\(\int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx\) [1202]

Optimal result

Integrand size = 22, antiderivative size = 162 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {c^2 \sqrt {a+b x^2}}{2 d \left (b c^2+a d^2\right ) (c+d x)^2}+\frac {c \left (b c^2+4 a d^2\right ) \sqrt {a+b x^2}}{2 d \left (b c^2+a d^2\right )^2 (c+d x)}+\frac {a \left (b c^2-2 a d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 \left (b c^2+a d^2\right )^{5/2}} \] Output:

-1/2*c^2*(b*x^2+a)^(1/2)/d/(a*d^2+b*c^2)/(d*x+c)^2+1/2*c*(4*a*d^2+b*c^2)*( 
b*x^2+a)^(1/2)/d/(a*d^2+b*c^2)^2/(d*x+c)+1/2*a*(-2*a*d^2+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)^(5/2)
 

Mathematica [A] (verified)

Time = 1.02 (sec) , antiderivative size = 134, normalized size of antiderivative = 0.83 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {c \sqrt {a+b x^2} \left (b c^2 x+a d (3 c+4 d x)\right )}{2 \left (b c^2+a d^2\right )^2 (c+d x)^2}+\frac {a \left (b c^2-2 a d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )}{\left (-b c^2-a d^2\right )^{5/2}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.55 (sec) , antiderivative size = 176, normalized size of antiderivative = 1.09, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.182, Rules used = {603, 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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 679

\(\displaystyle -\frac {\frac {a \left (b c^2-2 a d^2\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{a d^2+b c^2}-\frac {c \sqrt {a+b x^2} \left (4 a d^2+b c^2\right )}{d (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {-\frac {a \left (b c^2-2 a d^2\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{a d^2+b c^2}-\frac {c \sqrt {a+b x^2} \left (4 a d^2+b c^2\right )}{d (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^2 \sqrt {a+b x^2}}{2 d (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

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

Input:

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

Output:

-1/2*(c^2*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)*(c + d*x)^2) - (-((c*(b*c^2 
+ 4*a*d^2)*Sqrt[a + b*x^2])/(d*(b*c^2 + a*d^2)*(c + d*x))) - (a*(b*c^2 - 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))
 

Defintions of rubi rules used

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 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

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]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(789\) vs. \(2(146)=292\).

Time = 0.39 (sec) , antiderivative size = 790, normalized size of antiderivative = 4.88

method	result	size
default	\(-\frac {\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 )}{d^{3} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {c^{2} \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^{5}}-\frac {2 c \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^{4}}\)	\(790\)

Input:

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

Output:

-1/d^3/((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))+c^2/d^5*(-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*c/d^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)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 341 vs. \(2 (147) = 294\).

Time = 0.20 (sec) , antiderivative size = 709, normalized size of antiderivative = 4.38 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\left [-\frac {{\left (a b c^{4} - 2 \, a^{2} c^{2} d^{2} + {\left (a b c^{2} d^{2} - 2 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b c^{3} d - 2 \, a^{2} c d^{3}\right )} x\right )} \sqrt {b c^{2} + a d^{2}} \log \left (\frac {2 \, a b c d x - a b c^{2} - 2 \, a^{2} d^{2} - {\left (2 \, b^{2} c^{2} + a b d^{2}\right )} x^{2} - 2 \, \sqrt {b c^{2} + a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{d^{2} x^{2} + 2 \, c d x + c^{2}}\right ) - 2 \, {\left (3 \, a b c^{4} d + 3 \, a^{2} c^{2} d^{3} + {\left (b^{2} c^{5} + 5 \, a b c^{3} d^{2} + 4 \, a^{2} c d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{4 \, {\left (b^{3} c^{8} + 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{4} d^{4} + a^{3} c^{2} d^{6} + {\left (b^{3} c^{6} d^{2} + 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{2} d^{6} + a^{3} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{7} d + 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{3} d^{5} + a^{3} c d^{7}\right )} x\right )}}, \frac {{\left (a b c^{4} - 2 \, a^{2} c^{2} d^{2} + {\left (a b c^{2} d^{2} - 2 \, a^{2} d^{4}\right )} x^{2} + 2 \, {\left (a b c^{3} d - 2 \, a^{2} c d^{3}\right )} x\right )} \sqrt {-b c^{2} - a d^{2}} \arctan \left (\frac {\sqrt {-b c^{2} - a d^{2}} {\left (b c x - a d\right )} \sqrt {b x^{2} + a}}{a b c^{2} + a^{2} d^{2} + {\left (b^{2} c^{2} + a b d^{2}\right )} x^{2}}\right ) + {\left (3 \, a b c^{4} d + 3 \, a^{2} c^{2} d^{3} + {\left (b^{2} c^{5} + 5 \, a b c^{3} d^{2} + 4 \, a^{2} c d^{4}\right )} x\right )} \sqrt {b x^{2} + a}}{2 \, {\left (b^{3} c^{8} + 3 \, a b^{2} c^{6} d^{2} + 3 \, a^{2} b c^{4} d^{4} + a^{3} c^{2} d^{6} + {\left (b^{3} c^{6} d^{2} + 3 \, a b^{2} c^{4} d^{4} + 3 \, a^{2} b c^{2} d^{6} + a^{3} d^{8}\right )} x^{2} + 2 \, {\left (b^{3} c^{7} d + 3 \, a b^{2} c^{5} d^{3} + 3 \, a^{2} b c^{3} d^{5} + a^{3} c d^{7}\right )} x\right )}}\right ] \] Input:

