3.694 \(\int \frac{x^4 (c+d x^2)^{5/2}}{a+b x^2} \, dx\)

Optimal. Leaf size=291 \[ \frac{x^3 \sqrt{c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{c+d x^2} \left (144 a^2 b c d^2-64 a^3 d^3-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac{\left (-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^5}+\frac{d x^5 \sqrt{c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \]

[Out]

((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*Sqrt[c + d*x^2])/(128*b^4*d) + ((59*b^2*c^2 - 1
04*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/(192*b^3) + (d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2])/(48*b^2) +
(d*x^5*(c + d*x^2)^(3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^
2])])/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[
d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2))

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Rubi [A]  time = 0.574524, antiderivative size = 291, normalized size of antiderivative = 1., number of steps used = 9, number of rules used = 8, integrand size = 24, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.333, Rules used = {477, 581, 582, 523, 217, 206, 377, 205} \[ \frac{x^3 \sqrt{c+d x^2} \left (48 a^2 d^2-104 a b c d+59 b^2 c^2\right )}{192 b^3}+\frac{x \sqrt{c+d x^2} \left (144 a^2 b c d^2-64 a^3 d^3-88 a b^2 c^2 d+5 b^3 c^3\right )}{128 b^4 d}-\frac{\left (-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4+40 a b^3 c^3 d+5 b^4 c^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 b^5 d^{3/2}}+\frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^5}+\frac{d x^5 \sqrt{c+d x^2} (11 b c-8 a d)}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b} \]

Antiderivative was successfully verified.

[In]

Int[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]

[Out]

((5*b^3*c^3 - 88*a*b^2*c^2*d + 144*a^2*b*c*d^2 - 64*a^3*d^3)*x*Sqrt[c + d*x^2])/(128*b^4*d) + ((59*b^2*c^2 - 1
04*a*b*c*d + 48*a^2*d^2)*x^3*Sqrt[c + d*x^2])/(192*b^3) + (d*(11*b*c - 8*a*d)*x^5*Sqrt[c + d*x^2])/(48*b^2) +
(d*x^5*(c + d*x^2)^(3/2))/(8*b) + (a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqrt[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^
2])])/b^5 - ((5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*ArcTanh[(Sqrt[
d]*x)/Sqrt[c + d*x^2]])/(128*b^5*d^(3/2))

