\(\int \frac {(a+b x^2)^{3/2} (A+B x+C x^2)}{x^6 (c+d x)} \, dx\) [84]

Optimal result

Integrand size = 32, antiderivative size = 404 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=-\frac {a A \sqrt {a+b x^2}}{5 c x^5}-\frac {a (B c-A d) \sqrt {a+b x^2}}{4 c^2 x^4}-\frac {\left (6 A b c+5 a c C-5 a B d+\frac {5 a A d^2}{c}\right ) \sqrt {a+b x^2}}{15 c^2 x^3}-\frac {\left (5 b c^2 (B c-A d)-4 a d \left (c^2 C-B c d+A d^2\right )\right ) \sqrt {a+b x^2}}{8 c^4 x^2}-\frac {\left (5 a c (c C-B d) \left (4 b c^2+3 a d^2\right )+A \left (3 b^2 c^4+20 a b c^2 d^2+15 a^2 d^4\right )\right ) \sqrt {a+b x^2}}{15 a c^5 x}-\frac {\left (b c^2+a d^2\right )^{3/2} \left (c^2 C-B c d+A d^2\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{c^6}-\frac {\left (3 b^2 c^4 (B c-A d)-12 a b c^2 d \left (c^2 C-B c d+A d^2\right )-8 a^2 d^3 \left (c^2 C-B c d+A d^2\right )\right ) \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right )}{8 \sqrt {a} c^6} \] Output:

-1/5*a*A*(b*x^2+a)^(1/2)/c/x^5-1/4*a*(-A*d+B*c)*(b*x^2+a)^(1/2)/c^2/x^4-1/ 
15*(6*A*b*c+5*C*a*c-5*B*a*d+5*a*A*d^2/c)*(b*x^2+a)^(1/2)/c^2/x^3-1/8*(5*b* 
c^2*(-A*d+B*c)-4*a*d*(A*d^2-B*c*d+C*c^2))*(b*x^2+a)^(1/2)/c^4/x^2-1/15*(5* 
a*c*(-B*d+C*c)*(3*a*d^2+4*b*c^2)+A*(15*a^2*d^4+20*a*b*c^2*d^2+3*b^2*c^4))* 
(b*x^2+a)^(1/2)/a/c^5/x-(a*d^2+b*c^2)^(3/2)*(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^6-1/8*(3*b^2*c^4*(-A*d+B 
*c)-12*a*b*c^2*d*(A*d^2-B*c*d+C*c^2)-8*a^2*d^3*(A*d^2-B*c*d+C*c^2))*arctan 
h((b*x^2+a)^(1/2)/a^(1/2))/a^(1/2)/c^6
 

Mathematica [A] (verified)

Time = 2.74 (sec) , antiderivative size = 449, normalized size of antiderivative = 1.11 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=-\frac {\frac {c \sqrt {a+b x^2} \left (24 A b^2 c^4 x^4+a b c^2 x^2 \left (5 c x (15 B c+32 c C x-32 B d x)+A \left (48 c^2-75 c d x+160 d^2 x^2\right )\right )+2 a^2 \left (A \left (12 c^4-15 c^3 d x+20 c^2 d^2 x^2-30 c d^3 x^3+60 d^4 x^4\right )+5 c x \left (2 c C x \left (2 c^2-3 c d x+6 d^2 x^2\right )+B \left (3 c^3-4 c^2 d x+6 c d^2 x^2-12 d^3 x^3\right )\right )\right )\right )}{a x^5}+240 \left (-b c^2-a d^2\right )^{3/2} \left (c^2 C-B c d+A d^2\right ) \arctan \left (\frac {\sqrt {-b c^2-a d^2} x}{\sqrt {a} (c+d x)-c \sqrt {a+b x^2}}\right )-\frac {15 \left (3 b^2 c^4 (-B c+A d)+12 a b c^2 d \left (c^2 C-B c d+A d^2\right )+8 a^2 d^3 \left (c^2 C-B c d+A d^2\right )\right ) \log (x)}{\sqrt {a}}+\frac {15 \left (3 b^2 c^4 (-B c+A d)+12 a b c^2 d \left (c^2 C-B c d+A d^2\right )+8 a^2 d^3 \left (c^2 C-B c d+A d^2\right )\right ) \log \left (-\sqrt {a}+\sqrt {a+b x^2}\right )}{\sqrt {a}}}{120 c^6} \] Input:

