3.62 \(\int \left (a+b x^2\right )^{5/2} \left (c+d x^2\right )^3 \, dx\)

Optimal. Leaf size=349 \[ \frac{d x \left (a+b x^2\right )^{7/2} \left (15 a^2 d^2-68 a b c d+152 b^2 c^2\right )}{960 b^3}+\frac{x \left (a+b x^2\right )^{5/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1920 b^3}+\frac{a x \left (a+b x^2\right )^{3/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1536 b^3}+\frac{a^2 x \sqrt{a+b x^2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^3}+\frac{a^3 \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right ) (16 b c-5 a d)}{120 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b} \]

[Out]

(a^2*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x
^2])/(1024*b^3) + (a*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3
)*x*(a + b*x^2)^(3/2))/(1536*b^3) + ((320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c
*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(5/2))/(1920*b^3) + (d*(152*b^2*c^2 - 68*a*b*c*d
+ 15*a^2*d^2)*x*(a + b*x^2)^(7/2))/(960*b^3) + (d*(16*b*c - 5*a*d)*x*(a + b*x^2
)^(7/2)*(c + d*x^2))/(120*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2)^2)/(12*b) +
(a^3*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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Rubi [A]  time = 0.56244, antiderivative size = 349, normalized size of antiderivative = 1., number of steps used = 8, number of rules used = 6, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.286 \[ \frac{d x \left (a+b x^2\right )^{7/2} \left (15 a^2 d^2-68 a b c d+152 b^2 c^2\right )}{960 b^3}+\frac{x \left (a+b x^2\right )^{5/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1920 b^3}+\frac{a x \left (a+b x^2\right )^{3/2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1536 b^3}+\frac{a^2 x \sqrt{a+b x^2} \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right )}{1024 b^3}+\frac{a^3 \left (-5 a^3 d^3+36 a^2 b c d^2-120 a b^2 c^2 d+320 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{1024 b^{7/2}}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right ) (16 b c-5 a d)}{120 b^2}+\frac{d x \left (a+b x^2\right )^{7/2} \left (c+d x^2\right )^2}{12 b} \]

Antiderivative was successfully verified.

[In]  Int[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]

[Out]

(a^2*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*x*Sqrt[a + b*x
^2])/(1024*b^3) + (a*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3
)*x*(a + b*x^2)^(3/2))/(1536*b^3) + ((320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c
*d^2 - 5*a^3*d^3)*x*(a + b*x^2)^(5/2))/(1920*b^3) + (d*(152*b^2*c^2 - 68*a*b*c*d
+ 15*a^2*d^2)*x*(a + b*x^2)^(7/2))/(960*b^3) + (d*(16*b*c - 5*a*d)*x*(a + b*x^2
)^(7/2)*(c + d*x^2))/(120*b^2) + (d*x*(a + b*x^2)^(7/2)*(c + d*x^2)^2)/(12*b) +
(a^3*(320*b^3*c^3 - 120*a*b^2*c^2*d + 36*a^2*b*c*d^2 - 5*a^3*d^3)*ArcTanh[(Sqrt[
b]*x)/Sqrt[a + b*x^2]])/(1024*b^(7/2))

