\(\int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx\) [177]

Optimal result

Integrand size = 24, antiderivative size = 258 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\frac {(B c-A d) \sqrt {a+b x^2}}{3 \left (b c^2+a d^2\right ) (c+d x)^3}-\frac {\left (3 a B d^2-b c (2 B c-5 A d)\right ) \sqrt {a+b x^2}}{6 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {b \left (b c^2 (2 B c-11 A d)-a d^2 (13 B c-4 A d)\right ) \sqrt {a+b x^2}}{6 \left (b c^2+a d^2\right )^3 (c+d x)}-\frac {b \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{7/2}} \] Output:

1/3*(-A*d+B*c)*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)/(d*x+c)^3-1/6*(3*a*B*d^2-b*c* 
(-5*A*d+2*B*c))*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^2/(d*x+c)^2+1/6*b*(b*c^2*(-1 
1*A*d+2*B*c)-a*d^2*(-4*A*d+13*B*c))*(b*x^2+a)^(1/2)/(a*d^2+b*c^2)^3/(d*x+c 
)-1/2*b*(A*b*c*(-3*a*d^2+2*b*c^2)+a*B*d*(-a*d^2+4*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)^(7/2)
 

Mathematica [A] (verified)

Time = 10.43 (sec) , antiderivative size = 286, normalized size of antiderivative = 1.11 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {b c^2+a d^2} \sqrt {a+b x^2} \left (2 (B c-A d) \left (b c^2+a d^2\right )^2+\left (b c^2+a d^2\right ) \left (-3 a B d^2+b c (2 B c-5 A d)\right ) (c+d x)+b \left (b c^2 (2 B c-11 A d)+a d^2 (-13 B c+4 A d)\right ) (c+d x)^2\right )+3 b \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) (c+d x)^3 \log (c+d x)-3 b \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) (c+d x)^3 \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{6 \left (b c^2+a d^2\right )^{7/2} (c+d x)^3} \] Input:

Integrate[(A + B*x)/((c + d*x)^4*Sqrt[a + b*x^2]),x]
 

Output:

(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]*(2*(B*c - A*d)*(b*c^2 + a*d^2)^2 + (b 
*c^2 + a*d^2)*(-3*a*B*d^2 + b*c*(2*B*c - 5*A*d))*(c + d*x) + b*(b*c^2*(2*B 
*c - 11*A*d) + a*d^2*(-13*B*c + 4*A*d))*(c + d*x)^2) + 3*b*(A*b*c*(2*b*c^2 
 - 3*a*d^2) + a*B*d*(4*b*c^2 - a*d^2))*(c + d*x)^3*Log[c + d*x] - 3*b*(A*b 
*c*(2*b*c^2 - 3*a*d^2) + a*B*d*(4*b*c^2 - a*d^2))*(c + d*x)^3*Log[a*d - b* 
c*x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(6*(b*c^2 + a*d^2)^(7/2)*(c + 
d*x)^3)
 

Rubi [A] (verified)

Time = 0.51 (sec) , antiderivative size = 288, normalized size of antiderivative = 1.12, number of steps used = 9, number of rules used = 8, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {688, 25, 688, 25, 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.

\(\displaystyle \int \frac {A+B x}{\sqrt {a+b x^2} (c+d x)^4} \, dx\)

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 679

\(\displaystyle \frac {\frac {b \left (\frac {3 \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}+\frac {\sqrt {a+b x^2} \left (b c^2 (2 B c-11 A d)-a d^2 (13 B c-4 A d)\right )}{(c+d x) \left (a d^2+b c^2\right )}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (3 a B d^2-b c (2 B c-5 A d)\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{3 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} (B c-A d)}{3 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle \frac {\frac {b \left (\frac {\sqrt {a+b x^2} \left (b c^2 (2 B c-11 A d)-a d^2 (13 B c-4 A d)\right )}{(c+d x) \left (a d^2+b c^2\right )}-\frac {3 \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (3 a B d^2-b c (2 B c-5 A d)\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{3 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} (B c-A d)}{3 (c+d x)^3 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle \frac {\frac {b \left (\frac {\sqrt {a+b x^2} \left (b c^2 (2 B c-11 A d)-a d^2 (13 B c-4 A d)\right )}{(c+d x) \left (a d^2+b c^2\right )}-\frac {3 \left (A b c \left (2 b c^2-3 a d^2\right )+a B d \left (4 b c^2-a d^2\right )\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{\left (a d^2+b c^2\right )^{3/2}}\right )}{2 \left (a d^2+b c^2\right )}-\frac {\sqrt {a+b x^2} \left (3 a B d^2-b c (2 B c-5 A d)\right )}{2 (c+d x)^2 \left (a d^2+b c^2\right )}}{3 \left (a d^2+b c^2\right )}+\frac {\sqrt {a+b x^2} (B c-A d)}{3 (c+d x)^3 \left (a d^2+b c^2\right )}\)

