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

Optimal. Leaf size=257 \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}-\frac{d x \sqrt{a+b x^2} \left (290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}+\frac{d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]

[Out]

-(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(48*a*b^4) - (d*(24*b^2*
c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x^2
]*(c + d*x^2)^2)/(6*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 - 144*a*b^2*
c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))

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Rubi [A]  time = 0.259272, antiderivative size = 257, normalized size of antiderivative = 1., number of steps used = 6, number of rules used = 5, integrand size = 21, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.238, Rules used = {413, 528, 388, 217, 206} \[ -\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right ) \left (35 a^2 d^2-64 a b c d+24 b^2 c^2\right )}{24 a b^3}-\frac{d x \sqrt{a+b x^2} \left (290 a^2 b c d^2-105 a^3 d^3-248 a b^2 c^2 d+48 b^3 c^3\right )}{48 a b^4}+\frac{d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}-\frac{d x \sqrt{a+b x^2} \left (c+d x^2\right )^2 (6 b c-7 a d)}{6 a b^2}+\frac{x \left (c+d x^2\right )^3 (b c-a d)}{a b \sqrt{a+b x^2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

-(d*(48*b^3*c^3 - 248*a*b^2*c^2*d + 290*a^2*b*c*d^2 - 105*a^3*d^3)*x*Sqrt[a + b*x^2])/(48*a*b^4) - (d*(24*b^2*
c^2 - 64*a*b*c*d + 35*a^2*d^2)*x*Sqrt[a + b*x^2]*(c + d*x^2))/(24*a*b^3) - (d*(6*b*c - 7*a*d)*x*Sqrt[a + b*x^2
]*(c + d*x^2)^2)/(6*a*b^2) + ((b*c - a*d)*x*(c + d*x^2)^3)/(a*b*Sqrt[a + b*x^2]) + (d*(64*b^3*c^3 - 144*a*b^2*
c^2*d + 120*a^2*b*c*d^2 - 35*a^3*d^3)*ArcTanh[(Sqrt[b]*x)/Sqrt[a + b*x^2]])/(16*b^(9/2))

Rule 413

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[((a*d - c*b)*x*(a + b*x^n)^
(p + 1)*(c + d*x^n)^(q - 1))/(a*b*n*(p + 1)), x] - Dist[1/(a*b*n*(p + 1)), Int[(a + b*x^n)^(p + 1)*(c + d*x^n)
^(q - 2)*Simp[c*(a*d - c*b*(n*(p + 1) + 1)) + d*(a*d*(n*(q - 1) + 1) - b*c*(n*(p + q) + 1))*x^n, x], x], x] /;
FreeQ[{a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && LtQ[p, -1] && GtQ[q, 1] && IntBinomialQ[a, b, c, d, n, p, q
, x]

Rule 528

Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.)*((e_) + (f_.)*(x_)^(n_)), x_Symbol] :> Simp[
(f*x*(a + b*x^n)^(p + 1)*(c + d*x^n)^q)/(b*(n*(p + q + 1) + 1)), x] + Dist[1/(b*(n*(p + q + 1) + 1)), Int[(a +
b*x^n)^p*(c + d*x^n)^(q - 1)*Simp[c*(b*e - a*f + b*e*n*(p + q + 1)) + (d*(b*e - a*f) + f*n*q*(b*c - a*d) + b*
d*e*n*(p + q + 1))*x^n, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && GtQ[q, 0] && NeQ[n*(p + q + 1) + 1
, 0]

Rule 388

Int[((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_)), x_Symbol] :> Simp[(d*x*(a + b*x^n)^(p + 1))/(b*(n*
(p + 1) + 1)), x] - Dist[(a*d - b*c*(n*(p + 1) + 1))/(b*(n*(p + 1) + 1)), Int[(a + b*x^n)^p, x], x] /; FreeQ[{
a, b, c, d, n}, x] && NeQ[b*c - a*d, 0] && NeQ[n*(p + 1) + 1, 0]

Rule 217

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 206

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(1*ArcTanh[(Rt[-b, 2]*x)/Rt[a, 2]])/(Rt[a, 2]*Rt[-b, 2]), x]
/; FreeQ[{a, b}, x] && NegQ[a/b] && (GtQ[a, 0] || LtQ[b, 0])