Integrate[((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)),x]
 

Output:

-1/120*((c*Sqrt[a + b*x^2]*(24*A*b^2*c^4*x^4 + a*b*c^2*x^2*(5*c*x*(15*B*c 
+ 32*c*C*x - 32*B*d*x) + A*(48*c^2 - 75*c*d*x + 160*d^2*x^2)) + 2*a^2*(A*( 
12*c^4 - 15*c^3*d*x + 20*c^2*d^2*x^2 - 30*c*d^3*x^3 + 60*d^4*x^4) + 5*c*x* 
(2*c*C*x*(2*c^2 - 3*c*d*x + 6*d^2*x^2) + B*(3*c^3 - 4*c^2*d*x + 6*c*d^2*x^ 
2 - 12*d^3*x^3)))))/(a*x^5) + 240*(-(b*c^2) - a*d^2)^(3/2)*(c^2*C - B*c*d 
+ A*d^2)*ArcTan[(Sqrt[-(b*c^2) - a*d^2]*x)/(Sqrt[a]*(c + d*x) - c*Sqrt[a + 
 b*x^2])] - (15*(3*b^2*c^4*(-(B*c) + A*d) + 12*a*b*c^2*d*(c^2*C - B*c*d + 
A*d^2) + 8*a^2*d^3*(c^2*C - B*c*d + A*d^2))*Log[x])/Sqrt[a] + (15*(3*b^2*c 
^4*(-(B*c) + A*d) + 12*a*b*c^2*d*(c^2*C - B*c*d + A*d^2) + 8*a^2*d^3*(c^2* 
C - B*c*d + A*d^2))*Log[-Sqrt[a] + Sqrt[a + b*x^2]])/Sqrt[a])/c^6
 

Rubi [A] (verified)

Time = 1.93 (sec) , antiderivative size = 763, normalized size of antiderivative = 1.89, 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.

\(\displaystyle \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx\)

\(\Big \downarrow \) 2353

\(\displaystyle \int \left (\frac {\left (a+b x^2\right )^{3/2} (B c-A d)}{c^2 x^5}+\frac {d^4 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^6 (c+d x)}-\frac {d^3 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^6 x}+\frac {d^2 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^5 x^2}-\frac {d \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^4 x^3}+\frac {\left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^3 x^4}+\frac {A \left (a+b x^2\right )^{3/2}}{c x^6}\right )dx\)

\(\Big \downarrow \) 2009

\(\displaystyle \frac {a^{3/2} d^3 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (A d^2-B c d+c^2 C\right )}{c^6}+\frac {b^{3/2} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (A d^2-B c d+c^2 C\right )}{c^3}-\frac {3 b^2 \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) (B c-A d)}{8 \sqrt {a} c^2}-\frac {\left (a d^2+b c^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{c^6}+\frac {3 a \sqrt {b} d^2 \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^5}-\frac {\sqrt {b} \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (3 a d^2+2 b c^2\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^5}+\frac {3 \sqrt {a} b d \text {arctanh}\left (\frac {\sqrt {a+b x^2}}{\sqrt {a}}\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^4}-\frac {3 b \sqrt {a+b x^2} (B c-A d)}{8 c^2 x^2}-\frac {\left (a+b x^2\right )^{3/2} (B c-A d)}{4 c^2 x^4}+\frac {d \sqrt {a+b x^2} \left (2 \left (a d^2+b c^2\right )-b c d x\right ) \left (A d^2-B c d+c^2 C\right )}{2 c^6}-\frac {a d^3 \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{c^6}-\frac {d^2 \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{c^5 x}+\frac {3 b d^2 x \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{2 c^5}+\frac {d \left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{2 c^4 x^2}-\frac {3 b d \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{2 c^4}-\frac {b \sqrt {a+b x^2} \left (A d^2-B c d+c^2 C\right )}{c^3 x}-\frac {\left (a+b x^2\right )^{3/2} \left (A d^2-B c d+c^2 C\right )}{3 c^3 x^3}-\frac {A \left (a+b x^2\right )^{5/2}}{5 a c x^5}\)

