\(\int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx\) [1015]

Optimal result

Integrand size = 22, antiderivative size = 277 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {6 c^2 \sqrt {a+b x^2}}{d^5}-\frac {3 c x \sqrt {a+b x^2}}{2 d^4}-\frac {c^4 \sqrt {a+b x^2}}{2 d^5 (c+d x)^2}+\frac {c^3 \left (9 b c^2+8 a d^2\right ) \sqrt {a+b x^2}}{2 d^5 \left (b c^2+a d^2\right ) (c+d x)}+\frac {\left (a+b x^2\right )^{3/2}}{3 b d^3}-\frac {c \left (20 b c^2+3 a d^2\right ) \text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right )}{2 \sqrt {b} d^6}-\frac {c^2 \left (20 b^2 c^4+33 a b c^2 d^2+12 a^2 d^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {b c^2+a d^2} \sqrt {a+b x^2}}\right )}{2 d^6 \left (b c^2+a d^2\right )^{3/2}} \] Output:

6*c^2*(b*x^2+a)^(1/2)/d^5-3/2*c*x*(b*x^2+a)^(1/2)/d^4-1/2*c^4*(b*x^2+a)^(1 
/2)/d^5/(d*x+c)^2+1/2*c^3*(8*a*d^2+9*b*c^2)*(b*x^2+a)^(1/2)/d^5/(a*d^2+b*c 
^2)/(d*x+c)+1/3*(b*x^2+a)^(3/2)/b/d^3-1/2*c*(3*a*d^2+20*b*c^2)*arctanh(b^( 
1/2)*x/(b*x^2+a)^(1/2))/b^(1/2)/d^6-1/2*c^2*(12*a^2*d^4+33*a*b*c^2*d^2+20* 
b^2*c^4)*arctanh((-b*c*x+a*d)/(a*d^2+b*c^2)^(1/2)/(b*x^2+a)^(1/2))/d^6/(a* 
d^2+b*c^2)^(3/2)
 

Mathematica [A] (verified)

Time = 10.56 (sec) , antiderivative size = 279, normalized size of antiderivative = 1.01 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {d \sqrt {a+b x^2} \left (36 c^2+\frac {2 a d^2}{b}-9 c d x+2 d^2 x^2-\frac {3 c^4}{(c+d x)^2}+\frac {3 c^3 \left (9 b c^2+8 a d^2\right )}{\left (b c^2+a d^2\right ) (c+d x)}\right )+\frac {3 c^2 \left (20 b^2 c^4+33 a b c^2 d^2+12 a^2 d^4\right ) \log (c+d x)}{\left (b c^2+a d^2\right )^{3/2}}-\frac {3 c \left (20 b c^2+3 a d^2\right ) \log \left (b x+\sqrt {b} \sqrt {a+b x^2}\right )}{\sqrt {b}}-\frac {3 c^2 \left (20 b^2 c^4+33 a b c^2 d^2+12 a^2 d^4\right ) \log \left (a d-b c x+\sqrt {b c^2+a d^2} \sqrt {a+b x^2}\right )}{\left (b c^2+a d^2\right )^{3/2}}}{6 d^6} \] Input:

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

Output:

(d*Sqrt[a + b*x^2]*(36*c^2 + (2*a*d^2)/b - 9*c*d*x + 2*d^2*x^2 - (3*c^4)/( 
c + d*x)^2 + (3*c^3*(9*b*c^2 + 8*a*d^2))/((b*c^2 + a*d^2)*(c + d*x))) + (3 
*c^2*(20*b^2*c^4 + 33*a*b*c^2*d^2 + 12*a^2*d^4)*Log[c + d*x])/(b*c^2 + a*d 
^2)^(3/2) - (3*c*(20*b*c^2 + 3*a*d^2)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/ 
Sqrt[b] - (3*c^2*(20*b^2*c^4 + 33*a*b*c^2*d^2 + 12*a^2*d^4)*Log[a*d - b*c* 
x + Sqrt[b*c^2 + a*d^2]*Sqrt[a + b*x^2]])/(b*c^2 + a*d^2)^(3/2))/(6*d^6)
 

Rubi [A] (verified)

Time = 1.73 (sec) , antiderivative size = 398, normalized size of antiderivative = 1.44, number of steps used = 12, number of rules used = 11, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.500, Rules used = {603, 2182, 2185, 27, 682, 27, 719, 224, 219, 488, 219}

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.

