\(\int \frac {(a+b x^2)^{3/2}}{c+d x^2} \, dx\) [71]

Optimal result

Integrand size = 21, antiderivative size = 113 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{c+d x^2} \, dx=\frac {b x \sqrt {a+b x^2}}{2 d}-\frac {\sqrt {b} (2 b c-3 a d) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 d^2}+\frac {(b c-a d)^{3/2} \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{\sqrt {c} d^2} \] Output:

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

Mathematica [A] (verified)

Time = 0.26 (sec) , antiderivative size = 126, normalized size of antiderivative = 1.12 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{c+d x^2} \, dx=\frac {b d x \sqrt {a+b x^2}-\frac {2 (-b c+a d)^{3/2} \arctan \left (\frac {-d x \sqrt {a+b x^2}+\sqrt {b} \left (c+d x^2\right )}{\sqrt {c} \sqrt {-b c+a d}}\right )}{\sqrt {c}}+\sqrt {b} (2 b c-3 a d) \log \left (-\sqrt {b} x+\sqrt {a+b x^2}\right )}{2 d^2} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.27 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05, number of steps used = 8, number of rules used = 7, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {318, 25, 398, 224, 219, 291, 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.

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

\(\Big \downarrow \) 318

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

\(\Big \downarrow \) 25

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

\(\Big \downarrow \) 398

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

\(\Big \downarrow \) 224

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

\(\Big \downarrow \) 219

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

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 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 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 291

Int[1/(Sqrt[(a_) + (b_.)*(x_)^2]*((c_) + (d_.)*(x_)^2)), x_Symbol] :> Subst 
[Int[1/(c - (b*c - a*d)*x^2), x], x, x/Sqrt[a + b*x^2]] /; FreeQ[{a, b, c, 
d}, x] && NeQ[b*c - a*d, 0]
 

rule 318

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_), x_Symbol] :> Sim 
p[d*x*(a + b*x^2)^(p + 1)*((c + d*x^2)^(q - 1)/(b*(2*(p + q) + 1))), x] + S 
imp[1/(b*(2*(p + q) + 1))   Int[(a + b*x^2)^p*(c + d*x^2)^(q - 2)*Simp[c*(b 
*c*(2*(p + q) + 1) - a*d) + d*(b*c*(2*(p + 2*q - 1) + 1) - a*d*(2*(q - 1) + 
 1))*x^2, x], x], x] /; FreeQ[{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && G 
tQ[q, 1] && NeQ[2*(p + q) + 1, 0] &&  !IGtQ[p, 1] && IntBinomialQ[a, b, c, 
d, 2, p, q, x]
 

rule 398

Int[((e_) + (f_.)*(x_)^2)/(((a_) + (b_.)*(x_)^2)*Sqrt[(c_) + (d_.)*(x_)^2]) 
, x_Symbol] :> Simp[f/b   Int[1/Sqrt[c + d*x^2], x], x] + Simp[(b*e - a*f)/ 
b   Int[1/((a + b*x^2)*Sqrt[c + d*x^2]), x], x] /; FreeQ[{a, b, c, d, e, f} 
, x]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 105, normalized size of antiderivative = 0.93

method	result	size
pseudoelliptic	\(-\frac {\frac {2 \left (a d -b c \right )^{2} \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}+b \left (-\sqrt {b \,x^{2}+a}\, d x -\frac {\left (3 a d -2 b c \right ) \operatorname {arctanh}\left (\frac {\sqrt {b \,x^{2}+a}}{x \sqrt {b}}\right )}{\sqrt {b}}\right )}{2 d^{2}}\)	\(105\)
risch	\(\frac {b x \sqrt {b \,x^{2}+a}}{2 d}+\frac {\frac {\sqrt {b}\, \left (3 a d -2 b c \right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d}+\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {\frac {2 a d -2 b c}{d}-\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x +\frac {\sqrt {-c d}}{d}\right )^{2} b -\frac {2 b \sqrt {-c d}\, \left (x +\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x +\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, d \sqrt {\frac {a d -b c}{d}}}-\frac {\left (a^{2} d^{2}-2 a b c d +b^{2} c^{2}\right ) \ln \left (\frac {\frac {2 a d -2 b c}{d}+\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+2 \sqrt {\frac {a d -b c}{d}}\, \sqrt {\left (x -\frac {\sqrt {-c d}}{d}\right )^{2} b +\frac {2 b \sqrt {-c d}\, \left (x -\frac {\sqrt {-c d}}{d}\right )}{d}+\frac {a d -b c}{d}}}{x -\frac {\sqrt {-c d}}{d}}\right )}{\sqrt {-c d}\, d \sqrt {\frac {a d -b c}{d}}}}{2 d}\)	\(401\)
default	\(\text {Expression too large to display}\)	\(1227\)

Input:

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

Output:

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

Fricas [A] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 721, normalized size of antiderivative = 6.38 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{c+d x^2} \, dx=\left [\frac {2 \, \sqrt {b x^{2} + a} b d x - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right ) - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac {2 \, \sqrt {b x^{2} + a} b d x + 2 \, {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (b c - a d\right )} \sqrt {\frac {b c - a d}{c}} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} - 4 \, {\left (a c^{2} x + {\left (2 \, b c^{2} - a c d\right )} x^{3}\right )} \sqrt {b x^{2} + a} \sqrt {\frac {b c - a d}{c}}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right )}{4 \, d^{2}}, \frac {2 \, \sqrt {b x^{2} + a} b d x - 2 \, {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right ) - {\left (2 \, b c - 3 \, a d\right )} \sqrt {b} \log \left (-2 \, b x^{2} - 2 \, \sqrt {b x^{2} + a} \sqrt {b} x - a\right )}{4 \, d^{2}}, \frac {\sqrt {b x^{2} + a} b d x + {\left (2 \, b c - 3 \, a d\right )} \sqrt {-b} \arctan \left (\frac {\sqrt {-b} x}{\sqrt {b x^{2} + a}}\right ) - {\left (b c - a d\right )} \sqrt {-\frac {b c - a d}{c}} \arctan \left (\frac {{\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a} \sqrt {-\frac {b c - a d}{c}}}{2 \, {\left ({\left (b^{2} c - a b d\right )} x^{3} + {\left (a b c - a^{2} d\right )} x\right )}}\right )}{2 \, d^{2}}\right ] \] Input:

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

Output:

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

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [F(-2)]

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

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

Output:

Exception raised: TypeError >> an error occurred running a Giac command:IN 
PUT:sage2:=int(sage0,sageVARx):;OUTPUT:index.cc index_m i_lex_is_greater E 
rror: Bad Argument Value
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.23 (sec) , antiderivative size = 287, normalized size of antiderivative = 2.54 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{c+d x^2} \, dx=\frac {-2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}-\sqrt {d}\, \sqrt {b \,x^{2}+a}-\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b c -2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) a d +2 \sqrt {c}\, \sqrt {a d -b c}\, \mathit {atan} \left (\frac {\sqrt {a d -b c}+\sqrt {d}\, \sqrt {b \,x^{2}+a}+\sqrt {d}\, \sqrt {b}\, x}{\sqrt {c}\, \sqrt {b}}\right ) b c +\sqrt {b \,x^{2}+a}\, b c d x +3 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) a c d -2 \sqrt {b}\, \mathrm {log}\left (\frac {\sqrt {b \,x^{2}+a}+\sqrt {b}\, x}{\sqrt {a}}\right ) b \,c^{2}}{2 c \,d^{2}} \] Input:

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

Output:

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