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Rubi in Sympy [A]  time = 65.8349, size = 352, normalized size = 1.01 \[ - \frac{a^{3} \left (5 a^{3} d^{3} - 36 a^{2} b c d^{2} + 120 a b^{2} c^{2} d - 320 b^{3} c^{3}\right ) \operatorname{atanh}{\left (\frac{\sqrt{b} x}{\sqrt{a + b x^{2}}} \right )}}{1024 b^{\frac{7}{2}}} - \frac{a^{2} x \sqrt{a + b x^{2}} \left (5 a^{3} d^{3} - 36 a^{2} b c d^{2} + 120 a b^{2} c^{2} d - 320 b^{3} c^{3}\right )}{1024 b^{3}} - \frac{a x \left (a + b x^{2}\right )^{\frac{3}{2}} \left (5 a^{3} d^{3} - 36 a^{2} b c d^{2} + 120 a b^{2} c^{2} d - 320 b^{3} c^{3}\right )}{1536 b^{3}} + \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (c + d x^{2}\right )^{2}}{12 b} - \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (c + d x^{2}\right ) \left (5 a d - 16 b c\right )}{120 b^{2}} + \frac{d x \left (a + b x^{2}\right )^{\frac{7}{2}} \left (15 a^{2} d^{2} - 68 a b c d + 152 b^{2} c^{2}\right )}{960 b^{3}} - \frac{x \left (a + b x^{2}\right )^{\frac{5}{2}} \left (5 a^{3} d^{3} - 36 a^{2} b c d^{2} + 120 a b^{2} c^{2} d - 320 b^{3} c^{3}\right )}{1920 b^{3}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  rubi_integrate((b*x**2+a)**(5/2)*(d*x**2+c)**3,x)

[Out]

-a**3*(5*a**3*d**3 - 36*a**2*b*c*d**2 + 120*a*b**2*c**2*d - 320*b**3*c**3)*atanh
(sqrt(b)*x/sqrt(a + b*x**2))/(1024*b**(7/2)) - a**2*x*sqrt(a + b*x**2)*(5*a**3*d
**3 - 36*a**2*b*c*d**2 + 120*a*b**2*c**2*d - 320*b**3*c**3)/(1024*b**3) - a*x*(a
+ b*x**2)**(3/2)*(5*a**3*d**3 - 36*a**2*b*c*d**2 + 120*a*b**2*c**2*d - 320*b**3
*c**3)/(1536*b**3) + d*x*(a + b*x**2)**(7/2)*(c + d*x**2)**2/(12*b) - d*x*(a + b
*x**2)**(7/2)*(c + d*x**2)*(5*a*d - 16*b*c)/(120*b**2) + d*x*(a + b*x**2)**(7/2)
*(15*a**2*d**2 - 68*a*b*c*d + 152*b**2*c**2)/(960*b**3) - x*(a + b*x**2)**(5/2)*
(5*a**3*d**3 - 36*a**2*b*c*d**2 + 120*a*b**2*c**2*d - 320*b**3*c**3)/(1920*b**3)

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Mathematica [A]  time = 0.30782, size = 270, normalized size = 0.77 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (75 a^5 d^3-10 a^4 b d^2 \left (54 c+5 d x^2\right )+40 a^3 b^2 d \left (45 c^2+9 c d x^2+d^2 x^4\right )+48 a^2 b^3 \left (220 c^3+295 c^2 d x^2+186 c d^2 x^4+45 d^3 x^6\right )+64 a b^4 x^2 \left (130 c^3+255 c^2 d x^2+189 c d^2 x^4+50 d^3 x^6\right )+128 b^5 x^4 \left (20 c^3+45 c^2 d x^2+36 c d^2 x^4+10 d^3 x^6\right )\right )-15 a^3 \left (5 a^3 d^3-36 a^2 b c d^2+120 a b^2 c^2 d-320 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{15360 b^{7/2}} \]

Antiderivative was successfully verified.

[In]  Integrate[(a + b*x^2)^(5/2)*(c + d*x^2)^3,x]

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(75*a^5*d^3 - 10*a^4*b*d^2*(54*c + 5*d*x^2) + 40*a^3*
b^2*d*(45*c^2 + 9*c*d*x^2 + d^2*x^4) + 128*b^5*x^4*(20*c^3 + 45*c^2*d*x^2 + 36*c
*d^2*x^4 + 10*d^3*x^6) + 48*a^2*b^3*(220*c^3 + 295*c^2*d*x^2 + 186*c*d^2*x^4 + 4
5*d^3*x^6) + 64*a*b^4*x^2*(130*c^3 + 255*c^2*d*x^2 + 189*c*d^2*x^4 + 50*d^3*x^6)
) - 15*a^3*(-320*b^3*c^3 + 120*a*b^2*c^2*d - 36*a^2*b*c*d^2 + 5*a^3*d^3)*Log[b*x
+ Sqrt[b]*Sqrt[a + b*x^2]])/(15360*b^(7/2))