Rule 477

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

Rule 581

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

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 \left (c+d x^2\right )^{5/2}}{a+b x^2} \, dx &=\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{\int \frac{x^4 \sqrt{c+d x^2} \left (c (8 b c-5 a d)+d (11 b c-8 a d) x^2\right )}{a+b x^2} \, dx}{8 b}\\ &=\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{\int \frac{x^4 \left (c \left (48 b^2 c^2-85 a b c d+40 a^2 d^2\right )+d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{48 b^2}\\ &=\frac{\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt{c+d x^2}}{192 b^3}+\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}-\frac{\int \frac{x^2 \left (3 a c d \left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right )-3 d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x^2\right )}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{192 b^3 d}\\ &=\frac{\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt{c+d x^2}}{128 b^4 d}+\frac{\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt{c+d x^2}}{192 b^3}+\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{\int \frac{-3 a c d \left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right )-3 d \left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) x^2}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{384 b^4 d^2}\\ &=\frac{\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt{c+d x^2}}{128 b^4 d}+\frac{\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt{c+d x^2}}{192 b^3}+\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{\left (a^2 (b c-a d)^3\right ) \int \frac{1}{\left (a+b x^2\right ) \sqrt{c+d x^2}} \, dx}{b^5}-\frac{\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \int \frac{1}{\sqrt{c+d x^2}} \, dx}{128 b^5 d}\\ &=\frac{\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt{c+d x^2}}{128 b^4 d}+\frac{\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt{c+d x^2}}{192 b^3}+\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{\left (a^2 (b c-a d)^3\right ) \operatorname{Subst}\left (\int \frac{1}{a-(-b c+a d) x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{b^5}-\frac{\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \operatorname{Subst}\left (\int \frac{1}{1-d x^2} \, dx,x,\frac{x}{\sqrt{c+d x^2}}\right )}{128 b^5 d}\\ &=\frac{\left (5 b^3 c^3-88 a b^2 c^2 d+144 a^2 b c d^2-64 a^3 d^3\right ) x \sqrt{c+d x^2}}{128 b^4 d}+\frac{\left (59 b^2 c^2-104 a b c d+48 a^2 d^2\right ) x^3 \sqrt{c+d x^2}}{192 b^3}+\frac{d (11 b c-8 a d) x^5 \sqrt{c+d x^2}}{48 b^2}+\frac{d x^5 \left (c+d x^2\right )^{3/2}}{8 b}+\frac{a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{\sqrt{b c-a d} x}{\sqrt{a} \sqrt{c+d x^2}}\right )}{b^5}-\frac{\left (5 b^4 c^4+40 a b^3 c^3 d-240 a^2 b^2 c^2 d^2+320 a^3 b c d^3-128 a^4 d^4\right ) \tanh ^{-1}\left (\frac{\sqrt{d} x}{\sqrt{c+d x^2}}\right )}{128 b^5 d^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.244448, size = 247, normalized size = 0.85 \[ \frac{\frac{b x \sqrt{c+d x^2} \left (48 a^2 b d^2 \left (9 c+2 d x^2\right )-192 a^3 d^3-8 a b^2 d \left (33 c^2+26 c d x^2+8 d^2 x^4\right )+b^3 \left (118 c^2 d x^2+15 c^3+136 c d^2 x^4+48 d^3 x^6\right )\right )}{d}+\frac{3 \left (240 a^2 b^2 c^2 d^2-320 a^3 b c d^3+128 a^4 d^4-40 a b^3 c^3 d-5 b^4 c^4\right ) \log \left (\sqrt{d} \sqrt{c+d x^2}+d x\right )}{d^{3/2}}+384 a^{3/2} (b c-a d)^{5/2} \tan ^{-1}\left (\frac{x \sqrt{b c-a d}}{\sqrt{a} \sqrt{c+d x^2}}\right )}{384 b^5} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^4*(c + d*x^2)^(5/2))/(a + b*x^2),x]

[Out]

((b*x*Sqrt[c + d*x^2]*(-192*a^3*d^3 + 48*a^2*b*d^2*(9*c + 2*d*x^2) - 8*a*b^2*d*(33*c^2 + 26*c*d*x^2 + 8*d^2*x^
4) + b^3*(15*c^3 + 118*c^2*d*x^2 + 136*c*d^2*x^4 + 48*d^3*x^6)))/d + 384*a^(3/2)*(b*c - a*d)^(5/2)*ArcTan[(Sqr
t[b*c - a*d]*x)/(Sqrt[a]*Sqrt[c + d*x^2])] + (3*(-5*b^4*c^4 - 40*a*b^3*c^3*d + 240*a^2*b^2*c^2*d^2 - 320*a^3*b
*c*d^3 + 128*a^4*d^4)*Log[d*x + Sqrt[d]*Sqrt[c + d*x^2]])/d^(3/2))/(384*b^5)

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Maple [B]  time = 0.019, size = 3373, normalized size = 11.6 \begin{align*} \text{output too large to display} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^4*(d*x^2+c)^(5/2)/(b*x^2+a),x)

[Out]