Input:

Int[(A + B*x)/((c + d*x)^4*Sqrt[a + b*x^2]),x]
 

Output:

((B*c - A*d)*Sqrt[a + b*x^2])/(3*(b*c^2 + a*d^2)*(c + d*x)^3) + (-1/2*((3* 
a*B*d^2 - b*c*(2*B*c - 5*A*d))*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x) 
^2) + (b*(((b*c^2*(2*B*c - 11*A*d) - a*d^2*(13*B*c - 4*A*d))*Sqrt[a + b*x^ 
2])/((b*c^2 + a*d^2)*(c + d*x)) - (3*(A*b*c*(2*b*c^2 - 3*a*d^2) + a*B*d*(4 
*b*c^2 - a*d^2))*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)))/(2*(b*c^2 + a*d^2)))/(3*(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 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. \(1214\) vs. \(2(238)=476\).

Time = 1.38 (sec) , antiderivative size = 1215, normalized size of antiderivative = 4.71

method	result	size
default	\(\text {Expression too large to display}\)	\(1215\)

Input:

int((B*x+A)/(d*x+c)^4/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

B/d^4*(-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)))+(A*d-B*c)/d^5*(-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...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 797 vs. \(2 (239) = 478\).

Time = 4.16 (sec) , antiderivative size = 1620, normalized size of antiderivative = 6.28 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(d*x+c)^4/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/12*(3*(2*A*b^3*c^6 + 4*B*a*b^2*c^5*d - 3*A*a*b^2*c^4*d^2 - B*a^2*b*c^3 
*d^3 + (2*A*b^3*c^3*d^3 + 4*B*a*b^2*c^2*d^4 - 3*A*a*b^2*c*d^5 - B*a^2*b*d^ 
6)*x^3 + 3*(2*A*b^3*c^4*d^2 + 4*B*a*b^2*c^3*d^3 - 3*A*a*b^2*c^2*d^4 - B*a^ 
2*b*c*d^5)*x^2 + 3*(2*A*b^3*c^5*d + 4*B*a*b^2*c^4*d^2 - 3*A*a*b^2*c^3*d^3 
- B*a^2*b*c^2*d^4)*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)*s 
qrt(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 2*(6*B*b^3*c^7 - 18*A*b^3*c^6 
*d - 4*B*a*b^2*c^5*d^2 - 23*A*a*b^2*c^4*d^3 - 11*B*a^2*b*c^3*d^4 - 7*A*a^2 
*b*c^2*d^5 - B*a^3*c*d^6 - 2*A*a^3*d^7 + (2*B*b^3*c^5*d^2 - 11*A*b^3*c^4*d 
^3 - 11*B*a*b^2*c^3*d^4 - 7*A*a*b^2*c^2*d^5 - 13*B*a^2*b*c*d^6 + 4*A*a^2*b 
*d^7)*x^2 + 3*(2*B*b^3*c^6*d - 9*A*b^3*c^5*d^2 - 7*B*a*b^2*c^4*d^3 - 8*A*a 
*b^2*c^3*d^4 - 10*B*a^2*b*c^2*d^5 + A*a^2*b*c*d^6 - B*a^3*d^7)*x)*sqrt(b*x 
^2 + a))/(b^4*c^11 + 4*a*b^3*c^9*d^2 + 6*a^2*b^2*c^7*d^4 + 4*a^3*b*c^5*d^6 
 + a^4*c^3*d^8 + (b^4*c^8*d^3 + 4*a*b^3*c^6*d^5 + 6*a^2*b^2*c^4*d^7 + 4*a^ 
3*b*c^2*d^9 + a^4*d^11)*x^3 + 3*(b^4*c^9*d^2 + 4*a*b^3*c^7*d^4 + 6*a^2*b^2 
*c^5*d^6 + 4*a^3*b*c^3*d^8 + a^4*c*d^10)*x^2 + 3*(b^4*c^10*d + 4*a*b^3*c^8 
*d^3 + 6*a^2*b^2*c^6*d^5 + 4*a^3*b*c^4*d^7 + a^4*c^2*d^9)*x), -1/6*(3*(2*A 
*b^3*c^6 + 4*B*a*b^2*c^5*d - 3*A*a*b^2*c^4*d^2 - B*a^2*b*c^3*d^3 + (2*A*b^ 
3*c^3*d^3 + 4*B*a*b^2*c^2*d^4 - 3*A*a*b^2*c*d^5 - B*a^2*b*d^6)*x^3 + 3*(2* 
A*b^3*c^4*d^2 + 4*B*a*b^2*c^3*d^3 - 3*A*a*b^2*c^2*d^4 - B*a^2*b*c*d^5)*...
 

