\(\int \frac {A+B x}{(c+d x)^2 (a+b x^2)^{5/2}} \, dx\) [193]

Optimal result

Integrand size = 24, antiderivative size = 308 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {B c-A d}{\left (b c^2+a d^2\right ) (c+d x) \left (a+b x^2\right )^{3/2}}+\frac {a \left (a B d^2-b c (4 B c-5 A d)\right )+b \left (A b c^2+5 a B c d-4 a A d^2\right ) x}{3 a \left (b c^2+a d^2\right )^2 \left (a+b x^2\right )^{3/2}}+\frac {3 a^2 d^2 \left (a B d^2-b c (4 B c-5 A d)\right )-b \left (a B c d \left (2 b c^2-13 a d^2\right )-A \left (2 b^2 c^4+9 a b c^2 d^2-8 a^2 d^4\right )\right ) x}{3 a^2 \left (b c^2+a d^2\right )^3 \sqrt {a+b x^2}}-\frac {d^3 \left (a B d^2-b c (4 B c-5 A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{7/2}} \] Output:

(-A*d+B*c)/(a*d^2+b*c^2)/(d*x+c)/(b*x^2+a)^(3/2)+1/3*(a*(a*B*d^2-b*c*(-5*A 
*d+4*B*c))+b*(-4*A*a*d^2+A*b*c^2+5*B*a*c*d)*x)/a/(a*d^2+b*c^2)^2/(b*x^2+a) 
^(3/2)+1/3*(3*a^2*d^2*(a*B*d^2-b*c*(-5*A*d+4*B*c))-b*(a*B*c*d*(-13*a*d^2+2 
*b*c^2)-A*(-8*a^2*d^4+9*a*b*c^2*d^2+2*b^2*c^4))*x)/a^2/(a*d^2+b*c^2)^3/(b* 
x^2+a)^(1/2)-d^3*(a*B*d^2-b*c*(-5*A*d+4*B*c))*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)^(7/2)
 

Mathematica [A] (verified)

Time = 11.90 (sec) , antiderivative size = 299, normalized size of antiderivative = 0.97 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\frac {\frac {d \left (a B c d \left (-2 b c^2+13 a d^2\right )+A \left (2 b^2 c^4+9 a b c^2 d^2-8 a^2 d^4\right )\right ) \sqrt {a+b x^2}}{a \left (b c^2+a d^2\right )^2}+\frac {A b c x+a (-B c+A d+B d x)}{\left (a+b x^2\right )^{3/2}}+\frac {2 A b^2 c^3 x+a^2 d^2 (-5 B c+4 A d+3 B d x)-a b c d (2 B c x+A (c-7 d x))}{a \left (b c^2+a d^2\right ) \sqrt {a+b x^2}}-\frac {3 a d^3 \left (a B d^2+b c (-4 B c+5 A d)\right ) (c+d x) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{\left (b c^2+a d^2\right )^{5/2}}}{3 a \left (b c^2+a d^2\right ) (c+d x)} \] Input:

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

Output:

((d*(a*B*c*d*(-2*b*c^2 + 13*a*d^2) + A*(2*b^2*c^4 + 9*a*b*c^2*d^2 - 8*a^2* 
d^4))*Sqrt[a + b*x^2])/(a*(b*c^2 + a*d^2)^2) + (A*b*c*x + a*(-(B*c) + A*d 
+ B*d*x))/(a + b*x^2)^(3/2) + (2*A*b^2*c^3*x + a^2*d^2*(-5*B*c + 4*A*d + 3 
*B*d*x) - a*b*c*d*(2*B*c*x + A*(c - 7*d*x)))/(a*(b*c^2 + a*d^2)*Sqrt[a + b 
*x^2]) - (3*a*d^3*(a*B*d^2 + b*c*(-4*B*c + 5*A*d))*(c + d*x)*ArcTanh[(a*d 
- b*c*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^(5/2))/(3 
*a*(b*c^2 + a*d^2)*(c + d*x))
 

Rubi [A] (verified)

Time = 0.58 (sec) , antiderivative size = 362, normalized size of antiderivative = 1.18, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {686, 25, 27, 686, 27, 679, 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.

