\(\int x (a+\frac {b}{c+d x^2})^{3/2} \, dx\) [178]

Optimal result

Integrand size = 19, antiderivative size = 93 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {b \sqrt {a+\frac {b}{c+d x^2}}}{d}+\frac {a \left (c+d x^2\right ) \sqrt {a+\frac {b}{c+d x^2}}}{2 d}+\frac {3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {a+\frac {b}{c+d x^2}}}{\sqrt {a}}\right )}{2 d} \] Output:

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

Mathematica [A] (verified)

Time = 0.15 (sec) , antiderivative size = 87, normalized size of antiderivative = 0.94 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}} \left (-2 b+a \left (c+d x^2\right )\right )+3 \sqrt {a} b \text {arctanh}\left (\frac {\sqrt {\frac {b+a c+a d x^2}{c+d x^2}}}{\sqrt {a}}\right )}{2 d} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.36 (sec) , antiderivative size = 90, normalized size of antiderivative = 0.97, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.316, Rules used = {2024, 773, 51, 60, 73, 221}

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.

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

\(\Big \downarrow \) 2024

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

\(\Big \downarrow \) 773

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

\(\Big \downarrow \) 51

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

rule 51

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + 1))), x] - Simp[d*(n/(b*(m + 1))) 
Int[(a + b*x)^(m + 1)*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, c, d, n}, x 
] && ILtQ[m, -1] && FractionQ[n] && GtQ[n, 0]
 

rule 60

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[ 
(a + b*x)^(m + 1)*((c + d*x)^n/(b*(m + n + 1))), x] + Simp[n*((b*c - a*d)/( 
b*(m + n + 1)))   Int[(a + b*x)^m*(c + d*x)^(n - 1), x], x] /; FreeQ[{a, b, 
 c, d}, x] && GtQ[n, 0] && NeQ[m + n + 1, 0] &&  !(IGtQ[m, 0] && ( !Integer 
Q[n] || (GtQ[m, 0] && LtQ[m - n, 0]))) &&  !ILtQ[m + n + 2, 0] && IntLinear 
Q[a, b, c, d, m, n, x]
 

rule 73

Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_), x_Symbol] :> With[ 
{p = Denominator[m]}, Simp[p/b   Subst[Int[x^(p*(m + 1) - 1)*(c - a*(d/b) + 
 d*(x^p/b))^n, x], x, (a + b*x)^(1/p)], x]] /; FreeQ[{a, b, c, d}, x] && Lt 
Q[-1, m, 0] && LeQ[-1, n, 0] && LeQ[Denominator[n], Denominator[m]] && IntL 
inearQ[a, b, c, d, m, n, x]
 

rule 221

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-a/b, 2]/a)*ArcTanh[x 
/Rt[-a/b, 2]], x] /; FreeQ[{a, b}, x] && NegQ[a/b]
 

rule 773

Int[((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> -Subst[Int[(a + b/x^n)^p/x^ 
2, x], x, 1/x] /; FreeQ[{a, b, p}, x] && ILtQ[n, 0] &&  !IntegerQ[p]
 

rule 2024

Int[((a_.) + (b_.)*(Pq_)^(n_.))^(p_.)*(Qr_), x_Symbol] :> With[{q = Expon[P 
q, x], r = Expon[Qr, x]}, Simp[Coeff[Qr, x, r]/(q*Coeff[Pq, x, q])   Subst[ 
Int[(a + b*x^n)^p, x], x, Pq], x] /; EqQ[r, q - 1] && EqQ[Coeff[Qr, x, r]*D 
[Pq, x], q*Coeff[Pq, x, q]*Qr]] /; FreeQ[{a, b, n, p}, x] && PolyQ[Pq, x] & 
& PolyQ[Qr, x]
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(186\) vs. \(2(79)=158\).

