\(\int \frac {x (a+b x^2)^{5/2}}{(c+d x)^5} \, dx\) [1147]

Optimal result

Integrand size = 20, antiderivative size = 370 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^5} \, dx=-\frac {5 b^2 c \sqrt {a+b x^2}}{d^6}+\frac {b^2 x \sqrt {a+b x^2}}{2 d^5}+\frac {c \left (b c^2+a d^2\right )^2 \sqrt {a+b x^2}}{4 d^6 (c+d x)^4}-\frac {\left (b c^2+a d^2\right ) \left (21 b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{12 d^6 (c+d x)^3}+\frac {b c \left (138 b c^2+79 a d^2\right ) \sqrt {a+b x^2}}{24 d^6 (c+d x)^2}-\frac {b \left (342 b^2 c^4+383 a b c^2 d^2+56 a^2 d^4\right ) \sqrt {a+b x^2}}{24 d^6 \left (b c^2+a d^2\right ) (c+d x)}+\frac {5 b^{3/2} \left (6 b c^2+a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^7}+\frac {5 b^2 c \left (24 b^2 c^4+40 a b c^2 d^2+15 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{8 d^7 \left (b c^2+a d^2\right )^{3/2}} \] Output:

-5*b^2*c*(b*x^2+a)^(1/2)/d^6+1/2*b^2*x*(b*x^2+a)^(1/2)/d^5+1/4*c*(a*d^2+b* 
c^2)^2*(b*x^2+a)^(1/2)/d^6/(d*x+c)^4-1/12*(a*d^2+b*c^2)*(4*a*d^2+21*b*c^2) 
*(b*x^2+a)^(1/2)/d^6/(d*x+c)^3+1/24*b*c*(79*a*d^2+138*b*c^2)*(b*x^2+a)^(1/ 
2)/d^6/(d*x+c)^2-1/24*b*(56*a^2*d^4+383*a*b*c^2*d^2+342*b^2*c^4)*(b*x^2+a) 
^(1/2)/d^6/(a*d^2+b*c^2)/(d*x+c)+5/2*b^(3/2)*(a*d^2+6*b*c^2)*arctanh(b^(1/ 
2)*x/(b*x^2+a)^(1/2))/d^7+5/8*b^2*c*(15*a^2*d^4+40*a*b*c^2*d^2+24*b^2*c^4) 
*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^7/(a*d^2+b*c^ 
2)^(3/2)
 

Mathematica [A] (verified)

Time = 10.55 (sec) , antiderivative size = 347, normalized size of antiderivative = 0.94 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^5} \, dx=\frac {d \sqrt {a+b x^2} \left (-120 b^2 c+12 b^2 d x+\frac {6 c \left (b c^2+a d^2\right )^2}{(c+d x)^4}-\frac {2 \left (21 b^2 c^4+25 a b c^2 d^2+4 a^2 d^4\right )}{(c+d x)^3}+\frac {b c \left (138 b c^2+79 a d^2\right )}{(c+d x)^2}-\frac {b \left (342 b^2 c^4+383 a b c^2 d^2+56 a^2 d^4\right )}{\left (b c^2+a d^2\right ) (c+d x)}\right )-\frac {15 b^2 c \left (24 b^2 c^4+40 a b c^2 d^2+15 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{3/2}}+60 b^{3/2} \left (6 b c^2+a d^2\right ) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )+\frac {15 b^2 c \left (24 b^2 c^4+40 a b c^2 d^2+15 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{3/2}}}{24 d^7} \] Input:

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

Output:

(d*Sqrt[a + b*x^2]*(-120*b^2*c + 12*b^2*d*x + (6*c*(b*c^2 + a*d^2)^2)/(c + 
 d*x)^4 - (2*(21*b^2*c^4 + 25*a*b*c^2*d^2 + 4*a^2*d^4))/(c + d*x)^3 + (b*c 
*(138*b*c^2 + 79*a*d^2))/(c + d*x)^2 - (b*(342*b^2*c^4 + 383*a*b*c^2*d^2 + 
 56*a^2*d^4))/((b*c^2 + a*d^2)*(c + d*x))) - (15*b^2*c*(24*b^2*c^4 + 40*a* 
b*c^2*d^2 + 15*a^2*d^4)*Log[c + d*x])/(b*c^2 + a*d^2)^(3/2) + 60*b^(3/2)*( 
6*b*c^2 + a*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]] + (15*b^2*c*(24*b^2*c^ 
4 + 40*a*b*c^2*d^2 + 15*a^2*d^4)*Log[a*d - b*c*x + Sqrt[b*c^2 + a*d^2]*Sqr 
t[a + b*x^2]])/(b*c^2 + a*d^2)^(3/2))/(24*d^7)
 