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 686

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

\(\displaystyle \frac {-\frac {d \left (\frac {\sqrt {a+b x^2} \left (a B c d \left (2 b c^2-13 a d^2\right )-A \left (-8 a^2 d^4+9 a b c^2 d^2+2 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}-\frac {3 a^2 d^2 \left (a B d^2-b c (4 B c-5 A d)\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (2 b c^2-3 a d^2\right )-A b c \left (7 a d^2+2 b c^2\right )\right )+a d \left (-4 a A d^2+5 a B c d+A b c^2\right )}{a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {-\frac {d \left (\frac {3 a^2 d^2 \left (a B d^2-b c (4 B c-5 A d)\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}+\frac {\sqrt {a+b x^2} \left (a B c d \left (2 b c^2-13 a d^2\right )-A \left (-8 a^2 d^4+9 a b c^2 d^2+2 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (2 b c^2-3 a d^2\right )-A b c \left (7 a d^2+2 b c^2\right )\right )+a d \left (-4 a A d^2+5 a B c d+A b c^2\right )}{a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {-\frac {d \left (\frac {3 a^2 d^2 \left (a B d^2-b c (4 B c-5 A d)\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}+\frac {\sqrt {a+b x^2} \left (a B c d \left (2 b c^2-13 a d^2\right )-A \left (-8 a^2 d^4+9 a b c^2 d^2+2 b^2 c^4\right )\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{a \left (a d^2+b c^2\right )}-\frac {x \left (a B d \left (2 b c^2-3 a d^2\right )-A b c \left (7 a d^2+2 b c^2\right )\right )+a d \left (-4 a A d^2+5 a B c d+A b c^2\right )}{a \sqrt {a+b x^2} (c+d x) \left (a d^2+b c^2\right )}}{3 a \left (a d^2+b c^2\right )}-\frac {a (B c-A d)-x (a B d+A b c)}{3 a \left (a+b x^2\right )^{3/2} (c+d x) \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/3*(a*(B*c - A*d) - (A*b*c + a*B*d)*x)/(a*(b*c^2 + a*d^2)*(c + d*x)*(a + 
 b*x^2)^(3/2)) + (-((a*d*(A*b*c^2 + 5*a*B*c*d - 4*a*A*d^2) + (a*B*d*(2*b*c 
^2 - 3*a*d^2) - A*b*c*(2*b*c^2 + 7*a*d^2))*x)/(a*(b*c^2 + a*d^2)*(c + d*x) 
*Sqrt[a + b*x^2])) - (d*(((a*B*c*d*(2*b*c^2 - 13*a*d^2) - A*(2*b^2*c^4 + 9 
*a*b*c^2*d^2 - 8*a^2*d^4))*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x)) + 
(3*a^2*d^2*(a*B*d^2 - b*c*(4*B*c - 5*A*d))*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c 
^2 + a*d^2]*Sqrt[a + b*x^2])])/(b*c^2 + a*d^2)^(3/2)))/(a*(b*c^2 + a*d^2)) 
)/(3*a*(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 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 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 686

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_), x_Symbol] :> Simp[(-(d + e*x)^(m + 1))*(f*a*c*e - a*g*c*d + c*(c*d*f + 
a*e*g)*x)*((a + c*x^2)^(p + 1)/(2*a*c*(p + 1)*(c*d^2 + a*e^2))), x] + Simp[ 
1/(2*a*c*(p + 1)*(c*d^2 + a*e^2))   Int[(d + e*x)^m*(a + c*x^2)^(p + 1)*Sim 
p[f*(c^2*d^2*(2*p + 3) + a*c*e^2*(m + 2*p + 3)) - a*c*d*e*g*m + c*e*(c*d*f 
+ a*e*g)*(m + 2*p + 4)*x, x], x], x] /; FreeQ[{a, c, d, e, f, g}, x] && LtQ 
[p, -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. \(1508\) vs. \(2(292)=584\).