Rubi steps

\begin{align*} \int \frac{\left (c+d x^2\right )^4}{\left (a+b x^2\right )^{3/2}} \, dx &=\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{\left (c+d x^2\right )^2 \left (a c d-d (6 b c-7 a d) x^2\right )}{\sqrt{a+b x^2}} \, dx}{a b}\\ &=-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{\left (c+d x^2\right ) \left (a c d (12 b c-7 a d)-d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x^2\right )}{\sqrt{a+b x^2}} \, dx}{6 a b^2}\\ &=-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\int \frac{a c d \left (72 b^2 c^2-92 a b c d+35 a^2 d^2\right )-d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x^2}{\sqrt{a+b x^2}} \, dx}{24 a b^3}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \int \frac{1}{\sqrt{a+b x^2}} \, dx}{16 b^4}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{\left (d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right )\right ) \operatorname{Subst}\left (\int \frac{1}{1-b x^2} \, dx,x,\frac{x}{\sqrt{a+b x^2}}\right )}{16 b^4}\\ &=-\frac{d \left (48 b^3 c^3-248 a b^2 c^2 d+290 a^2 b c d^2-105 a^3 d^3\right ) x \sqrt{a+b x^2}}{48 a b^4}-\frac{d \left (24 b^2 c^2-64 a b c d+35 a^2 d^2\right ) x \sqrt{a+b x^2} \left (c+d x^2\right )}{24 a b^3}-\frac{d (6 b c-7 a d) x \sqrt{a+b x^2} \left (c+d x^2\right )^2}{6 a b^2}+\frac{(b c-a d) x \left (c+d x^2\right )^3}{a b \sqrt{a+b x^2}}+\frac{d \left (64 b^3 c^3-144 a b^2 c^2 d+120 a^2 b c d^2-35 a^3 d^3\right ) \tanh ^{-1}\left (\frac{\sqrt{b} x}{\sqrt{a+b x^2}}\right )}{16 b^{9/2}}\\ \end{align*}

Mathematica [A]  time = 5.19608, size = 172, normalized size = 0.67 \[ \frac{\sqrt{b} x \sqrt{a+b x^2} \left (3 d^2 \left (19 a^2 d^2-56 a b c d+48 b^2 c^2\right )+2 b d^3 x^2 (24 b c-11 a d)+\frac{48 (b c-a d)^4}{a \left (a+b x^2\right )}+8 b^2 d^4 x^4\right )+3 d \left (120 a^2 b c d^2-35 a^3 d^3-144 a b^2 c^2 d+64 b^3 c^3\right ) \log \left (\sqrt{b} \sqrt{a+b x^2}+b x\right )}{48 b^{9/2}} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(Sqrt[b]*x*Sqrt[a + b*x^2]*(3*d^2*(48*b^2*c^2 - 56*a*b*c*d + 19*a^2*d^2) + 2*b*d^3*(24*b*c - 11*a*d)*x^2 + 8*b
^2*d^4*x^4 + (48*(b*c - a*d)^4)/(a*(a + b*x^2))) + 3*d*(64*b^3*c^3 - 144*a*b^2*c^2*d + 120*a^2*b*c*d^2 - 35*a^
3*d^3)*Log[b*x + Sqrt[b]*Sqrt[a + b*x^2]])/(48*b^(9/2))

