\(\int \frac {x^3 (c+d x^2)^{3/2}}{a+b x^2} \, dx\) [930]

Optimal result

Integrand size = 24, antiderivative size = 115 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=-\frac {a (b c-a d) \sqrt {c+d x^2}}{b^3}-\frac {a \left (c+d x^2\right )^{3/2}}{3 b^2}+\frac {\left (c+d x^2\right )^{5/2}}{5 b d}+\frac {a (b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{7/2}} \] Output:

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

Mathematica [A] (verified)

Time = 0.20 (sec) , antiderivative size = 109, normalized size of antiderivative = 0.95 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {\sqrt {c+d x^2} \left (15 a^2 d^2+3 b^2 \left (c+d x^2\right )^2-5 a b d \left (4 c+d x^2\right )\right )}{15 b^3 d}-\frac {a (-b c+a d)^{3/2} \arctan \left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {-b c+a d}}\right )}{b^{7/2}} \] Input:

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

Output:

(Sqrt[c + d*x^2]*(15*a^2*d^2 + 3*b^2*(c + d*x^2)^2 - 5*a*b*d*(4*c + d*x^2) 
))/(15*b^3*d) - (a*(-(b*c) + a*d)^(3/2)*ArcTan[(Sqrt[b]*Sqrt[c + d*x^2])/S 
qrt[-(b*c) + a*d]])/b^(7/2)
 

Rubi [A] (verified)

Time = 0.24 (sec) , antiderivative size = 129, normalized size of antiderivative = 1.12, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.250, Rules used = {354, 90, 60, 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.

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

\(\Big \downarrow \) 354

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

\(\Big \downarrow \) 90

\(\displaystyle \frac {1}{2} \left (\frac {2 \left (c+d x^2\right )^{5/2}}{5 b d}-\frac {a \int \frac {\left (d x^2+c\right )^{3/2}}{b x^2+a}dx^2}{b}\right )\)

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 60

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

\(\Big \downarrow \) 73

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

\(\Big \downarrow \) 221

\(\displaystyle \frac {1}{2} \left (\frac {2 \left (c+d x^2\right )^{5/2}}{5 b d}-\frac {a \left (\frac {(b c-a d) \left (\frac {2 \sqrt {c+d x^2}}{b}-\frac {2 \sqrt {b c-a d} \text {arctanh}\left (\frac {\sqrt {b} \sqrt {c+d x^2}}{\sqrt {b c-a d}}\right )}{b^{3/2}}\right )}{b}+\frac {2 \left (c+d x^2\right )^{3/2}}{3 b}\right )}{b}\right )\)

Input:

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

Output:

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

Defintions of rubi rules used

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 90

Int[((a_.) + (b_.)*(x_))*((c_.) + (d_.)*(x_))^(n_.)*((e_.) + (f_.)*(x_))^(p 
_.), x_] :> Simp[b*(c + d*x)^(n + 1)*((e + f*x)^(p + 1)/(d*f*(n + p + 2))), 
 x] + Simp[(a*d*f*(n + p + 2) - b*(d*e*(n + 1) + c*f*(p + 1)))/(d*f*(n + p 
+ 2))   Int[(c + d*x)^n*(e + f*x)^p, x], x] /; FreeQ[{a, b, c, d, e, f, n, 
p}, x] && NeQ[n + p + 2, 0]
 

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 354

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^2)^(p_.)*((c_) + (d_.)*(x_)^2)^(q_.), x_S 
ymbol] :> Simp[1/2   Subst[Int[x^((m - 1)/2)*(a + b*x)^p*(c + d*x)^q, x], x 
, x^2], x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IntegerQ 
[(m - 1)/2]
 

Maple [A] (verified)

Time = 0.75 (sec) , antiderivative size = 116, normalized size of antiderivative = 1.01

