\(\int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx\) [1029]

Optimal result

Integrand size = 20, antiderivative size = 180 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {c \sqrt {a+b x^2}}{2 d^2 (c+d x)^2}-\frac {\left (3 b c^2+2 a d^2\right ) \sqrt {a+b x^2}}{2 d^2 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{d^3}+\frac {b c \left (2 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^3 \left (b c^2+a d^2\right )^{3/2}} \] Output:

1/2*c*(b*x^2+a)^(1/2)/d^2/(d*x+c)^2-1/2*(2*a*d^2+3*b*c^2)*(b*x^2+a)^(1/2)/ 
d^2/(a*d^2+b*c^2)/(d*x+c)+b^(1/2)*arctanh(b^(1/2)*x/(b*x^2+a)^(1/2))/d^3+1 
/2*b*c*(3*a*d^2+2*b*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a 
)^(1/2))/d^3/(a*d^2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 0.23 (sec) , antiderivative size = 173, normalized size of antiderivative = 0.96 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=-\frac {\frac {d \sqrt {a+b x^2} \left (a d^2 (c+2 d x)+b c^2 (2 c+3 d x)\right )}{\left (b c^2+a d^2\right ) (c+d x)^2}+\frac {2 b c \left (2 b c^2+3 a d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\left (-b c^2-a d^2\right )^{3/2}}+2 \sqrt {b} \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^3} \] Input:

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

Output:

-1/2*((d*Sqrt[a + b*x^2]*(a*d^2*(c + 2*d*x) + b*c^2*(2*c + 3*d*x)))/((b*c^ 
2 + a*d^2)*(c + d*x)^2) + (2*b*c*(2*b*c^2 + 3*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)^(3/2 
) + 2*Sqrt[b]*Log[-(Sqrt[b]*x) + Sqrt[a + b*x^2]])/d^3
 

Rubi [A] (verified)

Time = 0.57 (sec) , antiderivative size = 203, normalized size of antiderivative = 1.13, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.350, Rules used = {589, 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.

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

\(\Big \downarrow \) 589

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

\(\Big \downarrow \) 27

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

\(\Big \downarrow \) 719

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 488

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

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*((c*(2*b*c^2 + a*d^2) + d*(3*b*c^2 + 2*a*d^2)*x)*Sqrt[a + b*x^2])/(d^ 
2*(b*c^2 + a*d^2)*(c + d*x)^2) - (b*((-2*(b*c^2 + a*d^2)*ArcTanh[(Sqrt[b]* 
x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d) - (c*(2*b*c^2 + 3*a*d^2)*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])))/(2 
*d^2*(b*c^2 + a*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 589

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*(a*d^2 + b*c^2*(2*p + 1)) - d*( 
a*d^2*(n + 1) + b*c^2*(n - 2*p + 1))*x)/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2 
))), x] + Simp[b*(p/(d^2*(n + 1)*(n + 2)*(b*c^2 + a*d^2)))   Int[(c + d*x)^ 
(n + 2)*(a + b*x^2)^(p - 1)*Simp[2*a*c*d*(n + 2) - (2*a*d^2*(n + 1) - 2*b*c 
^2*(2*p + 1))*x, x], x], x] /; FreeQ[{a, b, c, d}, x] && GtQ[p, 0] && LtQ[n 
, -2] && LtQ[n + 2*p, 0] &&  !ILtQ[n + 2*p + 3, 0]
 

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. \(1437\) vs. \(2(158)=316\).

Time = 0.41 (sec) , antiderivative size = 1438, normalized size of antiderivative = 7.99

method	result	size
default	\(\text {Expression too large to display}\)	\(1438\)

Input:

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

Output:

1/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)-b*c*d/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b 
*c^2)/d^2)^(1/2)-b^(1/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2* 
b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))-(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*b/( 
a*d^2+b*c^2)*d^2*(1/4*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d) 
+(a*d^2+b*c^2)/d^2)^(1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2 
)*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2) 
/d^2)^(1/2))))-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)^(3/2)+1/2*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)^(3/2)-b*c*d 
/(a*d^2+b*c^2)*((b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b^(1 
/2)*c/d*ln((-b*c/d+b*(x+c/d))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+ 
b*c^2)/d^2)^(1/2))-(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*b/(a*d^2+b*c^2)*d^2*(1/4 
*(2*b*(x+c/d)-2*b*c/d)/b*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^( 
1/2)+1/8*(4*b*(a*d^2+b*c^2)/d^2-4*b^2*c^2/d^2)/b^(3/2)*ln((-b*c/d+b*(x+c/d 
))/b^(1/2)+(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2))))+1/2...
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 436 vs. \(2 (159) = 318\).

