\(\int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx\) [176]

Optimal result

Integrand size = 24, antiderivative size = 177 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {(B c-A d) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right ) (c+d x)^2}-\frac {\left (2 a B d^2-b c (B c-3 A d)\right ) \sqrt {a+b x^2}}{2 \left (b c^2+a d^2\right )^2 (c+d x)}-\frac {b \left (2 A b c^2+3 a B c d-a A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{5/2}} \] Output:

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

Mathematica [A] (verified)

Time = 1.25 (sec) , antiderivative size = 167, normalized size of antiderivative = 0.94 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {\sqrt {a+b x^2} \left (-a d^2 (A d+B (c+2 d x))+b c (B c (2 c+d x)-A d (4 c+3 d x))\right )}{2 \left (b c^2+a d^2\right )^2 (c+d x)^2}-\frac {b \left (2 A b c^2+3 a B c d-a A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

Integrate[(A + B*x)/((c + d*x)^3*Sqrt[a + b*x^2]),x]
 

Output:

(Sqrt[a + b*x^2]*(-(a*d^2*(A*d + B*(c + 2*d*x))) + b*c*(B*c*(2*c + d*x) - 
A*d*(4*c + 3*d*x))))/(2*(b*c^2 + a*d^2)^2*(c + d*x)^2) - (b*(2*A*b*c^2 + 3 
*a*B*c*d - a*A*d^2)*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-( 
b*c^2) - a*d^2]])/(-(b*c^2) - a*d^2)^(5/2)
 

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 191, normalized size of antiderivative = 1.08, number of steps used = 6, number of rules used = 5, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.208, Rules used = {688, 25, 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.

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

\(\Big \downarrow \) 688

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 679

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

Int[(A + B*x)/((c + d*x)^3*Sqrt[a + b*x^2]),x]
 

Output:

((B*c - A*d)*Sqrt[a + b*x^2])/(2*(b*c^2 + a*d^2)*(c + d*x)^2) + (-(((2*a*B 
*d^2 - b*c*(B*c - 3*A*d))*Sqrt[a + b*x^2])/((b*c^2 + a*d^2)*(c + d*x))) - 
(b*(2*A*b*c^2 + 3*a*B*c*d - a*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))
 

Defintions of rubi rules used

rule 25

Int[-(Fx_), x_Symbol] :> Simp[Identity[-1]   Int[Fx, x], 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. \(667\) vs. \(2(161)=322\).

Time = 1.37 (sec) , antiderivative size = 668, normalized size of antiderivative = 3.77

method	result	size
default	\(\frac {B \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}+\frac {\left (A d -B c \right ) \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \left (-\frac {d^{2} \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,d^{2} \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\)	\(668\)

Input:

int((B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

B/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)^(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)))+(A*d-B*c)/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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 444 vs. \(2 (162) = 324\).

Time = 1.18 (sec) , antiderivative size = 914, normalized size of antiderivative = 5.16 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

