3.954 \(\int \frac{x^5 \left (a+b x^2\right )^{5/2}}{\sqrt{c+d x^2}} \, dx\)

Optimal. Leaf size=340 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{256 b^2 d^5}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{384 b^2 d^4}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{480 b^2 d^3}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}-\frac{3 \left (a+b x^2\right )^{7/2} \sqrt{c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac{x^2 \left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}}{10 b d} \]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/(256*b^2*d^5) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*
x^2)^(3/2)*Sqrt[c + d*x^2])/(384*b^2*d^4) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^
2)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(480*b^2*d^3) - (3*(3*b*c + a*d)*(a + b*x^
2)^(7/2)*Sqrt[c + d*x^2])/(80*b^2*d^2) + (x^2*(a + b*x^2)^(7/2)*Sqrt[c + d*x^2])
/(10*b*d) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(256*b^(5/2)*d^(11/2))

_______________________________________________________________________________________

Rubi [A]  time = 0.930944, antiderivative size = 340, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 26, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.231 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} (b c-a d)^2 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{256 b^2 d^5}-\frac{\left (a+b x^2\right )^{3/2} \sqrt{c+d x^2} (b c-a d) \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{384 b^2 d^4}+\frac{\left (a+b x^2\right )^{5/2} \sqrt{c+d x^2} \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right )}{480 b^2 d^3}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \tanh ^{-1}\left (\frac{\sqrt{d} \sqrt{a+b x^2}}{\sqrt{b} \sqrt{c+d x^2}}\right )}{256 b^{5/2} d^{11/2}}-\frac{3 \left (a+b x^2\right )^{7/2} \sqrt{c+d x^2} (a d+3 b c)}{80 b^2 d^2}+\frac{x^2 \left (a+b x^2\right )^{7/2} \sqrt{c+d x^2}}{10 b d} \]

Antiderivative was successfully verified.

[In]  Int[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

((b*c - a*d)^2*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Sqrt[a + b*x^2]*Sqrt[c + d*
x^2])/(256*b^2*d^5) - ((b*c - a*d)*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*(a + b*
x^2)^(3/2)*Sqrt[c + d*x^2])/(384*b^2*d^4) + ((63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^
2)*(a + b*x^2)^(5/2)*Sqrt[c + d*x^2])/(480*b^2*d^3) - (3*(3*b*c + a*d)*(a + b*x^
2)^(7/2)*Sqrt[c + d*x^2])/(80*b^2*d^2) + (x^2*(a + b*x^2)^(7/2)*Sqrt[c + d*x^2])
/(10*b*d) - ((b*c - a*d)^3*(63*b^2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*ArcTanh[(Sqrt[d
]*Sqrt[a + b*x^2])/(Sqrt[b]*Sqrt[c + d*x^2])])/(256*b^(5/2)*d^(11/2))

_______________________________________________________________________________________

Rubi in Sympy [A]  time = 81.7891, size = 323, normalized size = 0.95 \[ \frac{x^{2} \left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}}}{10 b d} - \frac{3 \left (a + b x^{2}\right )^{\frac{7}{2}} \sqrt{c + d x^{2}} \left (a d + 3 b c\right )}{80 b^{2} d^{2}} + \frac{\left (a + b x^{2}\right )^{\frac{5}{2}} \sqrt{c + d x^{2}} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{480 b^{2} d^{3}} + \frac{\left (a + b x^{2}\right )^{\frac{3}{2}} \sqrt{c + d x^{2}} \left (a d - b c\right ) \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{384 b^{2} d^{4}} + \frac{\sqrt{a + b x^{2}} \sqrt{c + d x^{2}} \left (a d - b c\right )^{2} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right )}{256 b^{2} d^{5}} + \frac{\left (a d - b c\right )^{3} \left (3 a^{2} d^{2} + 14 a b c d + 63 b^{2} c^{2}\right ) \operatorname{atanh}{\left (\frac{\sqrt{d} \sqrt{a + b x^{2}}}{\sqrt{b} \sqrt{c + d x^{2}}} \right )}}{256 b^{\frac{5}{2}} d^{\frac{11}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate(x**5*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

