\(\int \frac {(a+b x^2)^{3/2}}{(c+d x^2)^3} \, dx\) [73]

Optimal result

Integrand size = 21, antiderivative size = 113 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\frac {x \left (a+b x^2\right )^{3/2}}{4 c \left (c+d x^2\right )^2}+\frac {3 a x \sqrt {a+b x^2}}{8 c^2 \left (c+d x^2\right )}+\frac {3 a^2 \text {arctanh}\left (\frac {\sqrt {b c-a d} x}{\sqrt {c} \sqrt {a+b x^2}}\right )}{8 c^{5/2} \sqrt {b c-a d}} \] Output:

1/4*x*(b*x^2+a)^(3/2)/c/(d*x^2+c)^2+3/8*a*x*(b*x^2+a)^(1/2)/c^2/(d*x^2+c)+ 
3/8*a^2*arctanh((-a*d+b*c)^(1/2)*x/c^(1/2)/(b*x^2+a)^(1/2))/c^(5/2)/(-a*d+ 
b*c)^(1/2)
 

Mathematica [A] (warning: unable to verify)

Time = 10.45 (sec) , antiderivative size = 163, normalized size of antiderivative = 1.44 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\frac {x \sqrt {a+b x^2} \left (\frac {\sqrt {\frac {c \left (a+b x^2\right )}{a \left (c+d x^2\right )}} \left (5 a c+2 b c x^2+3 a d x^2\right )}{\left (c+d x^2\right ) \sqrt {1+\frac {d x^2}{c}}}+\frac {3 a \arcsin \left (\frac {\sqrt {\left (-\frac {b}{a}+\frac {d}{c}\right ) x^2}}{\sqrt {1+\frac {d x^2}{c}}}\right )}{\sqrt {\frac {(-b c+a d) x^2}{a c}}}\right )}{8 c^3 \sqrt {1+\frac {b x^2}{a}}} \] Input:

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

Output:

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

Rubi [A] (verified)

Time = 0.22 (sec) , antiderivative size = 119, normalized size of antiderivative = 1.05, number of steps used = 5, number of rules used = 4, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.190, Rules used = {292, 292, 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.

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

\(\Big \downarrow \) 292

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

\(\Big \downarrow \) 292

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

\(\Big \downarrow \) 291

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

\(\Big \downarrow \) 221

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

Input:

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

Output:

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

Defintions of rubi rules used

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 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 292

Int[((a_) + (b_.)*(x_)^2)^(p_)*((c_) + (d_.)*(x_)^2)^(q_.), x_Symbol] :> Si 
mp[(-x)*(a + b*x^2)^(p + 1)*((c + d*x^2)^q/(2*a*(p + 1))), x] - Simp[c*(q/( 
a*(p + 1)))   Int[(a + b*x^2)^(p + 1)*(c + d*x^2)^(q - 1), x], x] /; FreeQ[ 
{a, b, c, d, p}, x] && NeQ[b*c - a*d, 0] && EqQ[2*(p + q + 1) + 1, 0] && Gt 
Q[q, 0] && NeQ[p, -1]
 

Maple [A] (verified)

Time = 0.70 (sec) , antiderivative size = 94, normalized size of antiderivative = 0.83

method	result	size
pseudoelliptic	\(-\frac {a^{2} \left (-\frac {\sqrt {b \,x^{2}+a}\, \left (3 a d \,x^{2}+2 x^{2} b c +5 a c \right ) x}{a^{2} \left (x^{2} d +c \right )^{2}}+\frac {3 \arctan \left (\frac {c \sqrt {b \,x^{2}+a}}{x \sqrt {\left (a d -b c \right ) c}}\right )}{\sqrt {\left (a d -b c \right ) c}}\right )}{8 c^{2}}\)	\(94\)
default	\(\text {Expression too large to display}\)	\(6921\)

Input:

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

Output:

-1/8*a^2/c^2*(-(b*x^2+a)^(1/2)*(3*a*d*x^2+2*b*c*x^2+5*a*c)*x/a^2/(d*x^2+c) 
^2+3/((a*d-b*c)*c)^(1/2)*arctan(c*(b*x^2+a)^(1/2)/x/((a*d-b*c)*c)^(1/2)))
 

Fricas [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 243 vs. \(2 (93) = 186\).