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Maple [A]  time = 0.018, size = 476, normalized size = 1.4 \[{\frac{{c}^{3}x}{6} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{5\,a{c}^{3}x}{24} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{5\,{a}^{2}{c}^{3}x}{16}\sqrt{b{x}^{2}+a}}+{\frac{5\,{c}^{3}{a}^{3}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){\frac{1}{\sqrt{b}}}}+{\frac{{d}^{3}{x}^{5}}{12\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{d}^{3}{x}^{3}}{24\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{{d}^{3}{a}^{2}x}{64\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{{a}^{3}{d}^{3}x}{384\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{d}^{3}{a}^{4}x}{1536\,{b}^{3}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{5\,{d}^{3}{a}^{5}x}{1024\,{b}^{3}}\sqrt{b{x}^{2}+a}}-{\frac{5\,{d}^{3}{a}^{6}}{1024}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+{\frac{3\,c{d}^{2}{x}^{3}}{10\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{9\,c{d}^{2}ax}{80\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}+{\frac{3\,{a}^{2}c{d}^{2}x}{160\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}+{\frac{3\,c{d}^{2}{a}^{3}x}{128\,{b}^{2}} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}+{\frac{9\,c{d}^{2}{a}^{4}x}{256\,{b}^{2}}\sqrt{b{x}^{2}+a}}+{\frac{9\,c{d}^{2}{a}^{5}}{256}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{5}{2}}}}+{\frac{3\,{c}^{2}dx}{8\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{7}{2}}}}-{\frac{a{c}^{2}dx}{16\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{5}{2}}}}-{\frac{5\,{c}^{2}d{a}^{2}x}{64\,b} \left ( b{x}^{2}+a \right ) ^{{\frac{3}{2}}}}-{\frac{15\,{c}^{2}d{a}^{3}x}{128\,b}\sqrt{b{x}^{2}+a}}-{\frac{15\,{c}^{2}d{a}^{4}}{128}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{3}{2}}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  int((b*x^2+a)^(5/2)*(d*x^2+c)^3,x)

[Out]

1/6*c^3*x*(b*x^2+a)^(5/2)+5/24*c^3*a*x*(b*x^2+a)^(3/2)+5/16*c^3*a^2*x*(b*x^2+a)^
(1/2)+5/16*c^3*a^3/b^(1/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+1/12*d^3*x^5*(b*x^2+a)^
(7/2)/b-1/24*d^3*a/b^2*x^3*(b*x^2+a)^(7/2)+1/64*d^3*a^2/b^3*x*(b*x^2+a)^(7/2)-1/
384*d^3*a^3/b^3*x*(b*x^2+a)^(5/2)-5/1536*d^3*a^4/b^3*x*(b*x^2+a)^(3/2)-5/1024*d^
3*a^5/b^3*x*(b*x^2+a)^(1/2)-5/1024*d^3*a^6/b^(7/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))
+3/10*c*d^2*x^3*(b*x^2+a)^(7/2)/b-9/80*c*d^2*a/b^2*x*(b*x^2+a)^(7/2)+3/160*c*d^2
*a^2/b^2*x*(b*x^2+a)^(5/2)+3/128*c*d^2*a^3/b^2*x*(b*x^2+a)^(3/2)+9/256*c*d^2*a^4
/b^2*x*(b*x^2+a)^(1/2)+9/256*c*d^2*a^5/b^(5/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+3/8
*c^2*d*x*(b*x^2+a)^(7/2)/b-1/16*c^2*d*a/b*x*(b*x^2+a)^(5/2)-5/64*c^2*d*a^2/b*x*(
b*x^2+a)^(3/2)-15/128*c^2*d*a^3/b*x*(b*x^2+a)^(1/2)-15/128*c^2*d*a^4/b^(3/2)*ln(
x*b^(1/2)+(b*x^2+a)^(1/2))