x**2*(a + b*x**2)**(7/2)*sqrt(c + d*x**2)/(10*b*d) - 3*(a + b*x**2)**(7/2)*sqrt(
c + d*x**2)*(a*d + 3*b*c)/(80*b**2*d**2) + (a + b*x**2)**(5/2)*sqrt(c + d*x**2)*
(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(480*b**2*d**3) + (a + b*x**2)**(3/2)*
sqrt(c + d*x**2)*(a*d - b*c)*(3*a**2*d**2 + 14*a*b*c*d + 63*b**2*c**2)/(384*b**2
*d**4) + sqrt(a + b*x**2)*sqrt(c + d*x**2)*(a*d - b*c)**2*(3*a**2*d**2 + 14*a*b*
c*d + 63*b**2*c**2)/(256*b**2*d**5) + (a*d - b*c)**3*(3*a**2*d**2 + 14*a*b*c*d +
63*b**2*c**2)*atanh(sqrt(d)*sqrt(a + b*x**2)/(sqrt(b)*sqrt(c + d*x**2)))/(256*b
**(5/2)*d**(11/2))

_______________________________________________________________________________________

Mathematica [A]  time = 0.305848, size = 274, normalized size = 0.81 \[ \frac{\sqrt{a+b x^2} \sqrt{c+d x^2} \left (-45 a^4 d^4+30 a^3 b d^3 \left (d x^2-3 c\right )+2 a^2 b^2 d^2 \left (782 c^2-481 c d x^2+372 d^2 x^4\right )+2 a b^3 d \left (-1155 c^3+749 c^2 d x^2-592 c d^2 x^4+504 d^3 x^6\right )+b^4 \left (945 c^4-630 c^3 d x^2+504 c^2 d^2 x^4-432 c d^3 x^6+384 d^4 x^8\right )\right )}{3840 b^2 d^5}-\frac{(b c-a d)^3 \left (3 a^2 d^2+14 a b c d+63 b^2 c^2\right ) \log \left (2 \sqrt{b} \sqrt{d} \sqrt{a+b x^2} \sqrt{c+d x^2}+a d+b c+2 b d x^2\right )}{512 b^{5/2} d^{11/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(x^5*(a + b*x^2)^(5/2))/Sqrt[c + d*x^2],x]

[Out]

(Sqrt[a + b*x^2]*Sqrt[c + d*x^2]*(-45*a^4*d^4 + 30*a^3*b*d^3*(-3*c + d*x^2) + 2*
a^2*b^2*d^2*(782*c^2 - 481*c*d*x^2 + 372*d^2*x^4) + 2*a*b^3*d*(-1155*c^3 + 749*c
^2*d*x^2 - 592*c*d^2*x^4 + 504*d^3*x^6) + b^4*(945*c^4 - 630*c^3*d*x^2 + 504*c^2
*d^2*x^4 - 432*c*d^3*x^6 + 384*d^4*x^8)))/(3840*b^2*d^5) - ((b*c - a*d)^3*(63*b^
2*c^2 + 14*a*b*c*d + 3*a^2*d^2)*Log[b*c + a*d + 2*b*d*x^2 + 2*Sqrt[b]*Sqrt[d]*Sq
rt[a + b*x^2]*Sqrt[c + d*x^2]])/(512*b^(5/2)*d^(11/2))

_______________________________________________________________________________________

Maple [B]  time = 0.05, size = 1054, normalized size = 3.1 \[ \text{result too large to display} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int(x^5*(b*x^2+a)^(5/2)/(d*x^2+c)^(1/2),x)

[Out]

1/7680*(b*x^2+a)^(1/2)*(d*x^2+c)^(1/2)*(768*x^8*b^4*d^4*(b*d*x^4+a*d*x^2+b*c*x^2
+a*c)^(1/2)*(b*d)^(1/2)+2016*x^6*a*b^3*d^4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(
b*d)^(1/2)-864*x^6*b^4*c*d^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+148
8*x^4*a^2*b^2*d^4*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)-2368*x^4*a*b^3
*c*d^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+1008*x^4*b^4*c^2*d^2*(b*d
*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+60*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/
2)*x^2*a^3*d^4*(b*d)^(1/2)*b-1924*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*a^2*c*
d^3*(b*d)^(1/2)*b^2+2996*b^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*a*c^2*d^2*(
b*d)^(1/2)-1260*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*x^2*c^3*b^4*d*(b*d)^(1/2)+45
*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b
*d)^(1/2))*a^5*d^5+75*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b
*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^4*c*d^4*b+450*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d
*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*a^3*c^2*d^3*b^2-2250*b
^3*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/
(b*d)^(1/2))*a^2*c^3*d^2+2625*b^4*ln(1/2*(2*b*d*x^2+2*(b*d*x^4+a*d*x^2+b*c*x^2+a
*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^4*a*d-945*b^5*ln(1/2*(2*b*d*x^2+2*
(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*(b*d)^(1/2)+a*d+b*c)/(b*d)^(1/2))*c^5-90*(b*
d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a^4*d^4*(b*d)^(1/2)-180*(b*d*x^4+a*d*x^2+b*c*x^
2+a*c)^(1/2)*a^3*c*d^3*(b*d)^(1/2)*b+3128*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a^
2*c^2*d^2*(b*d)^(1/2)*b^2-4620*b^3*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*a*c^3*d*(
b*d)^(1/2)+1890*(b*d*x^4+a*d*x^2+b*c*x^2+a*c)^(1/2)*c^4*b^4*(b*d)^(1/2))/(b*d*x^
4+a*d*x^2+b*c*x^2+a*c)^(1/2)/d^5/(b*d)^(1/2)/b^2

