3.695 \(\int \frac{x^3 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx\)
Optimal. Leaf size=144 \[ \frac{a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{9/2}}-\frac{a \sqrt{c+d x^2} (b c-a d)^2}{b^4}-\frac{a \left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^3}-\frac{a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac{\left (c+d x^2\right )^{7/2}}{7 b d} \]
[Out]
-((a*(b*c - a*d)^2*Sqrt[c + d*x^2])/b^4) - (a*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*
b^3) - (a*(c + d*x^2)^(5/2))/(5*b^2) + (c + d*x^2)^(7/2)/(7*b*d) + (a*(b*c - a*d
)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(9/2)
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Rubi [A] time = 0.377182, antiderivative size = 144, normalized size of antiderivative = 1.,
number of steps used = 7, number of rules used = 5, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.208
\[ \frac{a (b c-a d)^{5/2} \tanh ^{-1}\left (\frac{\sqrt{b} \sqrt{c+d x^2}}{\sqrt{b c-a d}}\right )}{b^{9/2}}-\frac{a \sqrt{c+d x^2} (b c-a d)^2}{b^4}-\frac{a \left (c+d x^2\right )^{3/2} (b c-a d)}{3 b^3}-\frac{a \left (c+d x^2\right )^{5/2}}{5 b^2}+\frac{\left (c+d x^2\right )^{7/2}}{7 b d} \]
Antiderivative was successfully verified.
[In] Int[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2),x]
[Out]
-((a*(b*c - a*d)^2*Sqrt[c + d*x^2])/b^4) - (a*(b*c - a*d)*(c + d*x^2)^(3/2))/(3*
b^3) - (a*(c + d*x^2)^(5/2))/(5*b^2) + (c + d*x^2)^(7/2)/(7*b*d) + (a*(b*c - a*d
)^(5/2)*ArcTanh[(Sqrt[b]*Sqrt[c + d*x^2])/Sqrt[b*c - a*d]])/b^(9/2)
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Rubi in Sympy [A] time = 43.9815, size = 122, normalized size = 0.85 \[ - \frac{a \left (c + d x^{2}\right )^{\frac{5}{2}}}{5 b^{2}} + \frac{a \left (c + d x^{2}\right )^{\frac{3}{2}} \left (a d - b c\right )}{3 b^{3}} - \frac{a \sqrt{c + d x^{2}} \left (a d - b c\right )^{2}}{b^{4}} + \frac{a \left (a d - b c\right )^{\frac{5}{2}} \operatorname{atan}{\left (\frac{\sqrt{b} \sqrt{c + d x^{2}}}{\sqrt{a d - b c}} \right )}}{b^{\frac{9}{2}}} + \frac{\left (c + d x^{2}\right )^{\frac{7}{2}}}{7 b d} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] rubi_integrate(x**3*(d*x**2+c)**(5/2)/(b*x**2+a),x)
[Out]
-a*(c + d*x**2)**(5/2)/(5*b**2) + a*(c + d*x**2)**(3/2)*(a*d - b*c)/(3*b**3) - a
*sqrt(c + d*x**2)*(a*d - b*c)**2/b**4 + a*(a*d - b*c)**(5/2)*atan(sqrt(b)*sqrt(c
+ d*x**2)/sqrt(a*d - b*c))/b**(9/2) + (c + d*x**2)**(7/2)/(7*b*d)
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Mathematica [C] time = 0.674259, size = 298, normalized size = 2.07 \[ \frac{105 a d (b c-a d)^{5/2} \log \left (-\frac{2 b^{9/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}-i \sqrt{a} d x+\sqrt{b} c\right )}{\left (a \sqrt{b} x+i a^{3/2}\right ) (b c-a d)^{7/2}}\right )+105 a d (b c-a d)^{5/2} \log \left (-\frac{2 b^{9/2} \left (\sqrt{c+d x^2} \sqrt{b c-a d}+i \sqrt{a} d x+\sqrt{b} c\right )}{\left (a \sqrt{b} x-i a^{3/2}\right ) (b c-a d)^{7/2}}\right )+2 \sqrt{b} \sqrt{c+d x^2} \left (-105 a^3 d^3+35 a^2 b d^2 \left (7 c+d x^2\right )-7 a b^2 d \left (23 c^2+11 c d x^2+3 d^2 x^4\right )+15 b^3 \left (c+d x^2\right )^3\right )}{210 b^{9/2} d} \]
Antiderivative was successfully verified.