integrate((B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[-1/4*((2*A*b^2*c^4 + 3*B*a*b*c^3*d - A*a*b*c^2*d^2 + (2*A*b^2*c^2*d^2 + 3 
*B*a*b*c*d^3 - A*a*b*d^4)*x^2 + 2*(2*A*b^2*c^3*d + 3*B*a*b*c^2*d^2 - A*a*b 
*c*d^3)*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*(2*B*b^2*c^5 - 4*A*b^2*c^4*d + B*a*b*c 
^3*d^2 - 5*A*a*b*c^2*d^3 - B*a^2*c*d^4 - A*a^2*d^5 + (B*b^2*c^4*d - 3*A*b^ 
2*c^3*d^2 - B*a*b*c^2*d^3 - 3*A*a*b*c*d^4 - 2*B*a^2*d^5)*x)*sqrt(b*x^2 + a 
))/(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 + (b^3*c^6*d 
^2 + 3*a*b^2*c^4*d^4 + 3*a^2*b*c^2*d^6 + a^3*d^8)*x^2 + 2*(b^3*c^7*d + 3*a 
*b^2*c^5*d^3 + 3*a^2*b*c^3*d^5 + a^3*c*d^7)*x), -1/2*((2*A*b^2*c^4 + 3*B*a 
*b*c^3*d - A*a*b*c^2*d^2 + (2*A*b^2*c^2*d^2 + 3*B*a*b*c*d^3 - A*a*b*d^4)*x 
^2 + 2*(2*A*b^2*c^3*d + 3*B*a*b*c^2*d^2 - A*a*b*c*d^3)*x)*sqrt(-b*c^2 - a* 
d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqrt(b*x^2 + a)/(a*b*c^2 + 
a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) - (2*B*b^2*c^5 - 4*A*b^2*c^4*d + B*a*b 
*c^3*d^2 - 5*A*a*b*c^2*d^3 - B*a^2*c*d^4 - A*a^2*d^5 + (B*b^2*c^4*d - 3*A* 
b^2*c^3*d^2 - B*a*b*c^2*d^3 - 3*A*a*b*c*d^4 - 2*B*a^2*d^5)*x)*sqrt(b*x^2 + 
 a))/(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 + (b^3*c^6 
*d^2 + 3*a*b^2*c^4*d^4 + 3*a^2*b*c^2*d^6 + a^3*d^8)*x^2 + 2*(b^3*c^7*d + 3 
*a*b^2*c^5*d^3 + 3*a^2*b*c^3*d^5 + a^3*c*d^7)*x)]
 

Sympy [F]

\[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\int \frac {A + B x}{\sqrt {a + b x^{2}} \left (c + d x\right )^{3}}\, dx \] Input:

integrate((B*x+A)/(d*x+c)**3/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x)/(sqrt(a + b*x**2)*(c + d*x)**3), x)
 

Maxima [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 538 vs. \(2 (162) = 324\).

Time = 0.08 (sec) , antiderivative size = 538, normalized size of antiderivative = 3.04 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {3 \, \sqrt {b x^{2} + a} B b c^{2}}{2 \, {\left (b^{2} c^{4} d x + 2 \, a b c^{2} d^{3} x + a^{2} d^{5} x + b^{2} c^{5} + 2 \, a b c^{3} d^{2} + a^{2} c d^{4}\right )}} - \frac {3 \, \sqrt {b x^{2} + a} A b c}{2 \, {\left (b^{2} c^{4} x + 2 \, a b c^{2} d^{2} x + a^{2} d^{4} x + \frac {b^{2} c^{5}}{d} + 2 \, a b c^{3} d + a^{2} c d^{3}\right )}} + \frac {\sqrt {b x^{2} + a} B c}{2 \, {\left (b c^{2} d^{2} x^{2} + a d^{4} x^{2} + 2 \, b c^{3} d x + 2 \, a c d^{3} x + b c^{4} + a c^{2} d^{2}\right )}} - \frac {\sqrt {b x^{2} + a} A}{2 \, {\left (b c^{2} d x^{2} + a d^{3} x^{2} + 2 \, b c^{3} x + 2 \, a c d^{2} x + \frac {b c^{4}}{d} + a c^{2} d\right )}} - \frac {\sqrt {b x^{2} + a} B}{b c^{2} d x + a d^{3} x + b c^{3} + a c d^{2}} - \frac {3 \, B b^{2} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{6}} + \frac {3 \, A b^{2} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{5}} + \frac {3 \, B b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{4}} - \frac {A b \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{3}} \] Input:

integrate((B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 583 vs. \(2 (162) = 324\).