Sympy [F]

\[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\int \frac {A + B x}{\sqrt {a + b x^{2}} \left (c + d x\right )^{4}}\, dx \] Input:

integrate((B*x+A)/(d*x+c)**4/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(sqrt(a + b*x**2)*(c + d*x)**4), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1097 vs. \(2 (239) = 478\).

Time = 0.11 (sec) , antiderivative size = 1097, normalized size of antiderivative = 4.25 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((B*x+A)/(d*x+c)^4/(b*x^2+a)^(1/2),x, algorithm="maxima")
                                                                                    
                                                                                    
 

Output:

5/2*sqrt(b*x^2 + a)*B*b^2*c^3/(b^3*c^6*d*x + 3*a*b^2*c^4*d^3*x + 3*a^2*b*c 
^2*d^5*x + a^3*d^7*x + b^3*c^7 + 3*a*b^2*c^5*d^2 + 3*a^2*b*c^3*d^4 + a^3*c 
*d^6) - 5/2*sqrt(b*x^2 + a)*A*b^2*c^2/(b^3*c^6*x + 3*a*b^2*c^4*d^2*x + 3*a 
^2*b*c^2*d^4*x + a^3*d^6*x + b^3*c^7/d + 3*a*b^2*c^5*d + 3*a^2*b*c^3*d^3 + 
 a^3*c*d^5) + 5/6*sqrt(b*x^2 + a)*B*b*c^2/(b^2*c^4*d^2*x^2 + 2*a*b*c^2*d^4 
*x^2 + a^2*d^6*x^2 + 2*b^2*c^5*d*x + 4*a*b*c^3*d^3*x + 2*a^2*c*d^5*x + b^2 
*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4) - 5/6*sqrt(b*x^2 + a)*A*b*c/(b^2*c^4*d 
*x^2 + 2*a*b*c^2*d^3*x^2 + a^2*d^5*x^2 + 2*b^2*c^5*x + 4*a*b*c^3*d^2*x + 2 
*a^2*c*d^4*x + b^2*c^6/d + 2*a*b*c^4*d + a^2*c^2*d^3) - 13/6*sqrt(b*x^2 + 
a)*B*b*c/(b^2*c^4*d*x + 2*a*b*c^2*d^3*x + a^2*d^5*x + b^2*c^5 + 2*a*b*c^3* 
d^2 + a^2*c*d^4) + 2/3*sqrt(b*x^2 + a)*A*b/(b^2*c^4*x + 2*a*b*c^2*d^2*x + 
a^2*d^4*x + b^2*c^5/d + 2*a*b*c^3*d + a^2*c*d^3) + 1/3*sqrt(b*x^2 + a)*B*c 
/(b*c^2*d^3*x^3 + a*d^5*x^3 + 3*b*c^3*d^2*x^2 + 3*a*c*d^4*x^2 + 3*b*c^4*d* 
x + 3*a*c^2*d^3*x + b*c^5 + a*c^3*d^2) - 1/3*sqrt(b*x^2 + a)*A/(b*c^2*d^2* 
x^3 + a*d^4*x^3 + 3*b*c^3*d*x^2 + 3*a*c*d^3*x^2 + 3*b*c^4*x + 3*a*c^2*d^2* 
x + b*c^5/d + a*c^3*d) - 1/2*sqrt(b*x^2 + a)*B/(b*c^2*d^2*x^2 + a*d^4*x^2 
+ 2*b*c^3*d*x + 2*a*c*d^3*x + b*c^4 + a*c^2*d^2) - 5/2*B*b^3*c^4*arcsinh(b 
*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/ 
d^2)^(7/2)*d^8) + 5/2*A*b^3*c^3*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a 
*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(7/2)*d^7) + 3*B*b^2*c^2*...
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1076 vs. \(2 (239) = 478\).