Time = 1.36 (sec) , antiderivative size = 1509, normalized size of antiderivative = 4.90

method	result	size
default	\(\text {Expression too large to display}\)	\(1509\)

Input:

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

Output:

B/d^2*(1/3/(a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^ 
2)^(3/2)+b*c*d/(a*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2) 
/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+ 
16/3*b/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x 
+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/( 
a*d^2+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b 
*c*d/(a*d^2+b*c^2)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/ 
d^2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(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))))+(A*d-B*c)/d^3*(-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)^(3/2)+5*b*c*d/(a*d^2+b*c^2)*(1/3/(a*d^2 
+b*c^2)*d^2/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+b*c*d/(a 
*d^2+b*c^2)*(2/3*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^ 
2)/(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(3/2)+16/3*b/(4*b*(a*d^ 
2+b*c^2)/d^2-4*b^2*c^2/d^2)^2*(2*b*(x+c/d)-2*b*c/d)/(b*(x+c/d)^2-2*b*c/d*( 
x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))+1/(a*d^2+b*c^2)*d^2*(1/(a*d^2+b*c^2)*d^2/ 
(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+2*b*c*d/(a*d^2+b*c^2 
)*(2*b*(x+c/d)-2*b*c/d)/(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/(b*(x+c/d)^2 
-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-1/(a*d^2+b*c^2)*d^2/((a*d^2+b...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1254 vs. \(2 (293) = 586\).

Time = 1.54 (sec) , antiderivative size = 2534, normalized size of antiderivative = 8.23 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

[-1/6*(3*(4*B*a^4*b*c^3*d^3 - 5*A*a^4*b*c^2*d^4 - B*a^5*c*d^5 + (4*B*a^2*b 
^3*c^2*d^4 - 5*A*a^2*b^3*c*d^5 - B*a^3*b^2*d^6)*x^5 + (4*B*a^2*b^3*c^3*d^3 
 - 5*A*a^2*b^3*c^2*d^4 - B*a^3*b^2*c*d^5)*x^4 + 2*(4*B*a^3*b^2*c^2*d^4 - 5 
*A*a^3*b^2*c*d^5 - B*a^4*b*d^6)*x^3 + 2*(4*B*a^3*b^2*c^3*d^3 - 5*A*a^3*b^2 
*c^2*d^4 - B*a^4*b*c*d^5)*x^2 + (4*B*a^4*b*c^2*d^4 - 5*A*a^4*b*c*d^5 - B*a 
^5*d^6)*x)*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*(B*a^2*b^3*c^7 - 2*A*a^2*b^3*c^6*d + 1 
0*B*a^3*b^2*c^5*d^2 - 16*A*a^3*b^2*c^4*d^3 + 2*B*a^4*b*c^3*d^4 - 11*A*a^4* 
b*c^2*d^5 - 7*B*a^5*c*d^6 + 3*A*a^5*d^7 - (2*A*b^5*c^6*d - 2*B*a*b^4*c^5*d 
^2 + 11*A*a*b^4*c^4*d^3 + 11*B*a^2*b^3*c^3*d^4 + A*a^2*b^3*c^2*d^5 + 13*B* 
a^3*b^2*c*d^6 - 8*A*a^3*b^2*d^7)*x^4 - (2*A*b^5*c^7 - 2*B*a*b^4*c^6*d + 11 
*A*a*b^4*c^5*d^2 - B*a^2*b^3*c^4*d^3 + 16*A*a^2*b^3*c^3*d^4 + 4*B*a^3*b^2* 
c^2*d^5 + 7*A*a^3*b^2*c*d^6 + 3*B*a^4*b*d^7)*x^3 - 3*(A*a*b^4*c^6*d - 3*B* 
a^2*b^3*c^5*d^2 + 8*A*a^2*b^3*c^4*d^3 + 4*B*a^3*b^2*c^3*d^4 + 3*A*a^3*b^2* 
c^2*d^5 + 7*B*a^4*b*c*d^6 - 4*A*a^4*b*d^7)*x^2 - (3*A*a*b^4*c^7 - B*a^2*b^ 
3*c^6*d + 14*A*a^2*b^3*c^5*d^2 + 2*B*a^3*b^2*c^4*d^3 + 19*A*a^3*b^2*c^3*d^ 
4 + 7*B*a^4*b*c^2*d^5 + 8*A*a^4*b*c*d^6 + 4*B*a^5*d^7)*x)*sqrt(b*x^2 + a)) 
/(a^4*b^4*c^9 + 4*a^5*b^3*c^7*d^2 + 6*a^6*b^2*c^5*d^4 + 4*a^7*b*c^3*d^6 + 
a^8*c*d^8 + (a^2*b^6*c^8*d + 4*a^3*b^5*c^6*d^3 + 6*a^4*b^4*c^4*d^5 + 4*...
 

