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

Optimal. Leaf size=150 \[ -\frac{\sqrt{a} (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{d}}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{x \sqrt{c+d x^2}}{b^2} \]

[Out]

(x*Sqrt[c + d*x^2])/b^2 - (x^3*Sqrt[c + d*x^2])/(2*b*(a + b*x^2)) - (Sqrt[a]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^3*Sqrt[b*c - a*d]) + ((b*c - 4*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*
x^2]])/(2*b^3*Sqrt[d])

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Rubi [A]  time = 0.168776, antiderivative size = 150, normalized size of antiderivative = 1., number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.292, Rules used = {467, 582, 523, 217, 206, 377, 205} \[ -\frac{\sqrt{a} (3 b c-4 a d) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{d}}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{x \sqrt{c+d x^2}}{b^2} \]

Antiderivative was successfully verified.

[In]

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

[Out]

(x*Sqrt[c + d*x^2])/b^2 - (x^3*Sqrt[c + d*x^2])/(2*b*(a + b*x^2)) - (Sqrt[a]*(3*b*c - 4*a*d)*ArcTan[(Sqrt[b*c
- a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])])/(2*b^3*Sqrt[b*c - a*d]) + ((b*c - 4*a*d)*ArcTanh[(Sqrt[d]*x)/Sqrt[c + d*
x^2]])/(2*b^3*Sqrt[d])

Rule 467

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

Rule 582

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

Rule 523

Int[((e_) + (f_.)*(x_)^(n_))/(((a_) + (b_.)*(x_)^(n_))*Sqrt[(c_) + (d_.)*(x_)^(n_)]), x_Symbol] :> Dist[f/b, I
nt[1/Sqrt[c + d*x^n], x], x] + Dist[(b*e - a*f)/b, Int[1/((a + b*x^n)*Sqrt[c + d*x^n]), x], x] /; FreeQ[{a, b,
c, d, e, f, n}, x]

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])

Rule 377

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

Rule 205

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[a/b, 2]*ArcTan[x/Rt[a/b, 2]])/a, x] /; FreeQ[{a, b}, x]
&& PosQ[a/b]

Rubi steps

\begin{align*} \int \frac{x^4 \sqrt{c+d x^2}}{\left (a+b x^2\right )^2} \, dx &=-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{\int \frac{x^2 \left (3 c+4 d x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 b}\\ &=\frac{x \sqrt{c+d x^2}}{b^2}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}-\frac{\int \frac{4 a c d-2 d (b c-4 a d) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{4 b^2 d}\\ &=\frac{x \sqrt{c+d x^2}}{b^2}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{(b c-4 a d) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{2 b^3}-\frac{(a (3 b c-4 a d)) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{2 b^3}\\ &=\frac{x \sqrt{c+d x^2}}{b^2}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}+\frac{(b c-4 a d) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 b^3}-\frac{(a (3 b c-4 a d)) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{2 b^3}\\ &=\frac{x \sqrt{c+d x^2}}{b^2}-\frac{x^3 \sqrt{c+d x^2}}{2 b \left (a+b x^2\right )}-\frac{\sqrt{a} (3 b c-4 a d) \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{b c-a d}}+\frac{(b c-4 a d) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{2 b^3 \sqrt{d}}\\ \end{align*}

Mathematica [A]  time = 0.174694, size = 134, normalized size = 0.89 \[ \frac{\frac{b x \left (2 a+b x^2\right ) \sqrt{c+d x^2}}{a+b x^2}+\frac{(b c-4 a d) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{\sqrt{d}}+\frac{\sqrt{a} (4 a d-3 b c) \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{\sqrt{b c-a d}}}{2 b^3} \]

Antiderivative was successfully verified.

[In]

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

[Out]

((b*x*(2*a + b*x^2)*Sqrt[c + d*x^2])/(a + b*x^2) + (Sqrt[a]*(-3*b*c + 4*a*d)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[
a]*Sqrt[c + d*x^2])])/Sqrt[b*c - a*d] + ((b*c - 4*a*d)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/Sqrt[d])/(2*b^3)