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

\(\Big \downarrow \) 603

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

\(\Big \downarrow \) 2182

\(\displaystyle -\frac {-\frac {\int \frac {\sqrt {b x^2+a} \left (a \left (\frac {5 b c^2}{d^2}+6 a\right ) c^2-2 \left (\frac {9 b^2 c^4}{d^3}+\frac {12 a b c^2}{d}+2 a^2 d\right ) x c+\frac {2 \left (b c^2+a d^2\right )^2 x^2}{d^2}\right )}{c+d x}dx}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 2185

\(\displaystyle -\frac {-\frac {\frac {\int \frac {3 b c \left (a c \left (5 b c^2+6 a d^2\right )-2 \left (\frac {10 b^2 c^4}{d}+14 a b d c^2+3 a^2 d^3\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{3 b d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {c \int \frac {\left (a c \left (5 b c^2+6 a d^2\right )-2 \left (\frac {10 b^2 c^4}{d}+14 a b d c^2+3 a^2 d^3\right ) x\right ) \sqrt {b x^2+a}}{c+d x}dx}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 682

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\int \frac {2 b \left (b c^2+a d^2\right ) \left (a c d \left (10 b c^2+9 a d^2\right )-\left (b c^2+a d^2\right ) \left (20 b c^2+3 a d^2\right ) x\right )}{d (c+d x) \sqrt {b x^2+a}}dx}{2 b d^2}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 27

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \int \frac {a c d \left (10 b c^2+9 a d^2\right )-\left (b c^2+a d^2\right ) \left (20 b c^2+3 a d^2\right ) x}{(c+d x) \sqrt {b x^2+a}}dx}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 719

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (12 a^2 d^4+33 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a d^2+b c^2\right ) \left (3 a d^2+20 b c^2\right ) \int \frac {1}{\sqrt {b x^2+a}}dx}{d}\right )}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 224

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (12 a^2 d^4+33 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\left (a d^2+b c^2\right ) \left (3 a d^2+20 b c^2\right ) \int \frac {1}{1-\frac {b x^2}{b x^2+a}}d\frac {x}{\sqrt {b x^2+a}}}{d}\right )}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \left (\frac {c \left (12 a^2 d^4+33 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {1}{(c+d x) \sqrt {b x^2+a}}dx}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (3 a d^2+20 b c^2\right )}{\sqrt {b} d}\right )}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 488

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \left (-\frac {c \left (12 a^2 d^4+33 a b c^2 d^2+20 b^2 c^4\right ) \int \frac {1}{b c^2+a d^2-\frac {(a d-b c x)^2}{b x^2+a}}d\frac {a d-b c x}{\sqrt {b x^2+a}}}{d}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (3 a d^2+20 b c^2\right )}{\sqrt {b} d}\right )}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

\(\Big \downarrow \) 219

\(\displaystyle -\frac {-\frac {\frac {c \left (\frac {\left (a d^2+b c^2\right ) \left (-\frac {c \left (12 a^2 d^4+33 a b c^2 d^2+20 b^2 c^4\right ) \text {arctanh}\left (\frac {a d-b c x}{\sqrt {a+b x^2} \sqrt {a d^2+b c^2}}\right )}{d \sqrt {a d^2+b c^2}}-\frac {\text {arctanh}\left (\frac {\sqrt {b} x}{\sqrt {a+b x^2}}\right ) \left (a d^2+b c^2\right ) \left (3 a d^2+20 b c^2\right )}{\sqrt {b} d}\right )}{d^3}+\frac {\sqrt {a+b x^2} \left (c \left (12 a^2 d^3+33 a b c^2 d+\frac {20 b^2 c^4}{d}\right )-x \left (3 a^2 d^4+14 a b c^2 d^2+10 b^2 c^4\right )\right )}{d^2}\right )}{d^2}+\frac {2 \left (a+b x^2\right )^{3/2} \left (a d^2+b c^2\right )^2}{3 b d^3}}{a d^2+b c^2}-\frac {c^3 \left (a+b x^2\right )^{3/2} \left (8 a d^2+7 b c^2\right )}{d^3 (c+d x) \left (a d^2+b c^2\right )}}{2 \left (a d^2+b c^2\right )}-\frac {c^4 \left (a+b x^2\right )^{3/2}}{2 d^3 (c+d x)^2 \left (a d^2+b c^2\right )}\)

