\(\int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx\) [122]

Optimal result

Integrand size = 32, antiderivative size = 251 \[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {A \sqrt {a+b x^2}}{3 a c x^3}-\frac {(B c-A d) \sqrt {a+b x^2}}{2 a c^2 x^2}-\frac {\left (3 a c (c C-B d)-A \left (2 b c^2-3 a d^2\right )\right ) \sqrt {a+b x^2}}{3 a^2 c^3 x}-\frac {d^2 \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^4 \sqrt {b c^2+a d^2}}+\frac {\left (b c^2 (B c-A d)+2 a d \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{2 a^{3/2} c^4} \] Output:

-1/3*A*(b*x^2+a)^(1/2)/a/c/x^3-1/2*(-A*d+B*c)*(b*x^2+a)^(1/2)/a/c^2/x^2-1/ 
3*(3*a*c*(-B*d+C*c)-A*(-3*a*d^2+2*b*c^2))*(b*x^2+a)^(1/2)/a^2/c^3/x-d^2*(A 
*d^2-B*c*d+C*c^2)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2) 
)/c^4/(a*d^2+b*c^2)^(1/2)+1/2*(b*c^2*(-A*d+B*c)+2*a*d*(A*d^2-B*c*d+C*c^2)) 
*arctanh((b*x^2+a)^(1/2)/a^(1/2))/a^(3/2)/c^4
 

Mathematica [A] (verified)

Time = 1.67 (sec) , antiderivative size = 260, normalized size of antiderivative = 1.04 \[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (-4 A b c^2 x^2+a A \left (2 c^2-3 c d x+6 d^2 x^2\right )+3 a c x (2 c C x+B (c-2 d x))\right )}{a^2 x^3}+\frac {12 d^2 \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {b} (c+d x)-d \sqrt {a+b x^2}}{\sqrt {-b c^2-a d^2}}\right )}{\sqrt {-b c^2-a d^2}}+\frac {12 A d^3 \text {arctanh}\left (\frac {\sqrt {b} x-\sqrt {a+b x^2}}{\sqrt {a}}\right )}{\sqrt {a}}-\frac {6 c (b c (B c-A d)+2 a d (c C-B d)) \text {arctanh}\left (\frac {-\sqrt {b} x+\sqrt {a+b x^2}}{\sqrt {a}}\right )}{a^{3/2}}}{6 c^4} \] Input:

Integrate[(A + B*x + C*x^2)/(x^4*(c + d*x)*Sqrt[a + b*x^2]),x]
 

Output:

-1/6*((c*Sqrt[a + b*x^2]*(-4*A*b*c^2*x^2 + a*A*(2*c^2 - 3*c*d*x + 6*d^2*x^ 
2) + 3*a*c*x*(2*c*C*x + B*(c - 2*d*x))))/(a^2*x^3) + (12*d^2*(c^2*C - B*c* 
d + A*d^2)*ArcTan[(Sqrt[b]*(c + d*x) - d*Sqrt[a + b*x^2])/Sqrt[-(b*c^2) - 
a*d^2]])/Sqrt[-(b*c^2) - a*d^2] + (12*A*d^3*ArcTanh[(Sqrt[b]*x - Sqrt[a + 
b*x^2])/Sqrt[a]])/Sqrt[a] - (6*c*(b*c*(B*c - A*d) + 2*a*d*(c*C - B*d))*Arc 
Tanh[(-(Sqrt[b]*x) + Sqrt[a + b*x^2])/Sqrt[a]])/a^(3/2))/c^4
 

Rubi [A] (verified)

Time = 0.99 (sec) , antiderivative size = 281, normalized size of antiderivative = 1.12, number of steps used = 2, number of rules used = 2, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.062, Rules used = {2353, 2009}

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.