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Maple [B]  time = 0.022, size = 2615, normalized size = 17.4 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*x*(d*x^2+c)^(1/2)/b^2+1/2/b^2*c/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c)^(1/2))-1/4/b^2*a/(a*d-b*c)/(x+1/b*(-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)-1/4/b^3*a*d*(-a*b)^(
1/2)/(a*d-b*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)-1/4/b^3*a^
2*d^(3/2)/(a*d-b*c)*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))-1/4/b^4*a^2*d^2*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))+1/4/b^3*a*d*(-a*b)^(1/2)/(a*d-
b*c)/(-(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+1/
4/b^2*a*d/(a*d-b*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+1/4
/b^2*a*d^(1/2)/(a*d-b*c)*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-1/4/b^2*a/(a*d-b*c)/(x-1/b*(-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)+1/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*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)-1/4/b^3*a^2*d^(3/2)/(a*d-b*
c)*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))+1/4/b^4*a^2*d^2*(-a*b)^(1/2)/(a*d-b*c)/(-(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)))-1/4/b^3*a*d*(-a*b)^(1/2)/(a*d-b*c)/(-(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+1/4/b^2*a*d/(a*d-b*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+1/4/b^2*a*d^(1/2)/(a*
d-b*c)*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+3/4/b^2*a/(-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)-3/4/b^3*a*d^(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))+3/4/b^3*a^2/(-a*b)^(1/2)/(
-(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)))*d-3/4/b^2*
a/(-a*b)^(1/2)/(-(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-3/4/b^2*a/(-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)-3/4/b^3*a*d^(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))-3/4/b^3*a^2/(-a*b)^(1/2)/(-(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)))*d+3/4/b^2*a/(-a*b)^(1/2)/(-(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

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Fricas [A]  time = 2.77704, size = 2172, normalized size = 14.48 \begin{align*} \text{result too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

[-1/8*(2*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + (
3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4*a*b*d^2)*x^2)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8*a*b*c*d + 8*a^2*d^
2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3 - (a*b*c^2 - a^2*c
*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b^2*d*x^3 + 2*a*b*d*x)*sqrt(d*x
^2 + c))/(b^4*d*x^2 + a*b^3*d), -1/8*(4*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(-d)*arctan(sqrt(-d)*x/s
qrt(d*x^2 + c)) + (3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4*a*b*d^2)*x^2)*sqrt(-a/(b*c - a*d))*log(((b^2*c^2 - 8
*a*b*c*d + 8*a^2*d^2)*x^4 + a^2*c^2 - 2*(3*a*b*c^2 - 4*a^2*c*d)*x^2 + 4*((b^2*c^2 - 3*a*b*c*d + 2*a^2*d^2)*x^3
- (a*b*c^2 - a^2*c*d)*x)*sqrt(d*x^2 + c)*sqrt(-a/(b*c - a*d)))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 4*(b^2*d*x^3 +
2*a*b*d*x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d), 1/4*((3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4*a*b*d^2)*x^2)*
sqrt(a/(b*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x
)) - (a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)*sqrt(d)*log(-2*d*x^2 + 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) + 2*(b^
2*d*x^3 + 2*a*b*d*x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d), -1/4*(2*(a*b*c - 4*a^2*d + (b^2*c - 4*a*b*d)*x^2)
*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) - (3*a*b*c*d - 4*a^2*d^2 + (3*b^2*c*d - 4*a*b*d^2)*x^2)*sqrt(a/(b
*c - a*d))*arctan(-1/2*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)*sqrt(a/(b*c - a*d))/(a*d*x^3 + a*c*x)) - 2*(b
^2*d*x^3 + 2*a*b*d*x)*sqrt(d*x^2 + c))/(b^4*d*x^2 + a*b^3*d)]

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

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

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

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Giac [B]  time = 1.20954, size = 389, normalized size = 2.59 \begin{align*} \frac{\sqrt{d x^{2} + c} x}{2 \, b^{2}} + \frac{{\left (3 \, a b c \sqrt{d} - 4 \, a^{2} d^{\frac{3}{2}}\right )} \arctan \left (\frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b - b c + 2 \, a d}{2 \, \sqrt{a b c d - a^{2} d^{2}}}\right )}{2 \, \sqrt{a b c d - a^{2} d^{2}} b^{3}} - \frac{{\left (b c \sqrt{d} - 4 \, a d^{\frac{3}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{4 \, b^{3} d} - \frac{{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a b c \sqrt{d} - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a^{2} d^{\frac{3}{2}} - a b c^{2} \sqrt{d}}{{\left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{4} b - 2 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} b c + 4 \,{\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2} a d + b c^{2}\right )} b^{3}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

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

[Out]

1/2*sqrt(d*x^2 + c)*x/b^2 + 1/2*(3*a*b*c*sqrt(d) - 4*a^2*d^(3/2))*arctan(1/2*((sqrt(d)*x - sqrt(d*x^2 + c))^2*
b - b*c + 2*a*d)/sqrt(a*b*c*d - a^2*d^2))/(sqrt(a*b*c*d - a^2*d^2)*b^3) - 1/4*(b*c*sqrt(d) - 4*a*d^(3/2))*log(
(sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^3*d) - ((sqrt(d)*x - sqrt(d*x^2 + c))^2*a*b*c*sqrt(d) - 2*(sqrt(d)*x - sqr
t(d*x^2 + c))^2*a^2*d^(3/2) - a*b*c^2*sqrt(d))/(((sqrt(d)*x - sqrt(d*x^2 + c))^4*b - 2*(sqrt(d)*x - sqrt(d*x^2
+ c))^2*b*c + 4*(sqrt(d)*x - sqrt(d*x^2 + c))^2*a*d + b*c^2)*b^3)