Input:

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

Output:

-1/2*(c^4*(a + b*x^2)^(3/2))/(d^3*(b*c^2 + a*d^2)*(c + d*x)^2) - (-((c^3*( 
7*b*c^2 + 8*a*d^2)*(a + b*x^2)^(3/2))/(d^3*(b*c^2 + a*d^2)*(c + d*x))) - ( 
(2*(b*c^2 + a*d^2)^2*(a + b*x^2)^(3/2))/(3*b*d^3) + (c*(((c*((20*b^2*c^4)/ 
d + 33*a*b*c^2*d + 12*a^2*d^3) - (10*b^2*c^4 + 14*a*b*c^2*d^2 + 3*a^2*d^4) 
*x)*Sqrt[a + b*x^2])/d^2 + ((b*c^2 + a*d^2)*(-(((b*c^2 + a*d^2)*(20*b*c^2 
+ 3*a*d^2)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(Sqrt[b]*d)) - (c*(20*b^2 
*c^4 + 33*a*b*c^2*d^2 + 12*a^2*d^4)*ArcTanh[(a*d - b*c*x)/(Sqrt[b*c^2 + a* 
d^2]*Sqrt[a + b*x^2])])/(d*Sqrt[b*c^2 + a*d^2])))/d^3))/d^2)/(b*c^2 + a*d^ 
2))/(2*(b*c^2 + a*d^2))
 

Defintions of rubi rules used

rule 27

Int[(a_)*(Fx_), x_Symbol] :> Simp[a   Int[Fx, x], x] /; FreeQ[a, x] &&  !Ma 
tchQ[Fx, (b_)*(Gx_) /; FreeQ[b, 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 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 488

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

rule 603

Int[(x_)^(m_)*((c_) + (d_.)*(x_))^(n_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol 
] :> With[{Qx = PolynomialQuotient[x^m, c + d*x, x], R = PolynomialRemainde 
r[x^m, c + d*x, x]}, Simp[d*R*(c + d*x)^(n + 1)*((a + b*x^2)^(p + 1)/((n + 
1)*(b*c^2 + a*d^2))), x] + Simp[1/((n + 1)*(b*c^2 + a*d^2))   Int[(c + d*x) 
^(n + 1)*(a + b*x^2)^p*ExpandToSum[(n + 1)*(b*c^2 + a*d^2)*Qx + b*c*R*(n + 
1) - b*d*R*(n + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, c, d, p}, x] && IGt 
Q[m, 1] && LtQ[n, -1] && NeQ[b*c^2 + a*d^2, 0]
 

rule 682

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[(d + e*x)^(m + 1)*(c*e*f*(m + 2*p + 2) - g*c*d*(2*p 
+ 1) + g*c*e*(m + 2*p + 1)*x)*((a + c*x^2)^p/(c*e^2*(m + 2*p + 1)*(m + 2*p 
+ 2))), x] + Simp[2*(p/(c*e^2*(m + 2*p + 1)*(m + 2*p + 2)))   Int[(d + e*x) 
^m*(a + c*x^2)^(p - 1)*Simp[f*a*c*e^2*(m + 2*p + 2) + a*c*d*e*g*m - (c^2*f* 
d*e*(m + 2*p + 2) - g*(c^2*d^2*(2*p + 1) + a*c*e^2*(m + 2*p + 1)))*x, x], x 
], x] /; FreeQ[{a, c, d, e, f, g, m}, x] && GtQ[p, 0] && (IntegerQ[p] ||  ! 
RationalQ[m] || (GeQ[m, -1] && LtQ[m, 0])) &&  !ILtQ[m + 2*p, 0] && (Intege 
rQ[m] || IntegerQ[p] || IntegersQ[2*m, 2*p])
 