method	result	size
pseudoelliptic	\(\frac {\left (\frac {b^{2} \left (x^{2} d +c \right )^{2}}{5}-\frac {4 a \left (\frac {x^{2} d}{4}+c \right ) d b}{3}+a^{2} d^{2}\right ) \sqrt {\left (a d -b c \right ) b}\, \sqrt {x^{2} d +c}-a d \left (a d -b c \right )^{2} \arctan \left (\frac {\sqrt {x^{2} d +c}\, b}{\sqrt {\left (a d -b c \right ) b}}\right )}{\sqrt {\left (a d -b c \right ) b}\, d \,b^{3}}\)	\(116\)
risch	\(\frac {\left (3 b^{2} d^{2} x^{4}-5 x^{2} a b \,d^{2}+6 x^{2} b^{2} c d +15 a^{2} d^{2}-20 a b c d +3 b^{2} c^{2}\right ) \sqrt {x^{2} d +c}}{15 d \,b^{3}}-\frac {a \left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \left (-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}+\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x -\frac {\sqrt {-a b}}{b}\right )^{2} d +\frac {2 d \sqrt {-a b}\, \left (x -\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x -\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}-\frac {\ln \left (\frac {-\frac {2 \left (a d -b c \right )}{b}-\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}+2 \sqrt {-\frac {a d -b c}{b}}\, \sqrt {\left (x +\frac {\sqrt {-a b}}{b}\right )^{2} d -\frac {2 d \sqrt {-a b}\, \left (x +\frac {\sqrt {-a b}}{b}\right )}{b}-\frac {a d -b c}{b}}}{x +\frac {\sqrt {-a b}}{b}}\right )}{2 b \sqrt {-\frac {a d -b c}{b}}}\right )}{b^{3}}\)	\(399\)
default	\(\text {Expression too large to display}\)	\(1256\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.12 (sec) , antiderivative size = 397, normalized size of antiderivative = 3.45 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\left [-\frac {15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {\frac {b c - a d}{b}} \log \left (\frac {b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \, {\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} - 4 \, {\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt {d x^{2} + c} \sqrt {\frac {b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) - 4 \, {\left (3 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{60 \, b^{3} d}, \frac {15 \, {\left (a b c d - a^{2} d^{2}\right )} \sqrt {-\frac {b c - a d}{b}} \arctan \left (-\frac {{\left (b d x^{2} + 2 \, b c - a d\right )} \sqrt {d x^{2} + c} \sqrt {-\frac {b c - a d}{b}}}{2 \, {\left (b c^{2} - a c d + {\left (b c d - a d^{2}\right )} x^{2}\right )}}\right ) + 2 \, {\left (3 \, b^{2} d^{2} x^{4} + 3 \, b^{2} c^{2} - 20 \, a b c d + 15 \, a^{2} d^{2} + {\left (6 \, b^{2} c d - 5 \, a b d^{2}\right )} x^{2}\right )} \sqrt {d x^{2} + c}}{30 \, b^{3} d}\right ] \] Input:

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

Output:

[-1/60*(15*(a*b*c*d - a^2*d^2)*sqrt((b*c - a*d)/b)*log((b^2*d^2*x^4 + 8*b^ 
2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 - 4*(b^2*d*x^2 
 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a*b* 
x^2 + a^2)) - 4*(3*b^2*d^2*x^4 + 3*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 + (6* 
b^2*c*d - 5*a*b*d^2)*x^2)*sqrt(d*x^2 + c))/(b^3*d), 1/30*(15*(a*b*c*d - a^ 
2*d^2)*sqrt(-(b*c - a*d)/b)*arctan(-1/2*(b*d*x^2 + 2*b*c - a*d)*sqrt(d*x^2 
 + c)*sqrt(-(b*c - a*d)/b)/(b*c^2 - a*c*d + (b*c*d - a*d^2)*x^2)) + 2*(3*b 
^2*d^2*x^4 + 3*b^2*c^2 - 20*a*b*c*d + 15*a^2*d^2 + (6*b^2*c*d - 5*a*b*d^2) 
*x^2)*sqrt(d*x^2 + c))/(b^3*d)]
                                                                                    
                                                                                    
 

Sympy [A] (verification not implemented)

Time = 8.14 (sec) , antiderivative size = 143, normalized size of antiderivative = 1.24 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\begin {cases} \frac {2 \left (- \frac {a d \left (c + d x^{2}\right )^{\frac {3}{2}}}{6 b^{2}} - \frac {a d \left (a d - b c\right )^{2} \operatorname {atan}{\left (\frac {\sqrt {c + d x^{2}}}{\sqrt {\frac {a d - b c}{b}}} \right )}}{2 b^{4} \sqrt {\frac {a d - b c}{b}}} + \frac {\left (c + d x^{2}\right )^{\frac {5}{2}}}{10 b} + \frac {\sqrt {c + d x^{2}} \left (a^{2} d^{2} - a b c d\right )}{2 b^{3}}\right )}{d} & \text {for}\: d \neq 0 \\c^{\frac {3}{2}} \left (- \frac {a \left (\begin {cases} \frac {x^{2}}{a} & \text {for}\: b = 0 \\\frac {\log {\left (a + b x^{2} \right )}}{b} & \text {otherwise} \end {cases}\right )}{2 b} + \frac {x^{2}}{2 b}\right ) & \text {otherwise} \end {cases} \] Input:

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

Output:

Piecewise((2*(-a*d*(c + d*x**2)**(3/2)/(6*b**2) - a*d*(a*d - b*c)**2*atan( 
sqrt(c + d*x**2)/sqrt((a*d - b*c)/b))/(2*b**4*sqrt((a*d - b*c)/b)) + (c + 
d*x**2)**(5/2)/(10*b) + sqrt(c + d*x**2)*(a**2*d**2 - a*b*c*d)/(2*b**3))/d 
, Ne(d, 0)), (c**(3/2)*(-a*Piecewise((x**2/a, Eq(b, 0)), (log(a + b*x**2)/ 
b, True))/(2*b) + x**2/(2*b)), True))
 

Maxima [F(-2)]

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

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

Output:

Exception raised: ValueError >> Computation failed since Maxima requested 
additional constraints; using the 'assume' command before evaluation *may* 
 help (example of legal syntax is 'assume(a*d-b*c>0)', see `assume?` for m 
ore detail
 

Giac [A] (verification not implemented)

Time = 0.13 (sec) , antiderivative size = 151, normalized size of antiderivative = 1.31 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=-\frac {{\left (a b^{2} c^{2} - 2 \, a^{2} b c d + a^{3} d^{2}\right )} \arctan \left (\frac {\sqrt {d x^{2} + c} b}{\sqrt {-b^{2} c + a b d}}\right )}{\sqrt {-b^{2} c + a b d} b^{3}} + \frac {3 \, {\left (d x^{2} + c\right )}^{\frac {5}{2}} b^{4} d^{4} - 5 \, {\left (d x^{2} + c\right )}^{\frac {3}{2}} a b^{3} d^{5} - 15 \, \sqrt {d x^{2} + c} a b^{3} c d^{5} + 15 \, \sqrt {d x^{2} + c} a^{2} b^{2} d^{6}}{15 \, b^{5} d^{5}} \] Input:

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

Output:

-(a*b^2*c^2 - 2*a^2*b*c*d + a^3*d^2)*arctan(sqrt(d*x^2 + c)*b/sqrt(-b^2*c 
+ a*b*d))/(sqrt(-b^2*c + a*b*d)*b^3) + 1/15*(3*(d*x^2 + c)^(5/2)*b^4*d^4 - 
 5*(d*x^2 + c)^(3/2)*a*b^3*d^5 - 15*sqrt(d*x^2 + c)*a*b^3*c*d^5 + 15*sqrt( 
d*x^2 + c)*a^2*b^2*d^6)/(b^5*d^5)
 

Mupad [B] (verification not implemented)

Time = 1.01 (sec) , antiderivative size = 179, normalized size of antiderivative = 1.56 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {{\left (d\,x^2+c\right )}^{5/2}}{5\,b\,d}-{\left (d\,x^2+c\right )}^{3/2}\,\left (\frac {c}{3\,b\,d}+\frac {a\,d^2-b\,c\,d}{3\,b^2\,d^2}\right )-\frac {a\,\mathrm {atan}\left (\frac {a\,\sqrt {b}\,\sqrt {d\,x^2+c}\,{\left (a\,d-b\,c\right )}^{3/2}}{a^3\,d^2-2\,a^2\,b\,c\,d+a\,b^2\,c^2}\right )\,{\left (a\,d-b\,c\right )}^{3/2}}{b^{7/2}}+\frac {\sqrt {d\,x^2+c}\,\left (a\,d^2-b\,c\,d\right )\,\left (\frac {c}{b\,d}+\frac {a\,d^2-b\,c\,d}{b^2\,d^2}\right )}{b\,d} \] Input:

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.19 (sec) , antiderivative size = 277, normalized size of antiderivative = 2.41 \[ \int \frac {x^3 \left (c+d x^2\right )^{3/2}}{a+b x^2} \, dx=\frac {-15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a^{2} d^{2}+15 \sqrt {b}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {d}\, \sqrt {d \,x^{2}+c}\, b x +b c +b d \,x^{2}}{\sqrt {b}\, \sqrt {d \,x^{2}+c}\, \sqrt {a d -b c}+\sqrt {d}\, \sqrt {b}\, \sqrt {a d -b c}\, x}\right ) a b c d +15 \sqrt {d \,x^{2}+c}\, a^{2} b \,d^{2}-20 \sqrt {d \,x^{2}+c}\, a \,b^{2} c d -5 \sqrt {d \,x^{2}+c}\, a \,b^{2} d^{2} x^{2}+3 \sqrt {d \,x^{2}+c}\, b^{3} c^{2}+6 \sqrt {d \,x^{2}+c}\, b^{3} c d \,x^{2}+3 \sqrt {d \,x^{2}+c}\, b^{3} d^{2} x^{4}}{15 b^{4} d} \] Input:

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

Output:

( - 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt(c + d*x**2)*b*x + b*c + 
b*d*x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c) + sqrt(d)*sqrt(b)*sqrt 
(a*d - b*c)*x))*a**2*d**2 + 15*sqrt(b)*sqrt(a*d - b*c)*atan((sqrt(d)*sqrt( 
c + d*x**2)*b*x + b*c + b*d*x**2)/(sqrt(b)*sqrt(c + d*x**2)*sqrt(a*d - b*c 
) + sqrt(d)*sqrt(b)*sqrt(a*d - b*c)*x))*a*b*c*d + 15*sqrt(c + d*x**2)*a**2 
*b*d**2 - 20*sqrt(c + d*x**2)*a*b**2*c*d - 5*sqrt(c + d*x**2)*a*b**2*d**2* 
x**2 + 3*sqrt(c + d*x**2)*b**3*c**2 + 6*sqrt(c + d*x**2)*b**3*c*d*x**2 + 3 
*sqrt(c + d*x**2)*b**3*d**2*x**4)/(15*b**4*d)