\(\displaystyle \int \frac {A+B x+C x^2}{x^4 \sqrt {a+b x^2} (c+d x)} \, dx\)

\(\Big \downarrow \) 2353

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

\(\Big \downarrow \) 2009

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

Input:

Int[(A + B*x + C*x^2)/(x^4*(c + d*x)*Sqrt[a + b*x^2]),x]
 

Output:

-1/3*(A*Sqrt[a + b*x^2])/(a*c*x^3) - ((B*c - A*d)*Sqrt[a + b*x^2])/(2*a*c^ 
2*x^2) + (2*A*b*Sqrt[a + b*x^2])/(3*a^2*c*x) - ((c^2*C - B*c*d + A*d^2)*Sq 
rt[a + b*x^2])/(a*c^3*x) - (d^2*(c^2*C - B*c*d + A*d^2)*ArcTanh[(a*d - b*c 
*x)/(Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2])])/(c^4*Sqrt[b*c^2 + a*d^2]) + (b 
*(B*c - A*d)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*a^(3/2)*c^2) + (d*(c^2*C 
 - B*c*d + A*d^2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(Sqrt[a]*c^4)
 

Defintions of rubi rules used

rule 2009

Int[u_, x_Symbol] :> Simp[IntSum[u, x], x] /; SumQ[u]
 

rule 2353

Int[(Px_)*((e_.)*(x_))^(m_.)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2) 
^(p_), x_Symbol] :> Int[ExpandIntegrand[Px*(e*x)^m*(c + d*x)^n*(a + b*x^2)^ 
p, x], x] /; FreeQ[{a, b, c, d, e, m, n, p}, x] && PolyQ[Px, x] && (Integer 
Q[p] || (IntegerQ[2*p] && IntegerQ[m] && ILtQ[n, 0]))
 

Maple [A] (verified)

Time = 0.31 (sec) , antiderivative size = 305, normalized size of antiderivative = 1.22

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (6 A a \,d^{2} x^{2}-4 A b \,c^{2} x^{2}-6 B a c d \,x^{2}+6 C a \,c^{2} x^{2}-3 A a c d x +3 B a \,c^{2} x +2 A \,c^{2} a \right )}{6 a^{2} c^{3} x^{3}}+\frac {\frac {\left (2 A a \,d^{3}-A b \,c^{2} d -2 B a c \,d^{2}+B b \,c^{3}+2 C a \,c^{2} d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {2 d a \left (A \,d^{2}-B c d +C \,c^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{2 c^{3} a}\)	\(305\)
default	\(\frac {A \left (-\frac {\sqrt {b \,x^{2}+a}}{3 a \,x^{3}}+\frac {2 b \sqrt {b \,x^{2}+a}}{3 a^{2} x}\right )}{c}-\frac {\left (A d -B c \right ) \left (-\frac {\sqrt {b \,x^{2}+a}}{2 a \,x^{2}}+\frac {b \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{2 a^{\frac {3}{2}}}\right )}{c^{2}}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{c^{3} a x}-\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) d \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c^{4} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {\left (A \,d^{2}-B c d +C \,c^{2}\right ) d \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c^{4} \sqrt {a}}\)	\(329\)

Input:

int((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x,method=_RETURNVERBOSE)
 

Output:

-1/6*(b*x^2+a)^(1/2)*(6*A*a*d^2*x^2-4*A*b*c^2*x^2-6*B*a*c*d*x^2+6*C*a*c^2* 
x^2-3*A*a*c*d*x+3*B*a*c^2*x+2*A*a*c^2)/a^2/c^3/x^3+1/2/c^3/a*((2*A*a*d^3-A 
*b*c^2*d-2*B*a*c*d^2+B*b*c^3+2*C*a*c^2*d)/c/a^(1/2)*ln((2*a+2*a^(1/2)*(b*x 
^2+a)^(1/2))/x)-2*d*a*(A*d^2-B*c*d+C*c^2)/c/((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 [A] (verification not implemented)