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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)*(d*x^2 + c)^3,x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 1.24011, size = 1, normalized size = 0. \[ \left [\frac{2 \,{\left (1280 \, b^{5} d^{3} x^{11} + 128 \,{\left (36 \, b^{5} c d^{2} + 25 \, a b^{4} d^{3}\right )} x^{9} + 144 \,{\left (40 \, b^{5} c^{2} d + 84 \, a b^{4} c d^{2} + 15 \, a^{2} b^{3} d^{3}\right )} x^{7} + 8 \,{\left (320 \, b^{5} c^{3} + 2040 \, a b^{4} c^{2} d + 1116 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{5} + 10 \,{\left (832 \, a b^{4} c^{3} + 1416 \, a^{2} b^{3} c^{2} d + 36 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{3} + 15 \,{\left (704 \, a^{2} b^{3} c^{3} + 120 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 5 \, a^{5} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{b} - 15 \,{\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \log \left (2 \, \sqrt{b x^{2} + a} b x -{\left (2 \, b x^{2} + a\right )} \sqrt{b}\right )}{30720 \, b^{\frac{7}{2}}}, \frac{{\left (1280 \, b^{5} d^{3} x^{11} + 128 \,{\left (36 \, b^{5} c d^{2} + 25 \, a b^{4} d^{3}\right )} x^{9} + 144 \,{\left (40 \, b^{5} c^{2} d + 84 \, a b^{4} c d^{2} + 15 \, a^{2} b^{3} d^{3}\right )} x^{7} + 8 \,{\left (320 \, b^{5} c^{3} + 2040 \, a b^{4} c^{2} d + 1116 \, a^{2} b^{3} c d^{2} + 5 \, a^{3} b^{2} d^{3}\right )} x^{5} + 10 \,{\left (832 \, a b^{4} c^{3} + 1416 \, a^{2} b^{3} c^{2} d + 36 \, a^{3} b^{2} c d^{2} - 5 \, a^{4} b d^{3}\right )} x^{3} + 15 \,{\left (704 \, a^{2} b^{3} c^{3} + 120 \, a^{3} b^{2} c^{2} d - 36 \, a^{4} b c d^{2} + 5 \, a^{5} d^{3}\right )} x\right )} \sqrt{b x^{2} + a} \sqrt{-b} + 15 \,{\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right )}{15360 \, \sqrt{-b} b^{3}}\right ] \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^3,x, algorithm="fricas")

[Out]

[1/30720*(2*(1280*b^5*d^3*x^11 + 128*(36*b^5*c*d^2 + 25*a*b^4*d^3)*x^9 + 144*(40
*b^5*c^2*d + 84*a*b^4*c*d^2 + 15*a^2*b^3*d^3)*x^7 + 8*(320*b^5*c^3 + 2040*a*b^4*
c^2*d + 1116*a^2*b^3*c*d^2 + 5*a^3*b^2*d^3)*x^5 + 10*(832*a*b^4*c^3 + 1416*a^2*b
^3*c^2*d + 36*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x^3 + 15*(704*a^2*b^3*c^3 + 120*a^3*b
^2*c^2*d - 36*a^4*b*c*d^2 + 5*a^5*d^3)*x)*sqrt(b*x^2 + a)*sqrt(b) - 15*(320*a^3*
b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c*d^2 - 5*a^6*d^3)*log(2*sqrt(b*x^2 + a)*
b*x - (2*b*x^2 + a)*sqrt(b)))/b^(7/2), 1/15360*((1280*b^5*d^3*x^11 + 128*(36*b^5
*c*d^2 + 25*a*b^4*d^3)*x^9 + 144*(40*b^5*c^2*d + 84*a*b^4*c*d^2 + 15*a^2*b^3*d^3
)*x^7 + 8*(320*b^5*c^3 + 2040*a*b^4*c^2*d + 1116*a^2*b^3*c*d^2 + 5*a^3*b^2*d^3)*
x^5 + 10*(832*a*b^4*c^3 + 1416*a^2*b^3*c^2*d + 36*a^3*b^2*c*d^2 - 5*a^4*b*d^3)*x
^3 + 15*(704*a^2*b^3*c^3 + 120*a^3*b^2*c^2*d - 36*a^4*b*c*d^2 + 5*a^5*d^3)*x)*sq
rt(b*x^2 + a)*sqrt(-b) + 15*(320*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c*d^
2 - 5*a^6*d^3)*arctan(sqrt(-b)*x/sqrt(b*x^2 + a)))/(sqrt(-b)*b^3)]