Input:

Int[((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)),x]
 

Output:

(-3*b*d*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(2*c^4) - (a*d^3*(c^2*C - 
 B*c*d + A*d^2)*Sqrt[a + b*x^2])/c^6 - (3*b*(B*c - A*d)*Sqrt[a + b*x^2])/( 
8*c^2*x^2) - (b*(c^2*C - B*c*d + A*d^2)*Sqrt[a + b*x^2])/(c^3*x) + (3*b*d^ 
2*(c^2*C - B*c*d + A*d^2)*x*Sqrt[a + b*x^2])/(2*c^5) + (d*(c^2*C - B*c*d + 
 A*d^2)*(2*(b*c^2 + a*d^2) - b*c*d*x)*Sqrt[a + b*x^2])/(2*c^6) - ((B*c - A 
*d)*(a + b*x^2)^(3/2))/(4*c^2*x^4) - ((c^2*C - B*c*d + A*d^2)*(a + b*x^2)^ 
(3/2))/(3*c^3*x^3) + (d*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/2))/(2*c^4* 
x^2) - (d^2*(c^2*C - B*c*d + A*d^2)*(a + b*x^2)^(3/2))/(c^5*x) - (A*(a + b 
*x^2)^(5/2))/(5*a*c*x^5) + (b^(3/2)*(c^2*C - B*c*d + A*d^2)*ArcTanh[(Sqrt[ 
b]*x)/Sqrt[a + b*x^2]])/c^3 + (3*a*Sqrt[b]*d^2*(c^2*C - B*c*d + A*d^2)*Arc 
Tanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^5) - (Sqrt[b]*(2*b*c^2 + 3*a*d^2)* 
(c^2*C - B*c*d + A*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(2*c^5) - (( 
b*c^2 + a*d^2)^(3/2)*(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^6 - (3*b^2*(B*c - A*d)*ArcTanh[Sqrt[a + 
 b*x^2]/Sqrt[a]])/(8*Sqrt[a]*c^2) + (3*Sqrt[a]*b*d*(c^2*C - B*c*d + A*d^2) 
*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/(2*c^4) + (a^(3/2)*d^3*(c^2*C - B*c*d + 
 A*d^2)*ArcTanh[Sqrt[a + b*x^2]/Sqrt[a]])/c^6
 

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 = 0.45 (sec) , antiderivative size = 596, normalized size of antiderivative = 1.48