Time = 3.66 (sec) , antiderivative size = 1809, normalized size of antiderivative = 10.05 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

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

Output:

[1/4*(2*(b^2*c^6 + 2*a*b*c^4*d^2 + a^2*c^2*d^4 + (b^2*c^4*d^2 + 2*a*b*c^2* 
d^4 + a^2*d^6)*x^2 + 2*(b^2*c^5*d + 2*a*b*c^3*d^3 + a^2*c*d^5)*x)*sqrt(b)* 
log(-2*b*x^2 - 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + (2*b^2*c^5 + 3*a*b*c^3*d 
^2 + (2*b^2*c^3*d^2 + 3*a*b*c*d^4)*x^2 + 2*(2*b^2*c^4*d + 3*a*b*c^2*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^2*c^5*d + 3*a*b*c^3*d^3 + a^2*c*d^5 + (3*b 
^2*c^4*d^2 + 5*a*b*c^2*d^4 + 2*a^2*d^6)*x)*sqrt(b*x^2 + a))/(b^2*c^6*d^3 + 
 2*a*b*c^4*d^5 + a^2*c^2*d^7 + (b^2*c^4*d^5 + 2*a*b*c^2*d^7 + a^2*d^9)*x^2 
 + 2*(b^2*c^5*d^4 + 2*a*b*c^3*d^6 + a^2*c*d^8)*x), 1/2*((2*b^2*c^5 + 3*a*b 
*c^3*d^2 + (2*b^2*c^3*d^2 + 3*a*b*c*d^4)*x^2 + 2*(2*b^2*c^4*d + 3*a*b*c^2* 
d^3)*x)*sqrt(-b*c^2 - a*d^2)*arctan(sqrt(-b*c^2 - a*d^2)*(b*c*x - a*d)*sqr 
t(b*x^2 + a)/(a*b*c^2 + a^2*d^2 + (b^2*c^2 + a*b*d^2)*x^2)) + (b^2*c^6 + 2 
*a*b*c^4*d^2 + a^2*c^2*d^4 + (b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*x^2 + 
 2*(b^2*c^5*d + 2*a*b*c^3*d^3 + a^2*c*d^5)*x)*sqrt(b)*log(-2*b*x^2 - 2*sqr 
t(b*x^2 + a)*sqrt(b)*x - a) - (2*b^2*c^5*d + 3*a*b*c^3*d^3 + a^2*c*d^5 + ( 
3*b^2*c^4*d^2 + 5*a*b*c^2*d^4 + 2*a^2*d^6)*x)*sqrt(b*x^2 + a))/(b^2*c^6*d^ 
3 + 2*a*b*c^4*d^5 + a^2*c^2*d^7 + (b^2*c^4*d^5 + 2*a*b*c^2*d^7 + a^2*d^9)* 
x^2 + 2*(b^2*c^5*d^4 + 2*a*b*c^3*d^6 + a^2*c*d^8)*x), -1/4*(4*(b^2*c^6 + 2 
*a*b*c^4*d^2 + a^2*c^2*d^4 + (b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*x^...
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.06 (sec) , antiderivative size = 294, normalized size of antiderivative = 1.63 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {\sqrt {b x^{2} + a} b c^{2}}{2 \, {\left (b c^{2} d^{3} x + a d^{5} x + b c^{3} d^{2} + a c d^{4}\right )}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c}{2 \, {\left (b c^{2} d^{2} x^{2} + a d^{4} x^{2} + 2 \, b c^{3} d x + 2 \, a c d^{3} x + b c^{4} + a c^{2} d^{2}\right )}} - \frac {\sqrt {b x^{2} + a} b c}{2 \, {\left (b c^{2} d^{2} + a d^{4}\right )}} - \frac {\sqrt {b x^{2} + a}}{d^{3} x + c d^{2}} + \frac {\sqrt {b} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{3}} + \frac {b^{2} c^{3} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{6}} - \frac {3 \, b c \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{4}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (159) = 318\).

Time = 0.15 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.28 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=-\frac {{\left (2 \, b^{2} c^{3} + 3 \, a b c d^{2}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} d^{3} + a d^{5}\right )} \sqrt {-b c^{2} - a d^{2}}} - \frac {\sqrt {b} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{d^{3}} - \frac {4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{3} d + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c d^{3} + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{4} + {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{2} d^{2} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} d^{4} - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{3} d - 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c d^{3} + 3 \, a^{2} b^{\frac {3}{2}} c^{2} d^{2} + 2 \, a^{3} \sqrt {b} d^{4}}{{\left (b c^{2} d^{3} + a d^{5}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} \] Input:

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

Output:

-(2*b^2*c^3 + 3*a*b*c*d^2)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + sqrt 
(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2*d^3 + a*d^5)*sqrt(-b*c^2 - a*d^2)) - 
sqrt(b)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/d^3 - (4*(sqrt(b)*x - sqrt( 
b*x^2 + a))^3*b^2*c^3*d + 3*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c*d^3 + 6* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c^4 + (sqrt(b)*x - sqrt(b*x^2 + a) 
)^2*a*b^(3/2)*c^2*d^2 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*d^4 
- 8*(sqrt(b)*x - sqrt(b*x^2 + a))*a*b^2*c^3*d - 5*(sqrt(b)*x - sqrt(b*x^2 
+ a))*a^2*b*c*d^3 + 3*a^2*b^(3/2)*c^2*d^2 + 2*a^3*sqrt(b)*d^4)/((b*c^2*d^3 
 + a*d^5)*((sqrt(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. \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\int \frac {x\,\sqrt {b\,x^2+a}}{{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 1200, normalized size of antiderivative = 6.67 \[ \int \frac {x \sqrt {a+b x^2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

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

Output:

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