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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((b*x**2+a)**(5/2)*(d*x**2+c)**3,x)

[Out]

Timed out

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GIAC/XCAS [A]  time = 0.240651, size = 433, normalized size = 1.24 \[ \frac{1}{15360} \,{\left (2 \,{\left (4 \,{\left (2 \,{\left (8 \,{\left (10 \, b^{2} d^{3} x^{2} + \frac{36 \, b^{12} c d^{2} + 25 \, a b^{11} d^{3}}{b^{10}}\right )} x^{2} + \frac{9 \,{\left (40 \, b^{12} c^{2} d + 84 \, a b^{11} c d^{2} + 15 \, a^{2} b^{10} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac{320 \, b^{12} c^{3} + 2040 \, a b^{11} c^{2} d + 1116 \, a^{2} b^{10} c d^{2} + 5 \, a^{3} b^{9} d^{3}}{b^{10}}\right )} x^{2} + \frac{5 \,{\left (832 \, a b^{11} c^{3} + 1416 \, a^{2} b^{10} c^{2} d + 36 \, a^{3} b^{9} c d^{2} - 5 \, a^{4} b^{8} d^{3}\right )}}{b^{10}}\right )} x^{2} + \frac{15 \,{\left (704 \, a^{2} b^{10} c^{3} + 120 \, a^{3} b^{9} c^{2} d - 36 \, a^{4} b^{8} c d^{2} + 5 \, a^{5} b^{7} d^{3}\right )}}{b^{10}}\right )} \sqrt{b x^{2} + a} x - \frac{{\left (320 \, a^{3} b^{3} c^{3} - 120 \, a^{4} b^{2} c^{2} d + 36 \, a^{5} b c d^{2} - 5 \, a^{6} d^{3}\right )}{\rm ln}\left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{1024 \, b^{\frac{7}{2}}} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]  integrate((b*x^2 + a)^(5/2)*(d*x^2 + c)^3,x, algorithm="giac")

[Out]

1/15360*(2*(4*(2*(8*(10*b^2*d^3*x^2 + (36*b^12*c*d^2 + 25*a*b^11*d^3)/b^10)*x^2
+ 9*(40*b^12*c^2*d + 84*a*b^11*c*d^2 + 15*a^2*b^10*d^3)/b^10)*x^2 + (320*b^12*c^
3 + 2040*a*b^11*c^2*d + 1116*a^2*b^10*c*d^2 + 5*a^3*b^9*d^3)/b^10)*x^2 + 5*(832*
a*b^11*c^3 + 1416*a^2*b^10*c^2*d + 36*a^3*b^9*c*d^2 - 5*a^4*b^8*d^3)/b^10)*x^2 +
15*(704*a^2*b^10*c^3 + 120*a^3*b^9*c^2*d - 36*a^4*b^8*c*d^2 + 5*a^5*b^7*d^3)/b^
10)*sqrt(b*x^2 + a)*x - 1/1024*(320*a^3*b^3*c^3 - 120*a^4*b^2*c^2*d + 36*a^5*b*c
*d^2 - 5*a^6*d^3)*ln(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(7/2)