method	result	size
risch	\(-\frac {\sqrt {b \,x^{2}+a}\, \left (120 A \,a^{2} d^{4} x^{4}+160 A a b \,c^{2} d^{2} x^{4}+24 A \,b^{2} c^{4} x^{4}-120 B \,a^{2} c \,d^{3} x^{4}-160 B a b \,c^{3} d \,x^{4}+120 C \,a^{2} c^{2} d^{2} x^{4}+160 C a b \,c^{4} x^{4}-60 A \,a^{2} c \,d^{3} x^{3}-75 A a b \,c^{3} d \,x^{3}+60 B \,a^{2} c^{2} d^{2} x^{3}+75 B a b \,c^{4} x^{3}-60 C \,a^{2} c^{3} d \,x^{3}+40 A \,a^{2} c^{2} d^{2} x^{2}+48 A a b \,c^{4} x^{2}-40 B \,a^{2} c^{3} d \,x^{2}+40 C \,a^{2} c^{4} x^{2}-30 A \,a^{2} c^{3} d x +30 B \,a^{2} c^{4} x +24 A \,a^{2} c^{4}\right )}{120 c^{5} x^{5} a}+\frac {\frac {\left (8 A \,a^{2} d^{5}+12 A a b \,c^{2} d^{3}+3 A \,b^{2} c^{4} d -8 B \,a^{2} c \,d^{4}-12 B a b \,c^{3} d^{2}-3 B \,b^{2} c^{5}+8 C \,a^{2} c^{2} d^{3}+12 C a b \,c^{4} d \right ) \ln \left (\frac {2 a +2 \sqrt {a}\, \sqrt {b \,x^{2}+a}}{x}\right )}{c \sqrt {a}}-\frac {8 \left (A \,a^{2} d^{6}+2 A a b \,c^{2} d^{4}+A \,b^{2} c^{4} d^{2}-B \,a^{2} c \,d^{5}-2 B a b \,c^{3} d^{3}-c^{5} B \,b^{2} d +C \,a^{2} c^{2} d^{4}+2 C a b \,c^{4} d^{2}+c^{6} C \,b^{2}\right ) \ln \left (\frac {\frac {2 a \,d^{2}+2 b \,c^{2}}{d^{2}}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+2 \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}\, \sqrt {b \left (x +\frac {c}{d}\right )^{2}-\frac {2 b c \left (x +\frac {c}{d}\right )}{d}+\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}{x +\frac {c}{d}}\right )}{c d \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}}{8 c^{5}}\)	\(596\)
default	\(\text {Expression too large to display}\)	\(1041\)

Input:

int((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x,method=_RETURNVERBOSE)
 

Output:

-1/120*(b*x^2+a)^(1/2)*(120*A*a^2*d^4*x^4+160*A*a*b*c^2*d^2*x^4+24*A*b^2*c 
^4*x^4-120*B*a^2*c*d^3*x^4-160*B*a*b*c^3*d*x^4+120*C*a^2*c^2*d^2*x^4+160*C 
*a*b*c^4*x^4-60*A*a^2*c*d^3*x^3-75*A*a*b*c^3*d*x^3+60*B*a^2*c^2*d^2*x^3+75 
*B*a*b*c^4*x^3-60*C*a^2*c^3*d*x^3+40*A*a^2*c^2*d^2*x^2+48*A*a*b*c^4*x^2-40 
*B*a^2*c^3*d*x^2+40*C*a^2*c^4*x^2-30*A*a^2*c^3*d*x+30*B*a^2*c^4*x+24*A*a^2 
*c^4)/c^5/x^5/a+1/8/c^5*((8*A*a^2*d^5+12*A*a*b*c^2*d^3+3*A*b^2*c^4*d-8*B*a 
^2*c*d^4-12*B*a*b*c^3*d^2-3*B*b^2*c^5+8*C*a^2*c^2*d^3+12*C*a*b*c^4*d)/c/a^ 
(1/2)*ln((2*a+2*a^(1/2)*(b*x^2+a)^(1/2))/x)-8*(A*a^2*d^6+2*A*a*b*c^2*d^4+A 
*b^2*c^4*d^2-B*a^2*c*d^5-2*B*a*b*c^3*d^3-B*b^2*c^5*d+C*a^2*c^2*d^4+2*C*a*b 
*c^4*d^2+C*b^2*c^6)/c/d/((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 = 14.68 (sec) , antiderivative size = 1974, normalized size of antiderivative = 4.89 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x, algorithm="fricas")
 

Output:

