^11 + 10*a^3*b^(5/2)*c^4*d^13 + 5*a^4*b^(3/2)*c^2*d^15 + a^5*sqrt(b) 
*d^17) + 5/48*(b*x^2 + a)^(3/2)*b^7*c^8/(b^6*c^12*d^4 + 6*a*b^5*c^10*d^6 + 
 15*a^2*b^4*c^8*d^8 + 20*a^3*b^3*c^6*d^10 + 15*a^4*b^2*c^4*d^12 + 6*a^5*b* 
c^2*d^14 + a^6*d^16) - 5/48*(b*x^2 + a)^(3/2)*b^7*c^7*x/(b^6*c^12*d^3 + 6* 
a*b^5*c^10*d^5 + 15*a^2*b^4*c^8*d^7 + 20*a^3*b^3*c^6*d^9 + 15*a^4*b^2*c^4* 
d^11 + 6*a^5*b*c^2*d^13 + a^6*d^15) - 5/32*sqrt(b*x^2 + a)*a*b^7*c^7*x/(b^ 
6*c^12*d^3 + 6*a*b^5*c^10*d^5 + 15*a^2*b^4*c^8*d^7 + 20*a^3*b^3*c^6*d^9 + 
15*a^4*b^2*c^4*d^11 + 6*a^5*b*c^2*d^13 + a^6*d^15) - 15/16*a*b^7*c^7*arcsi 
nh(b*x/sqrt(a*b))/(b^(11/2)*c^10*d^5 + 5*a*b^(9/2)*c^8*d^7 + 10*a^2*b^(7/2 
)*c^6*d^9 + 10*a^3*b^(5/2)*c^4*d^11 + 5*a^4*b^(3/2)*c^2*d^13 + a^5*sqrt(b) 
*d^15) + 1/16*(b*x^2 + a)^(5/2)*b^6*c^7/(b^6*c^12*d^3*x + 6*a*b^5*c^10*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2812 vs. \(2 (267) = 534\).

Time = 0.22 (sec) , antiderivative size = 2812, normalized size of antiderivative = 9.56 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\text {Too large to display} \] Input:

integrate(x*(b*x^2+a)^(5/2)/(d*x+c)^8,x, algorithm="giac")
 

Output:

5/8*a^4*b^3*d*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(- 
b*c^2 - a*d^2))/((b^4*c^8 + 4*a*b^3*c^6*d^2 + 6*a^2*b^2*c^4*d^4 + 4*a^3*b* 
c^2*d^6 + a^4*d^8)*sqrt(-b*c^2 - a*d^2)) + 1/168*(336*(sqrt(b)*x - sqrt(b* 
x^2 + a))^13*b^7*c^8*d^6 + 1344*(sqrt(b)*x - sqrt(b*x^2 + a))^13*a*b^6*c^6 
*d^8 + 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^13*a^2*b^5*c^4*d^10 + 1344*(sqrt 
(b)*x - sqrt(b*x^2 + a))^13*a^3*b^4*c^2*d^12 + 231*(sqrt(b)*x - sqrt(b*x^2 
 + a))^13*a^4*b^3*d^14 + 2016*(sqrt(b)*x - sqrt(b*x^2 + a))^12*b^(15/2)*c^ 
9*d^5 + 8064*(sqrt(b)*x - sqrt(b*x^2 + a))^12*a*b^(13/2)*c^7*d^7 + 12096*( 
sqrt(b)*x - sqrt(b*x^2 + a))^12*a^2*b^(11/2)*c^5*d^9 + 8064*(sqrt(b)*x - s 
qrt(b*x^2 + a))^12*a^3*b^(9/2)*c^3*d^11 + 651*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^12*a^4*b^(7/2)*c*d^13 + 6720*(sqrt(b)*x - sqrt(b*x^2 + a))^11*b^8*c^10*d 
^4 + 25984*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a*b^7*c^8*d^6 + 36736*(sqrt(b) 
*x - sqrt(b*x^2 + a))^11*a^2*b^6*c^6*d^8 + 21504*(sqrt(b)*x - sqrt(b*x^2 + 
 a))^11*a^3*b^5*c^4*d^10 - 4354*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a^4*b^4*c 
^2*d^12 - 196*(sqrt(b)*x - sqrt(b*x^2 + a))^11*a^5*b^3*d^14 + 13440*(sqrt( 
b)*x - sqrt(b*x^2 + a))^10*b^(17/2)*c^11*d^3 + 45920*(sqrt(b)*x - sqrt(b*x 
^2 + a))^10*a*b^(15/2)*c^9*d^5 + 49280*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^ 
2*b^(13/2)*c^7*d^7 + 6720*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^3*b^(11/2)*c^ 
5*d^9 - 40250*(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^4*b^(9/2)*c^3*d^11 - 140* 
(sqrt(b)*x - sqrt(b*x^2 + a))^10*a^5*b^(7/2)*c*d^13 + 16128*(sqrt(b)*x ...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx=\int \frac {x\,{\left (b\,x^2+a\right )}^{5/2}}{{\left (c+d\,x\right )}^8} \,d x \] Input:

int((x*(a + b*x^2)^(5/2))/(c + d*x)^8,x)
 

Output:

int((x*(a + b*x^2)^(5/2))/(c + d*x)^8, x)
 

Reduce [B] (verification not implemented)

Time = 6.09 (sec) , antiderivative size = 2204, normalized size of antiderivative = 7.50 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^8} \, dx =\text {Too large to display} \] Input:

int(x*(b*x^2+a)^(5/2)/(d*x+c)^8,x)
 

Output:

(105*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**3*c**7*d + 735*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**3*c**6*d**2*x + 2205*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**3*c**5*d**3*x**2 + 3675*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**3*c**4*d**4*x**3 + 3675*sqr 
t(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**3*c**3*d**5*x**4 + 2205*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**3*c**2*d**6*x**5 + 735*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**3*c*d**7*x**6 + 105*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**3*d**8*x**7 - 105*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*a**4*b**3*c**7*d - 735*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a**4*b**3*c**6*d**2*x - 2205*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a 
**4*b**3*c**5*d**3*x**2 - 3675*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b** 
3*c**4*d**4*x**3 - 3675*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b**3*c**3* 
d**5*x**4 - 2205*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b**3*c**2*d**6*x* 
*5 - 735*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b**3*c*d**7*x**6 - 105*sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a**4*b**3*d**8*x**7 - 8*sqrt(a + b*x**2)* 
a**7*c*d**8 - 56*sqrt(a + b*x**2)*a**7*d**9*x - 46*sqrt(a + b*x**2)*a**...