Rubi [A] (verified)

Time = 1.02 (sec) , antiderivative size = 359, normalized size of antiderivative = 0.97, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.550, Rules used = {590, 27, 680, 27, 681, 27, 719, 224, 219, 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.

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

\(\Big \downarrow \) 590

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 680

\(\displaystyle \frac {5 \left (\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}-\frac {\int -\frac {2 b \left (2 a d \left (3 b c^2+2 a d^2\right )-b c \left (12 b c^2+11 a d^2\right ) x\right ) \sqrt {b x^2+a}}{(c+d x)^2}dx}{4 d^2 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {b \int \frac {\left (2 a d \left (3 b c^2+2 a d^2\right )-b c \left (12 b c^2+11 a d^2\right ) x\right ) \sqrt {b x^2+a}}{(c+d x)^2}dx}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 681

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {\int \frac {2 b \left (a c d \left (12 b c^2+11 a d^2\right )-4 \left (b c^2+a d^2\right ) \left (6 b c^2+a d^2\right ) x\right )}{(c+d x) \sqrt {b x^2+a}}dx}{2 d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 27

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \int \frac {a c d \left (12 b c^2+11 a d^2\right )-4 \left (b c^2+a d^2\right ) \left (6 b c^2+a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 719

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \left (\frac {c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 224

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \left (\frac {c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \left (\frac {c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )}{\sqrt {b} d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \left (-\frac {c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )}{\sqrt {b} d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {5 \left (\frac {b \left (-\frac {b \left (-\frac {c \left (15 a^2 d^4+40 a b c^2 d^2+24 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}-\frac {4 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )}{\sqrt {b} d}\right )}{d^2}-\frac {\sqrt {a+b x^2} \left (b c d x \left (11 a d^2+12 b c^2\right )+4 \left (a d^2+b c^2\right ) \left (a d^2+6 b c^2\right )\right )}{d^2 (c+d x)}\right )}{2 d^2 \left (a d^2+b c^2\right )}+\frac {\left (a+b x^2\right )^{3/2} \left (-4 a^2 d^4+3 b c d x \left (5 a d^2+6 b c^2\right )+5 a b c^2 d^2+12 b^2 c^4\right )}{6 d^2 (c+d x)^3 \left (a d^2+b c^2\right )}\right )}{4 d^2}+\frac {\left (a+b x^2\right )^{5/2} (3 c+2 d x)}{4 d^2 (c+d x)^4}\)

Input:

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

Output:

((3*c + 2*d*x)*(a + b*x^2)^(5/2))/(4*d^2*(c + d*x)^4) + (5*(((12*b^2*c^4 + 
 5*a*b*c^2*d^2 - 4*a^2*d^4 + 3*b*c*d*(6*b*c^2 + 5*a*d^2)*x)*(a + b*x^2)^(3 
/2))/(6*d^2*(b*c^2 + a*d^2)*(c + d*x)^3) + (b*(-(((4*(b*c^2 + a*d^2)*(6*b* 
c^2 + a*d^2) + b*c*d*(12*b*c^2 + 11*a*d^2)*x)*Sqrt[a + b*x^2])/(d^2*(c + d 
*x))) - (b*((-4*(b*c^2 + a*d^2)*(6*b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt 
[a + b*x^2]])/(Sqrt[b]*d) - (c*(24*b^2*c^4 + 40*a*b*c^2*d^2 + 15*a^2*d^4)* 
ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b*c^ 
2 + a*d^2])))/d^2))/(2*d^2*(b*c^2 + a*d^2))))/(4*d^2)
 

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 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 224