3/2/b^4*a^4/(-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^2*c-3/2/b^4*a^4/(-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^2*c+3/2/b^3*a^3/(-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*c^2+1/8/b^3*a^2*d*((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-1/48/b*c/d*x*(d*x^2+c)^(5/2)-1/2/b^5*a^5/(-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+1/2/
b^2*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)))*c^3-1/2/b^2*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)))*c^3-1/b^3*a^3/(-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)*d*c+1/2/b^5*a^5/(-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+7/16/b^3*a^2*d*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/b^3*a^2*d*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/b^3*a^3/(-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)*d*c-3/2/b^3*a^3/(-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*c^2+1/10/b^2*a^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)^(5/2)+1/2/b^5*a^4*d^(5/2)*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))-1/10/b^2*a^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)^(5/2)+1/2/b^5*a^4*d^(5/2)*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))+1/8/b*x*(d*x^2+c)^(7/2)/d-5/128/b*c^4/d^(3/2)*ln(
x*d^(1/2)+(d*x^2+c)^(1/2))-1/6/b^2*a*x*(d*x^2+c)^(5/2)+1/2/b^2*a^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)*c^2-1/6/b^3*a^3/(-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)*d+1/6/b^2*a^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)^(3/2)*c+15/16/b^3*a^2*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))*c^2-1/6/b^2*a^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
)^(3/2)*c-1/4/b^4*a^3*d^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/b^4*a^3*d^(3/2)*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-1/2/b^4*a^4/(-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)*d^2-1/2/b^2*a^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)*c^2+1/8/b^3*a^2*d*((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/b^3*a^2*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))*c^2-
1/4/b^4*a^3*d^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/b^4
*a^3*d^(3/2)*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+1/2/b^4*a^4/(-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)*d^2-5/192/b*c^2/d*x*(d*x^2+c)^(3/2)-5/128/b*c^3/d*x*(d*x^2+c)^(1/2
)-5/24/b^2*a*c*x*(d*x^2+c)^(3/2)-5/16/b^2*a*c^2*x*(d*x^2+c)^(1/2)-5/16/b^2*a*c^3/d^(1/2)*ln(x*d^(1/2)+(d*x^2+c
)^(1/2))+1/6/b^3*a^3/(-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)*d

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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(x^4*(d*x^2+c)^(5/2)/(b*x^2+a),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [A]  time = 48.843, size = 3162, normalized size = 10.87 \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)^(5/2)/(b*x^2+a),x, algorithm="fricas")

[Out]

[-1/768*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(d)*log(-2*d
*x^2 - 2*sqrt(d*x^2 + c)*sqrt(d)*x - c) - 192*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a*b*c + a^2*d)*l
og(((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*c - 2*a*d)*x^3 - a
*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4 + 2*a*b*x^2 + a^2)) - 2*(48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3
- 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^
2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqrt(d*x^2 + c))/(b^5*d^2), 1/384*(3*(5*b^4*c^4 + 40*a*b^3*c^3*d
- 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(-d)*arctan(sqrt(-d)*x/sqrt(d*x^2 + c)) + 96*(a*b^2
*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(-a*b*c + a^2*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*c - 2*a*d)*x^3 - a*c*x)*sqrt(-a*b*c + a^2*d)*sqrt(d*x^2 + c))/(b^2*x^4
+ 2*a*b*x^2 + a^2)) + (48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*b^3*c*
d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqrt(d*x^
2 + c))/(b^5*d^2), 1/768*(384*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(a*b*c - a^2*d)*arctan(1/2*sqrt(a*
b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d)*x)) - 3*
(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(d)*log(-2*d*x^2 - 2*sq
rt(d*x^2 + c)*sqrt(d)*x - c) + 2*(48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 10
4*a*b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)
*sqrt(d*x^2 + c))/(b^5*d^2), 1/384*(192*(a*b^2*c^2*d^2 - 2*a^2*b*c*d^3 + a^3*d^4)*sqrt(a*b*c - a^2*d)*arctan(1
/2*sqrt(a*b*c - a^2*d)*((b*c - 2*a*d)*x^2 - a*c)*sqrt(d*x^2 + c)/((a*b*c*d - a^2*d^2)*x^3 + (a*b*c^2 - a^2*c*d
)*x)) + 3*(5*b^4*c^4 + 40*a*b^3*c^3*d - 240*a^2*b^2*c^2*d^2 + 320*a^3*b*c*d^3 - 128*a^4*d^4)*sqrt(-d)*arctan(s
qrt(-d)*x/sqrt(d*x^2 + c)) + (48*b^4*d^4*x^7 + 8*(17*b^4*c*d^3 - 8*a*b^3*d^4)*x^5 + 2*(59*b^4*c^2*d^2 - 104*a*
b^3*c*d^3 + 48*a^2*b^2*d^4)*x^3 + 3*(5*b^4*c^3*d - 88*a*b^3*c^2*d^2 + 144*a^2*b^2*c*d^3 - 64*a^3*b*d^4)*x)*sqr
t(d*x^2 + c))/(b^5*d^2)]