Sympy [F(-1)]

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

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

Output:

Timed out
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1360 vs. \(2 (293) = 586\).

Time = 0.17 (sec) , antiderivative size = 1360, normalized size of antiderivative = 4.42 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-5*B*b^2*c^3*x/(sqrt(b*x^2 + a)*a*b^3*c^6/d + 3*sqrt(b*x^2 + a)*a^2*b^2*c^ 
4*d + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^3 + sqrt(b*x^2 + a)*a^4*d^5) - 5/3*B*b 
^2*c^3*x/((b*x^2 + a)^(3/2)*a*b^2*c^4*d + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d^ 
3 + (b*x^2 + a)^(3/2)*a^3*d^5) - 10/3*B*b^2*c^3*x/(sqrt(b*x^2 + a)*a^2*b^2 
*c^4*d + 2*sqrt(b*x^2 + a)*a^3*b*c^2*d^3 + sqrt(b*x^2 + a)*a^4*d^5) + 5/3* 
A*b^2*c^2*x/((b*x^2 + a)^(3/2)*a*b^2*c^4 + 2*(b*x^2 + a)^(3/2)*a^2*b*c^2*d 
^2 + (b*x^2 + a)^(3/2)*a^3*d^4) + 5*A*b^2*c^2*x/(3*sqrt(b*x^2 + a)*a^2*b^2 
*c^4 + sqrt(b*x^2 + a)*a*b^3*c^6/d^2 + 3*sqrt(b*x^2 + a)*a^3*b*c^2*d^2 + s 
qrt(b*x^2 + a)*a^4*d^4) + 10/3*A*b^2*c^2*x/(sqrt(b*x^2 + a)*a^2*b^2*c^4 + 
2*sqrt(b*x^2 + a)*a^3*b*c^2*d^2 + sqrt(b*x^2 + a)*a^4*d^4) - 5/3*B*b*c^2/( 
(b*x^2 + a)^(3/2)*b^2*c^4 + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d^2 + (b*x^2 + a)^ 
(3/2)*a^2*d^4) - 5*B*b*c^2/(3*sqrt(b*x^2 + a)*a*b^2*c^4 + sqrt(b*x^2 + a)* 
b^3*c^6/d^2 + 3*sqrt(b*x^2 + a)*a^2*b*c^2*d^2 + sqrt(b*x^2 + a)*a^3*d^4) + 
 B*b*c*x/(sqrt(b*x^2 + a)*a*b^2*c^4/d + 2*sqrt(b*x^2 + a)*a^2*b*c^2*d + sq 
rt(b*x^2 + a)*a^3*d^3) + 5/3*B*b*c*x/((b*x^2 + a)^(3/2)*a*b*c^2*d + (b*x^2 
 + a)^(3/2)*a^2*d^3) + 10/3*B*b*c*x/(sqrt(b*x^2 + a)*a^2*b*c^2*d + sqrt(b* 
x^2 + a)*a^3*d^3) + 5*A*b*c/(sqrt(b*x^2 + a)*b^3*c^6/d^3 + 3*sqrt(b*x^2 + 
a)*a*b^2*c^4/d + 3*sqrt(b*x^2 + a)*a^2*b*c^2*d + sqrt(b*x^2 + a)*a^3*d^3) 
+ 5/3*A*b*c/((b*x^2 + a)^(3/2)*b^2*c^4/d + 2*(b*x^2 + a)^(3/2)*a*b*c^2*d + 
 (b*x^2 + a)^(3/2)*a^2*d^3) - 4/3*A*b*x/((b*x^2 + a)^(3/2)*a*b*c^2 + (b...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1957 vs. \(2 (293) = 586\).