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

Output:

[-1/4*((a*b*c^4 - 2*a^2*c^2*d^2 + (a*b*c^2*d^2 - 2*a^2*d^4)*x^2 + 2*(a*b*c 
^3*d - 2*a^2*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*(3*a*b*c^4*d + 3*a^2*c^2*d 
^3 + (b^2*c^5 + 5*a*b*c^3*d^2 + 4*a^2*c*d^4)*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*((a*b*c^4 - 2*a^2*c^2*d^2 + (a*b*c 
^2*d^2 - 2*a^2*d^4)*x^2 + 2*(a*b*c^3*d - 2*a^2*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)) + (3*a*b*c^4*d + 3*a^2*c^2*d^3 + (b^2*c 
^5 + 5*a*b*c^3*d^2 + 4*a^2*c*d^4)*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]

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

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

Output:

Integral(x**2/(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. 353 vs. \(2 (147) = 294\).

Time = 0.08 (sec) , antiderivative size = 353, normalized size of antiderivative = 2.18 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=-\frac {3 \, \sqrt {b x^{2} + a} b c^{3}}{2 \, {\left (b^{2} c^{4} d^{2} x + 2 \, a b c^{2} d^{4} x + a^{2} d^{6} x + b^{2} c^{5} d + 2 \, a b c^{3} d^{3} + a^{2} c d^{5}\right )}} - \frac {\sqrt {b x^{2} + a} c^{2}}{2 \, {\left (b c^{2} d^{3} x^{2} + a d^{5} x^{2} + 2 \, b c^{3} d^{2} x + 2 \, a c d^{4} x + b c^{4} d + a c^{2} d^{3}\right )}} + \frac {2 \, \sqrt {b x^{2} + a} c}{b c^{2} d^{2} x + a d^{4} x + b c^{3} d + a c d^{3}} + \frac {3 \, b^{2} c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {5}{2}} d^{7}} - \frac {5 \, b c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{5}} + \frac {\operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{\sqrt {a + \frac {b c^{2}}{d^{2}}} d^{3}} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 411 vs. \(2 (147) = 294\).

Time = 0.13 (sec) , antiderivative size = 411, normalized size of antiderivative = 2.54 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {{\left (a b c^{2} - 2 \, a^{2} d^{2}\right )} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b^{2} c^{4} + 2 \, a b c^{2} d^{2} + a^{2} d^{4}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{4} d + 5 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{2} d^{3} + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{5} + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{3} d^{2} - 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c d^{4} - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{4} d - 11 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{2} d^{3} + a^{2} b^{\frac {3}{2}} c^{3} d^{2} + 4 \, a^{3} \sqrt {b} c d^{4}}{{\left (b^{2} c^{4} d^{2} + 2 \, a b c^{2} d^{4} + a^{2} d^{6}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} \] Input:

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

Output:

(a*b*c^2 - 2*a^2*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*b^2*c^4*d + 5*(sqrt(b)*x - sq 
rt(b*x^2 + a))^3*a*b*c^2*d^3 + 2*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b^(5/2)*c 
^5 + 7*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c^3*d^2 - 4*(sqrt(b)*x - 
sqrt(b*x^2 + a))^2*a^2*sqrt(b)*c*d^4 - 2*(sqrt(b)*x - sqrt(b*x^2 + a))*a*b 
^2*c^4*d - 11*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b*c^2*d^3 + a^2*b^(3/2)*c^ 
3*d^2 + 4*a^3*sqrt(b)*c*d^4)/((b^2*c^4*d^2 + 2*a*b*c^2*d^4 + a^2*d^6)*((sq 
rt(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. \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\int \frac {x^2}{\sqrt {b\,x^2+a}\,{\left (c+d\,x\right )}^3} \,d x \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.21 (sec) , antiderivative size = 723, normalized size of antiderivative = 4.46 \[ \int \frac {x^2}{(c+d x)^3 \sqrt {a+b x^2}} \, dx=\frac {2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} c^{2} d^{2}+4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} c \,d^{3} x +2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a^{2} d^{4} x^{2}-\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{4}-2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{3} d x -\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (\sqrt {b \,x^{2}+a}\, \sqrt {a \,d^{2}+b \,c^{2}}-a d +b c x \right ) a b \,c^{2} d^{2} x^{2}-2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} c^{2} d^{2}-4 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} c \,d^{3} x -2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a^{2} d^{4} x^{2}+\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{4}+2 \sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{3} d x +\sqrt {a \,d^{2}+b \,c^{2}}\, \mathrm {log}\left (d x +c \right ) a b \,c^{2} d^{2} x^{2}+3 \sqrt {b \,x^{2}+a}\, a^{2} c^{2} d^{3}+4 \sqrt {b \,x^{2}+a}\, a^{2} c \,d^{4} x +3 \sqrt {b \,x^{2}+a}\, a b \,c^{4} d +5 \sqrt {b \,x^{2}+a}\, a b \,c^{3} d^{2} x +\sqrt {b \,x^{2}+a}\, b^{2} c^{5} x}{2 a^{3} d^{8} x^{2}+6 a^{2} b \,c^{2} d^{6} x^{2}+6 a \,b^{2} c^{4} d^{4} x^{2}+2 b^{3} c^{6} d^{2} x^{2}+4 a^{3} c \,d^{7} x +12 a^{2} b \,c^{3} d^{5} x +12 a \,b^{2} c^{5} d^{3} x +4 b^{3} c^{7} d x +2 a^{3} c^{2} d^{6}+6 a^{2} b \,c^{4} d^{4}+6 a \,b^{2} c^{6} d^{2}+2 b^{3} c^{8}} \] Input:

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

Output:

(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*c**2*d**2 + 4*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqr 
t(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*c*d**3*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)*a**2*d**4*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)*a*b*c**4 - 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*c**3*d*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*b*c**2*d**2*x**2 - 2*sqrt 
(a*d**2 + b*c**2)*log(c + d*x)*a**2*c**2*d**2 - 4*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*a**2*c*d**3*x - 2*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*d**4* 
x**2 + sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**4 + 2*sqrt(a*d**2 + b*c** 
2)*log(c + d*x)*a*b*c**3*d*x + sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b*c**2 
*d**2*x**2 + 3*sqrt(a + b*x**2)*a**2*c**2*d**3 + 4*sqrt(a + b*x**2)*a**2*c 
*d**4*x + 3*sqrt(a + b*x**2)*a*b*c**4*d + 5*sqrt(a + b*x**2)*a*b*c**3*d**2 
*x + sqrt(a + b*x**2)*b**2*c**5*x)/(2*(a**3*c**2*d**6 + 2*a**3*c*d**7*x + 
a**3*d**8*x**2 + 3*a**2*b*c**4*d**4 + 6*a**2*b*c**3*d**5*x + 3*a**2*b*c**2 
*d**6*x**2 + 3*a*b**2*c**6*d**2 + 6*a*b**2*c**5*d**3*x + 3*a*b**2*c**4*d** 
4*x**2 + b**3*c**8 + 2*b**3*c**7*d*x + b**3*c**6*d**2*x**2))