Time = 0.24 (sec) , antiderivative size = 187, normalized size of antiderivative = 2.01

method	result	size
derivativedivides	\(\frac {\sqrt {\frac {\left (d \,x^{2}+c \right ) a +b}{d \,x^{2}+c}}\, \left (6 a^{\frac {3}{2}} \sqrt {a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )}\, \left (d \,x^{2}+c \right )^{2}+3 \ln \left (\frac {2 \sqrt {a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )}\, \sqrt {a}+2 \left (d \,x^{2}+c \right ) a +b}{2 \sqrt {a}}\right ) a b \left (d \,x^{2}+c \right )^{2}-4 {\left (a \left (d \,x^{2}+c \right )^{2}+b \left (d \,x^{2}+c \right )\right )}^{\frac {3}{2}} \sqrt {a}\right )}{4 d \left (d \,x^{2}+c \right ) \sqrt {\left (d \,x^{2}+c \right ) \left (\left (d \,x^{2}+c \right ) a +b \right )}\, \sqrt {a}}\)	\(187\)
risch	\(\frac {\left (d \,x^{2}+c \right ) a \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{2 d}+\frac {b \left (\frac {3 a \ln \left (\frac {a c d +\frac {1}{2} b d +a \,d^{2} x^{2}}{\sqrt {a \,d^{2}}}+\sqrt {a \,c^{2}+b c +\left (2 a c d +b d \right ) x^{2}+a \,d^{2} x^{4}}\right )}{2 \sqrt {a \,d^{2}}}-\frac {2 \left (a d \,x^{2}+a c +b \right )}{d \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}}\right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}}{2 a d \,x^{2}+2 a c +2 b}\)	\(217\)
default	\(\frac {\left (3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b \,d^{2} x^{2}+2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a d \,x^{2}+3 \ln \left (\frac {2 a \,d^{2} x^{2}+2 a c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}+b d}{2 \sqrt {a \,d^{2}}}\right ) a b c d +2 \sqrt {a \,d^{2} x^{4}+2 a d \,x^{2} c +b d \,x^{2}+a \,c^{2}+b c}\, \sqrt {a \,d^{2}}\, a c -4 \sqrt {a \,d^{2}}\, \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, b \right ) \sqrt {\frac {a d \,x^{2}+a c +b}{d \,x^{2}+c}}}{4 d \sqrt {\left (d \,x^{2}+c \right ) \left (a d \,x^{2}+a c +b \right )}\, \sqrt {a \,d^{2}}}\)	\(336\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 269, normalized size of antiderivative = 2.89 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\left [\frac {3 \, \sqrt {a} b \log \left (8 \, a^{2} d^{2} x^{4} + 8 \, a^{2} c^{2} + 8 \, {\left (2 \, a^{2} c + a b\right )} d x^{2} + 8 \, a b c + b^{2} + 4 \, {\left (2 \, a d^{2} x^{4} + {\left (4 \, a c + b\right )} d x^{2} + 2 \, a c^{2} + b c\right )} \sqrt {a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}\right ) + 4 \, {\left (a d x^{2} + a c - 2 \, b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{8 \, d}, -\frac {3 \, \sqrt {-a} b \arctan \left (\frac {{\left (2 \, a d x^{2} + 2 \, a c + b\right )} \sqrt {-a} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a^{2} d x^{2} + a^{2} c + a b\right )}}\right ) - 2 \, {\left (a d x^{2} + a c - 2 \, b\right )} \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{4 \, d}\right ] \] Input:

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

Output:

[1/8*(3*sqrt(a)*b*log(8*a^2*d^2*x^4 + 8*a^2*c^2 + 8*(2*a^2*c + a*b)*d*x^2 
+ 8*a*b*c + b^2 + 4*(2*a*d^2*x^4 + (4*a*c + b)*d*x^2 + 2*a*c^2 + b*c)*sqrt 
(a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))) + 4*(a*d*x^2 + a*c - 2*b)*sqrt( 
(a*d*x^2 + a*c + b)/(d*x^2 + c)))/d, -1/4*(3*sqrt(-a)*b*arctan(1/2*(2*a*d* 
x^2 + 2*a*c + b)*sqrt(-a)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 + c))/(a^2*d*x^2 
 + a^2*c + a*b)) - 2*(a*d*x^2 + a*c - 2*b)*sqrt((a*d*x^2 + a*c + b)/(d*x^2 
 + c)))/d]
 

Sympy [F]

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

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

Output:

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

Maxima [A] (verification not implemented)

Time = 0.14 (sec) , antiderivative size = 156, normalized size of antiderivative = 1.68 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {a b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{2 \, {\left (a d - \frac {{\left (a d x^{2} + a c + b\right )} d}{d x^{2} + c}\right )}} - \frac {3 \, \sqrt {a} b \log \left (-\frac {\sqrt {a} - \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{\sqrt {a} + \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}\right )}{4 \, d} - \frac {b \sqrt {\frac {a d x^{2} + a c + b}{d x^{2} + c}}}{d} \] Input:

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 381 vs. \(2 (79) = 158\).