rule 719

Int[((d_.) + (e_.)*(x_))^(m_)*((f_.) + (g_.)*(x_))*((a_) + (c_.)*(x_)^2)^(p 
_.), x_Symbol] :> Simp[g/e   Int[(d + e*x)^(m + 1)*(a + c*x^2)^p, x], x] + 
Simp[(e*f - d*g)/e   Int[(d + e*x)^m*(a + c*x^2)^p, x], x] /; FreeQ[{a, c, 
d, e, f, g, m, p}, x] &&  !IGtQ[m, 0]
 

rule 2182

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] :> 
 With[{Qx = PolynomialQuotient[Pq, d + e*x, x], R = PolynomialRemainder[Pq, 
 d + e*x, x]}, Simp[e*R*(d + e*x)^(m + 1)*((a + b*x^2)^(p + 1)/((m + 1)*(b* 
d^2 + a*e^2))), x] + Simp[1/((m + 1)*(b*d^2 + a*e^2))   Int[(d + e*x)^(m + 
1)*(a + b*x^2)^p*ExpandToSum[(m + 1)*(b*d^2 + a*e^2)*Qx + b*d*R*(m + 1) - b 
*e*R*(m + 2*p + 3)*x, x], x], x]] /; FreeQ[{a, b, d, e, p}, x] && PolyQ[Pq, 
 x] && NeQ[b*d^2 + a*e^2, 0] && LtQ[m, -1]
 

rule 2185

Int[(Pq_)*((d_) + (e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^2)^(p_), x_Symbol] : 
> With[{q = Expon[Pq, x], f = Coeff[Pq, x, Expon[Pq, x]]}, Simp[f*(d + e*x) 
^(m + q - 1)*((a + b*x^2)^(p + 1)/(b*e^(q - 1)*(m + q + 2*p + 1))), x] + Si 
mp[1/(b*e^q*(m + q + 2*p + 1))   Int[(d + e*x)^m*(a + b*x^2)^p*ExpandToSum[ 
b*e^q*(m + q + 2*p + 1)*Pq - b*f*(m + q + 2*p + 1)*(d + e*x)^q - f*(d + e*x 
)^(q - 2)*(a*e^2*(m + q - 1) - b*d^2*(m + q + 2*p + 1) - 2*b*d*e*(m + q + p 
)*x), x], x], x] /; GtQ[q, 1] && NeQ[m + q + 2*p + 1, 0]] /; FreeQ[{a, b, d 
, e, m, p}, x] && PolyQ[Pq, x] && NeQ[b*d^2 + a*e^2, 0] &&  !(EqQ[d, 0] && 
True) &&  !(IGtQ[m, 0] && RationalQ[a, b, d, e] && (IntegerQ[p] || ILtQ[p + 
 1/2, 0]))
 

Maple [B] (verified)

Leaf count of result is larger than twice the leaf count of optimal. \(918\) vs. \(2(243)=486\).

Time = 0.44 (sec) , antiderivative size = 919, normalized size of antiderivative = 3.32