Time = 6.31 (sec) , antiderivative size = 1653, normalized size of antiderivative = 6.59 \[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\text {Too large to display} \] Input:

integrate((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="fricas")
 

Output:

[1/12*(6*(C*a^2*c^2*d^2 - B*a^2*c*d^3 + A*a^2*d^4)*sqrt(b*c^2 + a*d^2)*x^3 
*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*sq 
rt(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) 
) + 3*(B*b^2*c^5 - B*a*b*c^3*d^2 - 2*B*a^2*c*d^4 + 2*A*a^2*d^5 + (2*C*a*b 
- A*b^2)*c^4*d + (2*C*a^2 + A*a*b)*c^2*d^3)*sqrt(a)*x^3*log(-(b*x^2 + 2*sq 
rt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) - 2*(2*A*a*b*c^5 + 2*A*a^2*c^3*d^2 - 2*( 
3*B*a*b*c^4*d + 3*B*a^2*c^2*d^3 - 3*A*a^2*c*d^4 - (3*C*a*b - 2*A*b^2)*c^5 
- (3*C*a^2 + A*a*b)*c^3*d^2)*x^2 + 3*(B*a*b*c^5 - A*a*b*c^4*d + B*a^2*c^3* 
d^2 - A*a^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/((a^2*b*c^6 + a^3*c^4*d^2)*x^3), 
-1/12*(12*(C*a^2*c^2*d^2 - B*a^2*c*d^3 + A*a^2*d^4)*sqrt(-b*c^2 - a*d^2)*x 
^3*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*(B*b^2*c^5 - B*a*b*c^3*d^2 - 2*B*a^2 
*c*d^4 + 2*A*a^2*d^5 + (2*C*a*b - A*b^2)*c^4*d + (2*C*a^2 + A*a*b)*c^2*d^3 
)*sqrt(a)*x^3*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(2*A 
*a*b*c^5 + 2*A*a^2*c^3*d^2 - 2*(3*B*a*b*c^4*d + 3*B*a^2*c^2*d^3 - 3*A*a^2* 
c*d^4 - (3*C*a*b - 2*A*b^2)*c^5 - (3*C*a^2 + A*a*b)*c^3*d^2)*x^2 + 3*(B*a* 
b*c^5 - A*a*b*c^4*d + B*a^2*c^3*d^2 - A*a^2*c^2*d^3)*x)*sqrt(b*x^2 + a))/( 
(a^2*b*c^6 + a^3*c^4*d^2)*x^3), -1/6*(3*(B*b^2*c^5 - B*a*b*c^3*d^2 - 2*B*a 
^2*c*d^4 + 2*A*a^2*d^5 + (2*C*a*b - A*b^2)*c^4*d + (2*C*a^2 + A*a*b)*c^2*d 
^3)*sqrt(-a)*x^3*arctan(sqrt(b*x^2 + a)*sqrt(-a)/a) - 3*(C*a^2*c^2*d^2 ...
 

Sympy [F]

\[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {A + B x + C x^{2}}{x^{4} \sqrt {a + b x^{2}} \left (c + d x\right )}\, dx \] Input:

integrate((C*x**2+B*x+A)/x**4/(d*x+c)/(b*x**2+a)**(1/2),x)
 

Output:

Integral((A + B*x + C*x**2)/(x**4*sqrt(a + b*x**2)*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\int { \frac {C x^{2} + B x + A}{\sqrt {b x^{2} + a} {\left (d x + c\right )} x^{4}} \,d x } \] Input:

integrate((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)/(sqrt(b*x^2 + a)*(d*x + c)*x^4), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 553 vs. \(2 (226) = 452\).