[1/240*(120*(C*a*b*c^4 - B*a*b*c^3*d - B*a^2*c*d^3 + A*a^2*d^4 + (C*a^2 + 
A*a*b)*c^2*d^2)*sqrt(b*c^2 + a*d^2)*x^5*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*sqrt(b*c^2 + a*d^2)*(b*c*x - a*d)*sqr 
t(b*x^2 + a))/(d^2*x^2 + 2*c*d*x + c^2)) - 15*(3*B*b^2*c^5 + 12*B*a*b*c^3* 
d^2 + 8*B*a^2*c*d^4 - 8*A*a^2*d^5 - 3*(4*C*a*b + A*b^2)*c^4*d - 4*(2*C*a^2 
 + 3*A*a*b)*c^2*d^3)*sqrt(a)*x^5*log(-(b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(a) + 
 2*a)/x^2) - 2*(24*A*a^2*c^5 - 8*(20*B*a*b*c^4*d + 15*B*a^2*c^2*d^3 - 15*A 
*a^2*c*d^4 - (20*C*a*b + 3*A*b^2)*c^5 - 5*(3*C*a^2 + 4*A*a*b)*c^3*d^2)*x^4 
 + 15*(5*B*a*b*c^5 + 4*B*a^2*c^3*d^2 - 4*A*a^2*c^2*d^3 - (4*C*a^2 + 5*A*a* 
b)*c^4*d)*x^3 - 8*(5*B*a^2*c^4*d - 5*A*a^2*c^3*d^2 - (5*C*a^2 + 6*A*a*b)*c 
^5)*x^2 + 30*(B*a^2*c^5 - A*a^2*c^4*d)*x)*sqrt(b*x^2 + a))/(a*c^6*x^5), -1 
/240*(240*(C*a*b*c^4 - B*a*b*c^3*d - B*a^2*c*d^3 + A*a^2*d^4 + (C*a^2 + A* 
a*b)*c^2*d^2)*sqrt(-b*c^2 - a*d^2)*x^5*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)) + 15 
*(3*B*b^2*c^5 + 12*B*a*b*c^3*d^2 + 8*B*a^2*c*d^4 - 8*A*a^2*d^5 - 3*(4*C*a* 
b + A*b^2)*c^4*d - 4*(2*C*a^2 + 3*A*a*b)*c^2*d^3)*sqrt(a)*x^5*log(-(b*x^2 
+ 2*sqrt(b*x^2 + a)*sqrt(a) + 2*a)/x^2) + 2*(24*A*a^2*c^5 - 8*(20*B*a*b*c^ 
4*d + 15*B*a^2*c^2*d^3 - 15*A*a^2*c*d^4 - (20*C*a*b + 3*A*b^2)*c^5 - 5*(3* 
C*a^2 + 4*A*a*b)*c^3*d^2)*x^4 + 15*(5*B*a*b*c^5 + 4*B*a^2*c^3*d^2 - 4*A*a^ 
2*c^2*d^3 - (4*C*a^2 + 5*A*a*b)*c^4*d)*x^3 - 8*(5*B*a^2*c^4*d - 5*A*a^2...
 

Sympy [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\int \frac {\left (a + b x^{2}\right )^{\frac {3}{2}} \left (A + B x + C x^{2}\right )}{x^{6} \left (c + d x\right )}\, dx \] Input:

integrate((b*x**2+a)**(3/2)*(C*x**2+B*x+A)/x**6/(d*x+c),x)
 

Output:

Integral((a + b*x**2)**(3/2)*(A + B*x + C*x**2)/(x**6*(c + d*x)), x)
 

Maxima [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\int { \frac {{\left (C x^{2} + B x + A\right )} {\left (b x^{2} + a\right )}^{\frac {3}{2}}}{{\left (d x + c\right )} x^{6}} \,d x } \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x, algorithm="maxima")
 

Output:

integrate((C*x^2 + B*x + A)*(b*x^2 + a)^(3/2)/((d*x + c)*x^6), x)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 1819 vs. \(2 (371) = 742\).