method	result	size
risch	\(\frac {\left (2 b \,x^{2} d^{2}-9 b c d x +2 a \,d^{2}+36 b \,c^{2}\right ) \sqrt {b \,x^{2}+a}}{6 b \,d^{5}}-\frac {c \left (\frac {\left (3 a \,d^{2}+20 b \,c^{2}\right ) \ln \left (\sqrt {b}\, x +\sqrt {b \,x^{2}+a}\right )}{d \sqrt {b}}+\frac {6 c \left (2 a \,d^{2}+5 b \,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 )}{d^{2} \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}+\frac {4 c^{2} \left (2 a \,d^{2}+3 b \,c^{2}\right ) \left (-\frac {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}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c 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 )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{3}}-\frac {2 c^{3} \left (a \,d^{2}+b \,c^{2}\right ) \left (-\frac {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}}}}{2 \left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )^{2}}+\frac {3 b c d \left (-\frac {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}}}}{\left (a \,d^{2}+b \,c^{2}\right ) \left (x +\frac {c}{d}\right )}-\frac {b c 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 )}{\left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{2 \left (a \,d^{2}+b \,c^{2}\right )}+\frac {b \,d^{2} \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 )}{2 \left (a \,d^{2}+b \,c^{2}\right ) \sqrt {\frac {a \,d^{2}+b \,c^{2}}{d^{2}}}}\right )}{d^{4}}\right )}{2 d^{5}}\)	\(919\)
default	\(\text {Expression too large to display}\)	\(1765\)

Input:

int(x^4*(b*x^2+a)^(1/2)/(d*x+c)^3,x,method=_RETURNVERBOSE)
 

Output:

1/6*(2*b*d^2*x^2-9*b*c*d*x+2*a*d^2+36*b*c^2)*(b*x^2+a)^(1/2)/b/d^5-1/2*c/d 
^5*((3*a*d^2+20*b*c^2)/d*ln(b^(1/2)*x+(b*x^2+a)^(1/2))/b^(1/2)+6*c/d^2*(2* 
a*d^2+5*b*c^2)/((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))+4*c^2/d^3*(2*a*d^2+3*b*c^2)*(-1/(a*d^2+b*c^2)*d^2 
/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)-b*c*d/(a*d^ 
2+b*c^2)/((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)))-2*c^3*(a*d^2+b*c^2)/d^4*(-1/2/(a*d^2+b*c^2)*d^2/(x+c/d 
)^2*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+b*c^2)/d^2)^(1/2)+3/2*b*c*d/(a*d^2 
+b*c^2)*(-1/(a*d^2+b*c^2)*d^2/(x+c/d)*(b*(x+c/d)^2-2*b*c/d*(x+c/d)+(a*d^2+ 
b*c^2)/d^2)^(1/2)-b*c*d/(a*d^2+b*c^2)/((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)))+1/2*b/(a*d^2+b*c^2)*d^2/( 
(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 [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 694 vs. \(2 (244) = 488\).

Time = 43.53 (sec) , antiderivative size = 2841, normalized size of antiderivative = 10.26 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\text {Too large to display} \] Input:

integrate(x^4*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="fricas")
 

Output:

[1/12*(3*(20*b^3*c^9 + 43*a*b^2*c^7*d^2 + 26*a^2*b*c^5*d^4 + 3*a^3*c^3*d^6 
 + (20*b^3*c^7*d^2 + 43*a*b^2*c^5*d^4 + 26*a^2*b*c^3*d^6 + 3*a^3*c*d^8)*x^ 
2 + 2*(20*b^3*c^8*d + 43*a*b^2*c^6*d^3 + 26*a^2*b*c^4*d^5 + 3*a^3*c^2*d^7) 
*x)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) + 3*(20*b^3*c^ 
8 + 33*a*b^2*c^6*d^2 + 12*a^2*b*c^4*d^4 + (20*b^3*c^6*d^2 + 33*a*b^2*c^4*d 
^4 + 12*a^2*b*c^2*d^6)*x^2 + 2*(20*b^3*c^7*d + 33*a*b^2*c^5*d^3 + 12*a^2*b 
*c^3*d^5)*x)*sqrt(b*c^2 + a*d^2)*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)*sqrt(b*x^2 
 + a))/(d^2*x^2 + 2*c*d*x + c^2)) + 2*(60*b^3*c^8*d + 119*a*b^2*c^6*d^3 + 
61*a^2*b*c^4*d^5 + 2*a^3*c^2*d^7 + 2*(b^3*c^4*d^5 + 2*a*b^2*c^2*d^7 + a^2* 
b*d^9)*x^4 - 5*(b^3*c^5*d^4 + 2*a*b^2*c^3*d^6 + a^2*b*c*d^8)*x^3 + 2*(10*b 
^3*c^6*d^3 + 21*a*b^2*c^4*d^5 + 12*a^2*b*c^2*d^7 + a^3*d^9)*x^2 + (90*b^3* 
c^7*d^2 + 181*a*b^2*c^5*d^4 + 95*a^2*b*c^3*d^6 + 4*a^3*c*d^8)*x)*sqrt(b*x^ 
2 + a))/(b^3*c^6*d^6 + 2*a*b^2*c^4*d^8 + a^2*b*c^2*d^10 + (b^3*c^4*d^8 + 2 
*a*b^2*c^2*d^10 + a^2*b*d^12)*x^2 + 2*(b^3*c^5*d^7 + 2*a*b^2*c^3*d^9 + a^2 
*b*c*d^11)*x), -1/12*(6*(20*b^3*c^8 + 33*a*b^2*c^6*d^2 + 12*a^2*b*c^4*d^4 
+ (20*b^3*c^6*d^2 + 33*a*b^2*c^4*d^4 + 12*a^2*b*c^2*d^6)*x^2 + 2*(20*b^3*c 
^7*d + 33*a*b^2*c^5*d^3 + 12*a^2*b*c^3*d^5)*x)*sqrt(-b*c^2 - a*d^2)*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*(20*b^3*c^9 + 43*a*b^2*c^7*d^2 + 26*a^2*b*...
 