Time = 0.14 (sec) , antiderivative size = 1076, normalized size of antiderivative = 4.17 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(d*x+c)^4/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

(2*A*b^3*c^3 + 4*B*a*b^2*c^2*d - 3*A*a*b^2*c*d^2 - B*a^2*b*d^3)*arctan(-(( 
sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^3*c^ 
6 + 3*a*b^2*c^4*d^2 + 3*a^2*b*c^2*d^4 + a^3*d^6)*sqrt(-b*c^2 - a*d^2)) - 1 
/3*(6*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*b^3*c^3*d^3 + 12*(sqrt(b)*x - sqrt 
(b*x^2 + a))^5*B*a*b^2*c^2*d^4 - 9*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*a*b^2 
*c*d^5 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*B*a^2*b*d^6 + 30*(sqrt(b)*x - s 
qrt(b*x^2 + a))^4*A*b^(7/2)*c^4*d^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B 
*a*b^(5/2)*c^3*d^3 - 45*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a*b^(5/2)*c^2*d^ 
4 - 15*(sqrt(b)*x - sqrt(b*x^2 + a))^4*B*a^2*b^(3/2)*c*d^5 - 8*(sqrt(b)*x 
- sqrt(b*x^2 + a))^3*B*b^4*c^6 + 44*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b^4* 
c^5*d + 64*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a*b^3*c^4*d^2 - 82*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^3*A*a*b^3*c^3*d^3 - 78*(sqrt(b)*x - sqrt(b*x^2 + a))^3 
*B*a^2*b^2*c^2*d^4 + 24*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*a^2*b^2*c*d^5 + 
24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a*b^(7/2)*c^5*d - 102*(sqrt(b)*x - sq 
rt(b*x^2 + a))^2*A*a*b^(7/2)*c^4*d^2 - 102*(sqrt(b)*x - sqrt(b*x^2 + a))^2 
*B*a^2*b^(5/2)*c^3*d^3 + 36*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^2*b^(5/2)* 
c^2*d^4 + 24*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^3*b^(3/2)*c*d^5 - 12*(sqr 
t(b)*x - sqrt(b*x^2 + a))^2*A*a^3*b^(3/2)*d^6 - 12*(sqrt(b)*x - sqrt(b*x^2 
 + a))*B*a^2*b^3*c^4*d^2 + 60*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a^2*b^3*c^3* 
d^3 + 66*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^3*b^2*c^2*d^4 - 15*(sqrt(b)*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^4} \,d x \] Input:

int((A + B*x)/((a + b*x^2)^(1/2)*(c + d*x)^4),x)
 

Output:

int((A + B*x)/((a + b*x^2)^(1/2)*(c + d*x)^4), x)
 

Reduce [B] (verification not implemented)

Time = 0.26 (sec) , antiderivative size = 2137, normalized size of antiderivative = 8.28 \[ \int \frac {A+B x}{(c+d x)^4 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(d*x+c)^4/(b*x^2+a)^(1/2),x)
 

Output:

(9*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**2 + 27*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**3*x + 3*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**2*b**2*c**3*d**3 + 27*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**2*d**4*x**2 + 9*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**2*d**4*x + 9*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**5*x**3 + 9*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**5*x**2 + 3*sqrt(a*d**2 + b*c**2)*log( - sqrt(a + b*x**2)*s 
qrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b**2*d**6*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*b** 
3*c**6 - 18*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*x - 12*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**5*d - 18*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**2*x**2 - 36*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**2*x - 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...