_______________________________________________________________________________________

Maxima [F]  time = 0., size = 0, normalized size = 0. \[ \text{Exception raised: ValueError} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^5/sqrt(d*x^2 + c),x, algorithm="maxima")

[Out]

Exception raised: ValueError

_______________________________________________________________________________________

Fricas [A]  time = 0.3067, size = 1, normalized size = 0. \[ \left [\frac{4 \,{\left (384 \, b^{4} d^{4} x^{8} + 945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} - 144 \,{\left (3 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + 8 \,{\left (63 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 93 \, a^{2} b^{2} d^{4}\right )} x^{4} - 2 \,{\left (315 \, b^{4} c^{3} d - 749 \, a b^{3} c^{2} d^{2} + 481 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{b d} - 15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \log \left (4 \,{\left (2 \, b^{2} d^{2} x^{2} + b^{2} c d + a b d^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} +{\left (8 \, b^{2} d^{2} x^{4} + b^{2} c^{2} + 6 \, a b c d + a^{2} d^{2} + 8 \,{\left (b^{2} c d + a b d^{2}\right )} x^{2}\right )} \sqrt{b d}\right )}{15360 \, \sqrt{b d} b^{2} d^{5}}, \frac{2 \,{\left (384 \, b^{4} d^{4} x^{8} + 945 \, b^{4} c^{4} - 2310 \, a b^{3} c^{3} d + 1564 \, a^{2} b^{2} c^{2} d^{2} - 90 \, a^{3} b c d^{3} - 45 \, a^{4} d^{4} - 144 \,{\left (3 \, b^{4} c d^{3} - 7 \, a b^{3} d^{4}\right )} x^{6} + 8 \,{\left (63 \, b^{4} c^{2} d^{2} - 148 \, a b^{3} c d^{3} + 93 \, a^{2} b^{2} d^{4}\right )} x^{4} - 2 \,{\left (315 \, b^{4} c^{3} d - 749 \, a b^{3} c^{2} d^{2} + 481 \, a^{2} b^{2} c d^{3} - 15 \, a^{3} b d^{4}\right )} x^{2}\right )} \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} \sqrt{-b d} - 15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )} \arctan \left (\frac{{\left (2 \, b d x^{2} + b c + a d\right )} \sqrt{-b d}}{2 \, \sqrt{b x^{2} + a} \sqrt{d x^{2} + c} b d}\right )}{7680 \, \sqrt{-b d} b^{2} d^{5}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^5/sqrt(d*x^2 + c),x, algorithm="fricas")

[Out]

[1/15360*(4*(384*b^4*d^4*x^8 + 945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2
*d^2 - 90*a^3*b*c*d^3 - 45*a^4*d^4 - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + 8*(63
*b^4*c^2*d^2 - 148*a*b^3*c*d^3 + 93*a^2*b^2*d^4)*x^4 - 2*(315*b^4*c^3*d - 749*a*
b^3*c^2*d^2 + 481*a^2*b^2*c*d^3 - 15*a^3*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2
+ c)*sqrt(b*d) - 15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3
*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*log(4*(2*b^2*d^2*x^2 + b^2*c*d + a*b*d
^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c) + (8*b^2*d^2*x^4 + b^2*c^2 + 6*a*b*c*d + a^2
*d^2 + 8*(b^2*c*d + a*b*d^2)*x^2)*sqrt(b*d)))/(sqrt(b*d)*b^2*d^5), 1/7680*(2*(38
4*b^4*d^4*x^8 + 945*b^4*c^4 - 2310*a*b^3*c^3*d + 1564*a^2*b^2*c^2*d^2 - 90*a^3*b
*c*d^3 - 45*a^4*d^4 - 144*(3*b^4*c*d^3 - 7*a*b^3*d^4)*x^6 + 8*(63*b^4*c^2*d^2 -
148*a*b^3*c*d^3 + 93*a^2*b^2*d^4)*x^4 - 2*(315*b^4*c^3*d - 749*a*b^3*c^2*d^2 + 4
81*a^2*b^2*c*d^3 - 15*a^3*b*d^4)*x^2)*sqrt(b*x^2 + a)*sqrt(d*x^2 + c)*sqrt(-b*d)
- 15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*d^2 - 30*a^3*b^2*c^2*d^3 -
5*a^4*b*c*d^4 - 3*a^5*d^5)*arctan(1/2*(2*b*d*x^2 + b*c + a*d)*sqrt(-b*d)/(sqrt(
b*x^2 + a)*sqrt(d*x^2 + c)*b*d)))/(sqrt(-b*d)*b^2*d^5)]