Time = 0.31 (sec) , antiderivative size = 1957, normalized size of antiderivative = 6.35 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx=\text {Too large to display} \] Input:

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

Output:

-1/3*((12*B*a^2*b^(3/2)*c^2*d^6*log(abs(-b*c*d + sqrt(b*c^2 + a*d^2)*sqrt( 
b)*abs(d))) - 15*A*a^2*b^(3/2)*c*d^7*log(abs(-b*c*d + sqrt(b*c^2 + a*d^2)* 
sqrt(b)*abs(d))) - 3*B*a^3*sqrt(b)*d^8*log(abs(-b*c*d + sqrt(b*c^2 + a*d^2 
)*sqrt(b)*abs(d))) + 2*sqrt(b*c^2 + a*d^2)*A*b^3*c^4*d^2*abs(d) - 2*sqrt(b 
*c^2 + a*d^2)*B*a*b^2*c^3*d^3*abs(d) + 9*sqrt(b*c^2 + a*d^2)*A*a*b^2*c^2*d 
^4*abs(d) + 13*sqrt(b*c^2 + a*d^2)*B*a^2*b*c*d^5*abs(d) - 8*sqrt(b*c^2 + a 
*d^2)*A*a^2*b*d^6*abs(d))*sgn(1/(d*x + c))*sgn(d)/(sqrt(b*c^2 + a*d^2)*a^2 
*b^(7/2)*c^6*abs(d) + 3*sqrt(b*c^2 + a*d^2)*a^3*b^(5/2)*c^4*d^2*abs(d) + 3 
*sqrt(b*c^2 + a*d^2)*a^4*b^(3/2)*c^2*d^4*abs(d) + sqrt(b*c^2 + a*d^2)*a^5* 
sqrt(b)*d^6*abs(d)) - ((2*A*b^5*c^4*d^13*sgn(1/(d*x + c))*sgn(d) - 2*B*a*b 
^4*c^3*d^14*sgn(1/(d*x + c))*sgn(d) + 9*A*a*b^4*c^2*d^15*sgn(1/(d*x + c))* 
sgn(d) + 13*B*a^2*b^3*c*d^16*sgn(1/(d*x + c))*sgn(d) - 8*A*a^2*b^3*d^17*sg 
n(1/(d*x + c))*sgn(d))/(a^2*b^4*c^6*d^11*sgn(1/(d*x + c))^2*sgn(d)^2 + 3*a 
^3*b^3*c^4*d^13*sgn(1/(d*x + c))^2*sgn(d)^2 + 3*a^4*b^2*c^2*d^15*sgn(1/(d* 
x + c))^2*sgn(d)^2 + a^5*b*d^17*sgn(1/(d*x + c))^2*sgn(d)^2) - (3*(2*A*b^5 
*c^5*d^14*sgn(1/(d*x + c))*sgn(d) - 2*B*a*b^4*c^4*d^15*sgn(1/(d*x + c))*sg 
n(d) + 9*A*a*b^4*c^3*d^16*sgn(1/(d*x + c))*sgn(d) + 17*B*a^2*b^3*c^2*d^17* 
sgn(1/(d*x + c))*sgn(d) - 13*A*a^2*b^3*c*d^18*sgn(1/(d*x + c))*sgn(d) - B* 
a^3*b^2*d^19*sgn(1/(d*x + c))*sgn(d))/(a^2*b^4*c^6*d^11*sgn(1/(d*x + c))^2 
*sgn(d)^2 + 3*a^3*b^3*c^4*d^13*sgn(1/(d*x + c))^2*sgn(d)^2 + 3*a^4*b^2*...
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 4.54 (sec) , antiderivative size = 2864, normalized size of antiderivative = 9.30 \[ \int \frac {A+B x}{(c+d x)^2 \left (a+b x^2\right )^{5/2}} \, dx =\text {Too large to display} \] Input:

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

Output:

(15*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*c**2*d**4 + 15*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*c*d**5*x + 3*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*c*d 
**5 + 3*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*d**6*x - 12*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**3*b**2*c**3*d**3 + 30*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**3 
*b**2*c**2*d**4*x**2 - 12*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**3*b**2*c**2*d**4*x + 30*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**3*b**2 
*c*d**5*x**3 + 6*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**3*b**2*c*d**5*x**2 + 6*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**3*b**2*d**6*x**3 
 - 24*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**3*c**3*d**3*x**2 + 15*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**3*c**2*d**4*x**4 - 
 24*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**3*c**2*d**4*x**3 + 15*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**3*c*d**5*x**5 + 3...