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Maple [A]  time = 0.017, size = 340, normalized size = 1.3 \begin{align*}{\frac{{d}^{4}{x}^{7}}{6\,b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{7\,a{d}^{4}{x}^{5}}{24\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{d}^{4}{a}^{2}{x}^{3}}{48\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{35\,{a}^{3}{d}^{4}x}{16\,{b}^{4}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{35\,{a}^{3}{d}^{4}}{16}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{9}{2}}}}+{\frac{c{d}^{3}{x}^{5}}{b}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{5\,c{d}^{3}a{x}^{3}}{2\,{b}^{2}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}-{\frac{15\,c{d}^{3}{a}^{2}x}{2\,{b}^{3}}{\frac{1}{\sqrt{b{x}^{2}+a}}}}+{\frac{15\,c{d}^{3}{a}^{2}}{2}\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ){b}^{-{\frac{7}{2}}}}+3\,{\frac{{c}^{2}{d}^{2}{x}^{3}}{b\sqrt{b{x}^{2}+a}}}+9\,{\frac{{c}^{2}{d}^{2}ax}{{b}^{2}\sqrt{b{x}^{2}+a}}}-9\,{\frac{{c}^{2}{d}^{2}a\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{5/2}}}-4\,{\frac{{c}^{3}dx}{b\sqrt{b{x}^{2}+a}}}+4\,{\frac{{c}^{3}d\ln \left ( x\sqrt{b}+\sqrt{b{x}^{2}+a} \right ) }{{b}^{3/2}}}+{\frac{{c}^{4}x}{a}{\frac{1}{\sqrt{b{x}^{2}+a}}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/6*d^4*x^7/b/(b*x^2+a)^(1/2)-7/24*d^4/b^2*a*x^5/(b*x^2+a)^(1/2)+35/48*d^4/b^3*a^2*x^3/(b*x^2+a)^(1/2)+35/16*d
^4/b^4*a^3*x/(b*x^2+a)^(1/2)-35/16*d^4/b^(9/2)*a^3*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+c*d^3*x^5/b/(b*x^2+a)^(1/2)-5
/2*c*d^3/b^2*a*x^3/(b*x^2+a)^(1/2)-15/2*c*d^3/b^3*a^2*x/(b*x^2+a)^(1/2)+15/2*c*d^3/b^(7/2)*a^2*ln(x*b^(1/2)+(b
*x^2+a)^(1/2))+3*c^2*d^2*x^3/b/(b*x^2+a)^(1/2)+9*c^2*d^2/b^2*a*x/(b*x^2+a)^(1/2)-9*c^2*d^2/b^(5/2)*a*ln(x*b^(1
/2)+(b*x^2+a)^(1/2))-4*c^3*d*x/b/(b*x^2+a)^(1/2)+4*c^3*d/b^(3/2)*ln(x*b^(1/2)+(b*x^2+a)^(1/2))+c^4*x/a/(b*x^2+
a)^(1/2)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