[In] Integrate[(x^3*(c + d*x^2)^(5/2))/(a + b*x^2),x]
[Out]
(2*Sqrt[b]*Sqrt[c + d*x^2]*(-105*a^3*d^3 + 15*b^3*(c + d*x^2)^3 + 35*a^2*b*d^2*(
7*c + d*x^2) - 7*a*b^2*d*(23*c^2 + 11*c*d*x^2 + 3*d^2*x^4)) + 105*a*d*(b*c - a*d
)^(5/2)*Log[(-2*b^(9/2)*(Sqrt[b]*c - I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*
x^2]))/((b*c - a*d)^(7/2)*(I*a^(3/2) + a*Sqrt[b]*x))] + 105*a*d*(b*c - a*d)^(5/2
)*Log[(-2*b^(9/2)*(Sqrt[b]*c + I*Sqrt[a]*d*x + Sqrt[b*c - a*d]*Sqrt[c + d*x^2]))
/((b*c - a*d)^(7/2)*((-I)*a^(3/2) + a*Sqrt[b]*x))])/(210*b^(9/2)*d)
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Maple [B] time = 0.022, size = 3127, normalized size = 21.7 \[ \text{output too large to display} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] int(x^3*(d*x^2+c)^(5/2)/(b*x^2+a),x)
[Out]
-3/2*a^2/b^3/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(
-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b
*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d*c^2-1/8*a/b^3*
d*(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))
-(a*d-b*c)/b)^(3/2)*x-15/16*a/b^3*d^(1/2)*(-a*b)^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1
/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*
(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c^2+1/2*a/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d
-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(
-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+
1/b*(-a*b)^(1/2)))*c^3+1/2*a/b^2/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a
*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*
d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2
)))*c^3+a^2/b^3*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2)
)-(a*d-b*c)/b)^(1/2)*d*c+a^2/b^3*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x
+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*d*c+1/2*a^3/b^5*d^(5/2)*(-a*b)^(1/2)*ln((-
d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))-1/2*a^4/b^5/(-(a*d-b*c)/b
)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)
/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d
-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d^3-1/10*a/b^2*((x-1/b*(-a*b)^(1/2))^2*d+2
*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)-1/10*a/b^2*((x+1/b*(-a
*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(5/2)+1/8*a/
b^3*d*(-a*b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1
/2))-(a*d-b*c)/b)^(3/2)*x+15/16*a/b^3*d^(1/2)*(-a*b)^(1/2)*ln((-d*(-a*b)^(1/2)/b
+(x+1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x
+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c^2-1/4*a^2/b^4*d^2*(-a*b)^(1/2)*((x+1/b*
(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x-5
/4*a^2/b^4*d^(3/2)*(-a*b)^(1/2)*ln((-d*(-a*b)^(1/2)/b+(x+1/b*(-a*b)^(1/2))*d)/d^
(1/2)+((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c
)/b)^(1/2))*c+3/2*a^3/b^4/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b-2*d*(-a*b)^(1/
2)/b*(x+1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(
-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x+1/b*(-a*b)^(1/2)))*d^2