Time = 0.18 (sec) , antiderivative size = 1819, normalized size of antiderivative = 4.50 \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\text {Too large to display} \] Input:

integrate((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x, algorithm="giac")
 

Output:

2*(C*b^2*c^6 - B*b^2*c^5*d + 2*C*a*b*c^4*d^2 + A*b^2*c^4*d^2 - 2*B*a*b*c^3 
*d^3 + C*a^2*c^2*d^4 + 2*A*a*b*c^2*d^4 - B*a^2*c*d^5 + A*a^2*d^6)*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^6) + 1/4*(3*B*b^2*c^5 - 12*C*a*b*c^4*d - 3*A*b^2*c^4*d + 
 12*B*a*b*c^3*d^2 - 8*C*a^2*c^2*d^3 - 12*A*a*b*c^2*d^3 + 8*B*a^2*c*d^4 - 8 
*A*a^2*d^5)*arctan(-(sqrt(b)*x - sqrt(b*x^2 + a))/sqrt(-a))/(sqrt(-a)*c^6) 
 + 1/60*(75*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*b^2*c^4 - 60*(sqrt(b)*x - sq 
rt(b*x^2 + a))^9*C*a*b*c^3*d - 75*(sqrt(b)*x - sqrt(b*x^2 + a))^9*A*b^2*c^ 
3*d + 60*(sqrt(b)*x - sqrt(b*x^2 + a))^9*B*a*b*c^2*d^2 - 60*(sqrt(b)*x - s 
qrt(b*x^2 + a))^9*A*a*b*c*d^3 + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*C*a*b^ 
(3/2)*c^4 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*b^(5/2)*c^4 - 240*(sqrt( 
b)*x - sqrt(b*x^2 + a))^8*B*a*b^(3/2)*c^3*d + 120*(sqrt(b)*x - sqrt(b*x^2 
+ a))^8*C*a^2*sqrt(b)*c^2*d^2 + 240*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a*b^ 
(3/2)*c^2*d^2 - 120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*B*a^2*sqrt(b)*c*d^3 + 
120*(sqrt(b)*x - sqrt(b*x^2 + a))^8*A*a^2*sqrt(b)*d^4 - 30*(sqrt(b)*x - sq 
rt(b*x^2 + a))^7*B*a*b^2*c^4 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^7*C*a^2*b 
*c^3*d + 30*(sqrt(b)*x - sqrt(b*x^2 + a))^7*A*a*b^2*c^3*d - 120*(sqrt(b)*x 
 - sqrt(b*x^2 + a))^7*B*a^2*b*c^2*d^2 + 120*(sqrt(b)*x - sqrt(b*x^2 + a))^ 
7*A*a^2*b*c*d^3 - 720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*C*a^2*b^(3/2)*c^4 + 
720*(sqrt(b)*x - sqrt(b*x^2 + a))^6*B*a^2*b^(3/2)*c^3*d - 480*(sqrt(b)*...
 

Mupad [F(-1)]

Timed out. \[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\int \frac {{\left (b\,x^2+a\right )}^{3/2}\,\left (C\,x^2+B\,x+A\right )}{x^6\,\left (c+d\,x\right )} \,d x \] Input:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)),x)
 

Output:

int(((a + b*x^2)^(3/2)*(A + B*x + C*x^2))/(x^6*(c + d*x)), x)
 

Reduce [F]

\[ \int \frac {\left (a+b x^2\right )^{3/2} \left (A+B x+C x^2\right )}{x^6 (c+d x)} \, dx=\int \frac {\left (b \,x^{2}+a \right )^{\frac {3}{2}} \left (C \,x^{2}+B x +A \right )}{x^{6} \left (d x +c \right )}d x \] Input:

int((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x)
 

Output:

int((b*x^2+a)^(3/2)*(C*x^2+B*x+A)/x^6/(d*x+c),x)