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Sympy [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int \frac{x^{4} \left (c + d x^{2}\right )^{\frac{5}{2}}}{a + b x^{2}}\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**4*(d*x**2+c)**(5/2)/(b*x**2+a),x)

[Out]

Integral(x**4*(c + d*x**2)**(5/2)/(a + b*x**2), x)

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Giac [A]  time = 1.68015, size = 483, normalized size = 1.66 \begin{align*} \frac{1}{384} \,{\left (2 \,{\left (4 \,{\left (\frac{6 \, d^{2} x^{2}}{b} + \frac{17 \, b^{14} c d^{7} - 8 \, a b^{13} d^{8}}{b^{15} d^{6}}\right )} x^{2} + \frac{59 \, b^{14} c^{2} d^{6} - 104 \, a b^{13} c d^{7} + 48 \, a^{2} b^{12} d^{8}}{b^{15} d^{6}}\right )} x^{2} + \frac{3 \,{\left (5 \, b^{14} c^{3} d^{5} - 88 \, a b^{13} c^{2} d^{6} + 144 \, a^{2} b^{12} c d^{7} - 64 \, a^{3} b^{11} d^{8}\right )}}{b^{15} d^{6}}\right )} \sqrt{d x^{2} + c} x - \frac{{\left (a^{2} b^{3} c^{3} \sqrt{d} - 3 \, a^{3} b^{2} c^{2} d^{\frac{3}{2}} + 3 \, a^{4} b c d^{\frac{5}{2}} - a^{5} d^{\frac{7}{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 )}{\sqrt{a b c d - a^{2} d^{2}} b^{5}} + \frac{{\left (5 \, b^{4} c^{4} \sqrt{d} + 40 \, a b^{3} c^{3} d^{\frac{3}{2}} - 240 \, a^{2} b^{2} c^{2} d^{\frac{5}{2}} + 320 \, a^{3} b c d^{\frac{7}{2}} - 128 \, a^{4} d^{\frac{9}{2}}\right )} \log \left ({\left (\sqrt{d} x - \sqrt{d x^{2} + c}\right )}^{2}\right )}{256 \, b^{5} d^{2}} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^4*(d*x^2+c)^(5/2)/(b*x^2+a),x, algorithm="giac")

[Out]

1/384*(2*(4*(6*d^2*x^2/b + (17*b^14*c*d^7 - 8*a*b^13*d^8)/(b^15*d^6))*x^2 + (59*b^14*c^2*d^6 - 104*a*b^13*c*d^
7 + 48*a^2*b^12*d^8)/(b^15*d^6))*x^2 + 3*(5*b^14*c^3*d^5 - 88*a*b^13*c^2*d^6 + 144*a^2*b^12*c*d^7 - 64*a^3*b^1
1*d^8)/(b^15*d^6))*sqrt(d*x^2 + c)*x - (a^2*b^3*c^3*sqrt(d) - 3*a^3*b^2*c^2*d^(3/2) + 3*a^4*b*c*d^(5/2) - a^5*
d^(7/2))*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^5) + 1/256*(5*b^4*c^4*sqrt(d) + 40*a*b^3*c^3*d^(3/2) - 240*a^2*b^2*c^2*d^(5/2) + 320*a^3*b*c*d^(7/
2) - 128*a^4*d^(9/2))*log((sqrt(d)*x - sqrt(d*x^2 + c))^2)/(b^5*d^2)