*c-1/2*a^3/b^5*d^(5/2)*(-a*b)^(1/2)*ln((d*(-a*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)
/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-
b*c)/b)^(1/2))-1/2*a^4/b^5/(-(a*d-b*c)/b)^(1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*
(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d^
3+1/4*a^2/b^4*d^2*(-a*b)^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1
/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x+5/4*a^2/b^4*d^(3/2)*(-a*b)^(1/2)*ln((d*(-a
*b)^(1/2)/b+(x-1/b*(-a*b)^(1/2))*d)/d^(1/2)+((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)
^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2))*c+3/2*a^3/b^4/(-(a*d-b*c)/b)^(
1/2)*ln((-2*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)
^(1/2)*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*
c)/b)^(1/2))/(x-1/b*(-a*b)^(1/2)))*d^2*c-3/2*a^2/b^3/(-(a*d-b*c)/b)^(1/2)*ln((-2
*(a*d-b*c)/b+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))+2*(-(a*d-b*c)/b)^(1/2)*((x-
1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)
)/(x-1/b*(-a*b)^(1/2)))*d*c^2+1/7*(d*x^2+c)^(7/2)/b/d+1/6*a^2/b^3*((x-1/b*(-a*b)
^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*d-1/6*a/b
^2*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b
)^(3/2)*c-1/2*a^3/b^4*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)
^(1/2))-(a*d-b*c)/b)^(1/2)*d^2-1/2*a/b^2*((x-1/b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1
/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*c^2-1/6*a/b^2*((x+1/b*(-a*b)^(1/2)
)^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*c-1/2*a^3/b^4*(
(x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1
/2)*d^2-1/2*a/b^2*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b*(-a*b)^(1/
2))-(a*d-b*c)/b)^(1/2)*c^2+1/6*a^2/b^3*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2
)/b*(x+1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(3/2)*d-7/16*a/b^3*d*(-a*b)^(1/2)*c*((x-1/
b*(-a*b)^(1/2))^2*d+2*d*(-a*b)^(1/2)/b*(x-1/b*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x
+7/16*a/b^3*d*(-a*b)^(1/2)*c*((x+1/b*(-a*b)^(1/2))^2*d-2*d*(-a*b)^(1/2)/b*(x+1/b
*(-a*b)^(1/2))-(a*d-b*c)/b)^(1/2)*x
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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((d*x^2 + c)^(5/2)*x^3/(b*x^2 + a),x, algorithm="maxima")
[Out]
Exception raised: ValueError
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Fricas [A] time = 0.294001, size = 1, normalized size = 0.01 \[ \left [\frac{105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt{\frac{b c - a d}{b}} \log \left (\frac{b^{2} d^{2} x^{4} + 8 \, b^{2} c^{2} - 8 \, a b c d + a^{2} d^{2} + 2 \,{\left (4 \, b^{2} c d - 3 \, a b d^{2}\right )} x^{2} + 4 \,{\left (b^{2} d x^{2} + 2 \, b^{2} c - a b d\right )} \sqrt{d x^{2} + c} \sqrt{\frac{b c - a d}{b}}}{b^{2} x^{4} + 2 \, a b x^{2} + a^{2}}\right ) + 4 \,{\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{420 \, b^{4} d}, \frac{105 \,{\left (a b^{2} c^{2} d - 2 \, a^{2} b c d^{2} + a^{3} d^{3}\right )} \sqrt{-\frac{b c - a d}{b}} \arctan \left (\frac{b d x^{2} + 2 \, b c - a d}{2 \, \sqrt{d x^{2} + c} b \sqrt{-\frac{b c - a d}{b}}}\right ) + 2 \,{\left (15 \, b^{3} d^{3} x^{6} + 15 \, b^{3} c^{3} - 161 \, a b^{2} c^{2} d + 245 \, a^{2} b c d^{2} - 105 \, a^{3} d^{3} + 3 \,{\left (15 \, b^{3} c d^{2} - 7 \, a b^{2} d^{3}\right )} x^{4} +{\left (45 \, b^{3} c^{2} d - 77 \, a b^{2} c d^{2} + 35 \, a^{2} b d^{3}\right )} x^{2}\right )} \sqrt{d x^{2} + c}}{210 \, b^{4} d}\right ] \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^3/(b*x^2 + a),x, algorithm="fricas")
[Out]
[1/420*(105*(a*b^2*c^2*d - 2*a^2*b*c*d^2 + a^3*d^3)*sqrt((b*c - a*d)/b)*log((b^2
*d^2*x^4 + 8*b^2*c^2 - 8*a*b*c*d + a^2*d^2 + 2*(4*b^2*c*d - 3*a*b*d^2)*x^2 + 4*(
b^2*d*x^2 + 2*b^2*c - a*b*d)*sqrt(d*x^2 + c)*sqrt((b*c - a*d)/b))/(b^2*x^4 + 2*a
*b*x^2 + a^2)) + 4*(15*b^3*d^3*x^6 + 15*b^3*c^3 - 161*a*b^2*c^2*d + 245*a^2*b*c*
d^2 - 105*a^3*d^3 + 3*(15*b^3*c*d^2 - 7*a*b^2*d^3)*x^4 + (45*b^3*c^2*d - 77*a*b^
2*c*d^2 + 35*a^2*b*d^3)*x^2)*sqrt(d*x^2 + c))/(b^4*d), 1/210*(105*(a*b^2*c^2*d -
2*a^2*b*c*d^2 + a^3*d^3)*sqrt(-(b*c - a*d)/b)*arctan(1/2*(b*d*x^2 + 2*b*c - a*d
)/(sqrt(d*x^2 + c)*b*sqrt(-(b*c - a*d)/b))) + 2*(15*b^3*d^3*x^6 + 15*b^3*c^3 - 1
61*a*b^2*c^2*d + 245*a^2*b*c*d^2 - 105*a^3*d^3 + 3*(15*b^3*c*d^2 - 7*a*b^2*d^3)*
x^4 + (45*b^3*c^2*d - 77*a*b^2*c*d^2 + 35*a^2*b*d^3)*x^2)*sqrt(d*x^2 + c))/(b^4*
d)]
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Sympy [F] time = 0., size = 0, normalized size = 0. \[ \int \frac{x^{3} \left (c + d x^{2}\right )^{\frac{5}{2}}}{a + b x^{2}}\, dx \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate(x**3*(d*x**2+c)**(5/2)/(b*x**2+a),x)
[Out]
Integral(x**3*(c + d*x**2)**(5/2)/(a + b*x**2), x)
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GIAC/XCAS [A] time = 0.248863, size = 308, normalized size = 2.14 \[ -\frac{{\left (a b^{3} c^{3} - 3 \, a^{2} b^{2} c^{2} d + 3 \, a^{3} b c d^{2} - a^{4} d^{3}\right )} \arctan \left (\frac{\sqrt{d x^{2} + c} b}{\sqrt{-b^{2} c + a b d}}\right )}{\sqrt{-b^{2} c + a b d} b^{4}} + \frac{15 \,{\left (d x^{2} + c\right )}^{\frac{7}{2}} b^{6} d^{6} - 21 \,{\left (d x^{2} + c\right )}^{\frac{5}{2}} a b^{5} d^{7} - 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a b^{5} c d^{7} - 105 \, \sqrt{d x^{2} + c} a b^{5} c^{2} d^{7} + 35 \,{\left (d x^{2} + c\right )}^{\frac{3}{2}} a^{2} b^{4} d^{8} + 210 \, \sqrt{d x^{2} + c} a^{2} b^{4} c d^{8} - 105 \, \sqrt{d x^{2} + c} a^{3} b^{3} d^{9}}{105 \, b^{7} d^{7}} \]
Verification of antiderivative is not currently implemented for this CAS.
[In] integrate((d*x^2 + c)^(5/2)*x^3/(b*x^2 + a),x, algorithm="giac")
[Out]
-(a*b^3*c^3 - 3*a^2*b^2*c^2*d + 3*a^3*b*c*d^2 - a^4*d^3)*arctan(sqrt(d*x^2 + c)*
b/sqrt(-b^2*c + a*b*d))/(sqrt(-b^2*c + a*b*d)*b^4) + 1/105*(15*(d*x^2 + c)^(7/2)
*b^6*d^6 - 21*(d*x^2 + c)^(5/2)*a*b^5*d^7 - 35*(d*x^2 + c)^(3/2)*a*b^5*c*d^7 - 1
05*sqrt(d*x^2 + c)*a*b^5*c^2*d^7 + 35*(d*x^2 + c)^(3/2)*a^2*b^4*d^8 + 210*sqrt(d
*x^2 + c)*a^2*b^4*c*d^8 - 105*sqrt(d*x^2 + c)*a^3*b^3*d^9)/(b^7*d^7)