Time = 0.32 (sec) , antiderivative size = 381, normalized size of antiderivative = 4.10 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {\sqrt {a} b {\left | d \right |} \log \left (24\right ) \mathrm {sgn}\left (d x^{2} + c\right )}{2 \, d^{2}} - \frac {\sqrt {a} b \log \left ({\left | 2 \, a^{2} c^{3} d + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} a^{\frac {3}{2}} c^{2} {\left | d \right |} + 6 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} a c d + a b c^{2} d + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{3} \sqrt {a} {\left | d \right |} + 2 \, {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )} \sqrt {a} b c {\left | d \right |} + {\left (\sqrt {a d^{2}} x^{2} - \sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c}\right )}^{2} b d \right |}\right ) \mathrm {sgn}\left (d x^{2} + c\right )}{4 \, {\left | d \right |}} + \frac {\sqrt {a d^{2} x^{4} + 2 \, a c d x^{2} + b d x^{2} + a c^{2} + b c} a \mathrm {sgn}\left (d x^{2} + c\right )}{2 \, d} \] Input:

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

Output:

-1/2*sqrt(a)*b*abs(d)*log(24)*sgn(d*x^2 + c)/d^2 - 1/4*sqrt(a)*b*log(abs(2 
*a^2*c^3*d + 6*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + 
 a*c^2 + b*c))*a^(3/2)*c^2*abs(d) + 6*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 
2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^2*a*c*d + a*b*c^2*d + 2*(sqrt(a*d^2) 
*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c))^3*sqrt(a)*ab 
s(d) + 2*(sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 
 + b*c))*sqrt(a)*b*c*abs(d) + (sqrt(a*d^2)*x^2 - sqrt(a*d^2*x^4 + 2*a*c*d* 
x^2 + b*d*x^2 + a*c^2 + b*c))^2*b*d))*sgn(d*x^2 + c)/abs(d) + 1/2*sqrt(a*d 
^2*x^4 + 2*a*c*d*x^2 + b*d*x^2 + a*c^2 + b*c)*a*sgn(d*x^2 + c)/d
 

Mupad [B] (verification not implemented)

Time = 10.42 (sec) , antiderivative size = 61, normalized size of antiderivative = 0.66 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=-\frac {{\left (a+\frac {b}{d\,x^2+c}\right )}^{3/2}\,\left (d\,x^2+c\right )\,{{}}_2{\mathrm {F}}_1\left (-\frac {3}{2},-\frac {1}{2};\ \frac {1}{2};\ -\frac {a\,\left (d\,x^2+c\right )}{b}\right )}{d\,{\left (\frac {a\,\left (d\,x^2+c\right )}{b}+1\right )}^{3/2}} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.27 (sec) , antiderivative size = 160, normalized size of antiderivative = 1.72 \[ \int x \left (a+\frac {b}{c+d x^2}\right )^{3/2} \, dx=\frac {\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a c +\sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, a d \,x^{2}-2 \sqrt {d \,x^{2}+c}\, \sqrt {a d \,x^{2}+a c +b}\, b +3 \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}-\sqrt {d \,x^{2}+c}\, a \right ) b c +3 \sqrt {a}\, \mathrm {log}\left (-\sqrt {a}\, \sqrt {a d \,x^{2}+a c +b}-\sqrt {d \,x^{2}+c}\, a \right ) b d \,x^{2}}{2 d \left (d \,x^{2}+c \right )} \] Input:

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

Output:

(sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*a*c + sqrt(c + d*x**2)*sqrt(a*c 
 + a*d*x**2 + b)*a*d*x**2 - 2*sqrt(c + d*x**2)*sqrt(a*c + a*d*x**2 + b)*b 
+ 3*sqrt(a)*log( - sqrt(a)*sqrt(a*c + a*d*x**2 + b) - sqrt(c + d*x**2)*a)* 
b*c + 3*sqrt(a)*log( - sqrt(a)*sqrt(a*c + a*d*x**2 + b) - sqrt(c + d*x**2) 
*a)*b*d*x**2)/(2*d*(c + d*x**2))