Sympy [F]

\[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\int \frac {x^{4} \sqrt {a + b x^{2}}}{\left (c + d x\right )^{3}}\, dx \] Input:

integrate(x**4*(b*x**2+a)**(1/2)/(d*x+c)**3,x)
 

Output:

Integral(x**4*sqrt(a + b*x**2)/(c + d*x)**3, x)
                                                                                    
                                                                                    
 

Maxima [A] (verification not implemented)

Time = 0.08 (sec) , antiderivative size = 437, normalized size of antiderivative = 1.58 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=-\frac {\sqrt {b x^{2} + a} b c^{5}}{2 \, {\left (b c^{2} d^{6} x + a d^{8} x + b c^{3} d^{5} + a c d^{7}\right )}} - \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}} c^{4}}{2 \, {\left (b c^{2} d^{5} x^{2} + a d^{7} x^{2} + 2 \, b c^{3} d^{4} x + 2 \, a c d^{6} x + b c^{4} d^{3} + a c^{2} d^{5}\right )}} + \frac {\sqrt {b x^{2} + a} b c^{4}}{2 \, {\left (b c^{2} d^{5} + a d^{7}\right )}} + \frac {4 \, \sqrt {b x^{2} + a} c^{3}}{d^{6} x + c d^{5}} - \frac {3 \, \sqrt {b x^{2} + a} c x}{2 \, d^{4}} - \frac {10 \, \sqrt {b} c^{3} \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{d^{6}} - \frac {3 \, a c \operatorname {arsinh}\left (\frac {b x}{\sqrt {a b}}\right )}{2 \, \sqrt {b} d^{4}} - \frac {b^{2} c^{6} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, {\left (a + \frac {b c^{2}}{d^{2}}\right )}^{\frac {3}{2}} d^{9}} + \frac {9 \, b c^{4} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{2 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} d^{7}} + \frac {6 \, \sqrt {a + \frac {b c^{2}}{d^{2}}} c^{2} \operatorname {arsinh}\left (\frac {b c x}{\sqrt {a b} {\left | d x + c \right |}} - \frac {a d}{\sqrt {a b} {\left | d x + c \right |}}\right )}{d^{5}} + \frac {6 \, \sqrt {b x^{2} + a} c^{2}}{d^{5}} + \frac {{\left (b x^{2} + a\right )}^{\frac {3}{2}}}{3 \, b d^{3}} \] Input:

integrate(x^4*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="maxima")
 

Output:

-1/2*sqrt(b*x^2 + a)*b*c^5/(b*c^2*d^6*x + a*d^8*x + b*c^3*d^5 + a*c*d^7) - 
 1/2*(b*x^2 + a)^(3/2)*c^4/(b*c^2*d^5*x^2 + a*d^7*x^2 + 2*b*c^3*d^4*x + 2* 
a*c*d^6*x + b*c^4*d^3 + a*c^2*d^5) + 1/2*sqrt(b*x^2 + a)*b*c^4/(b*c^2*d^5 
+ a*d^7) + 4*sqrt(b*x^2 + a)*c^3/(d^6*x + c*d^5) - 3/2*sqrt(b*x^2 + a)*c*x 
/d^4 - 10*sqrt(b)*c^3*arcsinh(b*x/sqrt(a*b))/d^6 - 3/2*a*c*arcsinh(b*x/sqr 
t(a*b))/(sqrt(b)*d^4) - 1/2*b^2*c^6*arcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) 
 - a*d/(sqrt(a*b)*abs(d*x + c)))/((a + b*c^2/d^2)^(3/2)*d^9) + 9/2*b*c^4*a 
rcsinh(b*c*x/(sqrt(a*b)*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/(sqr 
t(a + b*c^2/d^2)*d^7) + 6*sqrt(a + b*c^2/d^2)*c^2*arcsinh(b*c*x/(sqrt(a*b) 
*abs(d*x + c)) - a*d/(sqrt(a*b)*abs(d*x + c)))/d^5 + 6*sqrt(b*x^2 + a)*c^2 
/d^5 + 1/3*(b*x^2 + a)^(3/2)/(b*d^3)
 

Giac [B] (verification not implemented)

Leaf count of result is larger than twice the leaf count of optimal. 497 vs. \(2 (244) = 488\).

Time = 0.17 (sec) , antiderivative size = 497, normalized size of antiderivative = 1.79 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\frac {1}{6} \, \sqrt {b x^{2} + a} {\left (x {\left (\frac {2 \, x}{d^{3}} - \frac {9 \, c}{d^{4}}\right )} + \frac {2 \, {\left (18 \, b c^{2} d^{13} + a d^{15}\right )}}{b d^{18}}\right )} + \frac {{\left (20 \, b^{2} c^{6} + 33 \, a b c^{4} d^{2} + 12 \, a^{2} c^{2} d^{4}\right )} \arctan \left (-\frac {{\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} d + \sqrt {b} c}{\sqrt {-b c^{2} - a d^{2}}}\right )}{{\left (b c^{2} d^{6} + a d^{8}\right )} \sqrt {-b c^{2} - a d^{2}}} + \frac {10 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} b^{2} c^{6} d + 9 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{3} a b c^{4} d^{3} + 18 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} b^{\frac {5}{2}} c^{7} + 7 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a b^{\frac {3}{2}} c^{5} d^{2} - 8 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} a^{2} \sqrt {b} c^{3} d^{4} - 26 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a b^{2} c^{6} d - 23 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} a^{2} b c^{4} d^{3} + 9 \, a^{2} b^{\frac {3}{2}} c^{5} d^{2} + 8 \, a^{3} \sqrt {b} c^{3} d^{4}}{{\left (b c^{2} d^{6} + a d^{8}\right )} {\left ({\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )}^{2} d + 2 \, {\left (\sqrt {b} x - \sqrt {b x^{2} + a}\right )} \sqrt {b} c - a d\right )}^{2}} + \frac {{\left (20 \, b c^{3} + 3 \, a c d^{2}\right )} \log \left ({\left | -\sqrt {b} x + \sqrt {b x^{2} + a} \right |}\right )}{2 \, \sqrt {b} d^{6}} \] Input:

integrate(x^4*(b*x^2+a)^(1/2)/(d*x+c)^3,x, algorithm="giac")
 

Output:

1/6*sqrt(b*x^2 + a)*(x*(2*x/d^3 - 9*c/d^4) + 2*(18*b*c^2*d^13 + a*d^15)/(b 
*d^18)) + (20*b^2*c^6 + 33*a*b*c^4*d^2 + 12*a^2*c^2*d^4)*arctan(-((sqrt(b) 
*x - sqrt(b*x^2 + a))*d + sqrt(b)*c)/sqrt(-b*c^2 - a*d^2))/((b*c^2*d^6 + a 
*d^8)*sqrt(-b*c^2 - a*d^2)) + (10*(sqrt(b)*x - sqrt(b*x^2 + a))^3*b^2*c^6* 
d + 9*(sqrt(b)*x - sqrt(b*x^2 + a))^3*a*b*c^4*d^3 + 18*(sqrt(b)*x - sqrt(b 
*x^2 + a))^2*b^(5/2)*c^7 + 7*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a*b^(3/2)*c^5 
*d^2 - 8*(sqrt(b)*x - sqrt(b*x^2 + a))^2*a^2*sqrt(b)*c^3*d^4 - 26*(sqrt(b) 
*x - sqrt(b*x^2 + a))*a*b^2*c^6*d - 23*(sqrt(b)*x - sqrt(b*x^2 + a))*a^2*b 
*c^4*d^3 + 9*a^2*b^(3/2)*c^5*d^2 + 8*a^3*sqrt(b)*c^3*d^4)/((b*c^2*d^6 + a* 
d^8)*((sqrt(b)*x - sqrt(b*x^2 + a))^2*d + 2*(sqrt(b)*x - sqrt(b*x^2 + a))* 
sqrt(b)*c - a*d)^2) + 1/2*(20*b*c^3 + 3*a*c*d^2)*log(abs(-sqrt(b)*x + sqrt 
(b*x^2 + a)))/(sqrt(b)*d^6)
 

Mupad [F(-1)]

Timed out. \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx=\int \frac {x^4\,\sqrt {b\,x^2+a}}{{\left (c+d\,x\right )}^3} \,d x \] Input:

int((x^4*(a + b*x^2)^(1/2))/(c + d*x)^3,x)
 

Output:

int((x^4*(a + b*x^2)^(1/2))/(c + d*x)^3, x)
 

Reduce [B] (verification not implemented)

Time = 0.22 (sec) , antiderivative size = 1943, normalized size of antiderivative = 7.01 \[ \int \frac {x^4 \sqrt {a+b x^2}}{(c+d x)^3} \, dx =\text {Too large to display} \] Input:

int(x^4*(b*x^2+a)^(1/2)/(d*x+c)^3,x)
 

Output:

(72*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d 
 + b*c*x)*a**2*b*c**4*d**4 + 144*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2 
)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2*b*c**3*d**5*x + 72*sqrt(a*d**2 
 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a**2* 
b*c**2*d**6*x**2 + 198*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d 
**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**6*d**2 + 396*sqrt(a*d**2 + b*c**2)* 
log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*a*b**2*c**5*d**3 
*x + 198*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) 
- a*d + b*c*x)*a*b**2*c**4*d**4*x**2 + 120*sqrt(a*d**2 + b*c**2)*log(sqrt( 
a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b**3*c**8 + 240*sqrt(a*d* 
*2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + b*c**2) - a*d + b*c*x)*b** 
3*c**7*d*x + 120*sqrt(a*d**2 + b*c**2)*log(sqrt(a + b*x**2)*sqrt(a*d**2 + 
b*c**2) - a*d + b*c*x)*b**3*c**6*d**2*x**2 - 72*sqrt(a*d**2 + b*c**2)*log( 
c + d*x)*a**2*b*c**4*d**4 - 144*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b* 
c**3*d**5*x - 72*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a**2*b*c**2*d**6*x**2 
- 198*sqrt(a*d**2 + b*c**2)*log(c + d*x)*a*b**2*c**6*d**2 - 396*sqrt(a*d** 
2 + b*c**2)*log(c + d*x)*a*b**2*c**5*d**3*x - 198*sqrt(a*d**2 + b*c**2)*lo 
g(c + d*x)*a*b**2*c**4*d**4*x**2 - 120*sqrt(a*d**2 + b*c**2)*log(c + d*x)* 
b**3*c**8 - 240*sqrt(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**7*d*x - 120*sqr 
t(a*d**2 + b*c**2)*log(c + d*x)*b**3*c**6*d**2*x**2 + 4*sqrt(a + b*x**2...