Time = 0.14 (sec) , antiderivative size = 553, normalized size of antiderivative = 2.20 \[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\frac {2 \, {\left (C c^{2} d^{2} - B c d^{3} + A d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{\sqrt {-b c^{2} - a d^{2}} c^{4}} - \frac {{\left (B b c^{3} + 2 \, C a c^{2} d - A b c^{2} d - 2 \, B a c d^{2} + 2 \, A a d^{3}\right )} \arctan \left (-\frac {\sqrt {b} x - \sqrt {b x^{2} + a}}{\sqrt {-a}}\right )}{\sqrt {-a} a c^{4}} + \frac {3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} B b c^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{5} A b c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} C a \sqrt {b} c^{2} - 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} B a \sqrt {b} c d + 6 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{4} A a \sqrt {b} d^{2} - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} C a^{2} \sqrt {b} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a b^{\frac {3}{2}} c^{2} + 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} B a^{2} \sqrt {b} c d - 12 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} A a^{2} \sqrt {b} d^{2} - 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} B a^{2} b c^{2} + 3 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} A a^{2} b c d + 6 \, C a^{3} \sqrt {b} c^{2} - 4 \, A a^{2} b^{\frac {3}{2}} c^{2} - 6 \, B a^{3} \sqrt {b} c d + 6 \, A a^{3} \sqrt {b} d^{2}}{3 \, {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} - a\right )}^{3} a c^{3}} \] Input:

integrate((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x, algorithm="giac")
 

Output:

2*(C*c^2*d^2 - B*c*d^3 + A*d^4)*arctan(-((sqrt(b)*x - sqrt(b*x^2 + a))*d + 
 sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/(sqrt(-b*c^2 - a*d^2)*c^4) - (B*b*c^3 + 
2*C*a*c^2*d - A*b*c^2*d - 2*B*a*c*d^2 + 2*A*a*d^3)*arctan(-(sqrt(b)*x - sq 
rt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*a*c^4) + 1/3*(3*(sqrt(b)*x - sqrt(b*x^2 
 + a))^5*B*b*c^2 - 3*(sqrt(b)*x - sqrt(b*x^2 + a))^5*A*b*c*d + 6*(sqrt(b)* 
x - sqrt(b*x^2 + a))^4*C*a*sqrt(b)*c^2 - 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4 
*B*a*sqrt(b)*c*d + 6*(sqrt(b)*x - sqrt(b*x^2 + a))^4*A*a*sqrt(b)*d^2 - 12* 
(sqrt(b)*x - sqrt(b*x^2 + a))^2*C*a^2*sqrt(b)*c^2 + 12*(sqrt(b)*x - sqrt(b 
*x^2 + a))^2*A*a*b^(3/2)*c^2 + 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*B*a^2*sq 
rt(b)*c*d - 12*(sqrt(b)*x - sqrt(b*x^2 + a))^2*A*a^2*sqrt(b)*d^2 - 3*(sqrt 
(b)*x - sqrt(b*x^2 + a))*B*a^2*b*c^2 + 3*(sqrt(b)*x - sqrt(b*x^2 + a))*A*a 
^2*b*c*d + 6*C*a^3*sqrt(b)*c^2 - 4*A*a^2*b^(3/2)*c^2 - 6*B*a^3*sqrt(b)*c*d 
 + 6*A*a^3*sqrt(b)*d^2)/(((sqrt(b)*x - sqrt(b*x^2 + a))^2 - a)^3*a*c^3)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {C\,x^2+B\,x+A}{x^4\,\sqrt {b\,x^2+a}\,\left (c+d\,x\right )} \,d x \] Input:

int((A + B*x + C*x^2)/(x^4*(a + b*x^2)^(1/2)*(c + d*x)),x)
 

Output:

int((A + B*x + C*x^2)/(x^4*(a + b*x^2)^(1/2)*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {A+B x+C x^2}{x^4 (c+d x) \sqrt {a+b x^2}} \, dx=\int \frac {C \,x^{2}+B x +A}{x^{4} \left (d x +c \right ) \sqrt {b \,x^{2}+a}}d x \] Input:

int((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x)
 

Output:

int((C*x^2+B*x+A)/x^4/(d*x+c)/(b*x^2+a)^(1/2),x)