Time = 0.16 (sec) , antiderivative size = 526, normalized size of antiderivative = 4.65 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=\left [\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {b c^{2} - a c d} \log \left (\frac {{\left (8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2}\right )} x^{4} + a^{2} c^{2} + 2 \, {\left (4 \, a b c^{2} - 3 \, a^{2} c d\right )} x^{2} + 4 \, {\left ({\left (2 \, b c - a d\right )} x^{3} + a c x\right )} \sqrt {b c^{2} - a c d} \sqrt {b x^{2} + a}}{d^{2} x^{4} + 2 \, c d x^{2} + c^{2}}\right ) + 4 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{32 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}, -\frac {3 \, {\left (a^{2} d^{2} x^{4} + 2 \, a^{2} c d x^{2} + a^{2} c^{2}\right )} \sqrt {-b c^{2} + a c d} \arctan \left (\frac {\sqrt {-b c^{2} + a c d} {\left ({\left (2 \, b c - a d\right )} x^{2} + a c\right )} \sqrt {b x^{2} + a}}{2 \, {\left ({\left (b^{2} c^{2} - a b c d\right )} x^{3} + {\left (a b c^{2} - a^{2} c d\right )} x\right )}}\right ) - 2 \, {\left ({\left (2 \, b^{2} c^{3} + a b c^{2} d - 3 \, a^{2} c d^{2}\right )} x^{3} + 5 \, {\left (a b c^{3} - a^{2} c^{2} d\right )} x\right )} \sqrt {b x^{2} + a}}{16 \, {\left (b c^{6} - a c^{5} d + {\left (b c^{4} d^{2} - a c^{3} d^{3}\right )} x^{4} + 2 \, {\left (b c^{5} d - a c^{4} d^{2}\right )} x^{2}\right )}}\right ] \] Input:

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

Output:

[1/32*(3*(a^2*d^2*x^4 + 2*a^2*c*d*x^2 + a^2*c^2)*sqrt(b*c^2 - a*c*d)*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*((2*b*c - a*d)*x^3 + a*c*x)*sqrt(b*c^2 - a*c*d)*sqrt(b*x^2 + a)) 
/(d^2*x^4 + 2*c*d*x^2 + c^2)) + 4*((2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2)*x 
^3 + 5*(a*b*c^3 - a^2*c^2*d)*x)*sqrt(b*x^2 + a))/(b*c^6 - a*c^5*d + (b*c^4 
*d^2 - a*c^3*d^3)*x^4 + 2*(b*c^5*d - a*c^4*d^2)*x^2), -1/16*(3*(a^2*d^2*x^ 
4 + 2*a^2*c*d*x^2 + a^2*c^2)*sqrt(-b*c^2 + a*c*d)*arctan(1/2*sqrt(-b*c^2 + 
 a*c*d)*((2*b*c - a*d)*x^2 + a*c)*sqrt(b*x^2 + a)/((b^2*c^2 - a*b*c*d)*x^3 
 + (a*b*c^2 - a^2*c*d)*x)) - 2*((2*b^2*c^3 + a*b*c^2*d - 3*a^2*c*d^2)*x^3 
+ 5*(a*b*c^3 - a^2*c^2*d)*x)*sqrt(b*x^2 + a))/(b*c^6 - a*c^5*d + (b*c^4*d^ 
2 - a*c^3*d^3)*x^4 + 2*(b*c^5*d - a*c^4*d^2)*x^2)]
 

Sympy [F]

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

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

Output:

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

Maxima [F]

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

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

Output:

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

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 451 vs. \(2 (93) = 186\).