Exception raised: ValueError

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Fricas [A]  time = 2.32665, size = 1272, normalized size = 4.95 \begin{align*} \left [-\frac{3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{b} \log \left (-2 \, b x^{2} + 2 \, \sqrt{b x^{2} + a} \sqrt{b} x - a\right ) - 2 \,{\left (8 \, a b^{4} d^{4} x^{7} + 2 \,{\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} +{\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt{b x^{2} + a}}{96 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}, -\frac{3 \,{\left (64 \, a^{2} b^{3} c^{3} d - 144 \, a^{3} b^{2} c^{2} d^{2} + 120 \, a^{4} b c d^{3} - 35 \, a^{5} d^{4} +{\left (64 \, a b^{4} c^{3} d - 144 \, a^{2} b^{3} c^{2} d^{2} + 120 \, a^{3} b^{2} c d^{3} - 35 \, a^{4} b d^{4}\right )} x^{2}\right )} \sqrt{-b} \arctan \left (\frac{\sqrt{-b} x}{\sqrt{b x^{2} + a}}\right ) -{\left (8 \, a b^{4} d^{4} x^{7} + 2 \,{\left (24 \, a b^{4} c d^{3} - 7 \, a^{2} b^{3} d^{4}\right )} x^{5} +{\left (144 \, a b^{4} c^{2} d^{2} - 120 \, a^{2} b^{3} c d^{3} + 35 \, a^{3} b^{2} d^{4}\right )} x^{3} + 3 \,{\left (16 \, b^{5} c^{4} - 64 \, a b^{4} c^{3} d + 144 \, a^{2} b^{3} c^{2} d^{2} - 120 \, a^{3} b^{2} c d^{3} + 35 \, a^{4} b d^{4}\right )} x\right )} \sqrt{b x^{2} + a}}{48 \,{\left (a b^{6} x^{2} + a^{2} b^{5}\right )}}\right ] \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/96*(3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a^5*d^4 + (64*a*b^4*c^3*d - 144*a^2*b
^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*sqrt(b)*log(-2*b*x^2 + 2*sqrt(b*x^2 + a)*sqrt(b)*x - a) -
2*(8*a*b^4*d^4*x^7 + 2*(24*a*b^4*c*d^3 - 7*a^2*b^3*d^4)*x^5 + (144*a*b^4*c^2*d^2 - 120*a^2*b^3*c*d^3 + 35*a^3*
b^2*d^4)*x^3 + 3*(16*b^5*c^4 - 64*a*b^4*c^3*d + 144*a^2*b^3*c^2*d^2 - 120*a^3*b^2*c*d^3 + 35*a^4*b*d^4)*x)*sqr
t(b*x^2 + a))/(a*b^6*x^2 + a^2*b^5), -1/48*(3*(64*a^2*b^3*c^3*d - 144*a^3*b^2*c^2*d^2 + 120*a^4*b*c*d^3 - 35*a
^5*d^4 + (64*a*b^4*c^3*d - 144*a^2*b^3*c^2*d^2 + 120*a^3*b^2*c*d^3 - 35*a^4*b*d^4)*x^2)*sqrt(-b)*arctan(sqrt(-
b)*x/sqrt(b*x^2 + a)) - (8*a*b^4*d^4*x^7 + 2*(24*a*b^4*c*d^3 - 7*a^2*b^3*d^4)*x^5 + (144*a*b^4*c^2*d^2 - 120*a
^2*b^3*c*d^3 + 35*a^3*b^2*d^4)*x^3 + 3*(16*b^5*c^4 - 64*a*b^4*c^3*d + 144*a^2*b^3*c^2*d^2 - 120*a^3*b^2*c*d^3
+ 35*a^4*b*d^4)*x)*sqrt(b*x^2 + a))/(a*b^6*x^2 + a^2*b^5)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{\left (c + d x^{2}\right )^{4}}{\left (a + b x^{2}\right )^{\frac{3}{2}}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [A]  time = 1.15544, size = 317, normalized size = 1.23 \begin{align*} \frac{{\left ({\left (2 \,{\left (\frac{4 \, d^{4} x^{2}}{b} + \frac{24 \, a b^{6} c d^{3} - 7 \, a^{2} b^{5} d^{4}}{a b^{7}}\right )} x^{2} + \frac{144 \, a b^{6} c^{2} d^{2} - 120 \, a^{2} b^{5} c d^{3} + 35 \, a^{3} b^{4} d^{4}}{a b^{7}}\right )} x^{2} + \frac{3 \,{\left (16 \, b^{7} c^{4} - 64 \, a b^{6} c^{3} d + 144 \, a^{2} b^{5} c^{2} d^{2} - 120 \, a^{3} b^{4} c d^{3} + 35 \, a^{4} b^{3} d^{4}\right )}}{a b^{7}}\right )} x}{48 \, \sqrt{b x^{2} + a}} - \frac{{\left (64 \, b^{3} c^{3} d - 144 \, a b^{2} c^{2} d^{2} + 120 \, a^{2} b c d^{3} - 35 \, a^{3} d^{4}\right )} \log \left ({\left | -\sqrt{b} x + \sqrt{b x^{2} + a} \right |}\right )}{16 \, b^{\frac{9}{2}}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/48*((2*(4*d^4*x^2/b + (24*a*b^6*c*d^3 - 7*a^2*b^5*d^4)/(a*b^7))*x^2 + (144*a*b^6*c^2*d^2 - 120*a^2*b^5*c*d^3
+ 35*a^3*b^4*d^4)/(a*b^7))*x^2 + 3*(16*b^7*c^4 - 64*a*b^6*c^3*d + 144*a^2*b^5*c^2*d^2 - 120*a^3*b^4*c*d^3 + 3
5*a^4*b^3*d^4)/(a*b^7))*x/sqrt(b*x^2 + a) - 1/16*(64*b^3*c^3*d - 144*a*b^2*c^2*d^2 + 120*a^2*b*c*d^3 - 35*a^3*
d^4)*log(abs(-sqrt(b)*x + sqrt(b*x^2 + a)))/b^(9/2)