_______________________________________________________________________________________

Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \[ \text{Timed out} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate(x**5*(b*x**2+a)**(5/2)/(d*x**2+c)**(1/2),x)

[Out]

Timed out

_______________________________________________________________________________________

GIAC/XCAS [A]  time = 0.265059, size = 536, normalized size = 1.58 \[ \frac{\sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \sqrt{b x^{2} + a}{\left (2 \,{\left (b x^{2} + a\right )}{\left (4 \,{\left (b x^{2} + a\right )}{\left (6 \,{\left (b x^{2} + a\right )}{\left (\frac{8 \,{\left (b x^{2} + a\right )}}{b d} - \frac{9 \, b^{3} c d^{7} + 11 \, a b^{2} d^{8}}{b^{3} d^{9}}\right )} + \frac{63 \, b^{4} c^{2} d^{6} + 14 \, a b^{3} c d^{7} + 3 \, a^{2} b^{2} d^{8}}{b^{3} d^{9}}\right )} - \frac{5 \,{\left (63 \, b^{5} c^{3} d^{5} - 49 \, a b^{4} c^{2} d^{6} - 11 \, a^{2} b^{3} c d^{7} - 3 \, a^{3} b^{2} d^{8}\right )}}{b^{3} d^{9}}\right )} + \frac{15 \,{\left (63 \, b^{6} c^{4} d^{4} - 112 \, a b^{5} c^{3} d^{5} + 38 \, a^{2} b^{4} c^{2} d^{6} + 8 \, a^{3} b^{3} c d^{7} + 3 \, a^{4} b^{2} d^{8}\right )}}{b^{3} d^{9}}\right )} + \frac{15 \,{\left (63 \, b^{5} c^{5} - 175 \, a b^{4} c^{4} d + 150 \, a^{2} b^{3} c^{3} d^{2} - 30 \, a^{3} b^{2} c^{2} d^{3} - 5 \, a^{4} b c d^{4} - 3 \, a^{5} d^{5}\right )}{\rm ln}\left ({\left | -\sqrt{b x^{2} + a} \sqrt{b d} + \sqrt{b^{2} c +{\left (b x^{2} + a\right )} b d - a b d} \right |}\right )}{\sqrt{b d} d^{5}}}{3840 \, b{\left | b \right |}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*x^5/sqrt(d*x^2 + c),x, algorithm="giac")

[Out]

1/3840*(sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)*sqrt(b*x^2 + a)*(2*(b*x^2 + a)*(4*
(b*x^2 + a)*(6*(b*x^2 + a)*(8*(b*x^2 + a)/(b*d) - (9*b^3*c*d^7 + 11*a*b^2*d^8)/(
b^3*d^9)) + (63*b^4*c^2*d^6 + 14*a*b^3*c*d^7 + 3*a^2*b^2*d^8)/(b^3*d^9)) - 5*(63
*b^5*c^3*d^5 - 49*a*b^4*c^2*d^6 - 11*a^2*b^3*c*d^7 - 3*a^3*b^2*d^8)/(b^3*d^9)) +
15*(63*b^6*c^4*d^4 - 112*a*b^5*c^3*d^5 + 38*a^2*b^4*c^2*d^6 + 8*a^3*b^3*c*d^7 +
3*a^4*b^2*d^8)/(b^3*d^9)) + 15*(63*b^5*c^5 - 175*a*b^4*c^4*d + 150*a^2*b^3*c^3*
d^2 - 30*a^3*b^2*c^2*d^3 - 5*a^4*b*c*d^4 - 3*a^5*d^5)*ln(abs(-sqrt(b*x^2 + a)*sq
rt(b*d) + sqrt(b^2*c + (b*x^2 + a)*b*d - a*b*d)))/(sqrt(b*d)*d^5))/(b*abs(b))