Time = 0.35 (sec) , antiderivative size = 451, normalized size of antiderivative = 3.99 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx=-\frac {3 \, a^{2} \sqrt {b} \arctan \left (\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, b c - a d}{2 \, \sqrt {-b^{2} c^{2} + a b c d}}\right )}{8 \, \sqrt {-b^{2} c^{2} + a b c d} c^{2}} + \frac {8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} b^{\frac {5}{2}} c^{2} d - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{6} a^{2} \sqrt {b} d^{3} + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} b^{\frac {7}{2}} c^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a b^{\frac {5}{2}} c^{2} d - 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{2} b^{\frac {3}{2}} c d^{2} + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} a^{3} \sqrt {b} d^{3} + 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} b^{\frac {5}{2}} c^{2} d + 16 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{3} b^{\frac {3}{2}} c d^{2} - 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{4} \sqrt {b} d^{3} + 2 \, a^{4} b^{\frac {3}{2}} c d^{2} + 3 \, a^{5} \sqrt {b} d^{3}}{4 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} d + 4 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b c - 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a d + a^{2} d\right )}^{2} c^{2} d^{2}} \] Input:

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

Output:

-3/8*a^2*sqrt(b)*arctan(1/2*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*b*c - a 
*d)/sqrt(-b^2*c^2 + a*b*c*d))/(sqrt(-b^2*c^2 + a*b*c*d)*c^2) + 1/4*(8*(sqr 
t(b)*x - sqrt(b*x^2 + a))^6*b^(5/2)*c^2*d - 3*(sqrt(b)*x - sqrt(b*x^2 + a) 
)^6*a^2*sqrt(b)*d^3 + 16*(sqrt(b)*x - sqrt(b*x^2 + a))^4*b^(7/2)*c^3 + 8*( 
sqrt(b)*x - sqrt(b*x^2 + a))^4*a*b^(5/2)*c^2*d - 18*(sqrt(b)*x - sqrt(b*x^ 
2 + a))^4*a^2*b^(3/2)*c*d^2 + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^4*a^3*sqrt(b 
)*d^3 + 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*b^(5/2)*c^2*d + 16*(sqrt(b)* 
x - sqrt(b*x^2 + a))^2*a^3*b^(3/2)*c*d^2 - 9*(sqrt(b)*x - sqrt(b*x^2 + a)) 
^2*a^4*sqrt(b)*d^3 + 2*a^4*b^(3/2)*c*d^2 + 3*a^5*sqrt(b)*d^3)/(((sqrt(b)*x 
 - sqrt(b*x^2 + a))^4*d + 4*(sqrt(b)*x - sqrt(b*x^2 + a))^2*b*c - 2*(sqrt( 
b)*x - sqrt(b*x^2 + a))^2*a*d + a^2*d)^2*c^2*d^2)
 

Mupad [F(-1)]

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

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

Output:

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

Reduce [B] (verification not implemented)

Time = 0.31 (sec) , antiderivative size = 1097, normalized size of antiderivative = 9.71 \[ \int \frac {\left (a+b x^2\right )^{3/2}}{\left (c+d x^2\right )^3} \, dx =\text {Too large to display} \] Input:

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

Output:

( - 6*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**3*c**2*d**2 - 12*sqrt(c)*s 
qrt(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**3*c*d**3*x**2 - 6*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)/(sq 
rt(c)*sqrt(b)))*a**3*d**4*x**4 + 12*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**2*b*c**3*d + 24*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**2*b*c**2*d** 
2*x**2 + 12*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**2*b*c*d**3*x**4 - 6* 
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**3*c**2*d**2 - 12*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**3*c*d**3*x**2 - 6*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**3*d**4*x**4 + 12*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**2* 
b*c**3*d + 24*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**2*b*c**2*d**2*x...