Time = 0.13 (sec) , antiderivative size = 583, normalized size of antiderivative = 3.29 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {{\left (2 \, A b^{2} c^{2} + 3 \, B a b c d - A a b d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A b^{2} c^{2} d^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} B a b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} A a b d^{4} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B b^{\frac {5}{2}} c^{4} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A b^{\frac {5}{2}} c^{3} d + 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a b^{\frac {3}{2}} c^{2} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} c d^{3} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} d^{4} + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a b^{2} c^{3} d - 10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a b^{2} c^{2} d^{2} - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c d^{3} - {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b d^{4} - B a^{2} b^{\frac {3}{2}} c^{2} d^{2} + 3 \, A a^{2} b^{\frac {3}{2}} c d^{3} + 2 \, B a^{3} \sqrt {b} d^{4}}{{\left (b^{2} c^{4} d + 2 \, a b c^{2} d^{3} + a^{2} d^{5}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} \] Input:

integrate((B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

-(2*A*b^2*c^2 + 3*B*a*b*c*d - A*a*b*d^2)*arctan(((sqrt(b)*x - sqrt(b*x^2 + 
 a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b^2*c^4 + 2*a*b*c^2*d^2 + a^2* 
d^4)*sqrt(-b*c^2 - a*d^2)) - (2*(sqrt(b)*x - sqrt(b*x^2 + a))^3*A*b^2*c^2* 
d^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*B*a*b*c*d^3 - (sqrt(b)*x - sqrt(b* 
x^2 + a))^3*A*a*b*d^4 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*b^(5/2)*c^4 + 
6*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*b^(5/2)*c^3*d + 5*(sqrt(b)*x - sqrt(b* 
x^2 + a))^2*B*a*b^(3/2)*c^2*d^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a*b^ 
(3/2)*c*d^3 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sqrt(b)*d^4 + 4*(sqr 
t(b)*x - sqrt(b*x^2 + a))*B*a*b^2*c^3*d - 10*(sqrt(b)*x - sqrt(b*x^2 + a)) 
*A*a*b^2*c^2*d^2 - 5*(sqrt(b)*x - sqrt(b*x^2 + a))*B*a^2*b*c*d^3 - (sqrt(b 
)*x - sqrt(b*x^2 + a))*A*a^2*b*d^4 - B*a^2*b^(3/2)*c^2*d^2 + 3*A*a^2*b^(3/ 
2)*c*d^3 + 2*B*a^3*sqrt(b)*d^4)/((b^2*c^4*d + 2*a*b*c^2*d^3 + a^2*d^5)*((s 
qrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))*sqrt(b)* 
c - a*d)^2)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\int \frac {A+B\,x}{\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

int((A + B*x)/((a + b*x^2)^(1/2)*(c + d*x)^3),x)
 

Output:

int((A + B*x)/((a + b*x^2)^(1/2)*(c + d*x)^3), x)
 

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 1120, normalized size of antiderivative = 6.33 \[ \int \frac {A+B x}{(c+d x)^3 \sqrt {a+b x^2}} \, dx =\text {Too large to display} \] Input:

int((B*x+A)/(d*x+c)^3/(b*x^2+a)^(1/2),x)
 

Output:

(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*c**2*d**2 + 2*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*c*d**3*x + 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* 
d**4*x**2 - 2*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**2*c**4 - 4*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**2*c**3*d*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*b**2*c**3*d - 2*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**2*c**2*d**2*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*b**2*c* 
*2*d**2*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*b**2*c*d**3*x**2 - sqrt(a*d**2 + b*c**2)*log(c + 
d*x)*a**2*b*c**2*d**2 - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c*d**3 
*x - sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*d**4*x**2 + 2*sqrt(a*d**2 + 
 b*c**2)*log(c + d*x)*a*b**2*c**4 + 4*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a 
*b**2*c**3*d*x + 3*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**3*d + 2*sq 
rt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**2*d**2*x**2 + 6*sqrt(a*d**2 + b 
*c**2)*log(c + d*x)*a*b**2*c**2*d**2*x + 3*sqrt(a*d**2 + b*c**2)*log(c + d 
*x)*a*b**2*c*d**3*x**2 - sqrt(a + b*x**2)*a**3*d**5 - 5*sqrt(a + b*x**2...