Int[1/Sqrt[(a_) + (b_.)*(x_)^2], x_Symbol] :> Subst[Int[1/(1 - b*x^2), x], 
x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b}, x] &&  !GtQ[a, 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 590

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

rule 680

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

rule 681

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(e*f*(m + 2*p + 2) - d*g*(2*p + 1) 
 + e*g*(m + 1)*x)*((a + c*x^2)^p/(e^2*(m + 1)*(m + 2*p + 2))), x] + Simp[p/ 
(e^2*(m + 1)*(m + 2*p + 2))   Int[(d + e*x)^(m + 1)*(a + c*x^2)^(p - 1)*Sim 
p[g*(2*a*e + 2*a*e*m) + (g*(2*c*d + 4*c*d*p) - 2*c*e*f*(m + 2*p + 2))*x, x] 
, x], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (LtQ[m, -1] || 
EqQ[p, 1] || (IntegerQ[p] &&  !RationalQ[m])) && NeQ[m, -1] &&  !ILtQ[m + 2 
*p + 1, 0] && (IntegerQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

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

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(3078\) vs. \(2(332)=664\).

Time = 0.61 (sec) , antiderivative size = 3079, normalized size of antiderivative = 8.32

method	result	size
risch	\(\text {Expression too large to display}\)	\(3079\)
default	\(\text {Expression too large to display}\)	\(9661\)

Input:

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

Output:

-1/2*b^2*(-d*x+10*c)*(b*x^2+a)^(1/2)/d^6+1/2/d^6*(2/d^5*(a^3*d^6+9*a^2*b*c 
^2*d^4+15*a*b^2*c^4*d^2+7*b^3*c^6)*(-1/3/(a*d^2+b*c^2)*d^2/(x+c/d)^3*(b*(x 
+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+5/3*b*c*d/(a*d^2+b*c^2)*( 
-1/2/(a*d^2+b*c^2)*d^2/(x+c/d)^2*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2 
)/d^2)^(1/2)+3/2*b*c*d/(a*d^2+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c 
/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((a*d^2 
+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2 
)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d 
)))+1/2*b/(a*d^2+b*c^2)*d^2/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/ 
d^2-2*b*c/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/ 
d)+(a*d^2+b*c^2)/d^2)^(1/2))/(x+c/d)))-2/3*b/(a*d^2+b*c^2)*d^2*(-1/(a*d^2+ 
b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b 
*c*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c 
/d*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2 
+b*c^2)/d^2)^(1/2))/(x+c/d))))+5*b^(3/2)*(a*d^2+6*b*c^2)/d*ln(b^(1/2)*x+(b 
*x^2+a)^(1/2))+2*b/d^3*(3*a^2*d^4+30*a*b*c^2*d^2+35*b^2*c^4)*(-1/(a*d^2+b* 
c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c 
*d/(a*d^2+b*c^2)/((a*d^2+b*c^2)/d^2)^(1/2)*ln((2*(a*d^2+b*c^2)/d^2-2*b*c/d 
*(x+c/d)+2*((a*d^2+b*c^2)/d^2)^(1/2)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2))/(x+c/d)))-2*c*(a^3*d^6+3*a^2*b*c^2*d^4+3*a*b^2*c^4*d^...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1042 vs. \(2 (333) = 666\).

Time = 29.04 (sec) , antiderivative size = 4233, normalized size of antiderivative = 11.44 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

Too large to include
 

Sympy [F]

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

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

Output:

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

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 2150 vs. \(2 (333) = 666\).

Time = 0.22 (sec) , antiderivative size = 2150, normalized size of antiderivative = 5.81 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^5} \, dx=\text {Too large to display} \] Input:

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

Output:

5/16*b^6*c^8*arcsinh(b*x/sqrt(a*b))/(b^(7/2)*c^6*d^7 + 3*a*b^(5/2)*c^4*d^9 
 + 3*a^2*b^(3/2)*c^2*d^11 + a^3*sqrt(b)*d^13) + 5/16*a*b^5*c^6*arcsinh(b*x 
/sqrt(a*b))/(b^(7/2)*c^6*d^5 + 3*a*b^(5/2)*c^4*d^7 + 3*a^2*b^(3/2)*c^2*d^9 
 + a^3*sqrt(b)*d^11) - 5/16*sqrt(b*x^2 + a)*b^5*c^6*x/(b^3*c^6*d^5 + 3*a*b 
^2*c^4*d^7 + 3*a^2*b*c^2*d^9 + a^3*d^11) - 65/16*b^5*c^6*arcsinh(b*x/sqrt( 
a*b))/(b^(5/2)*c^4*d^7 + 2*a*b^(3/2)*c^2*d^9 + a^2*sqrt(b)*d^11) + 5/24*(b 
*x^2 + a)^(3/2)*b^4*c^5/(b^3*c^6*d^4 + 3*a*b^2*c^4*d^6 + 3*a^2*b*c^2*d^8 + 
 a^3*d^10) - 5/24*(b*x^2 + a)^(3/2)*b^4*c^4*x/(b^3*c^6*d^3 + 3*a*b^2*c^4*d 
^5 + 3*a^2*b*c^2*d^7 + a^3*d^9) - 5/16*sqrt(b*x^2 + a)*a*b^4*c^4*x/(b^3*c^ 
6*d^3 + 3*a*b^2*c^4*d^5 + 3*a^2*b*c^2*d^7 + a^3*d^9) - 15/4*a*b^4*c^4*arcs 
inh(b*x/sqrt(a*b))/(b^(5/2)*c^4*d^5 + 2*a*b^(3/2)*c^2*d^7 + a^2*sqrt(b)*d^ 
9) + 1/8*(b*x^2 + a)^(5/2)*b^3*c^4/(b^3*c^6*d^3*x + 3*a*b^2*c^4*d^5*x + 3* 
a^2*b*c^2*d^7*x + a^3*d^9*x + b^3*c^7*d^2 + 3*a*b^2*c^5*d^4 + 3*a^2*b*c^3* 
d^6 + a^3*c*d^8) + 5/8*sqrt(b*x^2 + a)*b^4*c^5/(b^2*c^4*d^6 + 2*a*b*c^2*d^ 
8 + a^2*d^10) + 25/8*sqrt(b*x^2 + a)*b^4*c^4*x/(b^2*c^4*d^5 + 2*a*b*c^2*d^ 
7 + a^2*d^9) + 75/16*b^4*c^4*arcsinh(b*x/sqrt(a*b))/(b^(3/2)*c^2*d^7 + a*s 
qrt(b)*d^9) - 1/24*(b*x^2 + a)^(7/2)*b^2*c^3/(b^3*c^6*d^2*x^2 + 3*a*b^2*c^ 
4*d^4*x^2 + 3*a^2*b*c^2*d^6*x^2 + a^3*d^8*x^2 + 2*b^3*c^7*d*x + 6*a*b^2*c^ 
5*d^3*x + 6*a^2*b*c^3*d^5*x + 2*a^3*c*d^7*x + b^3*c^8 + 3*a*b^2*c^6*d^2 + 
3*a^2*b*c^4*d^4 + a^3*c^2*d^6) + 1/24*(b*x^2 + a)^(5/2)*b^3*c^3/(b^3*c^...
 

Giac [F(-1)]

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

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

Output:

Timed out
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.59 (sec) , antiderivative size = 3255, normalized size of antiderivative = 8.80 \[ \int \frac {x \left (a+b x^2\right )^{5/2}}{(c+d x)^5} \, dx =\text {Too large to display} \] Input:

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

Output:

(225*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**2*b**2*c**5*d**4 + 900*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**2*b**2*c**4*d**5*x + 1 
350*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**2*b**2*c**3*d**6*x**2 + 900*sqrt(a*d**2 + b*c**2)*log( - s 
qrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*c**2*d**7*x 
**3 + 225*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**2*b**2*c*d**8*x**4 + 600*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*b**3*c**7*d**2 + 
 2400*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*b**3*c**6*d**3*x + 3600*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*b**3*c**5*d**4*x**2 + 
2400*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*b**3*c**4*d**5*x**3 + 600*sqrt(a*d**2 + b*c**2)*log( - sqr 
t(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**3*c**3*d**6*x**4 + 
 360*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)*b**4*c**9 + 1440*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)*b**4*c**8*d*x + 2160*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)*b**4 
*c**7*d**2*x**2 + 1440*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*sq...